Research ArticleA Normalized Transfer Matrix Method for theFree Vibration of Stepped Beams Comparison withExperimental and FE(3D) Methods
Tamer Ahmed El-Sayed and Said Hamed Farghaly
Department ofMechanical Design Faculty of EngineeringMataria HelwanUniversity PO Box 11718 Helmeiat-Elzaton Cairo Egypt
Correspondence should be addressed to Tamer Ahmed El-Sayed tamer alsayedm-enghelwanedueg
Received 2 June 2017 Revised 22 August 2017 Accepted 15 October 2017 Published 28 November 2017
Academic Editor Toshiaki Natsuki
Copyright copy 2017 Tamer Ahmed El-Sayed and Said Hamed Farghaly This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited
The exact solution for multistepped Timoshenko beam is derived using a set of fundamental solutions This set of solutions isderived to normalize the solution at the origin of the coordinates The start end and intermediate boundary conditions involveconcentrated masses and linear and rotational elastic supports The beam start end and intermediate equations are assembledusing the present normalized transfer matrix (NTM) The advantage of this method is that it is quicker than the standard methodbecause the size of the complete system coefficient matrix is 4 times 4 In addition during the assembly of this matrix there are noinversematrix steps requiredThe validity of thismethod is tested by comparing the results of the currentmethodwith the literatureThen the validity of the exact stepped analysis is checked using experimental and FE(3D) methods The experimental results forstepped beams with single step and two steps for sixteen different test samples are in excellent agreement with those of the three-dimensional finite element FE(3D) The comparison between the NTM method and the finite element method results shows thatthe modal percentage deviation is increased when a beam step location coincides with a peak point in the mode shape Meanwhilethe deviation decreases when a beam step location coincides with a straight portion in the mode shape
1 Introduction
Stepped beam-like structure plays an important role in theconstruction of mechanical and civil engineering systemsFlexural vibrations were first investigated by Euler-Bernoulliin the eighteenth century The rotary inertia effect wasconsidered by Rayleigh [1] Almost 95 years ago Timoshenkointroduced a correction for the beam theory to include theshear deformation effect [2] The effect of rotary inertiaand shear deformations on the beam natural frequencies issmall at the lower normal modes and large at the highernormal modes Cowper [3] derived formulae for the preciseevaluation of the shear coefficient for rectangular and roundcross sections as a function of Poissonrsquos ratio ]
The free vibrations of beams with discontinuities canbe solved using either exact or approximate solution Theexact methods include the derivation of the transcendentaleigenvalue equations in order to evaluate the beam natural
frequencies In the case of relatively simple problems closedform solutions for the eigenvalue problem were obtained[4 5]
For complicated problems the beam eigenvalues areobtained by the decomposition of the domain Numericalassembly technique (NAT) is one of the common methodsused for the evaluation of the eigenvalue problem for beamswith multiple discontinuity [6ndash8] In this method the sizeof the frequency equation determinant is 4119899 times 4119899 for beamwith 119899 segments Dynamic stiffness matrix is one of methodsthat is similar to the finite element method in assemblingof the elements but with exact element rather than approx-imate element [9 10] The frequency equation determinantsize for 119899 segments is (2119899 + 2) times (2119899 + 2) The Laplacetransformation method is used to obtain a solution for aTimoshenko beam mounted on elastic foundation with sev-eral combinations of discrete in-span attachments and withseveral combinations of attachments at the boundaries [11]
HindawiShock and VibrationVolume 2017 Article ID 8186976 23 pageshttpsdoiorg10115520178186976
2 Shock and Vibration
The attachments include translation and rotational springsmasses and undamped single degree of freedom systemThe characteristics equation for beam with 119899 segments is2119899 times 2119899 The transfer matrix method (TMM) is one of thefavorable methods in the analysis of multispan beams Manyresearchers used it to derive the frequency equation of a com-plicated beam system [12ndash14]The advantage of thismethod isthat for 119899 segments beam the size of the frequency equationdeterminant is 4 times 4 which reduces the computational timeOn the other hand during the formulation of the beamfrequency equation 119899 minus 1 inverse matrix steps are requiredto form the final system transcendental eigenvalue problemThe use of linearly independent fundamental set of solutionin solving the buckling and free vibrations of nonuniformrods was introduced by Li [15] and Li et al [16] Thismethod enables obtaining the closed form solution of a mul-tispan beam A set of fundamental solutions which suit theanalysis of single-span Timoshenko beams was introduced[17]
On the other hand there are numerous approximatemethods to approximate the eigenvalue problem for thetransverse vibrations of beams [21ndash23] Finite elementmethod is one of the most dominant methods for solvingthe free vibration of beams The effect of step ratios andeccentricity on the free vibration of arbitrarily beam wasinvestigated by Ju et al [21] The lowest three naturalfrequencies of a multistep up and down cantilever beamusing a global Rayleigh-Ritz formulation component modalanalyses (CMA) ANSYS and experimental are evaluated[24] Adomian decomposition method (ADM) was used toobtain the effect of step ratio and step location on the beamnatural frequencies [25 26]The free and forced vibrations ofbeams with either single- or multiple-step changes using thecomposite element method (CEM) were introduced by Lu etal [27] The accuracy and convergence of CEM were com-pared with existing theoretical and experimental results Dif-ferential transformationmethod (DTM) was applied in orderto analyze the natural frequencies for different geometricallyand material parameters stepped Bernoulli-Euler beam [28]The differential quadrature element method (DQEM) wasproposed to analyze the free vibration problem of beams withany discontinuities in cross-section [20] Discrete singularconvolution (DSC) was proposed for solving the free vibra-tion analysis of stepped beams [29]The solution formultistepTimoshenko beam using both numerical assembly techniqueanddifferential transformationmethod is proposed byYesilce[19] Experimental measurements were considered as a goodtool for the validation of the analytical results
The exact free vibration of two-span Timoshenko steppedbeams has been investigated by Gutierrez et al [18] andRossi et al [32] Farghaly [33] derived the exact solution forfour-span Timoshenko beam with attachments Yesilce [19]investigated the free vibration of stepped beams using exactnumerical assembly technique (NAT) and using approximatedifferential transformation method Recently Farghaly andEl-Sayed [6] drive the exact solution for the lateral vibra-tion of Timoshenko beam with generalized start end andintermediate conditions using numerical assembly technique(NAT)
The exact free vibration of amechanical system composedof two elastic Timoshenko segments carried on an intermedi-ate eccentric rigid body or on elastic supports was introducedby Farghaly and El-Sayed [34] Their analysis was based onboth analytical and experimental methods They claimed agood agreement between the analytical and experimentalresults Experimental setup using electromagnetic-acoustictransducer (EMAT) was introduced by Dıaz-De-Anda et al[35] They compared the experimental results with thoseobtained theoretically using Timoshenko beam theory (TBT)with one and two shear coefficients The flexural frequenciesand amplitudes for cylindrical and rectangular Timoshenkobeams were examined experimentally [35] They found thatthe experimental results coincide very well with theoreti-cal predictions The transverse vibration of Bernoulli-Eulerbeams with discontinuous geometry and elastic support wasinvestigated experimentally and analytically [36]
During the last decades many literatures were focusedon the problem of free and forced vibration analysis ofTimoshenko beam and the accuracy of the natural frequencypredictions To the authorsrsquo knowledge there is not enoughresearch that has tackled the experimental modal frequenciesof stepped thick beams computationally and experimentallyTherefore the main aim of this work is to investigate theresults of the modal frequencies for such beams usinganalytical experimental and the three-dimensional finiteelement FE(3D) An analytical analysis is proposed which isbased on the derivation of a set of fundamental solutions thatsuits the analysis of Timoshenko beams This set of solutionsis used to modify the TMM to include no inverse matrixprocedure which may be called normalized transfer matrixmethod (NTM) The comparison between the experimentalNTM and FE(3D) is done for selected single-step and two-step application models The percentage deviations betweenNTM and FE(3D) are investigated The results show thatthe finite element results are very close to the experimentalresults The study includes the effect of increasing the stepratio (119889) step location parameter (120583) and the length ratioFinally the capability of the present analysis to solve the freevibration of tapered beams has been investigated
2 Mathematical Model
The mathematical model for beam with multiple-steppedsections is shown in Figure 1 The total length of the beamis 119871 The beam model is divided into 119899 segments Thebeam has (119899 + 1) stations as shown in Figure 1 The stationnumbering corresponding to the start intermediate and endlocation is represented by (1 119894 + 1 119899 + 1) respectively Ateach station there are linear and rotational elastic supportsand concentrated mass with mass moment of inertia Asshown in Figure 1 the beam segments are described bytheir material and cross-sectional properties 120588119894 119864119894 119866119894 119860 119894 and119868119894 (119894 = 1 2 119899) which are the density Youngrsquos modulus ofelasticity rigidity modulus cross-sectional area and secondmoment of inertia respectively In this section the frequencyequation of the model is driven using the proposed normal-ized transfer matrix method Since the current analysis isbased on Timoshenko beam theory the rotary inertia and
Shock and Vibration 3
1 2 i+1n n+1
L1 Li Li+1 Ln
m1 m2 mi+1
mn mn+1
J1 J2pan i
Station 1Station(n + 1)
Ji+1Jn
Jn+1
k1 k2 ki+1 kn kn+1
L
1 2
tation i + 1S
pan i + 1SS
Figure 1 Stepped multispan model
shear deformations effects are considered In the currentanalysis the analytical solution is subject to the assumptionsthat the shear strain is assumed constant over the cross-section therefore a shear coefficient 1198961015840119894 is used to compensatethis assumption In addition the effect of stress concentrationat the beam steps is neglected
21 AnalyticalMethod andFrequency Equation Theobjectiveof this section is to derive the system frequency equationwhich represents the model shown in Figure 1 Timoshenkodifferential coupled equations of motion may be written herefor 119894th span as follows
(1198961015840119866119860)119894(11991010158401015840119894 minus 1205951015840119894 ) (119909119894 119905) minus 120588119894119860 119894 119910119894 (119909119894 119905) = 0
(119864119868)119894 12059510158401015840119894 (119909119894 119905) minus 120588119894119868119894119894 (119909119894 119905)+ (1198961015840119866119860)
119894(1199101015840119894 minus 120595119894) (119909119894 119905) = 0
where 119884119894 is the normal function of 119910119894 Ψ119894 is the normalfunction of120595119894 120585119894 is nondimensional length of each beam span119894 and 119895 = radicminus1
Substituting (2) into (1) and omitting the factor 119890119895120596119905 thefollowing equations can be derived
After decoupling the functions 119884119894(120585119894) and 119871 119894Ψ119894(120585119894) in ((3)-(4)) the decoupled fourth-order differential equations in thenondimensional form can be written as
Here 119894 denotes the 119894th span 119894 = 1 2 119899 in the case ofmultispan model
In order to introduce the current analysis the linearlyindependent fundamental solutions 1198781198841119894(120585) 1198781198842119894(120585) 1198781198843119894(120585)1198781198844119894(120585) and the corresponding 119878Ψ1119894(120585) 119878Ψ2119894(120585) 119878Ψ3119894(120585)119878Ψ4119894(120585) are derived In order to simplify the solution ofTimoshenko beam the following dependent functions aredefined
Γ119894 (120585) = 1198841015840119894 (120585) minus 119871 119894Ψ119894 (120585) (11a)
119878Γ1119894 (120585) = 11987811988410158401119894 (120585) minus 119878Ψ1119894 (120585) (11b)
119878Γ2119894 (120585) = 11987811988410158402119894 (120585) minus 119878Ψ2119894 (120585) (11c)
119878Γ3119894 (120585) = 11987811988410158403119894 (120585) minus 119878Ψ3119894 (120585) (11d)
119878Γ4119894 (120585) = 11987811988410158404119894 (120585) minus 119878Ψ4119894 (120585) (11e)
The Timoshenko solution will be normalized at the origin ofcoordinates as follows
where 1199101198940 = 119884119894(0) 1205951198940 = 119871 119894Ψ119894(0) 12059510158401198940 = 119871 119894Ψ1015840119894 (0) and1205741198940= Γ119894(0)The beam start boundary conditions at the point of
attachment 1 can be presented in nondimensional form as
where 1198691 is the mass moment of inertia at station 1 1198961 and1206011 are the linear and rotational elastic supports at station 1respectively and1198981 is the concentrated mass at station 1 seeFigure 1 for details
Substituting the solutions presented in (13a) (13b) (13c)(13d) (13e) (13f) (13g) (13h)-(14) into (15) the followingequations are obtained
(119869112058241 minus Φ1) 12059510 + 120595101584010 = 011990421 (119898112058241 minus 1198851) 11991010 + 12057410 = 0 (17)
The start boundary conditions in (17) can be presented inmatrix form as
[[
0 (119869112058241 minus 1206011) 1 011990421 (119898112058241 minus 1198851) 0 0 1]]
where 119869119899+1 is the mass moment of inertia at station (119899 + 1)119896119899+1 and 120601119899+1 are the linear and rotational elastic supports atstation (119899+1) respectively and119898119899+1 is the concentratedmassat station (119899 + 1)
Substituting the solutions in (13a) (13b) (13c) (13d)(13e) (13f) (13g) (13h)-(14) into (21) the following equationsare obtained
1199101198990 (119878Ψ10158401119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ1119899)+ 1205951198990 (119878Ψ10158402119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ2119899)+ 12059510158401198990 (119878Ψ10158403119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ3119899)+ 1205741198990 (119878Ψ10158404119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ4119899) = 0
where 119869119894+1 is the mass moment of inertia at station (119894+1) 119896119894+1and 120601119894+1 are the linear and rotational stiffness at station (119894 +1) respectively and 119898119894+1 is the concentrated mass at station(119894 + 1)
Substituting the solutions in (13a) (13b) (13c) (13d)(13e) (13f) (13g) (13h)-(14) into (27) we get the followingequations
Substituting (36) into the end condition of (25) results in thefollowing equations
[UE]2times4
[T]4times4 Δ1 = 0 (37)
[UIE]2times4
= [UE]2times4
[T]4times4 (38)
[UIE]2times4
Δ1 = 0 (39)
The general beam equation can be presented using the startboundary condition in (19) and the beam intermediate andend condition in (39) as shown below
[Utot]4times4
Δ1 = 0 (40)
where
[Utot]4times4
= [[[US]2times4[UIE]2times4
]] (41)
Equating the determinant of [Utot] by zero results in thesystem frequency equation In general the TMM has advan-tages over the traditional methods in that the final frequencyequation is 4 times 4 for any number of beam segments Theadvantage of the current method NTM over the TMM issignificant in the using of tailored solution that is normalizedat the origin of coordinates This type of solution enables theformulation of the system equations without the need to anyinverse matrix procedures as shown previously This reducesthe computational time comparing with the TMM
22 Finite Element Method Among the numerical toolsfinite element method is considered one of most efficientmethods to perform the vibration analysis of mechanical andstructural components In this section finite element is usedto obtain the natural frequencies andmode shapes of uniformand stepped beams ANSYS finite element commercial pack-age is used to perform the finite element analysisThe analysisis done using three-dimensional (3D) solid element modelsand SOLID95 elements are used for meshing Since all theexperimentally investigated samples in the current work are
round and stepped The beam cross-section is free meshedusing 87 SOLID95 elements for smaller cross-section and171 elements for the larger cross-section This mesh is thenextruded using 40 elements along the length of the beamThe total number of the element is ranging from 3480 (40times 87) to 6840 (40 times 171) elements based on the locationof the step see Figure 2 Modal analysis module is usedin this analysis and Block Lanczos method is used for themode extraction method The finite element model FE(3D)results are compared with those obtained experimentally andanalytically
3 Results and Discussion
31 Verification and Validation of NTM Results
311 Verification Example 1 In this example the first fivenondimensional natural frequencies (120582lowast4119894 = 120588111986011198714120596211986411198681)of stepped beam are compared with the exact solutionpresented by Gutierrez et al [18] see Figure 3 The modelis solved at two different step locations and several valuesof 11988721 and ℎ21 as shown in Table 1 Three different valuesof rotary inertia 11990321 = V119904 00036 and 001 are consideredin order to validate the current model in case of Bernoulli-Euler and Timoshenko beams The value of 11990421 = 31211990321 isconsidered in order to evaluate the shear deformation [18]The values of nondimensional linear and rotational elasticsupports stiffness at start and the end are (1198851 = 10Φ1 = V119897)(1198853 = V119904 Φ3 = V119897) respectively The results of Table 1 showthat the present NTM results are in good agreement with theexact solution presented by Gutierrez et al [18]
312 Verification Example 2 In this example Timoshenkobeam with three-step round cross-section presented in [19]is investigated see Figure 4 An intermediate lumped massof 119898lowast3 = 119898312058811198601119871 = 1 is located at a distance of 750mmfrom point 1 The input data for this example is listed inthe caption of Figure 4 Table 2 shows the results of the firstfive natural frequencies 120596119894 in (radsec) for pinned-pinnedfree-clamped clamped-free clamped-pinned and clamped-clamped configurations The results of reference [19] are
8 Shock and Vibration
Table 1 The first five natural frequencies 120582lowast2119894 for stepped Timoshenko beam in the case where the beam is rigidly restrained against rotation120601lowast1 = V119897 and elastically restrained in translation 119885lowast1 = 10 in comparison with [18]
Figure 3 Verification Example 1 [18] two-span stepped beam
Shock and Vibration 9
Table 2 Comparison of the first five natural frequencies in radsec using the present NTM results with those obtained in [19] for the exampleshown in Figure 4
Figure 4 Verification Example 2 [19] three-step beam having 119864 = 2069GPa 120588 = 78369 kgm3 119866 = 795769GPa and 1198961015840 = 075
10 Shock and Vibration
Table 3 Comparison of the present NTM results with those obtained in [5 20] for F-F beam shown in Figure 5
Mode number DQEM [20] Exp [5] NTM(40) FE(3D) error
I II III IV (II amp IV)1 292440 291 290316 290120 03022 1181300 1165 1162158 1167100 minus01803 1804100 1771 1767321 1772200 minus0068
1
254 140
32
549
254mm19
05
mm
Figure 5 Verification Example 3 [5 20] single-step (F-F) aluminum beam having 119864 = 717GPa 120588 = 2830Kgm3 and ] = 033
L20
L
1
1
mi
2 3 ik1
19 2021
21
k21
Figure 6 Verification Example 4 twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general endflexibilities conditions
calculated using two methods The first method is numericalassembly technique (NAT) and this method is the exactsolution The second method is differential transformationmethod and thismethod is approximatemethodThepinned-pinned and free-clamped boundary conditions results arefound in [19] The results of the current analysis are in goodagreement with the results presented in [19] As can be seenfromTable 2 the rotary inertia and shear deformation reducethemodal frequency especially for highermodes Comparingthe results of the clamped-free beam with the results of thefree-clamped beam shows that the natural frequency resultsof the clamped-free beam are lower than that of free-clampedbeam This may be explained by the fact that fixing thebeam from the thinner span results in lowering of the beamstiffness
313 Verification Example 3 The third verification exampleis shown in Figure 5 with identical dimensions to that usedin [5 20] The rotary inertia and shear deformation areconsidered The shear coefficient 1198961015840 is calculated based on[3] keeping Poissonrsquosrsquo ratio ] = 033 Table 3 shows thefirst three nonzero free-free (F-F) eigenvalues in Hz and incomparison with [5 20] The present results are computedusing both numerically (40) and FE(3D)methodsThe resultsof Table 3 show that the present analysis results are very closeto the experimental resultsThe percentage error between the
present FE(3D) results and the experimental results of [5] isless than 0302This represents the importance of includingthe effect of rotary inertia and shear deformation
314 Verification Example 4 The fourth example is shownin Figure 6 It is for a twenty-span uniform beam carrying19 equally spaced concentrated masses with 119898lowast119894 = 01Several beam start and end conditions are investigated aslisted in Table 4 The results are evaluated using the presentNTM method and previously published numerical assemblytechniqueNAT [6]methodThe computational time requiredto obtain the first three frequency parameters 120582lowast1 120582lowast2 and 120582lowast3using both NAT and NTM methods is calculated and listedin Table 4 Considerable reduction in computational time isobserved in all the investigated cases as shown in Table 4
32 Test Samples and Experimental Procedures In order tomeasure the natural frequencies of the system under studythe free-free test samples were put in free oscillations byusing an instrumented hammer model B and K 8202 Anaccelerometer model B and K 4366 is fixed to the shaft inorder to capture the vibration signal The output of thecharge amplifier B and K 2635 is connected to NI 6216data acquisition card This card is connected to the PC andmanaged by Lab VIEW software Figure 10 shows a photo ofthe current experimental setup The used card settings are
Shock and Vibration 11
Table 4 Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated masses119898lowast119894 = 01 (119903lowast21 = 119904lowast21 = V119904)in comparison with [6]
Figure 7 Schematic drawings for the experimental samples Group S2
sample frequency of 20 kHz sampling time of 3 5 8 sec andthe size of samples block to read is 1 k The time domaindata is captured and transformed into frequency domainThe resonant frequencies were obtained by the average of theresults of 10 impacts applied in three different locations ofthe sample Figures 7ndash9 show two groups of different steppedtest samples S2 and S3 respectively All the test samplesare manufactured from steel rods of dimensions Oslash 80 andOslash 40mm The common data for the steel test samples are120588 = 7850 kgm3 119864 = 2068GPa and 1198961015840 = 0845
Two-span twelve test samples are shown in group S2namely S2-4040 S2-40100 and S2-40140 shown inFigure 7 The dimensions of these samples are presentedin Table 5 The natural frequency results of these samples
are presented in Table 7 using experimental analytical andFE(3D) methods The captured signal for the free vibrationresponse for selected case from Table 7 is shown in Figure 11This case is italicized in the table The results of Table 7show that the three-dimensional FE solution is closer tothe experimental results than the analytical results Themaximum percentage error between the experimental andthe FE results is less than 05
Group S3 consists of four different samples namely S3-80250 S3-80200 S3-80150 and S3-80100 More detailsabout the geometrical dimensions and material propertiesfor these samples are shown in Table 6 and Figure 8 Theresults of Table 8 show that the deviation between the FE(3D)and the experimental is less than 116 The free vibration
12 Shock and Vibration
Table 6 The sample dimensions of group S3 as shown in Figure 8
Group S3 (Four cylindrical samples have two steps of steel rod Oslash 80 119871 = 500mm)119871 1198711 1198712 1198891 1198892 119898119905 (kg) 119871lowast2 119903lowast21
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
The attachments include translation and rotational springsmasses and undamped single degree of freedom systemThe characteristics equation for beam with 119899 segments is2119899 times 2119899 The transfer matrix method (TMM) is one of thefavorable methods in the analysis of multispan beams Manyresearchers used it to derive the frequency equation of a com-plicated beam system [12ndash14]The advantage of thismethod isthat for 119899 segments beam the size of the frequency equationdeterminant is 4 times 4 which reduces the computational timeOn the other hand during the formulation of the beamfrequency equation 119899 minus 1 inverse matrix steps are requiredto form the final system transcendental eigenvalue problemThe use of linearly independent fundamental set of solutionin solving the buckling and free vibrations of nonuniformrods was introduced by Li [15] and Li et al [16] Thismethod enables obtaining the closed form solution of a mul-tispan beam A set of fundamental solutions which suit theanalysis of single-span Timoshenko beams was introduced[17]
On the other hand there are numerous approximatemethods to approximate the eigenvalue problem for thetransverse vibrations of beams [21ndash23] Finite elementmethod is one of the most dominant methods for solvingthe free vibration of beams The effect of step ratios andeccentricity on the free vibration of arbitrarily beam wasinvestigated by Ju et al [21] The lowest three naturalfrequencies of a multistep up and down cantilever beamusing a global Rayleigh-Ritz formulation component modalanalyses (CMA) ANSYS and experimental are evaluated[24] Adomian decomposition method (ADM) was used toobtain the effect of step ratio and step location on the beamnatural frequencies [25 26]The free and forced vibrations ofbeams with either single- or multiple-step changes using thecomposite element method (CEM) were introduced by Lu etal [27] The accuracy and convergence of CEM were com-pared with existing theoretical and experimental results Dif-ferential transformationmethod (DTM) was applied in orderto analyze the natural frequencies for different geometricallyand material parameters stepped Bernoulli-Euler beam [28]The differential quadrature element method (DQEM) wasproposed to analyze the free vibration problem of beams withany discontinuities in cross-section [20] Discrete singularconvolution (DSC) was proposed for solving the free vibra-tion analysis of stepped beams [29]The solution formultistepTimoshenko beam using both numerical assembly techniqueanddifferential transformationmethod is proposed byYesilce[19] Experimental measurements were considered as a goodtool for the validation of the analytical results
The exact free vibration of two-span Timoshenko steppedbeams has been investigated by Gutierrez et al [18] andRossi et al [32] Farghaly [33] derived the exact solution forfour-span Timoshenko beam with attachments Yesilce [19]investigated the free vibration of stepped beams using exactnumerical assembly technique (NAT) and using approximatedifferential transformation method Recently Farghaly andEl-Sayed [6] drive the exact solution for the lateral vibra-tion of Timoshenko beam with generalized start end andintermediate conditions using numerical assembly technique(NAT)
The exact free vibration of amechanical system composedof two elastic Timoshenko segments carried on an intermedi-ate eccentric rigid body or on elastic supports was introducedby Farghaly and El-Sayed [34] Their analysis was based onboth analytical and experimental methods They claimed agood agreement between the analytical and experimentalresults Experimental setup using electromagnetic-acoustictransducer (EMAT) was introduced by Dıaz-De-Anda et al[35] They compared the experimental results with thoseobtained theoretically using Timoshenko beam theory (TBT)with one and two shear coefficients The flexural frequenciesand amplitudes for cylindrical and rectangular Timoshenkobeams were examined experimentally [35] They found thatthe experimental results coincide very well with theoreti-cal predictions The transverse vibration of Bernoulli-Eulerbeams with discontinuous geometry and elastic support wasinvestigated experimentally and analytically [36]
During the last decades many literatures were focusedon the problem of free and forced vibration analysis ofTimoshenko beam and the accuracy of the natural frequencypredictions To the authorsrsquo knowledge there is not enoughresearch that has tackled the experimental modal frequenciesof stepped thick beams computationally and experimentallyTherefore the main aim of this work is to investigate theresults of the modal frequencies for such beams usinganalytical experimental and the three-dimensional finiteelement FE(3D) An analytical analysis is proposed which isbased on the derivation of a set of fundamental solutions thatsuits the analysis of Timoshenko beams This set of solutionsis used to modify the TMM to include no inverse matrixprocedure which may be called normalized transfer matrixmethod (NTM) The comparison between the experimentalNTM and FE(3D) is done for selected single-step and two-step application models The percentage deviations betweenNTM and FE(3D) are investigated The results show thatthe finite element results are very close to the experimentalresults The study includes the effect of increasing the stepratio (119889) step location parameter (120583) and the length ratioFinally the capability of the present analysis to solve the freevibration of tapered beams has been investigated
2 Mathematical Model
The mathematical model for beam with multiple-steppedsections is shown in Figure 1 The total length of the beamis 119871 The beam model is divided into 119899 segments Thebeam has (119899 + 1) stations as shown in Figure 1 The stationnumbering corresponding to the start intermediate and endlocation is represented by (1 119894 + 1 119899 + 1) respectively Ateach station there are linear and rotational elastic supportsand concentrated mass with mass moment of inertia Asshown in Figure 1 the beam segments are described bytheir material and cross-sectional properties 120588119894 119864119894 119866119894 119860 119894 and119868119894 (119894 = 1 2 119899) which are the density Youngrsquos modulus ofelasticity rigidity modulus cross-sectional area and secondmoment of inertia respectively In this section the frequencyequation of the model is driven using the proposed normal-ized transfer matrix method Since the current analysis isbased on Timoshenko beam theory the rotary inertia and
Shock and Vibration 3
1 2 i+1n n+1
L1 Li Li+1 Ln
m1 m2 mi+1
mn mn+1
J1 J2pan i
Station 1Station(n + 1)
Ji+1Jn
Jn+1
k1 k2 ki+1 kn kn+1
L
1 2
tation i + 1S
pan i + 1SS
Figure 1 Stepped multispan model
shear deformations effects are considered In the currentanalysis the analytical solution is subject to the assumptionsthat the shear strain is assumed constant over the cross-section therefore a shear coefficient 1198961015840119894 is used to compensatethis assumption In addition the effect of stress concentrationat the beam steps is neglected
21 AnalyticalMethod andFrequency Equation Theobjectiveof this section is to derive the system frequency equationwhich represents the model shown in Figure 1 Timoshenkodifferential coupled equations of motion may be written herefor 119894th span as follows
(1198961015840119866119860)119894(11991010158401015840119894 minus 1205951015840119894 ) (119909119894 119905) minus 120588119894119860 119894 119910119894 (119909119894 119905) = 0
(119864119868)119894 12059510158401015840119894 (119909119894 119905) minus 120588119894119868119894119894 (119909119894 119905)+ (1198961015840119866119860)
119894(1199101015840119894 minus 120595119894) (119909119894 119905) = 0
where 119884119894 is the normal function of 119910119894 Ψ119894 is the normalfunction of120595119894 120585119894 is nondimensional length of each beam span119894 and 119895 = radicminus1
Substituting (2) into (1) and omitting the factor 119890119895120596119905 thefollowing equations can be derived
After decoupling the functions 119884119894(120585119894) and 119871 119894Ψ119894(120585119894) in ((3)-(4)) the decoupled fourth-order differential equations in thenondimensional form can be written as
Here 119894 denotes the 119894th span 119894 = 1 2 119899 in the case ofmultispan model
In order to introduce the current analysis the linearlyindependent fundamental solutions 1198781198841119894(120585) 1198781198842119894(120585) 1198781198843119894(120585)1198781198844119894(120585) and the corresponding 119878Ψ1119894(120585) 119878Ψ2119894(120585) 119878Ψ3119894(120585)119878Ψ4119894(120585) are derived In order to simplify the solution ofTimoshenko beam the following dependent functions aredefined
Γ119894 (120585) = 1198841015840119894 (120585) minus 119871 119894Ψ119894 (120585) (11a)
119878Γ1119894 (120585) = 11987811988410158401119894 (120585) minus 119878Ψ1119894 (120585) (11b)
119878Γ2119894 (120585) = 11987811988410158402119894 (120585) minus 119878Ψ2119894 (120585) (11c)
119878Γ3119894 (120585) = 11987811988410158403119894 (120585) minus 119878Ψ3119894 (120585) (11d)
119878Γ4119894 (120585) = 11987811988410158404119894 (120585) minus 119878Ψ4119894 (120585) (11e)
The Timoshenko solution will be normalized at the origin ofcoordinates as follows
where 1199101198940 = 119884119894(0) 1205951198940 = 119871 119894Ψ119894(0) 12059510158401198940 = 119871 119894Ψ1015840119894 (0) and1205741198940= Γ119894(0)The beam start boundary conditions at the point of
attachment 1 can be presented in nondimensional form as
where 1198691 is the mass moment of inertia at station 1 1198961 and1206011 are the linear and rotational elastic supports at station 1respectively and1198981 is the concentrated mass at station 1 seeFigure 1 for details
Substituting the solutions presented in (13a) (13b) (13c)(13d) (13e) (13f) (13g) (13h)-(14) into (15) the followingequations are obtained
(119869112058241 minus Φ1) 12059510 + 120595101584010 = 011990421 (119898112058241 minus 1198851) 11991010 + 12057410 = 0 (17)
The start boundary conditions in (17) can be presented inmatrix form as
[[
0 (119869112058241 minus 1206011) 1 011990421 (119898112058241 minus 1198851) 0 0 1]]
where 119869119899+1 is the mass moment of inertia at station (119899 + 1)119896119899+1 and 120601119899+1 are the linear and rotational elastic supports atstation (119899+1) respectively and119898119899+1 is the concentratedmassat station (119899 + 1)
Substituting the solutions in (13a) (13b) (13c) (13d)(13e) (13f) (13g) (13h)-(14) into (21) the following equationsare obtained
1199101198990 (119878Ψ10158401119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ1119899)+ 1205951198990 (119878Ψ10158402119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ2119899)+ 12059510158401198990 (119878Ψ10158403119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ3119899)+ 1205741198990 (119878Ψ10158404119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ4119899) = 0
where 119869119894+1 is the mass moment of inertia at station (119894+1) 119896119894+1and 120601119894+1 are the linear and rotational stiffness at station (119894 +1) respectively and 119898119894+1 is the concentrated mass at station(119894 + 1)
Substituting the solutions in (13a) (13b) (13c) (13d)(13e) (13f) (13g) (13h)-(14) into (27) we get the followingequations
Substituting (36) into the end condition of (25) results in thefollowing equations
[UE]2times4
[T]4times4 Δ1 = 0 (37)
[UIE]2times4
= [UE]2times4
[T]4times4 (38)
[UIE]2times4
Δ1 = 0 (39)
The general beam equation can be presented using the startboundary condition in (19) and the beam intermediate andend condition in (39) as shown below
[Utot]4times4
Δ1 = 0 (40)
where
[Utot]4times4
= [[[US]2times4[UIE]2times4
]] (41)
Equating the determinant of [Utot] by zero results in thesystem frequency equation In general the TMM has advan-tages over the traditional methods in that the final frequencyequation is 4 times 4 for any number of beam segments Theadvantage of the current method NTM over the TMM issignificant in the using of tailored solution that is normalizedat the origin of coordinates This type of solution enables theformulation of the system equations without the need to anyinverse matrix procedures as shown previously This reducesthe computational time comparing with the TMM
22 Finite Element Method Among the numerical toolsfinite element method is considered one of most efficientmethods to perform the vibration analysis of mechanical andstructural components In this section finite element is usedto obtain the natural frequencies andmode shapes of uniformand stepped beams ANSYS finite element commercial pack-age is used to perform the finite element analysisThe analysisis done using three-dimensional (3D) solid element modelsand SOLID95 elements are used for meshing Since all theexperimentally investigated samples in the current work are
round and stepped The beam cross-section is free meshedusing 87 SOLID95 elements for smaller cross-section and171 elements for the larger cross-section This mesh is thenextruded using 40 elements along the length of the beamThe total number of the element is ranging from 3480 (40times 87) to 6840 (40 times 171) elements based on the locationof the step see Figure 2 Modal analysis module is usedin this analysis and Block Lanczos method is used for themode extraction method The finite element model FE(3D)results are compared with those obtained experimentally andanalytically
3 Results and Discussion
31 Verification and Validation of NTM Results
311 Verification Example 1 In this example the first fivenondimensional natural frequencies (120582lowast4119894 = 120588111986011198714120596211986411198681)of stepped beam are compared with the exact solutionpresented by Gutierrez et al [18] see Figure 3 The modelis solved at two different step locations and several valuesof 11988721 and ℎ21 as shown in Table 1 Three different valuesof rotary inertia 11990321 = V119904 00036 and 001 are consideredin order to validate the current model in case of Bernoulli-Euler and Timoshenko beams The value of 11990421 = 31211990321 isconsidered in order to evaluate the shear deformation [18]The values of nondimensional linear and rotational elasticsupports stiffness at start and the end are (1198851 = 10Φ1 = V119897)(1198853 = V119904 Φ3 = V119897) respectively The results of Table 1 showthat the present NTM results are in good agreement with theexact solution presented by Gutierrez et al [18]
312 Verification Example 2 In this example Timoshenkobeam with three-step round cross-section presented in [19]is investigated see Figure 4 An intermediate lumped massof 119898lowast3 = 119898312058811198601119871 = 1 is located at a distance of 750mmfrom point 1 The input data for this example is listed inthe caption of Figure 4 Table 2 shows the results of the firstfive natural frequencies 120596119894 in (radsec) for pinned-pinnedfree-clamped clamped-free clamped-pinned and clamped-clamped configurations The results of reference [19] are
8 Shock and Vibration
Table 1 The first five natural frequencies 120582lowast2119894 for stepped Timoshenko beam in the case where the beam is rigidly restrained against rotation120601lowast1 = V119897 and elastically restrained in translation 119885lowast1 = 10 in comparison with [18]
Figure 3 Verification Example 1 [18] two-span stepped beam
Shock and Vibration 9
Table 2 Comparison of the first five natural frequencies in radsec using the present NTM results with those obtained in [19] for the exampleshown in Figure 4
Figure 4 Verification Example 2 [19] three-step beam having 119864 = 2069GPa 120588 = 78369 kgm3 119866 = 795769GPa and 1198961015840 = 075
10 Shock and Vibration
Table 3 Comparison of the present NTM results with those obtained in [5 20] for F-F beam shown in Figure 5
Mode number DQEM [20] Exp [5] NTM(40) FE(3D) error
I II III IV (II amp IV)1 292440 291 290316 290120 03022 1181300 1165 1162158 1167100 minus01803 1804100 1771 1767321 1772200 minus0068
1
254 140
32
549
254mm19
05
mm
Figure 5 Verification Example 3 [5 20] single-step (F-F) aluminum beam having 119864 = 717GPa 120588 = 2830Kgm3 and ] = 033
L20
L
1
1
mi
2 3 ik1
19 2021
21
k21
Figure 6 Verification Example 4 twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general endflexibilities conditions
calculated using two methods The first method is numericalassembly technique (NAT) and this method is the exactsolution The second method is differential transformationmethod and thismethod is approximatemethodThepinned-pinned and free-clamped boundary conditions results arefound in [19] The results of the current analysis are in goodagreement with the results presented in [19] As can be seenfromTable 2 the rotary inertia and shear deformation reducethemodal frequency especially for highermodes Comparingthe results of the clamped-free beam with the results of thefree-clamped beam shows that the natural frequency resultsof the clamped-free beam are lower than that of free-clampedbeam This may be explained by the fact that fixing thebeam from the thinner span results in lowering of the beamstiffness
313 Verification Example 3 The third verification exampleis shown in Figure 5 with identical dimensions to that usedin [5 20] The rotary inertia and shear deformation areconsidered The shear coefficient 1198961015840 is calculated based on[3] keeping Poissonrsquosrsquo ratio ] = 033 Table 3 shows thefirst three nonzero free-free (F-F) eigenvalues in Hz and incomparison with [5 20] The present results are computedusing both numerically (40) and FE(3D)methodsThe resultsof Table 3 show that the present analysis results are very closeto the experimental resultsThe percentage error between the
present FE(3D) results and the experimental results of [5] isless than 0302This represents the importance of includingthe effect of rotary inertia and shear deformation
314 Verification Example 4 The fourth example is shownin Figure 6 It is for a twenty-span uniform beam carrying19 equally spaced concentrated masses with 119898lowast119894 = 01Several beam start and end conditions are investigated aslisted in Table 4 The results are evaluated using the presentNTM method and previously published numerical assemblytechniqueNAT [6]methodThe computational time requiredto obtain the first three frequency parameters 120582lowast1 120582lowast2 and 120582lowast3using both NAT and NTM methods is calculated and listedin Table 4 Considerable reduction in computational time isobserved in all the investigated cases as shown in Table 4
32 Test Samples and Experimental Procedures In order tomeasure the natural frequencies of the system under studythe free-free test samples were put in free oscillations byusing an instrumented hammer model B and K 8202 Anaccelerometer model B and K 4366 is fixed to the shaft inorder to capture the vibration signal The output of thecharge amplifier B and K 2635 is connected to NI 6216data acquisition card This card is connected to the PC andmanaged by Lab VIEW software Figure 10 shows a photo ofthe current experimental setup The used card settings are
Shock and Vibration 11
Table 4 Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated masses119898lowast119894 = 01 (119903lowast21 = 119904lowast21 = V119904)in comparison with [6]
Figure 7 Schematic drawings for the experimental samples Group S2
sample frequency of 20 kHz sampling time of 3 5 8 sec andthe size of samples block to read is 1 k The time domaindata is captured and transformed into frequency domainThe resonant frequencies were obtained by the average of theresults of 10 impacts applied in three different locations ofthe sample Figures 7ndash9 show two groups of different steppedtest samples S2 and S3 respectively All the test samplesare manufactured from steel rods of dimensions Oslash 80 andOslash 40mm The common data for the steel test samples are120588 = 7850 kgm3 119864 = 2068GPa and 1198961015840 = 0845
Two-span twelve test samples are shown in group S2namely S2-4040 S2-40100 and S2-40140 shown inFigure 7 The dimensions of these samples are presentedin Table 5 The natural frequency results of these samples
are presented in Table 7 using experimental analytical andFE(3D) methods The captured signal for the free vibrationresponse for selected case from Table 7 is shown in Figure 11This case is italicized in the table The results of Table 7show that the three-dimensional FE solution is closer tothe experimental results than the analytical results Themaximum percentage error between the experimental andthe FE results is less than 05
Group S3 consists of four different samples namely S3-80250 S3-80200 S3-80150 and S3-80100 More detailsabout the geometrical dimensions and material propertiesfor these samples are shown in Table 6 and Figure 8 Theresults of Table 8 show that the deviation between the FE(3D)and the experimental is less than 116 The free vibration
12 Shock and Vibration
Table 6 The sample dimensions of group S3 as shown in Figure 8
Group S3 (Four cylindrical samples have two steps of steel rod Oslash 80 119871 = 500mm)119871 1198711 1198712 1198891 1198892 119898119905 (kg) 119871lowast2 119903lowast21
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
shear deformations effects are considered In the currentanalysis the analytical solution is subject to the assumptionsthat the shear strain is assumed constant over the cross-section therefore a shear coefficient 1198961015840119894 is used to compensatethis assumption In addition the effect of stress concentrationat the beam steps is neglected
21 AnalyticalMethod andFrequency Equation Theobjectiveof this section is to derive the system frequency equationwhich represents the model shown in Figure 1 Timoshenkodifferential coupled equations of motion may be written herefor 119894th span as follows
(1198961015840119866119860)119894(11991010158401015840119894 minus 1205951015840119894 ) (119909119894 119905) minus 120588119894119860 119894 119910119894 (119909119894 119905) = 0
(119864119868)119894 12059510158401015840119894 (119909119894 119905) minus 120588119894119868119894119894 (119909119894 119905)+ (1198961015840119866119860)
119894(1199101015840119894 minus 120595119894) (119909119894 119905) = 0
where 119884119894 is the normal function of 119910119894 Ψ119894 is the normalfunction of120595119894 120585119894 is nondimensional length of each beam span119894 and 119895 = radicminus1
Substituting (2) into (1) and omitting the factor 119890119895120596119905 thefollowing equations can be derived
After decoupling the functions 119884119894(120585119894) and 119871 119894Ψ119894(120585119894) in ((3)-(4)) the decoupled fourth-order differential equations in thenondimensional form can be written as
Here 119894 denotes the 119894th span 119894 = 1 2 119899 in the case ofmultispan model
In order to introduce the current analysis the linearlyindependent fundamental solutions 1198781198841119894(120585) 1198781198842119894(120585) 1198781198843119894(120585)1198781198844119894(120585) and the corresponding 119878Ψ1119894(120585) 119878Ψ2119894(120585) 119878Ψ3119894(120585)119878Ψ4119894(120585) are derived In order to simplify the solution ofTimoshenko beam the following dependent functions aredefined
Γ119894 (120585) = 1198841015840119894 (120585) minus 119871 119894Ψ119894 (120585) (11a)
119878Γ1119894 (120585) = 11987811988410158401119894 (120585) minus 119878Ψ1119894 (120585) (11b)
119878Γ2119894 (120585) = 11987811988410158402119894 (120585) minus 119878Ψ2119894 (120585) (11c)
119878Γ3119894 (120585) = 11987811988410158403119894 (120585) minus 119878Ψ3119894 (120585) (11d)
119878Γ4119894 (120585) = 11987811988410158404119894 (120585) minus 119878Ψ4119894 (120585) (11e)
The Timoshenko solution will be normalized at the origin ofcoordinates as follows
where 1199101198940 = 119884119894(0) 1205951198940 = 119871 119894Ψ119894(0) 12059510158401198940 = 119871 119894Ψ1015840119894 (0) and1205741198940= Γ119894(0)The beam start boundary conditions at the point of
attachment 1 can be presented in nondimensional form as
where 1198691 is the mass moment of inertia at station 1 1198961 and1206011 are the linear and rotational elastic supports at station 1respectively and1198981 is the concentrated mass at station 1 seeFigure 1 for details
Substituting the solutions presented in (13a) (13b) (13c)(13d) (13e) (13f) (13g) (13h)-(14) into (15) the followingequations are obtained
(119869112058241 minus Φ1) 12059510 + 120595101584010 = 011990421 (119898112058241 minus 1198851) 11991010 + 12057410 = 0 (17)
The start boundary conditions in (17) can be presented inmatrix form as
[[
0 (119869112058241 minus 1206011) 1 011990421 (119898112058241 minus 1198851) 0 0 1]]
where 119869119899+1 is the mass moment of inertia at station (119899 + 1)119896119899+1 and 120601119899+1 are the linear and rotational elastic supports atstation (119899+1) respectively and119898119899+1 is the concentratedmassat station (119899 + 1)
Substituting the solutions in (13a) (13b) (13c) (13d)(13e) (13f) (13g) (13h)-(14) into (21) the following equationsare obtained
1199101198990 (119878Ψ10158401119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ1119899)+ 1205951198990 (119878Ψ10158402119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ2119899)+ 12059510158401198990 (119878Ψ10158403119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ3119899)+ 1205741198990 (119878Ψ10158404119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ4119899) = 0
where 119869119894+1 is the mass moment of inertia at station (119894+1) 119896119894+1and 120601119894+1 are the linear and rotational stiffness at station (119894 +1) respectively and 119898119894+1 is the concentrated mass at station(119894 + 1)
Substituting the solutions in (13a) (13b) (13c) (13d)(13e) (13f) (13g) (13h)-(14) into (27) we get the followingequations
Substituting (36) into the end condition of (25) results in thefollowing equations
[UE]2times4
[T]4times4 Δ1 = 0 (37)
[UIE]2times4
= [UE]2times4
[T]4times4 (38)
[UIE]2times4
Δ1 = 0 (39)
The general beam equation can be presented using the startboundary condition in (19) and the beam intermediate andend condition in (39) as shown below
[Utot]4times4
Δ1 = 0 (40)
where
[Utot]4times4
= [[[US]2times4[UIE]2times4
]] (41)
Equating the determinant of [Utot] by zero results in thesystem frequency equation In general the TMM has advan-tages over the traditional methods in that the final frequencyequation is 4 times 4 for any number of beam segments Theadvantage of the current method NTM over the TMM issignificant in the using of tailored solution that is normalizedat the origin of coordinates This type of solution enables theformulation of the system equations without the need to anyinverse matrix procedures as shown previously This reducesthe computational time comparing with the TMM
22 Finite Element Method Among the numerical toolsfinite element method is considered one of most efficientmethods to perform the vibration analysis of mechanical andstructural components In this section finite element is usedto obtain the natural frequencies andmode shapes of uniformand stepped beams ANSYS finite element commercial pack-age is used to perform the finite element analysisThe analysisis done using three-dimensional (3D) solid element modelsand SOLID95 elements are used for meshing Since all theexperimentally investigated samples in the current work are
round and stepped The beam cross-section is free meshedusing 87 SOLID95 elements for smaller cross-section and171 elements for the larger cross-section This mesh is thenextruded using 40 elements along the length of the beamThe total number of the element is ranging from 3480 (40times 87) to 6840 (40 times 171) elements based on the locationof the step see Figure 2 Modal analysis module is usedin this analysis and Block Lanczos method is used for themode extraction method The finite element model FE(3D)results are compared with those obtained experimentally andanalytically
3 Results and Discussion
31 Verification and Validation of NTM Results
311 Verification Example 1 In this example the first fivenondimensional natural frequencies (120582lowast4119894 = 120588111986011198714120596211986411198681)of stepped beam are compared with the exact solutionpresented by Gutierrez et al [18] see Figure 3 The modelis solved at two different step locations and several valuesof 11988721 and ℎ21 as shown in Table 1 Three different valuesof rotary inertia 11990321 = V119904 00036 and 001 are consideredin order to validate the current model in case of Bernoulli-Euler and Timoshenko beams The value of 11990421 = 31211990321 isconsidered in order to evaluate the shear deformation [18]The values of nondimensional linear and rotational elasticsupports stiffness at start and the end are (1198851 = 10Φ1 = V119897)(1198853 = V119904 Φ3 = V119897) respectively The results of Table 1 showthat the present NTM results are in good agreement with theexact solution presented by Gutierrez et al [18]
312 Verification Example 2 In this example Timoshenkobeam with three-step round cross-section presented in [19]is investigated see Figure 4 An intermediate lumped massof 119898lowast3 = 119898312058811198601119871 = 1 is located at a distance of 750mmfrom point 1 The input data for this example is listed inthe caption of Figure 4 Table 2 shows the results of the firstfive natural frequencies 120596119894 in (radsec) for pinned-pinnedfree-clamped clamped-free clamped-pinned and clamped-clamped configurations The results of reference [19] are
8 Shock and Vibration
Table 1 The first five natural frequencies 120582lowast2119894 for stepped Timoshenko beam in the case where the beam is rigidly restrained against rotation120601lowast1 = V119897 and elastically restrained in translation 119885lowast1 = 10 in comparison with [18]
Figure 3 Verification Example 1 [18] two-span stepped beam
Shock and Vibration 9
Table 2 Comparison of the first five natural frequencies in radsec using the present NTM results with those obtained in [19] for the exampleshown in Figure 4
Figure 4 Verification Example 2 [19] three-step beam having 119864 = 2069GPa 120588 = 78369 kgm3 119866 = 795769GPa and 1198961015840 = 075
10 Shock and Vibration
Table 3 Comparison of the present NTM results with those obtained in [5 20] for F-F beam shown in Figure 5
Mode number DQEM [20] Exp [5] NTM(40) FE(3D) error
I II III IV (II amp IV)1 292440 291 290316 290120 03022 1181300 1165 1162158 1167100 minus01803 1804100 1771 1767321 1772200 minus0068
1
254 140
32
549
254mm19
05
mm
Figure 5 Verification Example 3 [5 20] single-step (F-F) aluminum beam having 119864 = 717GPa 120588 = 2830Kgm3 and ] = 033
L20
L
1
1
mi
2 3 ik1
19 2021
21
k21
Figure 6 Verification Example 4 twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general endflexibilities conditions
calculated using two methods The first method is numericalassembly technique (NAT) and this method is the exactsolution The second method is differential transformationmethod and thismethod is approximatemethodThepinned-pinned and free-clamped boundary conditions results arefound in [19] The results of the current analysis are in goodagreement with the results presented in [19] As can be seenfromTable 2 the rotary inertia and shear deformation reducethemodal frequency especially for highermodes Comparingthe results of the clamped-free beam with the results of thefree-clamped beam shows that the natural frequency resultsof the clamped-free beam are lower than that of free-clampedbeam This may be explained by the fact that fixing thebeam from the thinner span results in lowering of the beamstiffness
313 Verification Example 3 The third verification exampleis shown in Figure 5 with identical dimensions to that usedin [5 20] The rotary inertia and shear deformation areconsidered The shear coefficient 1198961015840 is calculated based on[3] keeping Poissonrsquosrsquo ratio ] = 033 Table 3 shows thefirst three nonzero free-free (F-F) eigenvalues in Hz and incomparison with [5 20] The present results are computedusing both numerically (40) and FE(3D)methodsThe resultsof Table 3 show that the present analysis results are very closeto the experimental resultsThe percentage error between the
present FE(3D) results and the experimental results of [5] isless than 0302This represents the importance of includingthe effect of rotary inertia and shear deformation
314 Verification Example 4 The fourth example is shownin Figure 6 It is for a twenty-span uniform beam carrying19 equally spaced concentrated masses with 119898lowast119894 = 01Several beam start and end conditions are investigated aslisted in Table 4 The results are evaluated using the presentNTM method and previously published numerical assemblytechniqueNAT [6]methodThe computational time requiredto obtain the first three frequency parameters 120582lowast1 120582lowast2 and 120582lowast3using both NAT and NTM methods is calculated and listedin Table 4 Considerable reduction in computational time isobserved in all the investigated cases as shown in Table 4
32 Test Samples and Experimental Procedures In order tomeasure the natural frequencies of the system under studythe free-free test samples were put in free oscillations byusing an instrumented hammer model B and K 8202 Anaccelerometer model B and K 4366 is fixed to the shaft inorder to capture the vibration signal The output of thecharge amplifier B and K 2635 is connected to NI 6216data acquisition card This card is connected to the PC andmanaged by Lab VIEW software Figure 10 shows a photo ofthe current experimental setup The used card settings are
Shock and Vibration 11
Table 4 Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated masses119898lowast119894 = 01 (119903lowast21 = 119904lowast21 = V119904)in comparison with [6]
Figure 7 Schematic drawings for the experimental samples Group S2
sample frequency of 20 kHz sampling time of 3 5 8 sec andthe size of samples block to read is 1 k The time domaindata is captured and transformed into frequency domainThe resonant frequencies were obtained by the average of theresults of 10 impacts applied in three different locations ofthe sample Figures 7ndash9 show two groups of different steppedtest samples S2 and S3 respectively All the test samplesare manufactured from steel rods of dimensions Oslash 80 andOslash 40mm The common data for the steel test samples are120588 = 7850 kgm3 119864 = 2068GPa and 1198961015840 = 0845
Two-span twelve test samples are shown in group S2namely S2-4040 S2-40100 and S2-40140 shown inFigure 7 The dimensions of these samples are presentedin Table 5 The natural frequency results of these samples
are presented in Table 7 using experimental analytical andFE(3D) methods The captured signal for the free vibrationresponse for selected case from Table 7 is shown in Figure 11This case is italicized in the table The results of Table 7show that the three-dimensional FE solution is closer tothe experimental results than the analytical results Themaximum percentage error between the experimental andthe FE results is less than 05
Group S3 consists of four different samples namely S3-80250 S3-80200 S3-80150 and S3-80100 More detailsabout the geometrical dimensions and material propertiesfor these samples are shown in Table 6 and Figure 8 Theresults of Table 8 show that the deviation between the FE(3D)and the experimental is less than 116 The free vibration
12 Shock and Vibration
Table 6 The sample dimensions of group S3 as shown in Figure 8
Group S3 (Four cylindrical samples have two steps of steel rod Oslash 80 119871 = 500mm)119871 1198711 1198712 1198891 1198892 119898119905 (kg) 119871lowast2 119903lowast21
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Here 119894 denotes the 119894th span 119894 = 1 2 119899 in the case ofmultispan model
In order to introduce the current analysis the linearlyindependent fundamental solutions 1198781198841119894(120585) 1198781198842119894(120585) 1198781198843119894(120585)1198781198844119894(120585) and the corresponding 119878Ψ1119894(120585) 119878Ψ2119894(120585) 119878Ψ3119894(120585)119878Ψ4119894(120585) are derived In order to simplify the solution ofTimoshenko beam the following dependent functions aredefined
Γ119894 (120585) = 1198841015840119894 (120585) minus 119871 119894Ψ119894 (120585) (11a)
119878Γ1119894 (120585) = 11987811988410158401119894 (120585) minus 119878Ψ1119894 (120585) (11b)
119878Γ2119894 (120585) = 11987811988410158402119894 (120585) minus 119878Ψ2119894 (120585) (11c)
119878Γ3119894 (120585) = 11987811988410158403119894 (120585) minus 119878Ψ3119894 (120585) (11d)
119878Γ4119894 (120585) = 11987811988410158404119894 (120585) minus 119878Ψ4119894 (120585) (11e)
The Timoshenko solution will be normalized at the origin ofcoordinates as follows
where 1199101198940 = 119884119894(0) 1205951198940 = 119871 119894Ψ119894(0) 12059510158401198940 = 119871 119894Ψ1015840119894 (0) and1205741198940= Γ119894(0)The beam start boundary conditions at the point of
attachment 1 can be presented in nondimensional form as
where 1198691 is the mass moment of inertia at station 1 1198961 and1206011 are the linear and rotational elastic supports at station 1respectively and1198981 is the concentrated mass at station 1 seeFigure 1 for details
Substituting the solutions presented in (13a) (13b) (13c)(13d) (13e) (13f) (13g) (13h)-(14) into (15) the followingequations are obtained
(119869112058241 minus Φ1) 12059510 + 120595101584010 = 011990421 (119898112058241 minus 1198851) 11991010 + 12057410 = 0 (17)
The start boundary conditions in (17) can be presented inmatrix form as
[[
0 (119869112058241 minus 1206011) 1 011990421 (119898112058241 minus 1198851) 0 0 1]]
where 119869119899+1 is the mass moment of inertia at station (119899 + 1)119896119899+1 and 120601119899+1 are the linear and rotational elastic supports atstation (119899+1) respectively and119898119899+1 is the concentratedmassat station (119899 + 1)
Substituting the solutions in (13a) (13b) (13c) (13d)(13e) (13f) (13g) (13h)-(14) into (21) the following equationsare obtained
1199101198990 (119878Ψ10158401119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ1119899)+ 1205951198990 (119878Ψ10158402119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ2119899)+ 12059510158401198990 (119878Ψ10158403119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ3119899)+ 1205741198990 (119878Ψ10158404119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ4119899) = 0
where 119869119894+1 is the mass moment of inertia at station (119894+1) 119896119894+1and 120601119894+1 are the linear and rotational stiffness at station (119894 +1) respectively and 119898119894+1 is the concentrated mass at station(119894 + 1)
Substituting the solutions in (13a) (13b) (13c) (13d)(13e) (13f) (13g) (13h)-(14) into (27) we get the followingequations
Substituting (36) into the end condition of (25) results in thefollowing equations
[UE]2times4
[T]4times4 Δ1 = 0 (37)
[UIE]2times4
= [UE]2times4
[T]4times4 (38)
[UIE]2times4
Δ1 = 0 (39)
The general beam equation can be presented using the startboundary condition in (19) and the beam intermediate andend condition in (39) as shown below
[Utot]4times4
Δ1 = 0 (40)
where
[Utot]4times4
= [[[US]2times4[UIE]2times4
]] (41)
Equating the determinant of [Utot] by zero results in thesystem frequency equation In general the TMM has advan-tages over the traditional methods in that the final frequencyequation is 4 times 4 for any number of beam segments Theadvantage of the current method NTM over the TMM issignificant in the using of tailored solution that is normalizedat the origin of coordinates This type of solution enables theformulation of the system equations without the need to anyinverse matrix procedures as shown previously This reducesthe computational time comparing with the TMM
22 Finite Element Method Among the numerical toolsfinite element method is considered one of most efficientmethods to perform the vibration analysis of mechanical andstructural components In this section finite element is usedto obtain the natural frequencies andmode shapes of uniformand stepped beams ANSYS finite element commercial pack-age is used to perform the finite element analysisThe analysisis done using three-dimensional (3D) solid element modelsand SOLID95 elements are used for meshing Since all theexperimentally investigated samples in the current work are
round and stepped The beam cross-section is free meshedusing 87 SOLID95 elements for smaller cross-section and171 elements for the larger cross-section This mesh is thenextruded using 40 elements along the length of the beamThe total number of the element is ranging from 3480 (40times 87) to 6840 (40 times 171) elements based on the locationof the step see Figure 2 Modal analysis module is usedin this analysis and Block Lanczos method is used for themode extraction method The finite element model FE(3D)results are compared with those obtained experimentally andanalytically
3 Results and Discussion
31 Verification and Validation of NTM Results
311 Verification Example 1 In this example the first fivenondimensional natural frequencies (120582lowast4119894 = 120588111986011198714120596211986411198681)of stepped beam are compared with the exact solutionpresented by Gutierrez et al [18] see Figure 3 The modelis solved at two different step locations and several valuesof 11988721 and ℎ21 as shown in Table 1 Three different valuesof rotary inertia 11990321 = V119904 00036 and 001 are consideredin order to validate the current model in case of Bernoulli-Euler and Timoshenko beams The value of 11990421 = 31211990321 isconsidered in order to evaluate the shear deformation [18]The values of nondimensional linear and rotational elasticsupports stiffness at start and the end are (1198851 = 10Φ1 = V119897)(1198853 = V119904 Φ3 = V119897) respectively The results of Table 1 showthat the present NTM results are in good agreement with theexact solution presented by Gutierrez et al [18]
312 Verification Example 2 In this example Timoshenkobeam with three-step round cross-section presented in [19]is investigated see Figure 4 An intermediate lumped massof 119898lowast3 = 119898312058811198601119871 = 1 is located at a distance of 750mmfrom point 1 The input data for this example is listed inthe caption of Figure 4 Table 2 shows the results of the firstfive natural frequencies 120596119894 in (radsec) for pinned-pinnedfree-clamped clamped-free clamped-pinned and clamped-clamped configurations The results of reference [19] are
8 Shock and Vibration
Table 1 The first five natural frequencies 120582lowast2119894 for stepped Timoshenko beam in the case where the beam is rigidly restrained against rotation120601lowast1 = V119897 and elastically restrained in translation 119885lowast1 = 10 in comparison with [18]
Figure 3 Verification Example 1 [18] two-span stepped beam
Shock and Vibration 9
Table 2 Comparison of the first five natural frequencies in radsec using the present NTM results with those obtained in [19] for the exampleshown in Figure 4
Figure 4 Verification Example 2 [19] three-step beam having 119864 = 2069GPa 120588 = 78369 kgm3 119866 = 795769GPa and 1198961015840 = 075
10 Shock and Vibration
Table 3 Comparison of the present NTM results with those obtained in [5 20] for F-F beam shown in Figure 5
Mode number DQEM [20] Exp [5] NTM(40) FE(3D) error
I II III IV (II amp IV)1 292440 291 290316 290120 03022 1181300 1165 1162158 1167100 minus01803 1804100 1771 1767321 1772200 minus0068
1
254 140
32
549
254mm19
05
mm
Figure 5 Verification Example 3 [5 20] single-step (F-F) aluminum beam having 119864 = 717GPa 120588 = 2830Kgm3 and ] = 033
L20
L
1
1
mi
2 3 ik1
19 2021
21
k21
Figure 6 Verification Example 4 twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general endflexibilities conditions
calculated using two methods The first method is numericalassembly technique (NAT) and this method is the exactsolution The second method is differential transformationmethod and thismethod is approximatemethodThepinned-pinned and free-clamped boundary conditions results arefound in [19] The results of the current analysis are in goodagreement with the results presented in [19] As can be seenfromTable 2 the rotary inertia and shear deformation reducethemodal frequency especially for highermodes Comparingthe results of the clamped-free beam with the results of thefree-clamped beam shows that the natural frequency resultsof the clamped-free beam are lower than that of free-clampedbeam This may be explained by the fact that fixing thebeam from the thinner span results in lowering of the beamstiffness
313 Verification Example 3 The third verification exampleis shown in Figure 5 with identical dimensions to that usedin [5 20] The rotary inertia and shear deformation areconsidered The shear coefficient 1198961015840 is calculated based on[3] keeping Poissonrsquosrsquo ratio ] = 033 Table 3 shows thefirst three nonzero free-free (F-F) eigenvalues in Hz and incomparison with [5 20] The present results are computedusing both numerically (40) and FE(3D)methodsThe resultsof Table 3 show that the present analysis results are very closeto the experimental resultsThe percentage error between the
present FE(3D) results and the experimental results of [5] isless than 0302This represents the importance of includingthe effect of rotary inertia and shear deformation
314 Verification Example 4 The fourth example is shownin Figure 6 It is for a twenty-span uniform beam carrying19 equally spaced concentrated masses with 119898lowast119894 = 01Several beam start and end conditions are investigated aslisted in Table 4 The results are evaluated using the presentNTM method and previously published numerical assemblytechniqueNAT [6]methodThe computational time requiredto obtain the first three frequency parameters 120582lowast1 120582lowast2 and 120582lowast3using both NAT and NTM methods is calculated and listedin Table 4 Considerable reduction in computational time isobserved in all the investigated cases as shown in Table 4
32 Test Samples and Experimental Procedures In order tomeasure the natural frequencies of the system under studythe free-free test samples were put in free oscillations byusing an instrumented hammer model B and K 8202 Anaccelerometer model B and K 4366 is fixed to the shaft inorder to capture the vibration signal The output of thecharge amplifier B and K 2635 is connected to NI 6216data acquisition card This card is connected to the PC andmanaged by Lab VIEW software Figure 10 shows a photo ofthe current experimental setup The used card settings are
Shock and Vibration 11
Table 4 Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated masses119898lowast119894 = 01 (119903lowast21 = 119904lowast21 = V119904)in comparison with [6]
Figure 7 Schematic drawings for the experimental samples Group S2
sample frequency of 20 kHz sampling time of 3 5 8 sec andthe size of samples block to read is 1 k The time domaindata is captured and transformed into frequency domainThe resonant frequencies were obtained by the average of theresults of 10 impacts applied in three different locations ofthe sample Figures 7ndash9 show two groups of different steppedtest samples S2 and S3 respectively All the test samplesare manufactured from steel rods of dimensions Oslash 80 andOslash 40mm The common data for the steel test samples are120588 = 7850 kgm3 119864 = 2068GPa and 1198961015840 = 0845
Two-span twelve test samples are shown in group S2namely S2-4040 S2-40100 and S2-40140 shown inFigure 7 The dimensions of these samples are presentedin Table 5 The natural frequency results of these samples
are presented in Table 7 using experimental analytical andFE(3D) methods The captured signal for the free vibrationresponse for selected case from Table 7 is shown in Figure 11This case is italicized in the table The results of Table 7show that the three-dimensional FE solution is closer tothe experimental results than the analytical results Themaximum percentage error between the experimental andthe FE results is less than 05
Group S3 consists of four different samples namely S3-80250 S3-80200 S3-80150 and S3-80100 More detailsabout the geometrical dimensions and material propertiesfor these samples are shown in Table 6 and Figure 8 Theresults of Table 8 show that the deviation between the FE(3D)and the experimental is less than 116 The free vibration
12 Shock and Vibration
Table 6 The sample dimensions of group S3 as shown in Figure 8
Group S3 (Four cylindrical samples have two steps of steel rod Oslash 80 119871 = 500mm)119871 1198711 1198712 1198891 1198892 119898119905 (kg) 119871lowast2 119903lowast21
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
where 1198691 is the mass moment of inertia at station 1 1198961 and1206011 are the linear and rotational elastic supports at station 1respectively and1198981 is the concentrated mass at station 1 seeFigure 1 for details
Substituting the solutions presented in (13a) (13b) (13c)(13d) (13e) (13f) (13g) (13h)-(14) into (15) the followingequations are obtained
(119869112058241 minus Φ1) 12059510 + 120595101584010 = 011990421 (119898112058241 minus 1198851) 11991010 + 12057410 = 0 (17)
The start boundary conditions in (17) can be presented inmatrix form as
[[
0 (119869112058241 minus 1206011) 1 011990421 (119898112058241 minus 1198851) 0 0 1]]
where 119869119899+1 is the mass moment of inertia at station (119899 + 1)119896119899+1 and 120601119899+1 are the linear and rotational elastic supports atstation (119899+1) respectively and119898119899+1 is the concentratedmassat station (119899 + 1)
Substituting the solutions in (13a) (13b) (13c) (13d)(13e) (13f) (13g) (13h)-(14) into (21) the following equationsare obtained
1199101198990 (119878Ψ10158401119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ1119899)+ 1205951198990 (119878Ψ10158402119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ2119899)+ 12059510158401198990 (119878Ψ10158403119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ3119899)+ 1205741198990 (119878Ψ10158404119899 (1) + (Φ119899+1 minus 1205824119899119869119899+1) 119878Ψ4119899) = 0
where 119869119894+1 is the mass moment of inertia at station (119894+1) 119896119894+1and 120601119894+1 are the linear and rotational stiffness at station (119894 +1) respectively and 119898119894+1 is the concentrated mass at station(119894 + 1)
Substituting the solutions in (13a) (13b) (13c) (13d)(13e) (13f) (13g) (13h)-(14) into (27) we get the followingequations
Substituting (36) into the end condition of (25) results in thefollowing equations
[UE]2times4
[T]4times4 Δ1 = 0 (37)
[UIE]2times4
= [UE]2times4
[T]4times4 (38)
[UIE]2times4
Δ1 = 0 (39)
The general beam equation can be presented using the startboundary condition in (19) and the beam intermediate andend condition in (39) as shown below
[Utot]4times4
Δ1 = 0 (40)
where
[Utot]4times4
= [[[US]2times4[UIE]2times4
]] (41)
Equating the determinant of [Utot] by zero results in thesystem frequency equation In general the TMM has advan-tages over the traditional methods in that the final frequencyequation is 4 times 4 for any number of beam segments Theadvantage of the current method NTM over the TMM issignificant in the using of tailored solution that is normalizedat the origin of coordinates This type of solution enables theformulation of the system equations without the need to anyinverse matrix procedures as shown previously This reducesthe computational time comparing with the TMM
22 Finite Element Method Among the numerical toolsfinite element method is considered one of most efficientmethods to perform the vibration analysis of mechanical andstructural components In this section finite element is usedto obtain the natural frequencies andmode shapes of uniformand stepped beams ANSYS finite element commercial pack-age is used to perform the finite element analysisThe analysisis done using three-dimensional (3D) solid element modelsand SOLID95 elements are used for meshing Since all theexperimentally investigated samples in the current work are
round and stepped The beam cross-section is free meshedusing 87 SOLID95 elements for smaller cross-section and171 elements for the larger cross-section This mesh is thenextruded using 40 elements along the length of the beamThe total number of the element is ranging from 3480 (40times 87) to 6840 (40 times 171) elements based on the locationof the step see Figure 2 Modal analysis module is usedin this analysis and Block Lanczos method is used for themode extraction method The finite element model FE(3D)results are compared with those obtained experimentally andanalytically
3 Results and Discussion
31 Verification and Validation of NTM Results
311 Verification Example 1 In this example the first fivenondimensional natural frequencies (120582lowast4119894 = 120588111986011198714120596211986411198681)of stepped beam are compared with the exact solutionpresented by Gutierrez et al [18] see Figure 3 The modelis solved at two different step locations and several valuesof 11988721 and ℎ21 as shown in Table 1 Three different valuesof rotary inertia 11990321 = V119904 00036 and 001 are consideredin order to validate the current model in case of Bernoulli-Euler and Timoshenko beams The value of 11990421 = 31211990321 isconsidered in order to evaluate the shear deformation [18]The values of nondimensional linear and rotational elasticsupports stiffness at start and the end are (1198851 = 10Φ1 = V119897)(1198853 = V119904 Φ3 = V119897) respectively The results of Table 1 showthat the present NTM results are in good agreement with theexact solution presented by Gutierrez et al [18]
312 Verification Example 2 In this example Timoshenkobeam with three-step round cross-section presented in [19]is investigated see Figure 4 An intermediate lumped massof 119898lowast3 = 119898312058811198601119871 = 1 is located at a distance of 750mmfrom point 1 The input data for this example is listed inthe caption of Figure 4 Table 2 shows the results of the firstfive natural frequencies 120596119894 in (radsec) for pinned-pinnedfree-clamped clamped-free clamped-pinned and clamped-clamped configurations The results of reference [19] are
8 Shock and Vibration
Table 1 The first five natural frequencies 120582lowast2119894 for stepped Timoshenko beam in the case where the beam is rigidly restrained against rotation120601lowast1 = V119897 and elastically restrained in translation 119885lowast1 = 10 in comparison with [18]
Figure 3 Verification Example 1 [18] two-span stepped beam
Shock and Vibration 9
Table 2 Comparison of the first five natural frequencies in radsec using the present NTM results with those obtained in [19] for the exampleshown in Figure 4
Figure 4 Verification Example 2 [19] three-step beam having 119864 = 2069GPa 120588 = 78369 kgm3 119866 = 795769GPa and 1198961015840 = 075
10 Shock and Vibration
Table 3 Comparison of the present NTM results with those obtained in [5 20] for F-F beam shown in Figure 5
Mode number DQEM [20] Exp [5] NTM(40) FE(3D) error
I II III IV (II amp IV)1 292440 291 290316 290120 03022 1181300 1165 1162158 1167100 minus01803 1804100 1771 1767321 1772200 minus0068
1
254 140
32
549
254mm19
05
mm
Figure 5 Verification Example 3 [5 20] single-step (F-F) aluminum beam having 119864 = 717GPa 120588 = 2830Kgm3 and ] = 033
L20
L
1
1
mi
2 3 ik1
19 2021
21
k21
Figure 6 Verification Example 4 twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general endflexibilities conditions
calculated using two methods The first method is numericalassembly technique (NAT) and this method is the exactsolution The second method is differential transformationmethod and thismethod is approximatemethodThepinned-pinned and free-clamped boundary conditions results arefound in [19] The results of the current analysis are in goodagreement with the results presented in [19] As can be seenfromTable 2 the rotary inertia and shear deformation reducethemodal frequency especially for highermodes Comparingthe results of the clamped-free beam with the results of thefree-clamped beam shows that the natural frequency resultsof the clamped-free beam are lower than that of free-clampedbeam This may be explained by the fact that fixing thebeam from the thinner span results in lowering of the beamstiffness
313 Verification Example 3 The third verification exampleis shown in Figure 5 with identical dimensions to that usedin [5 20] The rotary inertia and shear deformation areconsidered The shear coefficient 1198961015840 is calculated based on[3] keeping Poissonrsquosrsquo ratio ] = 033 Table 3 shows thefirst three nonzero free-free (F-F) eigenvalues in Hz and incomparison with [5 20] The present results are computedusing both numerically (40) and FE(3D)methodsThe resultsof Table 3 show that the present analysis results are very closeto the experimental resultsThe percentage error between the
present FE(3D) results and the experimental results of [5] isless than 0302This represents the importance of includingthe effect of rotary inertia and shear deformation
314 Verification Example 4 The fourth example is shownin Figure 6 It is for a twenty-span uniform beam carrying19 equally spaced concentrated masses with 119898lowast119894 = 01Several beam start and end conditions are investigated aslisted in Table 4 The results are evaluated using the presentNTM method and previously published numerical assemblytechniqueNAT [6]methodThe computational time requiredto obtain the first three frequency parameters 120582lowast1 120582lowast2 and 120582lowast3using both NAT and NTM methods is calculated and listedin Table 4 Considerable reduction in computational time isobserved in all the investigated cases as shown in Table 4
32 Test Samples and Experimental Procedures In order tomeasure the natural frequencies of the system under studythe free-free test samples were put in free oscillations byusing an instrumented hammer model B and K 8202 Anaccelerometer model B and K 4366 is fixed to the shaft inorder to capture the vibration signal The output of thecharge amplifier B and K 2635 is connected to NI 6216data acquisition card This card is connected to the PC andmanaged by Lab VIEW software Figure 10 shows a photo ofthe current experimental setup The used card settings are
Shock and Vibration 11
Table 4 Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated masses119898lowast119894 = 01 (119903lowast21 = 119904lowast21 = V119904)in comparison with [6]
Figure 7 Schematic drawings for the experimental samples Group S2
sample frequency of 20 kHz sampling time of 3 5 8 sec andthe size of samples block to read is 1 k The time domaindata is captured and transformed into frequency domainThe resonant frequencies were obtained by the average of theresults of 10 impacts applied in three different locations ofthe sample Figures 7ndash9 show two groups of different steppedtest samples S2 and S3 respectively All the test samplesare manufactured from steel rods of dimensions Oslash 80 andOslash 40mm The common data for the steel test samples are120588 = 7850 kgm3 119864 = 2068GPa and 1198961015840 = 0845
Two-span twelve test samples are shown in group S2namely S2-4040 S2-40100 and S2-40140 shown inFigure 7 The dimensions of these samples are presentedin Table 5 The natural frequency results of these samples
are presented in Table 7 using experimental analytical andFE(3D) methods The captured signal for the free vibrationresponse for selected case from Table 7 is shown in Figure 11This case is italicized in the table The results of Table 7show that the three-dimensional FE solution is closer tothe experimental results than the analytical results Themaximum percentage error between the experimental andthe FE results is less than 05
Group S3 consists of four different samples namely S3-80250 S3-80200 S3-80150 and S3-80100 More detailsabout the geometrical dimensions and material propertiesfor these samples are shown in Table 6 and Figure 8 Theresults of Table 8 show that the deviation between the FE(3D)and the experimental is less than 116 The free vibration
12 Shock and Vibration
Table 6 The sample dimensions of group S3 as shown in Figure 8
Group S3 (Four cylindrical samples have two steps of steel rod Oslash 80 119871 = 500mm)119871 1198711 1198712 1198891 1198892 119898119905 (kg) 119871lowast2 119903lowast21
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
where 119869119894+1 is the mass moment of inertia at station (119894+1) 119896119894+1and 120601119894+1 are the linear and rotational stiffness at station (119894 +1) respectively and 119898119894+1 is the concentrated mass at station(119894 + 1)
Substituting the solutions in (13a) (13b) (13c) (13d)(13e) (13f) (13g) (13h)-(14) into (27) we get the followingequations
Substituting (36) into the end condition of (25) results in thefollowing equations
[UE]2times4
[T]4times4 Δ1 = 0 (37)
[UIE]2times4
= [UE]2times4
[T]4times4 (38)
[UIE]2times4
Δ1 = 0 (39)
The general beam equation can be presented using the startboundary condition in (19) and the beam intermediate andend condition in (39) as shown below
[Utot]4times4
Δ1 = 0 (40)
where
[Utot]4times4
= [[[US]2times4[UIE]2times4
]] (41)
Equating the determinant of [Utot] by zero results in thesystem frequency equation In general the TMM has advan-tages over the traditional methods in that the final frequencyequation is 4 times 4 for any number of beam segments Theadvantage of the current method NTM over the TMM issignificant in the using of tailored solution that is normalizedat the origin of coordinates This type of solution enables theformulation of the system equations without the need to anyinverse matrix procedures as shown previously This reducesthe computational time comparing with the TMM
22 Finite Element Method Among the numerical toolsfinite element method is considered one of most efficientmethods to perform the vibration analysis of mechanical andstructural components In this section finite element is usedto obtain the natural frequencies andmode shapes of uniformand stepped beams ANSYS finite element commercial pack-age is used to perform the finite element analysisThe analysisis done using three-dimensional (3D) solid element modelsand SOLID95 elements are used for meshing Since all theexperimentally investigated samples in the current work are
round and stepped The beam cross-section is free meshedusing 87 SOLID95 elements for smaller cross-section and171 elements for the larger cross-section This mesh is thenextruded using 40 elements along the length of the beamThe total number of the element is ranging from 3480 (40times 87) to 6840 (40 times 171) elements based on the locationof the step see Figure 2 Modal analysis module is usedin this analysis and Block Lanczos method is used for themode extraction method The finite element model FE(3D)results are compared with those obtained experimentally andanalytically
3 Results and Discussion
31 Verification and Validation of NTM Results
311 Verification Example 1 In this example the first fivenondimensional natural frequencies (120582lowast4119894 = 120588111986011198714120596211986411198681)of stepped beam are compared with the exact solutionpresented by Gutierrez et al [18] see Figure 3 The modelis solved at two different step locations and several valuesof 11988721 and ℎ21 as shown in Table 1 Three different valuesof rotary inertia 11990321 = V119904 00036 and 001 are consideredin order to validate the current model in case of Bernoulli-Euler and Timoshenko beams The value of 11990421 = 31211990321 isconsidered in order to evaluate the shear deformation [18]The values of nondimensional linear and rotational elasticsupports stiffness at start and the end are (1198851 = 10Φ1 = V119897)(1198853 = V119904 Φ3 = V119897) respectively The results of Table 1 showthat the present NTM results are in good agreement with theexact solution presented by Gutierrez et al [18]
312 Verification Example 2 In this example Timoshenkobeam with three-step round cross-section presented in [19]is investigated see Figure 4 An intermediate lumped massof 119898lowast3 = 119898312058811198601119871 = 1 is located at a distance of 750mmfrom point 1 The input data for this example is listed inthe caption of Figure 4 Table 2 shows the results of the firstfive natural frequencies 120596119894 in (radsec) for pinned-pinnedfree-clamped clamped-free clamped-pinned and clamped-clamped configurations The results of reference [19] are
8 Shock and Vibration
Table 1 The first five natural frequencies 120582lowast2119894 for stepped Timoshenko beam in the case where the beam is rigidly restrained against rotation120601lowast1 = V119897 and elastically restrained in translation 119885lowast1 = 10 in comparison with [18]
Figure 3 Verification Example 1 [18] two-span stepped beam
Shock and Vibration 9
Table 2 Comparison of the first five natural frequencies in radsec using the present NTM results with those obtained in [19] for the exampleshown in Figure 4
Figure 4 Verification Example 2 [19] three-step beam having 119864 = 2069GPa 120588 = 78369 kgm3 119866 = 795769GPa and 1198961015840 = 075
10 Shock and Vibration
Table 3 Comparison of the present NTM results with those obtained in [5 20] for F-F beam shown in Figure 5
Mode number DQEM [20] Exp [5] NTM(40) FE(3D) error
I II III IV (II amp IV)1 292440 291 290316 290120 03022 1181300 1165 1162158 1167100 minus01803 1804100 1771 1767321 1772200 minus0068
1
254 140
32
549
254mm19
05
mm
Figure 5 Verification Example 3 [5 20] single-step (F-F) aluminum beam having 119864 = 717GPa 120588 = 2830Kgm3 and ] = 033
L20
L
1
1
mi
2 3 ik1
19 2021
21
k21
Figure 6 Verification Example 4 twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general endflexibilities conditions
calculated using two methods The first method is numericalassembly technique (NAT) and this method is the exactsolution The second method is differential transformationmethod and thismethod is approximatemethodThepinned-pinned and free-clamped boundary conditions results arefound in [19] The results of the current analysis are in goodagreement with the results presented in [19] As can be seenfromTable 2 the rotary inertia and shear deformation reducethemodal frequency especially for highermodes Comparingthe results of the clamped-free beam with the results of thefree-clamped beam shows that the natural frequency resultsof the clamped-free beam are lower than that of free-clampedbeam This may be explained by the fact that fixing thebeam from the thinner span results in lowering of the beamstiffness
313 Verification Example 3 The third verification exampleis shown in Figure 5 with identical dimensions to that usedin [5 20] The rotary inertia and shear deformation areconsidered The shear coefficient 1198961015840 is calculated based on[3] keeping Poissonrsquosrsquo ratio ] = 033 Table 3 shows thefirst three nonzero free-free (F-F) eigenvalues in Hz and incomparison with [5 20] The present results are computedusing both numerically (40) and FE(3D)methodsThe resultsof Table 3 show that the present analysis results are very closeto the experimental resultsThe percentage error between the
present FE(3D) results and the experimental results of [5] isless than 0302This represents the importance of includingthe effect of rotary inertia and shear deformation
314 Verification Example 4 The fourth example is shownin Figure 6 It is for a twenty-span uniform beam carrying19 equally spaced concentrated masses with 119898lowast119894 = 01Several beam start and end conditions are investigated aslisted in Table 4 The results are evaluated using the presentNTM method and previously published numerical assemblytechniqueNAT [6]methodThe computational time requiredto obtain the first three frequency parameters 120582lowast1 120582lowast2 and 120582lowast3using both NAT and NTM methods is calculated and listedin Table 4 Considerable reduction in computational time isobserved in all the investigated cases as shown in Table 4
32 Test Samples and Experimental Procedures In order tomeasure the natural frequencies of the system under studythe free-free test samples were put in free oscillations byusing an instrumented hammer model B and K 8202 Anaccelerometer model B and K 4366 is fixed to the shaft inorder to capture the vibration signal The output of thecharge amplifier B and K 2635 is connected to NI 6216data acquisition card This card is connected to the PC andmanaged by Lab VIEW software Figure 10 shows a photo ofthe current experimental setup The used card settings are
Shock and Vibration 11
Table 4 Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated masses119898lowast119894 = 01 (119903lowast21 = 119904lowast21 = V119904)in comparison with [6]
Figure 7 Schematic drawings for the experimental samples Group S2
sample frequency of 20 kHz sampling time of 3 5 8 sec andthe size of samples block to read is 1 k The time domaindata is captured and transformed into frequency domainThe resonant frequencies were obtained by the average of theresults of 10 impacts applied in three different locations ofthe sample Figures 7ndash9 show two groups of different steppedtest samples S2 and S3 respectively All the test samplesare manufactured from steel rods of dimensions Oslash 80 andOslash 40mm The common data for the steel test samples are120588 = 7850 kgm3 119864 = 2068GPa and 1198961015840 = 0845
Two-span twelve test samples are shown in group S2namely S2-4040 S2-40100 and S2-40140 shown inFigure 7 The dimensions of these samples are presentedin Table 5 The natural frequency results of these samples
are presented in Table 7 using experimental analytical andFE(3D) methods The captured signal for the free vibrationresponse for selected case from Table 7 is shown in Figure 11This case is italicized in the table The results of Table 7show that the three-dimensional FE solution is closer tothe experimental results than the analytical results Themaximum percentage error between the experimental andthe FE results is less than 05
Group S3 consists of four different samples namely S3-80250 S3-80200 S3-80150 and S3-80100 More detailsabout the geometrical dimensions and material propertiesfor these samples are shown in Table 6 and Figure 8 Theresults of Table 8 show that the deviation between the FE(3D)and the experimental is less than 116 The free vibration
12 Shock and Vibration
Table 6 The sample dimensions of group S3 as shown in Figure 8
Group S3 (Four cylindrical samples have two steps of steel rod Oslash 80 119871 = 500mm)119871 1198711 1198712 1198891 1198892 119898119905 (kg) 119871lowast2 119903lowast21
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Substituting (36) into the end condition of (25) results in thefollowing equations
[UE]2times4
[T]4times4 Δ1 = 0 (37)
[UIE]2times4
= [UE]2times4
[T]4times4 (38)
[UIE]2times4
Δ1 = 0 (39)
The general beam equation can be presented using the startboundary condition in (19) and the beam intermediate andend condition in (39) as shown below
[Utot]4times4
Δ1 = 0 (40)
where
[Utot]4times4
= [[[US]2times4[UIE]2times4
]] (41)
Equating the determinant of [Utot] by zero results in thesystem frequency equation In general the TMM has advan-tages over the traditional methods in that the final frequencyequation is 4 times 4 for any number of beam segments Theadvantage of the current method NTM over the TMM issignificant in the using of tailored solution that is normalizedat the origin of coordinates This type of solution enables theformulation of the system equations without the need to anyinverse matrix procedures as shown previously This reducesthe computational time comparing with the TMM
22 Finite Element Method Among the numerical toolsfinite element method is considered one of most efficientmethods to perform the vibration analysis of mechanical andstructural components In this section finite element is usedto obtain the natural frequencies andmode shapes of uniformand stepped beams ANSYS finite element commercial pack-age is used to perform the finite element analysisThe analysisis done using three-dimensional (3D) solid element modelsand SOLID95 elements are used for meshing Since all theexperimentally investigated samples in the current work are
round and stepped The beam cross-section is free meshedusing 87 SOLID95 elements for smaller cross-section and171 elements for the larger cross-section This mesh is thenextruded using 40 elements along the length of the beamThe total number of the element is ranging from 3480 (40times 87) to 6840 (40 times 171) elements based on the locationof the step see Figure 2 Modal analysis module is usedin this analysis and Block Lanczos method is used for themode extraction method The finite element model FE(3D)results are compared with those obtained experimentally andanalytically
3 Results and Discussion
31 Verification and Validation of NTM Results
311 Verification Example 1 In this example the first fivenondimensional natural frequencies (120582lowast4119894 = 120588111986011198714120596211986411198681)of stepped beam are compared with the exact solutionpresented by Gutierrez et al [18] see Figure 3 The modelis solved at two different step locations and several valuesof 11988721 and ℎ21 as shown in Table 1 Three different valuesof rotary inertia 11990321 = V119904 00036 and 001 are consideredin order to validate the current model in case of Bernoulli-Euler and Timoshenko beams The value of 11990421 = 31211990321 isconsidered in order to evaluate the shear deformation [18]The values of nondimensional linear and rotational elasticsupports stiffness at start and the end are (1198851 = 10Φ1 = V119897)(1198853 = V119904 Φ3 = V119897) respectively The results of Table 1 showthat the present NTM results are in good agreement with theexact solution presented by Gutierrez et al [18]
312 Verification Example 2 In this example Timoshenkobeam with three-step round cross-section presented in [19]is investigated see Figure 4 An intermediate lumped massof 119898lowast3 = 119898312058811198601119871 = 1 is located at a distance of 750mmfrom point 1 The input data for this example is listed inthe caption of Figure 4 Table 2 shows the results of the firstfive natural frequencies 120596119894 in (radsec) for pinned-pinnedfree-clamped clamped-free clamped-pinned and clamped-clamped configurations The results of reference [19] are
8 Shock and Vibration
Table 1 The first five natural frequencies 120582lowast2119894 for stepped Timoshenko beam in the case where the beam is rigidly restrained against rotation120601lowast1 = V119897 and elastically restrained in translation 119885lowast1 = 10 in comparison with [18]
Figure 3 Verification Example 1 [18] two-span stepped beam
Shock and Vibration 9
Table 2 Comparison of the first five natural frequencies in radsec using the present NTM results with those obtained in [19] for the exampleshown in Figure 4
Figure 4 Verification Example 2 [19] three-step beam having 119864 = 2069GPa 120588 = 78369 kgm3 119866 = 795769GPa and 1198961015840 = 075
10 Shock and Vibration
Table 3 Comparison of the present NTM results with those obtained in [5 20] for F-F beam shown in Figure 5
Mode number DQEM [20] Exp [5] NTM(40) FE(3D) error
I II III IV (II amp IV)1 292440 291 290316 290120 03022 1181300 1165 1162158 1167100 minus01803 1804100 1771 1767321 1772200 minus0068
1
254 140
32
549
254mm19
05
mm
Figure 5 Verification Example 3 [5 20] single-step (F-F) aluminum beam having 119864 = 717GPa 120588 = 2830Kgm3 and ] = 033
L20
L
1
1
mi
2 3 ik1
19 2021
21
k21
Figure 6 Verification Example 4 twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general endflexibilities conditions
calculated using two methods The first method is numericalassembly technique (NAT) and this method is the exactsolution The second method is differential transformationmethod and thismethod is approximatemethodThepinned-pinned and free-clamped boundary conditions results arefound in [19] The results of the current analysis are in goodagreement with the results presented in [19] As can be seenfromTable 2 the rotary inertia and shear deformation reducethemodal frequency especially for highermodes Comparingthe results of the clamped-free beam with the results of thefree-clamped beam shows that the natural frequency resultsof the clamped-free beam are lower than that of free-clampedbeam This may be explained by the fact that fixing thebeam from the thinner span results in lowering of the beamstiffness
313 Verification Example 3 The third verification exampleis shown in Figure 5 with identical dimensions to that usedin [5 20] The rotary inertia and shear deformation areconsidered The shear coefficient 1198961015840 is calculated based on[3] keeping Poissonrsquosrsquo ratio ] = 033 Table 3 shows thefirst three nonzero free-free (F-F) eigenvalues in Hz and incomparison with [5 20] The present results are computedusing both numerically (40) and FE(3D)methodsThe resultsof Table 3 show that the present analysis results are very closeto the experimental resultsThe percentage error between the
present FE(3D) results and the experimental results of [5] isless than 0302This represents the importance of includingthe effect of rotary inertia and shear deformation
314 Verification Example 4 The fourth example is shownin Figure 6 It is for a twenty-span uniform beam carrying19 equally spaced concentrated masses with 119898lowast119894 = 01Several beam start and end conditions are investigated aslisted in Table 4 The results are evaluated using the presentNTM method and previously published numerical assemblytechniqueNAT [6]methodThe computational time requiredto obtain the first three frequency parameters 120582lowast1 120582lowast2 and 120582lowast3using both NAT and NTM methods is calculated and listedin Table 4 Considerable reduction in computational time isobserved in all the investigated cases as shown in Table 4
32 Test Samples and Experimental Procedures In order tomeasure the natural frequencies of the system under studythe free-free test samples were put in free oscillations byusing an instrumented hammer model B and K 8202 Anaccelerometer model B and K 4366 is fixed to the shaft inorder to capture the vibration signal The output of thecharge amplifier B and K 2635 is connected to NI 6216data acquisition card This card is connected to the PC andmanaged by Lab VIEW software Figure 10 shows a photo ofthe current experimental setup The used card settings are
Shock and Vibration 11
Table 4 Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated masses119898lowast119894 = 01 (119903lowast21 = 119904lowast21 = V119904)in comparison with [6]
Figure 7 Schematic drawings for the experimental samples Group S2
sample frequency of 20 kHz sampling time of 3 5 8 sec andthe size of samples block to read is 1 k The time domaindata is captured and transformed into frequency domainThe resonant frequencies were obtained by the average of theresults of 10 impacts applied in three different locations ofthe sample Figures 7ndash9 show two groups of different steppedtest samples S2 and S3 respectively All the test samplesare manufactured from steel rods of dimensions Oslash 80 andOslash 40mm The common data for the steel test samples are120588 = 7850 kgm3 119864 = 2068GPa and 1198961015840 = 0845
Two-span twelve test samples are shown in group S2namely S2-4040 S2-40100 and S2-40140 shown inFigure 7 The dimensions of these samples are presentedin Table 5 The natural frequency results of these samples
are presented in Table 7 using experimental analytical andFE(3D) methods The captured signal for the free vibrationresponse for selected case from Table 7 is shown in Figure 11This case is italicized in the table The results of Table 7show that the three-dimensional FE solution is closer tothe experimental results than the analytical results Themaximum percentage error between the experimental andthe FE results is less than 05
Group S3 consists of four different samples namely S3-80250 S3-80200 S3-80150 and S3-80100 More detailsabout the geometrical dimensions and material propertiesfor these samples are shown in Table 6 and Figure 8 Theresults of Table 8 show that the deviation between the FE(3D)and the experimental is less than 116 The free vibration
12 Shock and Vibration
Table 6 The sample dimensions of group S3 as shown in Figure 8
Group S3 (Four cylindrical samples have two steps of steel rod Oslash 80 119871 = 500mm)119871 1198711 1198712 1198891 1198892 119898119905 (kg) 119871lowast2 119903lowast21
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Table 1 The first five natural frequencies 120582lowast2119894 for stepped Timoshenko beam in the case where the beam is rigidly restrained against rotation120601lowast1 = V119897 and elastically restrained in translation 119885lowast1 = 10 in comparison with [18]
Figure 3 Verification Example 1 [18] two-span stepped beam
Shock and Vibration 9
Table 2 Comparison of the first five natural frequencies in radsec using the present NTM results with those obtained in [19] for the exampleshown in Figure 4
Figure 4 Verification Example 2 [19] three-step beam having 119864 = 2069GPa 120588 = 78369 kgm3 119866 = 795769GPa and 1198961015840 = 075
10 Shock and Vibration
Table 3 Comparison of the present NTM results with those obtained in [5 20] for F-F beam shown in Figure 5
Mode number DQEM [20] Exp [5] NTM(40) FE(3D) error
I II III IV (II amp IV)1 292440 291 290316 290120 03022 1181300 1165 1162158 1167100 minus01803 1804100 1771 1767321 1772200 minus0068
1
254 140
32
549
254mm19
05
mm
Figure 5 Verification Example 3 [5 20] single-step (F-F) aluminum beam having 119864 = 717GPa 120588 = 2830Kgm3 and ] = 033
L20
L
1
1
mi
2 3 ik1
19 2021
21
k21
Figure 6 Verification Example 4 twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general endflexibilities conditions
calculated using two methods The first method is numericalassembly technique (NAT) and this method is the exactsolution The second method is differential transformationmethod and thismethod is approximatemethodThepinned-pinned and free-clamped boundary conditions results arefound in [19] The results of the current analysis are in goodagreement with the results presented in [19] As can be seenfromTable 2 the rotary inertia and shear deformation reducethemodal frequency especially for highermodes Comparingthe results of the clamped-free beam with the results of thefree-clamped beam shows that the natural frequency resultsof the clamped-free beam are lower than that of free-clampedbeam This may be explained by the fact that fixing thebeam from the thinner span results in lowering of the beamstiffness
313 Verification Example 3 The third verification exampleis shown in Figure 5 with identical dimensions to that usedin [5 20] The rotary inertia and shear deformation areconsidered The shear coefficient 1198961015840 is calculated based on[3] keeping Poissonrsquosrsquo ratio ] = 033 Table 3 shows thefirst three nonzero free-free (F-F) eigenvalues in Hz and incomparison with [5 20] The present results are computedusing both numerically (40) and FE(3D)methodsThe resultsof Table 3 show that the present analysis results are very closeto the experimental resultsThe percentage error between the
present FE(3D) results and the experimental results of [5] isless than 0302This represents the importance of includingthe effect of rotary inertia and shear deformation
314 Verification Example 4 The fourth example is shownin Figure 6 It is for a twenty-span uniform beam carrying19 equally spaced concentrated masses with 119898lowast119894 = 01Several beam start and end conditions are investigated aslisted in Table 4 The results are evaluated using the presentNTM method and previously published numerical assemblytechniqueNAT [6]methodThe computational time requiredto obtain the first three frequency parameters 120582lowast1 120582lowast2 and 120582lowast3using both NAT and NTM methods is calculated and listedin Table 4 Considerable reduction in computational time isobserved in all the investigated cases as shown in Table 4
32 Test Samples and Experimental Procedures In order tomeasure the natural frequencies of the system under studythe free-free test samples were put in free oscillations byusing an instrumented hammer model B and K 8202 Anaccelerometer model B and K 4366 is fixed to the shaft inorder to capture the vibration signal The output of thecharge amplifier B and K 2635 is connected to NI 6216data acquisition card This card is connected to the PC andmanaged by Lab VIEW software Figure 10 shows a photo ofthe current experimental setup The used card settings are
Shock and Vibration 11
Table 4 Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated masses119898lowast119894 = 01 (119903lowast21 = 119904lowast21 = V119904)in comparison with [6]
Figure 7 Schematic drawings for the experimental samples Group S2
sample frequency of 20 kHz sampling time of 3 5 8 sec andthe size of samples block to read is 1 k The time domaindata is captured and transformed into frequency domainThe resonant frequencies were obtained by the average of theresults of 10 impacts applied in three different locations ofthe sample Figures 7ndash9 show two groups of different steppedtest samples S2 and S3 respectively All the test samplesare manufactured from steel rods of dimensions Oslash 80 andOslash 40mm The common data for the steel test samples are120588 = 7850 kgm3 119864 = 2068GPa and 1198961015840 = 0845
Two-span twelve test samples are shown in group S2namely S2-4040 S2-40100 and S2-40140 shown inFigure 7 The dimensions of these samples are presentedin Table 5 The natural frequency results of these samples
are presented in Table 7 using experimental analytical andFE(3D) methods The captured signal for the free vibrationresponse for selected case from Table 7 is shown in Figure 11This case is italicized in the table The results of Table 7show that the three-dimensional FE solution is closer tothe experimental results than the analytical results Themaximum percentage error between the experimental andthe FE results is less than 05
Group S3 consists of four different samples namely S3-80250 S3-80200 S3-80150 and S3-80100 More detailsabout the geometrical dimensions and material propertiesfor these samples are shown in Table 6 and Figure 8 Theresults of Table 8 show that the deviation between the FE(3D)and the experimental is less than 116 The free vibration
12 Shock and Vibration
Table 6 The sample dimensions of group S3 as shown in Figure 8
Group S3 (Four cylindrical samples have two steps of steel rod Oslash 80 119871 = 500mm)119871 1198711 1198712 1198891 1198892 119898119905 (kg) 119871lowast2 119903lowast21
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Table 2 Comparison of the first five natural frequencies in radsec using the present NTM results with those obtained in [19] for the exampleshown in Figure 4
Figure 4 Verification Example 2 [19] three-step beam having 119864 = 2069GPa 120588 = 78369 kgm3 119866 = 795769GPa and 1198961015840 = 075
10 Shock and Vibration
Table 3 Comparison of the present NTM results with those obtained in [5 20] for F-F beam shown in Figure 5
Mode number DQEM [20] Exp [5] NTM(40) FE(3D) error
I II III IV (II amp IV)1 292440 291 290316 290120 03022 1181300 1165 1162158 1167100 minus01803 1804100 1771 1767321 1772200 minus0068
1
254 140
32
549
254mm19
05
mm
Figure 5 Verification Example 3 [5 20] single-step (F-F) aluminum beam having 119864 = 717GPa 120588 = 2830Kgm3 and ] = 033
L20
L
1
1
mi
2 3 ik1
19 2021
21
k21
Figure 6 Verification Example 4 twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general endflexibilities conditions
calculated using two methods The first method is numericalassembly technique (NAT) and this method is the exactsolution The second method is differential transformationmethod and thismethod is approximatemethodThepinned-pinned and free-clamped boundary conditions results arefound in [19] The results of the current analysis are in goodagreement with the results presented in [19] As can be seenfromTable 2 the rotary inertia and shear deformation reducethemodal frequency especially for highermodes Comparingthe results of the clamped-free beam with the results of thefree-clamped beam shows that the natural frequency resultsof the clamped-free beam are lower than that of free-clampedbeam This may be explained by the fact that fixing thebeam from the thinner span results in lowering of the beamstiffness
313 Verification Example 3 The third verification exampleis shown in Figure 5 with identical dimensions to that usedin [5 20] The rotary inertia and shear deformation areconsidered The shear coefficient 1198961015840 is calculated based on[3] keeping Poissonrsquosrsquo ratio ] = 033 Table 3 shows thefirst three nonzero free-free (F-F) eigenvalues in Hz and incomparison with [5 20] The present results are computedusing both numerically (40) and FE(3D)methodsThe resultsof Table 3 show that the present analysis results are very closeto the experimental resultsThe percentage error between the
present FE(3D) results and the experimental results of [5] isless than 0302This represents the importance of includingthe effect of rotary inertia and shear deformation
314 Verification Example 4 The fourth example is shownin Figure 6 It is for a twenty-span uniform beam carrying19 equally spaced concentrated masses with 119898lowast119894 = 01Several beam start and end conditions are investigated aslisted in Table 4 The results are evaluated using the presentNTM method and previously published numerical assemblytechniqueNAT [6]methodThe computational time requiredto obtain the first three frequency parameters 120582lowast1 120582lowast2 and 120582lowast3using both NAT and NTM methods is calculated and listedin Table 4 Considerable reduction in computational time isobserved in all the investigated cases as shown in Table 4
32 Test Samples and Experimental Procedures In order tomeasure the natural frequencies of the system under studythe free-free test samples were put in free oscillations byusing an instrumented hammer model B and K 8202 Anaccelerometer model B and K 4366 is fixed to the shaft inorder to capture the vibration signal The output of thecharge amplifier B and K 2635 is connected to NI 6216data acquisition card This card is connected to the PC andmanaged by Lab VIEW software Figure 10 shows a photo ofthe current experimental setup The used card settings are
Shock and Vibration 11
Table 4 Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated masses119898lowast119894 = 01 (119903lowast21 = 119904lowast21 = V119904)in comparison with [6]
Figure 7 Schematic drawings for the experimental samples Group S2
sample frequency of 20 kHz sampling time of 3 5 8 sec andthe size of samples block to read is 1 k The time domaindata is captured and transformed into frequency domainThe resonant frequencies were obtained by the average of theresults of 10 impacts applied in three different locations ofthe sample Figures 7ndash9 show two groups of different steppedtest samples S2 and S3 respectively All the test samplesare manufactured from steel rods of dimensions Oslash 80 andOslash 40mm The common data for the steel test samples are120588 = 7850 kgm3 119864 = 2068GPa and 1198961015840 = 0845
Two-span twelve test samples are shown in group S2namely S2-4040 S2-40100 and S2-40140 shown inFigure 7 The dimensions of these samples are presentedin Table 5 The natural frequency results of these samples
are presented in Table 7 using experimental analytical andFE(3D) methods The captured signal for the free vibrationresponse for selected case from Table 7 is shown in Figure 11This case is italicized in the table The results of Table 7show that the three-dimensional FE solution is closer tothe experimental results than the analytical results Themaximum percentage error between the experimental andthe FE results is less than 05
Group S3 consists of four different samples namely S3-80250 S3-80200 S3-80150 and S3-80100 More detailsabout the geometrical dimensions and material propertiesfor these samples are shown in Table 6 and Figure 8 Theresults of Table 8 show that the deviation between the FE(3D)and the experimental is less than 116 The free vibration
12 Shock and Vibration
Table 6 The sample dimensions of group S3 as shown in Figure 8
Group S3 (Four cylindrical samples have two steps of steel rod Oslash 80 119871 = 500mm)119871 1198711 1198712 1198891 1198892 119898119905 (kg) 119871lowast2 119903lowast21
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Table 3 Comparison of the present NTM results with those obtained in [5 20] for F-F beam shown in Figure 5
Mode number DQEM [20] Exp [5] NTM(40) FE(3D) error
I II III IV (II amp IV)1 292440 291 290316 290120 03022 1181300 1165 1162158 1167100 minus01803 1804100 1771 1767321 1772200 minus0068
1
254 140
32
549
254mm19
05
mm
Figure 5 Verification Example 3 [5 20] single-step (F-F) aluminum beam having 119864 = 717GPa 120588 = 2830Kgm3 and ] = 033
L20
L
1
1
mi
2 3 ik1
19 2021
21
k21
Figure 6 Verification Example 4 twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general endflexibilities conditions
calculated using two methods The first method is numericalassembly technique (NAT) and this method is the exactsolution The second method is differential transformationmethod and thismethod is approximatemethodThepinned-pinned and free-clamped boundary conditions results arefound in [19] The results of the current analysis are in goodagreement with the results presented in [19] As can be seenfromTable 2 the rotary inertia and shear deformation reducethemodal frequency especially for highermodes Comparingthe results of the clamped-free beam with the results of thefree-clamped beam shows that the natural frequency resultsof the clamped-free beam are lower than that of free-clampedbeam This may be explained by the fact that fixing thebeam from the thinner span results in lowering of the beamstiffness
313 Verification Example 3 The third verification exampleis shown in Figure 5 with identical dimensions to that usedin [5 20] The rotary inertia and shear deformation areconsidered The shear coefficient 1198961015840 is calculated based on[3] keeping Poissonrsquosrsquo ratio ] = 033 Table 3 shows thefirst three nonzero free-free (F-F) eigenvalues in Hz and incomparison with [5 20] The present results are computedusing both numerically (40) and FE(3D)methodsThe resultsof Table 3 show that the present analysis results are very closeto the experimental resultsThe percentage error between the
present FE(3D) results and the experimental results of [5] isless than 0302This represents the importance of includingthe effect of rotary inertia and shear deformation
314 Verification Example 4 The fourth example is shownin Figure 6 It is for a twenty-span uniform beam carrying19 equally spaced concentrated masses with 119898lowast119894 = 01Several beam start and end conditions are investigated aslisted in Table 4 The results are evaluated using the presentNTM method and previously published numerical assemblytechniqueNAT [6]methodThe computational time requiredto obtain the first three frequency parameters 120582lowast1 120582lowast2 and 120582lowast3using both NAT and NTM methods is calculated and listedin Table 4 Considerable reduction in computational time isobserved in all the investigated cases as shown in Table 4
32 Test Samples and Experimental Procedures In order tomeasure the natural frequencies of the system under studythe free-free test samples were put in free oscillations byusing an instrumented hammer model B and K 8202 Anaccelerometer model B and K 4366 is fixed to the shaft inorder to capture the vibration signal The output of thecharge amplifier B and K 2635 is connected to NI 6216data acquisition card This card is connected to the PC andmanaged by Lab VIEW software Figure 10 shows a photo ofthe current experimental setup The used card settings are
Shock and Vibration 11
Table 4 Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated masses119898lowast119894 = 01 (119903lowast21 = 119904lowast21 = V119904)in comparison with [6]
Figure 7 Schematic drawings for the experimental samples Group S2
sample frequency of 20 kHz sampling time of 3 5 8 sec andthe size of samples block to read is 1 k The time domaindata is captured and transformed into frequency domainThe resonant frequencies were obtained by the average of theresults of 10 impacts applied in three different locations ofthe sample Figures 7ndash9 show two groups of different steppedtest samples S2 and S3 respectively All the test samplesare manufactured from steel rods of dimensions Oslash 80 andOslash 40mm The common data for the steel test samples are120588 = 7850 kgm3 119864 = 2068GPa and 1198961015840 = 0845
Two-span twelve test samples are shown in group S2namely S2-4040 S2-40100 and S2-40140 shown inFigure 7 The dimensions of these samples are presentedin Table 5 The natural frequency results of these samples
are presented in Table 7 using experimental analytical andFE(3D) methods The captured signal for the free vibrationresponse for selected case from Table 7 is shown in Figure 11This case is italicized in the table The results of Table 7show that the three-dimensional FE solution is closer tothe experimental results than the analytical results Themaximum percentage error between the experimental andthe FE results is less than 05
Group S3 consists of four different samples namely S3-80250 S3-80200 S3-80150 and S3-80100 More detailsabout the geometrical dimensions and material propertiesfor these samples are shown in Table 6 and Figure 8 Theresults of Table 8 show that the deviation between the FE(3D)and the experimental is less than 116 The free vibration
12 Shock and Vibration
Table 6 The sample dimensions of group S3 as shown in Figure 8
Group S3 (Four cylindrical samples have two steps of steel rod Oslash 80 119871 = 500mm)119871 1198711 1198712 1198891 1198892 119898119905 (kg) 119871lowast2 119903lowast21
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Figure 7 Schematic drawings for the experimental samples Group S2
sample frequency of 20 kHz sampling time of 3 5 8 sec andthe size of samples block to read is 1 k The time domaindata is captured and transformed into frequency domainThe resonant frequencies were obtained by the average of theresults of 10 impacts applied in three different locations ofthe sample Figures 7ndash9 show two groups of different steppedtest samples S2 and S3 respectively All the test samplesare manufactured from steel rods of dimensions Oslash 80 andOslash 40mm The common data for the steel test samples are120588 = 7850 kgm3 119864 = 2068GPa and 1198961015840 = 0845
Two-span twelve test samples are shown in group S2namely S2-4040 S2-40100 and S2-40140 shown inFigure 7 The dimensions of these samples are presentedin Table 5 The natural frequency results of these samples
are presented in Table 7 using experimental analytical andFE(3D) methods The captured signal for the free vibrationresponse for selected case from Table 7 is shown in Figure 11This case is italicized in the table The results of Table 7show that the three-dimensional FE solution is closer tothe experimental results than the analytical results Themaximum percentage error between the experimental andthe FE results is less than 05
Group S3 consists of four different samples namely S3-80250 S3-80200 S3-80150 and S3-80100 More detailsabout the geometrical dimensions and material propertiesfor these samples are shown in Table 6 and Figure 8 Theresults of Table 8 show that the deviation between the FE(3D)and the experimental is less than 116 The free vibration
12 Shock and Vibration
Table 6 The sample dimensions of group S3 as shown in Figure 8
Group S3 (Four cylindrical samples have two steps of steel rod Oslash 80 119871 = 500mm)119871 1198711 1198712 1198891 1198892 119898119905 (kg) 119871lowast2 119903lowast21
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Table 6 The sample dimensions of group S3 as shown in Figure 8
Group S3 (Four cylindrical samples have two steps of steel rod Oslash 80 119871 = 500mm)119871 1198711 1198712 1198891 1198892 119898119905 (kg) 119871lowast2 119903lowast21
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Figure 12 Experimental first four natural frequencies for test sample S3-80200
signals of selected case italicized in Table 8 are plotted inFigure 12 In general the results of the three-span samplesreveal the same conclusion drawn from the two-span samplesthat the FE(3D) results are closer to the experimental resultsthan the analytical results The conclusion driven from theinvestigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction ofthe natural frequency of stepped beam
33 Percentage Modal Deviation between FE(3D) and NTMMethod The results of the previous section show theaccuracy of the FE(3D) model in evaluating the natural
frequencies of stepped beam Therefore in this section afree-free two-span model is deeply investigated using (40)and FE(3D) methods to justify the validity of the analyticalsolution in predicting the stepped beam results Two cate-gories of samples are considered as shown in Figure 13 Thefirst category includes 600mm length samples with 119903lowast21 =278 times 10minus4 and the second category includes 200mm lengthsamples with 119903lowast21 = 205 times 10minus3 The effects of changing 120583and 119889 are investigated 120583 varies from 0 to 1 and meanwhileonly three values are listed in Table 9 Four different valuesof 119889 are investigated 05 0625 075 and 0875 The study in
14 Shock and Vibration
Table 7 Percentage error between the computational and experimental results for single-step test samples (Group S2) Lowest four nonzerofree-free modes
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
(D) error (A B) minus2531 minus5532 minus3932 minus2565(E) error (A C) 0023 0565 0133 minus0192
16 Shock and Vibration
Oslash40
Oslash20
60
600
Oslash25
Oslash30
Oslash35
= 01
240
Oslash40
600
= 04
420
= 07
d = 05
d = 0625
d = 075
d = 0875
Long samples
Oslash40
80
Oslash25
Oslash30
Oslash35
= 01
Oslash40
Oslash20
200
= 04
140
= 07
d = 05
d = 0625
d = 075
d = 0875
Short samples
200
20
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Figure 13 Long and short stepped samples at three values of 120583 = 01 04 and 07 and four values of 119889 = ℎ21 = 05 0625 075 and 0875
this section focuses only on the first three nonzero free-freemodes The percentage modal deviation in analytical NTMsolution prediction in reference to the FE(3D) solution ispresented in Figure 13 for long samples and in Figure 14 forshort samples This percentage deviation is calculated usingthe following formula
Dev119894 = 119891(NTM)119894 minus 119891(FE)119894119891(FE)119894 lowast 100 (42)
whereDev119894 is the percentage deviation in the 119894th modalfrequency119891(FE)119894 is the 119894th mode natural frequency using three-dimensional finite element119891(NTM)119894 is the 119894th mode natural frequency usinganalytical NTMmethod
The results of Table 9 show that for the investigatedexamples with step ratio smaller than one the increase in120583 andor 119889 increases the modal frequencies The percentagedeviations in the analytical NTM results for the short samplesare higher than those for the longer samples The percentagedeviations in the analytical natural frequency prediction areplotted for the long and short examples in Figures 14 and15 respectively Figures 14(a) and 15(a) present a plot forDev1 Figures 14(b) and 15(b) present a plot for Dev2 andFigures 14(c) and 15(c) present a plot for Dev3 Figure 16(a)presents the first mode shape at the conditions of the peakpoint in Figure 15(a) Figure 16(b) presents the second modeshape at the conditions of the peak point in Figure 15(b) andFigure 16(c) presents the third mode shape at the conditionsof the peak point in Figure 15(c)
Figures 14(a) and 15(a) show that for long and shortstepped samples the Dev1 attains the peak when 120583 lies
Shock and Vibration 17
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Table 9 Typical relative deviation between present NTM and finite element (3D) results Lowest three nonzero modes for two categories ofsingle-step F-F beam see Figure 13
between 05 and 07 that is the step in the beam is locatedaround the peak of the first free-free mode shape seeFigure 16(a) Figures 14(b) and 15(b) show that there are twopeaks in Dev2 prediction The location of these peaks isfound to be near the position of the peaks of the secondfree-free mode shape see Figure 16(b) In addition the valueof Dev2 approaches zero when the step location lies in thesemistraight line between the two peaks in the second F-Fmode shape The same trend is repeated in the third modeshape as shown in Figures 14(c) 15(c) and 16(c) In additionwhen the location of diametric step in shaft coincides with a
peak point in 119894 mode shape the value of Dev119894 is increasedMeanwhile when the step lies in a straight portion of themode shape the Dev119894 approaches zero
34 Tapered Beam Approach Due to the importance oftapered or conical beams in many engineering applicationsthe current section is devoted to show how to use thepresent analysis to solve the problem of taper beam Thecurrent analysis is based on uniform beams while the partialdifferential equation which represents the lateral vibration oftapered or conical beams is fourth-order Bessel equation [30
18 Shock and Vibration
(0675 minus2307) +
minus3minus25minus2
minus15minus1
minus050
05D
evia
tion
()
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(a)
(0825 minus26941) +minus3
minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(b)
(0875 minus2867)
+minus3minus25minus2
minus15minus1
minus050
05
Dev
iatio
n (
)
01 02 03 04 05 06 07 08 09 10
d = 05
d = 0625
d = 075
d = 0875
(c)
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Figure 14 Variation of the percentage modal deviation between FE(3D) and NTM results as a function of 120583 and 119889 for free-free long samples119871 = 600mm (a) 1st mode (b) 2nd mode and (c) 3rd mode
31 37 38] To simulate nonuniform beam using the currentanalysis the beam is divided into multiple equal length spansas shown in Figure 17The height andor width of these spansare varying linearly between the start and the end to simulatethe tapered or conical beamThe height andor width ratio of119894 span can be calculated from the following formula
ℎ119894 is the height of 119894 beam segment and 119887119894 is the width of 119894 beamsegment
To verify the suitability of the current model to representconical beams the results of the current model are comparedwith the exact solution for cantilevered (C-F) conical beamwith variable taper ratio ℎ1198991 = 1198871198991 = 02 05 and 07 as shownin Table 10 The model was investigated using the presentNTMandusing several number of spans 119899 = 20 100 200 and1000 The first three eigenvalues are evaluated for each taperratio and number of spans The time used for computing thefirst three natural frequencies is evaluated The model resultsare also evaluated using NAT previously published in [6] at119899 = 20 in order to compare the time saving when using thepresent method
The results of Table 10 show that increasing the numberof spans results in increasing the accuracy of the evaluatedbeam eigenvalues in comparison with the exact solution in[30] On the other hand the results show that increasing thenumber of spans results in increasing the computational timeThe comparison between the computational time using thepresent NTM and previously published NAT shows that the
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Figure 15 Percentage modal deviation between FE(3D) and NTM as a function of 120583 and 119889 for free-free short samples 119871 = 200mm (a) 1stmode (b) 2nd mode and (c) 3rd mode
= 0675
minus1minus05
0
Y
051
152
253
02 04 06 08 10(a) First mode
= 0825
minus1minus05
0
Y
051
152
253
02 04 06 08 10(b) Second mode
= 0875
02 04 06 08 10minus1
minus05
Y
005
115
225
3
(c) Third mode
Figure 16 Typical first three modal shapes at the peak points shown in Figure 15 for free-free short sample and 119889 = 05
20 Shock and Vibration
Table10Th
efirstthree
eigenvaluesfor
C-Fconicalbeam
usingNTM
incomparis
onwith
thoseo
fthe
exactsolutionpresentedin
[30]
andNAT
results
usingthep
rogram
of[6]
ℎ 1198991=119887 1198991
[30]
NAT
[6]
119899=20
NTM 119899=20
NTM 119899=100
NTM 119899=200
NTM 119899=1000
Time(s)
Time(s)
Time(s)
Time(s)
Time(s)
02
120582lowast2 161964
61683
661706
1861954
53
61962
105
61963
326
120582lowast2 2183855
182513
182513
183801
183840
183853
120582lowast2 3398336
394809
394809
398194
398300
398336
05
120582lowast2 146252
46155
62
46175
1946249
54
46250
109
46251
34120582lowast2 2
195476
195074
195074
19546
0195472
195477
120582lowast2 3485789
484725
484725
485746
485779
485788
07
120582lowast2 1406
6940615
62
40635
19406
6856
406
69111
40670
35120582lowast2 2
205554
2053
602053
61205547
205553
205554
120582lowast2 3mdash
539625
539625
540131
540147
540152
Shock and Vibration 21
ℎ0
1
k1
xi
V
V
i ℎi
L
n+1
ℎn
kn+1
ℎi
bi
Sec VV
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
Figure 17 Stepped tapered beam with varying depth and width
present NTM method is quicker than the NAT method [6]for the same number of spans
To investigate the capability of the present model toevaluate the natural frequencies of taper or conical beamwithvariable boundary conditions some cases of nonuniformbeams are selected form [31]The analysis in [31] is numericaland based on solving the taper beam partial differential equa-tion using Runge-Kutta method The boundary conditionsand the taper ratio are listed in Table 11 The number of spansused in evaluating the current example is 119899 = 1000 Theeigenvalue results using the presentNTMmethod are in goodagreement with numerical results of [31]
4 Conclusion
A new proposed normalized transfer matrix NTM uses newset of fundamental solution in combination with the transfermatrix method This method has the advantage of the TMMin that the determinant of the frequency equation is 4 times 4for 119899 number of spans In addition the formation of thesystem frequency equation determinant is not included in anyinverse matrix steps which reduces the computational time
The current work introduces a comparison between theexperimental and analytical NTM and three-dimensionalfinite element analyses for stepped thick beams Differentsystem parameters such as the step diameters ratio 119889 the steplocation parameter 120583 and the elastic segment length ratio 119871lowast2are consideredThe following conclusions can be drawn fromthe present work
(1) An excellent agreement between the present experi-mental results using several test samples and thoseof FE(3D) predictions has been recorded
(2) The results show that the three-dimensional finiteelement can be trusted in the prediction of the modalfrequencies of stepped thick beams in structural andmechanical system
(3) An interesting percentage of modal deviationsbetween the FE(3D) results and those obtained
Table 11 The first four eigenvalues for taper or conical beamsubjected to variable end conditions using the presentNTMand [31]
Beam parameters Frequency parameter 120582lowast21[31] Present NTM
using the present frequency equation (40) has beenillustrated An increase in the step down locationparameter (120583) andor (119889) increases the modaldeviations The percentage deviations are higher forshorter beam than for longer one The maximumdeviation value in a specific normal mode is obtainedwhen the step location is adjacent to a peak locationin this normal mode Moreover the minimumpercentage deviation in a specific mode shape isobtained when the step location lies in a straightportion in the mode shape This may be explained bythe smaller effect of stress concentration at the stepunder such conditions
22 Shock and Vibration
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
(4) Typical example shows that the computational timeusing normalized transfer matrix (NTM) is greatlyreduced in comparison with numerical assemblytechnique (NAT)In addition to the above importance of conclusionone can obtain accurate results for any combinationof classical and elastic end conditions
(5) Based on the stepped beam the current analysis canbe used to evaluate the results of tapered beam Thepresent NTM show a very good stability at verylarge number of spans which enables the accurateevaluation of the tapered beam results using steppedbeam analysis
Nomenclature
119860 Cross-sectional area of the beam119886 119887 Polynomial roots119889 Segment diameter119889 Segment diameter ratio119864 Youngrsquos modulus of elasticity119891 Frequency (Hz)119866 Shear modulus of rigidity119868 Moment of inertia of the beam cross-sectionabout the neutral axis119869119894 Rotational moment of inertia of the stationmass119869119894+1 119869119894+1(1205881198601198713)119894 Shear deformation shape coefficient119896 120601 Elastic stiffness1198961 119896119899+1 End translational spring stiffness119871 Length of the beam (between points1 and 119899 + 1)119871lowast119894 Ratio 119871 119894119871119898119894 Concentrated mass at 119894 point119898119894+1 119898119894120588119894119860 119894119871 119894119898119905 Total mass of beam119903lowast21 Rotary inertia parameter 119868111986011198712119904lowast21 Shear deformation parameter 1198641119903lowast21 11986611198961119884 Nondimensional lateral deflection119909 119910 System coordinate of the beam1198851 119885119899+1 Nondimensional stiffness parameters definedas 11989611198713111986411198681 and 119896119899+11198713119899119864119899119868119899 respectively119885lowast1 119885lowast119899+1 Nondimensional stiffness parameters definedas 1198961119871311986411198681 and 119896119899+1119871311986411198681 respectivelyΓ119894 Nondimensional term (1198841015840119894 minus 119871 119894Ψ119894)1205751 1205752 Set of nondimensional terms defined as in (10a)and (10b)120582lowast41 Frequency parameter (120588111986011198714120596211986411198681)12058241 Frequency parameter (1205881119860111987141120596211986411198681)120583 Nondimensional beam length 1198711119871
] Poissonrsquos ratio120585 Nondimensional beam length 119909119871120588 Mass density of the beam material (kgm3)1206011 120601119899+1 End rotational spring stiffnessΦ1 Φ119899+1 Nondimensional rotational spring parametersdefined as 1206011119871111986411198681 and 120601119899+1119871119899119864119899119868119899respectively
Φlowast1 Φlowast119899+1 Nondimensional rotational springparameters defined as 120601111987111986411198681 and120601119899+111987111986411198681 respectively120595 Slope due to bending(sdot)1015840 1st derivative with respect to 119909 or 120585(sdot)10158401015840 2nd derivative with respect to119909 or 120585(sdot)101584010158401015840 3rd derivative with respect to 119909 or 120585(sdot)1015840101584010158401015840 4th derivative with respect to 119909 or 120585119862 Clamped (fixed) end119865 Free end119875 Pinned (hinged) end
S2 Two-span with single-step sampleS3 Three-span with two-step sampleV119897 10119864 + 12V119904 10119864 minus 12Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
The authors gratefully thank the assistants Eng M Zainworkshop technician and office secretary for their technicalsupport during all stages of this work
References
[1] J W S B RayleighTheTheory of Sound Macmillan 1896[2] S P Timoshenko ldquoLXVI On the correction for shear of the
differential equation for transverse vibrations of prismatic barsrdquoTheLondon Edinburgh andDublin PhilosophicalMagazine andJournal of Science vol 41 pp 744ndash746 1921
[3] G R Cowper ldquoThe Shear Coefficient in Timoshenkorsquos BeamTheoryrdquo Journal of AppliedMechanics vol 33 no 2 p 335 1966
[4] S K Jang and C W Bert ldquoFree vibration of stepped beamsHigher mode frequencies and effects of steps on frequencyrdquoJournal of Sound and Vibration vol 132 no 1 pp 164ndash168 1989
[5] M A Koplow A Bhattacharyya and B P Mann ldquoClosedform solutions for the dynamic response of Euler-Bernoullibeams with step changes in cross sectionrdquo Journal of Sound andVibration vol 295 no 1-2 pp 214ndash225 2006
[6] S H Farghaly and T A El-Sayed ldquoExact free vibration ofmulti-step Timoshenko beam systemwith several attachmentsrdquoMechanical Systems and Signal Processing vol 72-73 pp 525ndash546 2016
[7] H-Y Lin ldquoDynamic analysis of a multi-span uniform beamcarrying a number of various concentrated elementsrdquo Journalof Sound and Vibration vol 309 no 1-2 pp 262ndash275 2008
[8] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012
[9] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997
[10] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007
Shock and Vibration 23
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
[11] E B Magrab ldquoNatural frequencies and mode shapes of Tim-oshenko beams with attachmentsrdquo Journal of Vibration andControl vol 13 no 7 pp 905ndash934 2007
[12] J-SWu andB-H Chang ldquoFree vibration of axial-loadedmulti-step Timoshenko beam carrying arbitrary concentrated ele-ments using continuous-mass transfer matrix methodrdquo Euro-pean Journal of Mechanics - ASolids vol 38 pp 20ndash37 2013
[13] M Attar ldquoA transfer matrix method for free vibration analysisand crack identification of stepped beams with multiple edgecracks anddifferent boundary conditionsrdquo International Journalof Mechanical Sciences vol 57 no 1 pp 19ndash33 2012
[14] G Ma M Xu L Chen and Z An ldquoTransverse free vibrationof axially moving stepped beam with different length and tipmassrdquo Shock and Vibration vol 2015 Article ID 507581 2015
[15] Q S Li ldquoExact solutions for free longitudinal vibration ofstepped non-uniform rodsrdquo Applied Acoustics vol 60 no 1 pp13ndash28 2000
[16] Q S Li J Q Fang and A P Jeary ldquoFree vibration analysis ofcantilevered tall structures under various axial loadsrdquo Engineer-ing Structures vol 22 no 5 pp 525ndash534 2000
[17] Q-P Vu J-M Wang G-Q Xu and S-P Yung ldquoSpectralanalysis and system of fundamental solutions for Timoshenkobeamsrdquo Applied Mathematics Letters vol 18 no 2 pp 127ndash1342005
[18] R H Gutierrez P A A Laura and R E Rossi ldquoNaturalfrequencies of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the otherrdquoJournal of Sound and Vibration vol 141 no 1 pp 174ndash179 1990
[19] Y Yesilce ldquoDifferential transform method and numericalassembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number ofintermediate lumped masses and rotary inertiasrdquo StructuralEngineering and Mechanics vol 53 no 3 pp 537ndash573 2015
[20] X Wang and Y Wang ldquoFree vibration analysis of multiple-stepped beams by the differential quadrature element methodrdquoApplied Mathematics and Computation vol 219 no 11 pp5802ndash5810 2013
[21] F Ju H P Lee and K H Lee ldquoOn the free vibration of steppedbeamsrdquo International Journal of Solids and Structures vol 31 no22 pp 3125ndash3137 1994
[22] P A A Laura and R H Gutierrez ldquoAnalysis of vibrating Tim-oshenko beams using the method of differential quadraturerdquoShock and Vibration vol 1 no 1 pp 89ndash93 1993
[23] A Shahba R Attarnejad and S Hajilar ldquoFree vibration andstability of axially functionally graded tapered Euler-Bernoullibeamsrdquo Shock amp Vibration vol 18 no 5 pp 683ndash696 2011
[24] JW Jaworski and EHDowell ldquoFree vibration of a cantileveredbeam with multiple steps Comparison of several theoreticalmethods with experimentrdquo Journal of Sound and Vibration vol312 no 4-5 pp 713ndash725 2008
[25] Q Mao and S Pietrzko ldquoFree vibration analysis of steppedbeams by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 217 no 7 pp 3429ndash34412010
[26] Q Mao ldquoFree vibration analysis of multiple-stepped beamsby using Adomian decomposition methodrdquo Mathematical andComputer Modelling vol 54 no 1-2 pp 756ndash764 2011
[27] Z R Lu M Huang J K Liu W H Chen and W YLiao ldquoVibration analysis of multiple-stepped beams with thecomposite element modelrdquo Journal of Sound and Vibration vol322 no 4-5 pp 1070ndash1080 2009
[28] K Suddoung J Charoensuk andNWattanasakulpong ldquoAppli-cation of the differential transformation method to vibrationanalysis of stepped beams with elastically constrained endsrdquoJournal of Vibration and Control vol 19 no 16 pp 2387ndash24002013
[29] G Duan and X Wang ldquoFree vibration analysis of multiple-stepped beams by the discrete singular convolutionrdquo AppliedMathematics and Computation vol 219 no 24 pp 11096ndash111092013
[30] S Naguleswaran ldquoA Direct Solution for the Transverse Vibra-tion of Euler-Bernoulli Wedge and Cone Beamsrdquo Journal ofSound and Vibration vol 172 no 3 pp 289ndash304 1994
[31] B K Lee J K Lee T E Lee and S G Kim ldquoFree vibrations oftapered beams with general boundary conditionrdquoKSCE Journalof Civil Engineering vol 6 no 3 pp 283ndash288 2002
[32] R E Rossi P A A Laura and R H Gutierrez ldquoA note ontransverse vibrations of a Timoshenko beam of non-uniformthickness clamped at one end and carrying a concentratedmassat the otherrdquo Journal of Sound and Vibration vol 143 no 3 pp491ndash502 1990
[33] S H Farghaly ldquoVibration and stability analysis of Timoshenkobeams with discontinuities in cross-sectionrdquo Journal of Soundand Vibration vol 174 no 5 pp 591ndash605 1994
[34] S H Farghaly and T A El-Sayed ldquoExact free vibration analysisfor mechanical system composed of Timoshenko beams withintermediate eccentric rigid body on elastic supports Anexperimental and analytical investigationrdquo Mechanical Systemsand Signal Processing vol 82 pp 376ndash393 2017
[35] A Dıaz-De-Anda J Flores L Gutierrez R A Mendez-Sanchez G Monsivais and A Morales ldquoExperimental study oftheTimoshenko beam theory predictionsrdquo Journal of Sound andVibration vol 331 no 26 pp 5732ndash5744 2012
[36] J n Vaz and J de Lima ldquoVibration analysis of Euler-Bernoullibeams in multiple steps and different shapes of cross sectionrdquoJournal of Vibration andControl vol 22 no 1 pp 193ndash204 2016
[37] H H Mabie and C B Rogers ldquoTransverse Vibrations ofTapered Cantilever Beamswith End SupportrdquoThe Journal of theAcoustical Society of America vol 44 no 6 pp 1739ndash1741 1968
[38] S M Ibrahim S H Alsayed H Abbas E Carrera Y A Al-Salloum and T H Almusallam ldquoFree vibration of taperedbeams and plates based on unified beam theoryrdquo Journal ofVibration and Control vol 20 no 16 pp 2450ndash2463 2014
![Free vibration analysis of piezoelectric cylindrical nanoshell ......vibration analysis of shallow shells [25]. Loy et al. presented the free vibration analysis of cylindrical shells](https://static.documents.pub/doc/80x56/5f736ba706f884064751d11e/free-vibration-analysis-of-piezoelectric-cylindrical-nanoshell-vibration.jpg)
