Proceedings of IOE Graduate Conference, 2019-SummerPeer Reviewed

Year: 2019 Month: May Volume: 6ISSN: 2350-8914 (Online), 2350-8906 (Print)

Transverse Vibration Modal Analysis on offset Rotor Shaft oflarge Centrifugal Fan

Shailendra Shah a, Mahesh Chandra Luintel b

a, b Department of Mechanical Engineering, Pulchowk Campus, IOE, Tribhuvan University, NepalCorresponding Email: a shailendrashah7100@gmail.com, b mcluintel@ioe.edu.np,

AbstractIn this paper, transfer matrix method is used as a tool for modal analysis of offset rotor of centrifugal fanto obtain the natural frequencies of the rotating systems. Mathematical model that govern the transversevibration mode of rotor shaft is developed by transfer matrix method as the rotor of the centrifugal fan is offsetfrom the centre, and the weight of shaft and impeller is of considerable amount. The model is solved by usingstandard matrix method in MATLAB to obtain the natural frequencies and the corresponding mode shape ofrotor system. The parameters obtained after the solution includes displacement, shear force, bending momentand slope at each station point. Numerical simulation of modal analysis of the offset rotor shaft is done inANSYS workbench. The CAD model of impeller is done in SolidWorks and imported in ANSYS for modalanalysis. The accuracy of the model and the solution technique has been demonstrated by the comparisonwith results from the mathematical model and numerical simulation from ANSYS.

KeywordsTransfer matrix method, Field matrix, Point matrix, ANSYS, critical speed, unbalance, offset

1. Introduction

Vibration of the rotating components in turbo machineplays a great role in the performance in industrialplants. Turbo machines also play important role inaviation, heavy industries, oil refining industries andother important areas[1]. Huge economic losses areencountered because of its failure in heavy industries.Fatigue failure of shaft and impeller are thefundamental problem in the industry. Moreover, thestudy of the vibration analysis of rotor shaft isbeneficial to guarantee the safe operation of turbomachines and gives support to intense research onrelevant fundamental scientific problems. Fanvibrations may lead to operational problems,shutdowns, and curtailed operations[2]. Properunderstanding of vibration dynamics are required fordesign and control of rotating equipment.

Various methods have been used in analyzing therotary systems. The Jeffcott model for rotors assumesa massless shaft having a rigid rotor on it [3].Standard transfer matrix method is given in manyhandbooks. Prohl in 1945, one of the pioneers,derived it for critical speed analysis of rotor system[4]. Lund and Orcutt in 1967 developed Shaft’s

transfer matrix in a continuous fashion but neglectedthe effect of both rotary inertia and gyroscopic effect[5]. Analytical results are being generated todemonstrate the need for and the advantage of transfermatrix method in modelling procedures [6]. Theirresearch work describes the analytical basis and themethod of application for direct representation ofconical sections and trunnions for a transfer matrixanalysis. Rotor unbalance and shaft misalignment arethe two major domain of study in rotating machinery.Xu and Marangoni in 1993 developed a theoreticalmodel of a complete motor-flexible coupling rotorsystem capable of describing the mechanical vibrationresulting misalignment and unbalance [7]. Aquantitative comparison is made between the FiniteElement method and four variants of the transfermatrix method performed the quantitative comparisonof Transfer Matrix Method, as applied to freevibration analysis of rotor systems [8]. The dynamicsof a rotor-bearing system considering the gyroscopiceffect has been analyzed [9]. The rotor response dueto imbalances and offsets are studied by deriving thegeneral transfer matrix method (TMM) for rotorscontaining global and local coupler offset [10]. Theunbalance that exists in any rotor due to eccentricity

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Transverse Vibration Modal Analysis on offset Rotor Shaft of large Centrifugal Fan

Figure 2.1: CAD model of offset rotor shaft ofcentrifugal fan

has been used as excitation to perform harmonicanalysis using ANSYS [11]. ANSYS parametricdesign language has been been implemented toachieve the results.

Transverse modal analysis of rotor of centrifugal fanwith offset disk under free vibration is performed bothby mathematical model development and numericalsimulation. A mathematical model governing thetransverse vibration of rotor shaft is determined bytransfer matrix method. From the mathematicalmodel, natural frequencies and mode shape of therotor system is determined. Further, Numericalsimulation is performed in ANSYS for modal analysisfor such system. Unbalance effect is incorporatednumerically by performing harmonic analysis of therotor system in ANSYS.

2. Mathematical Model Development

The commonly used tools for rotor dynamic analysisemploy D’Alembert principle, the Lagrange’sequations, the transfer matrix method and the finiteelement method of multi-dof system. Since the rotorof the centrifugal fan is offset from the centre, and theweight of shaft and impeller is of considerableamount, Transfer Matrix Method (TMM) is used forthe modal analysis. The advantage of TMM is highprocessing speed and low memory requirement [8].The main focus of the analysis is to estimate the rotorsystem natural frequencies, mode shape and forcedresponses.

2.1 Modeling Assumption

In this analysis, the vibration amplitude is assumed tobe small and linear. The shaft is considered to be

Figure 2.2: Schematic diagram of rotor system

isotropic and of having uniform cross section. Simplysupported system is considered while deriving themathematical model for the transverse vibration.The transfer matrix of the shaft and the disk areobtained assuming that there is no gyroscopic effect,whirling effect and cross coupled terms are ignored.Bearing is assumed to be rigid and have low dampingeffect which is modeled by giving high stiffnesscoefficient to the bearing at both ends. Transfer matrixmethod obtained for one plane of motion is analogousto the perpendicular plane because of symmetry. Thesimple schematic diagram under consideration forrotor dynamic analysis is as shown in Figure 2.2.

2.2 Standard Transfer Matrix Method

Transfer matrix method (TMM), also calledMysklestand and Prohl method [4], the shaft isdivided into a number of imaginary smaller beamelements and the governing equations are derived foreach of these elements in order to determine theoverall system behavior. This method is relativelysimple and straightforward in application. For therotor system under analysis, rotor is considered tohave considerable mass and shaft is considered to bemass less. When the mass of the shaft is appreciablethen it is divided up into a number of smaller massesconcentrated (or lumped) at junctions or stations ofbeam segments so that concentrated masses and theshaft is modeled [12]. The station number is assignedwhenever there is change in the state vector whichincludes the translational and rotational (tilting)displacement, shear forces and the bending moments.

2.2.1 Shaft transfer matrix or field matrix

The free body diagram of general element of shaft isshown in Figure 2.4. Translational displacement,

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Figure 2.3: Schematic model for rotor model

Figure 2.4: Free body diagram of general shaftelement

rotational displacement, shear force and bendingmoment relation is obtained between (i−1)th and ith

station points. The displacement and the slope at thefree end are related to the applied moment and theshear force at free end by considering the beam asthough it were a cantilever and then the displacementand the slope of the fixed end is considered byconsidering the beam as a rigid. On assumption ofsmall displacements, the two steps could besuperimposed to get the total displacement and slopeof the ith shaft segment at the left of ith station. Thestandard notation for the sign convention of theequation is used as followed in [13] and [14].

Lyi = Ryi−1−Rϕi−1l−(LMyzi−1l2 + LSyi−1)

2EI+

LSyi−1l3

3EI(1)

Lϕi = Rϕi−1−(LMyzi−1l + LSyi−1)

EI+

LSyi−1l2

2EI(2)

LSi = RSi−1 (3)

LMyzi = LMyzi−1 + RSi−1 (4)

These equations can be rearranged and expressed inmatrix form as L

{S}

i = [F ]i R{

S}

i−1 with

Figure 2.5: Free body diagram of disk element

L{

S}

i =

L

−yϕx

Myz

Sy

i

; [F ]i =

1 l l2

2EIl3

6EI0 1 l

EIl2

EI0 0 1 l0 0 0 1

i

;

R{

S}

i−1 =

R

−yϕx

Myz

Sy

i−1

2.2.2 Disk transfer matrix or point matrix

The free body diagram of the point mass mi is shownin Figure 2.5 at ith location. The relationship betweenforces and displacements at thin disc is given by itsequation of motion

Ryi = Lyi (5)

Rϕi = Lϕi (6)

RMyzi =−Idiω2

Lϕi + LMyzi (7)

RSyi =−miω2

Ryi + LSyi −uyi (8)

On combining these equations in a matrix form asR{

S}

i = [P]i L{

S}

i +{

u}

i with

R{

S}

i =

R

−yϕx

Myz

Sy

i

; [P]i =

1 0 0 00 1 0 00 −ω2Id 1 0

mω2 0 0 1

i

;

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Transverse Vibration Modal Analysis on offset Rotor Shaft of large Centrifugal Fan

L{

S}

i =

L

−yϕx

Myz

Sy

i

;{

u}

i =

000

−uy

i

2.2.3 Overall transfer matrix

The point and field matrices can be used to form theoverall transfer matrix to relate the state vector at oneextreme end station (i.e., the left) to the other extremeend (i.e., the right). When there is no coupling betweenthe vertical and horizontal planes (gyroscopic effectsneglected) and no damping in the system is considered,then the overall transfer matrix [T] takes the size of5x5. The equation can be written as

R{

S}

i = [T ]i R{

S}

0 (9)

−yϕx

Myz

Sy

1

n

=

t1,1 t1,2 t1,3 t1,4 t1,5t2,1 t2,2 t2,3 t2,4 t2,5t3,1 t3,2 t3,3 t3,4 t3,5t4,1 t4,2 t4,3 t4,4 t4,5t5,1 t5,2 t5,3 t5,4 t5,5

−yϕx

Myz

Sy

1

0

(10)

On the application of boundary condition of simplysupported system the Equation11,

0ϕx

0Sy

1

n

=

t1,1 t1,2 t1,3 t1,4 t1,5t2,1 t2,2 t2,3 t2,4 t2,5t3,1 t3,2 t3,3 t3,4 t3,5t4,1 t4,2 t4,3 t4,4 t4,5t5,1 t5,2 t5,3 t5,4 t5,5

0ϕx

0Sy

1

0

(11)

2.3 Governing mathematical model

The Governing equations for the Simply supportedsystem (Pinned roller supports) is given by 12and 13with the application of boundary condition in the givensystem[

t1,2 t1,4t3,2 t3,4

]{ϕx

Sy

}=

{−t1,5−t3,5

}(12)

{ϕx

Sy

}R=

[t1,2 t1,4t3,2 t3,4

]{ϕx

Sy

}+

{t2,5t4,5

}(13)

3. Rotor dynamic analysis in ANSYS

ANSYS softwares are extensively used finite elementsimulation tool to solve different varieties of problem

Table 1: Parameters for modal analysis

Characteristic ValueRotational Speed (rpm) 990Material EN8Density (kg/m3) 7850Elastic Modulus (Gpa) 2.1Poisson’s Ratio 0.3

Figure 3.1: Geometry model used in ANSYS

in many engineering industries [11].In recent years,the rotor dynamic capabilities of ANSYS program hasbeen improved much subjected to the analysis need,feasible method and computational time [15].

3.1 Geometry

The modal analysis theory as described is used toobtain the various modes of vibration of the rotorsystem. The geometry of rotor system is thesimplified by assuming the shaft of rotor to be ofuniform cross-section. The impeller is designed insolidworks and is imported in ANSYS Designmodeler. One dimensional line sketch is done where acircular cross-section is attached. This decreases thecomplexity in the analysis. Rotor Geometry model isshown in Figure 3.1.

3.2 Model development

Before Building up the modal analysis of the rotor ofthe preheater fan, various constraints were imposed asto determine the critical speed and modal frequency.Boundary Conditions:

• Bearing support:Ground to Line Body connection is establishedat the extreme end of the shaft for modeling thebearing support in the rotor system. Theconnection type is body-ground. Stiffness ofthe both bearing K11 and K22 is given a value

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Proceedings of IOE Graduate Conference, 2019-Summer

Figure 3.2: Model development for rotor dynamicsimulation

Figure 3.3: Mesh development for rotor dynamicshaft

of 1.e+005N/mm for the rigid body motion.Rotation plane of the bearing support is of X-Zplane.

• Remote displacement:The rotor system is supposed to be simplysupported such that the bearing axis end pointshave zero displacement and zero moment.

• Rotational velocity:In order to obtain Campbell diagram, variousrotational speed has to be specified for thevarious mode.

3.3 Mesh

Automatic mesh is generated in ANSYS 15. Becauseof the dynamic loads, the three-dimensional solidelement is needed to analysis the response of thestructural with large dynamic loads in order toimprove the accuracy of analysis.

3.4 Analysis setting

Solution and Post processing: The solution of therotor dynamic system is done as per the programmed

Table 2: Rotor dynamic analysis setting

Object name Analysis settingMax no of modes 8Solver controlsDamped yesSolver type Program controlledRotor dynamic controlsCoriolis Effect OnCampbell diagram OnNumber of points 4

Figure 4.1: Variation of slope at station 0 with spinspeed

controlled shown in the analysis Table 2. Since, onlythe transverse displacement is considered for analysisin this thesis, the solution of mode 2 gives the desiredresult.

4. Results

4.1 Mathematical model solution

A mathematical model that governs the transversemode of vibration of offset rotor is obtained bytransfer matrix method. The obtained governingequation is solved by the standard matrix method forits solution. The standard matrix method is coded inMATLAB and the result is generated as shown inFigures 4.1 – 4.8.

A plot of the state variable and spin speed of the rotorat station point 0, 1 and 2 is shown in Figures. Theresonant condition can be seen as large amplitudes ofvibration and it indicate critical speed.

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Transverse Vibration Modal Analysis on offset Rotor Shaft of large Centrifugal Fan

Figure 4.2: Variation of Shear force at station 0 withspin speed

Figure 4.3: Variation of displacement at station 1with spin speed

Figure 4.4: Variation of Shear force at station 1 withspin speed

Figure 4.5: Variation of moment at station 1 withspin speed

Figure 4.6: Variation of slope at station 1 with spinspeed

Figure 4.7: Variation of slope at station 2 with spinspeed

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Proceedings of IOE Graduate Conference, 2019-Summer

Figure 4.8: Variation of Shear force at station 2 withspin speed

4.2 Rotor dynamic solution in ANSYS

4.2.1 Campbell diagram

Determination of eigen frequencies of rotating systemfor different operating speed is carried out in thisanalysis. In this, modal analysis is performed startingfrom 0 rad/s to 1000 rad/s.

4.2.2 Deflection results

Mode shape 2 is the transverse vibration of the shapeas shown in the Figure 4.10 and Figure 4.11.

5. Discussion

From the mathematical model of offset rotor shaft bytransfer matrix method, only transverse mode ofvibration is obtained. Other modes of vibration is notconsidered in analysis. The governing mathematicalequation is given by 12 and 13. A MATLAB code isdeveloped for the solution of governing equation bystandard matrix method. From the solution, twonatural frequencies 144.2495 rad/s and 338.7165 rad/sis obtained and the corresponding mode of vibrationis also obtained. The displacement, slope, shear forceand moment forces are obtained at various stationpoint. The resonant condition can be seen as largeamplitudes of vibration and indicate critical speed.For the numerical solution ANSYS 15 is used. Thegeometry required for analysis is obtained fromSolidWorks and imported to ANSYS workbench. Themodal analysis is done with the setting as shown inthe Table 2. The results obtained from modal analysis

Figure 4.9: Campbell diagram

Figure 4.10: Displacement at 144.2495 rad/s

Figure 4.11: Displacement at 335.0 rad/s

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Transverse Vibration Modal Analysis on offset Rotor Shaft of large Centrifugal Fan

showns transverse vibration, axial vibration andtorsional vibration and its higher modes. Butcomparison is done only for the transverse mode ofvibration with the mathematical model solution. Fromthe numerical analysis, the campbell diagram showsthe critical speed of 151.98 rad/s for the mode 2. Thismode shows the transverse mode which is used forcomparison.The obtained natural frequencies of transversevibration from mathematical model by transfer matrixmethod is 144.2495 rad/s and critical speed fromcampbell diagram from numerical simulation inANSYS is 151.98 rad/s. The obtained results is ofcomparable range. The error in both the results is 5.08%. This may be due to the various assumption donewhile deriving the mathematical model of transversevibration which includes the neglecting the effect ofgyroscopic effect, tilting of rotor and ignoring thecross coupled terms by transfer matrix method. Butthese effects are taken in consideration in solutionwhile numerical solution in ANSYS. Thus, obtainedmathematical model for offset rotor shaft ofcentrifugal fan by transfer matrix method can beconsidered consistent with the result.

6. Conclusion

A mathematical model that governs the vibration ofoffset rotor shaft of centrifugal fan is developed bytransfer matrix method. The obtained mathematicalmodel is solved by standard matrix method to obtainnatural frequencies and corresponding modes ofvibration. The natural frequencies obtained are144.2495 rad/s and 338.7165 rad/s. Modal analysis isperformed in ANSYS 15.0 and simulated numerically.The geometry required for simulation is generated inSolidWorks and imported in ANSYS. From the modalanalysis, campbell diagram is obtained whichdetermines various critical speed. Various modes ofvibration is also obtained from numerical analysis andfrom which mode 2 shows the transverse mode ofvibration. This mode is in our consideration and fromthe campbell diagram, critical speed obtained is151.98 rad/s. The mathematical model results andsimulation results are compared and result is obtained.

7. Recommendation

Only transverse mode of vibration is obtained whileobtaining mathematical model development of offsetrotor shaft by transfer matrix method. So other modes

of vibration can also be considered for the furtherstudy. Similarly more generalized analysis of offsetrotor can be analyzed by considering the gyroscopiceffect and tilting of the rotor while whirling motion.Moreover, the effect of unbalance force can also beincorporated with the increase in spin speed of shaft.For further generalized study the effect of thermal loadand aerodynamic load can also be considered.

8. Acknowledgments

The authors would like to acknowlege Institute ofEngineering [IOE] and Department of MechanicalEngineering, Pulchowk Campus for their support andopportunity.

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