The Adomian modified decomposition method (AMDM) is employed in this paper for dynamic analysis of a rotating Euler-Bernoulli beam under various boundary conditions. Based on AMDM, the governing differential equation for the rotating beambecomes a recursive algebraic equation. By using the boundary condition equations, the dimensionless natural frequencies andcorresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions aswell as different offset length and rotational speeds are presented. The accuracy is assured from the convergence and comparisonpublished results. It is shown that the AMDM offers an accurate and effective method of free vibration analysis of rotating beamswith arbitrary boundary conditions.
1. Introduction
The rotating Euler-Bernoulli beams have been the subjectof numerous investigations because they are widely usedin various aeronautical, robotic, and helicopter blade andwind turbine engineering fields. The free vibration analysisof rotating beams has been extensively studied by manyresearchers [1–10] with great success. Different numerical oranalysis methods such as differential transformation method[1, 2], the Frobenius method [3], finite element method [4, 5],and dynamic stiffness method [6] have been used in solvingfree vibration problems of such structures. References in [4, 5]give an exhaustive literature survey on the free vibrationanalysis of rotating beams. References in [11–13] discusseddynamic response of rotating beams with piezoceramic actu-ation and localized damages. No attempt will be made hereto present a bibliographical account of previous work in thisarea. Few selective recent papers [1–10] which provide furtherreferences on the subject are quoted.
Until now, most of the vibration analysis of rotatingbeams has been limited to classical boundary conditions(i.e., which are either clamped, free, simply supported, orsliding). In practice, however, the characteristics of a teststructure may be very well depart from these classicalboundary conditions. In this paper, a relatively new computed
approach called Adomian modified decomposition method(AMDM) [14–21] is used to analyze the free vibration forthe rotating Euler-Bernoulli beams under various boundaryconditions, rotating speeds, and offset lengths. The AMDMis a useful and powerful method for solving linear andnonlinear differential equations. The goal of the AMDM isto find the solution of linear and nonlinear, ordinary, orpartial differential equationwithout dependence on any smallparameter like perturbation method. The main advantagesof AMDM are computational simplicity and do not involveany linearization, discretization, perturbation, or unjustifiedassumptions which may alter the physics of the problems[14]. In AMDM, the solution is considered as a sum of aninfinite series and rapid convergence to an accurate solution[15]. Recently, AMDM has been applied to the problem ofvibration of structural and mechanical systems, and thismethod has shown reliable results in providing analyticalapproximation that converges rapidly [16–21].
Using the AMDM, the governing differential equationfor the rotating beam becomes a recursive algebraic equation[14–17]. The boundary conditions become simple algebraicfrequency equations which are suitable for symbolic com-putation. Moreover, after some simple algebraic operationson these frequency equations, we can obtain the naturalfrequency and corresponding closed-form series solution of
2 Mathematical Problems in Engineering
𝑘𝐿0
𝑘𝐿1
𝑘𝑅0
𝑘𝑅1
𝐿𝑟
Ω
Figure 1: A rotating beam elastically restrained at both ends.
mode shape simultaneously. Finally, some numerical exam-ples are studied to demonstrate the accuracy and efficiency ofthe proposed method.
2. AMDM for the Rotating Beams
Consider the free vibration of a rotating Euler-Bernoullibeam with length 𝐿, constant thickness ℎ, and width 𝑏, asshown in Figure 1.The partial differential equation describingthe free vibration of a rotating beam is as follows [1, 2]:
𝐸𝐼𝑑4𝑤 (𝑥, 𝑡)
𝑑𝑥4+ 𝜌𝐴
𝑑2𝑤 (𝑥, 𝑡)
𝑑𝑡2−𝑑
𝑑𝑥[𝑇 (𝑥)
𝑑𝑤 (𝑥, 𝑡)
𝑑𝑥] = 0,
(1)
where 𝐸 is Young’s modulus, 𝐼(𝑥) = 𝑏ℎ3/12 is the cross-
sectional moment of inertia of the beam, 𝐴 = 𝑏ℎ is the cross-sectional area, and 𝜌 is the density of the beam. 𝑇(𝑥) is theaxial force due to the centrifugal stiffening and is given by thefollowing:
𝑇 (𝑥) = ∫
𝐿
𝑥
[𝜌𝐴Ω2(𝑟 + 𝑥)] 𝑑𝑥
= 0.5𝜌𝐴Ω2(𝐿2+ 2𝑟𝐿 − 2𝑟𝑥 − 𝑥
2) ,
(2)
where Ω is the angular rotating speed of the beam and 𝑟 isoffset length between beam and rotating hub.
According tomodal analysis approach (for harmonic freevibration), the 𝑤(𝑥, 𝑡) can be separable in space and time:
𝑤 (𝑥, 𝑡) = 𝜙 (𝑥) 𝑒𝑖𝜔𝑡, (3)
where 𝑖 = √−1, 𝜙(𝑥) and𝜔 are the structural mode shape andthe natural frequency, respectively.
Substituting (3) into (1), then separating variable for time𝑡 and space 𝑥, the ordinary differential equation for therotating beam can be obtained:
𝐸𝐼𝑑4𝜙 (𝑥)
𝑑𝑥4−𝑑
𝑑𝑥[𝑇 (𝑥)
𝑑𝜙 (𝑥)
𝑑𝑥] − 𝜌𝐴𝜔
2𝜙 (𝑥) = 0. (4)
0 10 20 30 40 50 60 700
50
100
150
200
250
1st natural frequency2nd natural frequency3rd natural frequency
4th natural frequency5th natural frequency
𝜆(𝑛)
𝑀
Figure 2:The first five dimensionless natural frequencies 𝜆(𝑛) as thefunction of the series summation limit𝑀.
Substituting (2) into (4), then rewriting (4) in dimension-less form,
is the dimensionless rotating speed and 𝜆 = √𝜌𝐴0𝜔2𝐿4/𝐸𝐼 is
the dimensionless natural frequency.According to the AMDM [11–18], Φ(𝑋) in (5) can be
expressed as an infinite series:
Φ (𝑋) =
∞
∑
𝑚=0
𝐶𝑚𝑋𝑚, (6)
where the unknown coefficients𝐶𝑚will be determined recur-
rently.Impose a linear operator 𝐺 = 𝑑4/𝑑𝑋4, then the inverse
operator of 𝐺 is therefore a 4-fold integral operator definedby the following:
𝐺−1= ∫∫∫∫
𝑥
0
(⋅ ⋅ ⋅) 𝑑𝑋𝑑𝑋𝑑𝑋𝑑𝑋, (7)
𝐺−1𝐺 [Φ (𝑋)] = Φ (𝑋) − 𝐶
0− 𝐶1𝑋 − 𝐶
2𝑋2− 𝐶3𝑋3. (8)
Mathematical Problems in Engineering 3
0 0.2 0.4 0.6 0.8 1
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2N
orm
aliz
ed m
ode s
hape
1st mode2nd mode
3rd mode4th mode
𝑋
(a)
0 0.2 0.4 0.6 0.8 1
−1.5
−1
−0.5
0
0.5
1
1.5
Nor
mal
ized
mod
e sha
pe
1st mode2nd mode
3rd mode4th mode
𝑋
(b)
0 0.2 0.4 0.6 0.8 1−1.5
−1
−0.5
0
0.5
1
1.5
Nor
mal
ized
mod
e sha
pe
1st mode2nd mode
3rd mode4th mode
𝑋
(c)
Figure 3: The first four normalized mode shapes for the (a) clamped-free beam, (b) clamped-clamped beam, and (c) clamped-simplysupported beam when dimensionless rotating speed 𝑈 = 4 and offset length 𝑅 = 3.
Applying both sides of (5) with𝐺−1, we get the following:
𝐺−1𝐺 [Φ (𝑋)] = −𝐺
−1{−0.5𝑈
2(1 + 2𝑅)
𝑑2Φ (𝑋)
𝑑𝑋2
+ 𝑈2𝑅𝑑
𝑑𝑋[𝑋𝑑Φ (𝑋)
𝑑𝑋] + 0.5𝑈
2
×𝑑
𝑑𝑋[𝑋2 𝑑Φ (𝑋)
𝑑𝑋] − 𝜆2Φ (𝑋)} .
(9)
Substituting (6) and (8) into (9), we get the following:
Φ (𝑋) =
3
∑
𝑚=0
𝐶𝑚𝑋𝑚
+
∞
∑
𝑚=0
0.5𝑈2(1 + 2𝑅) (𝑚 + 1) (𝑚 + 2) 𝐶
𝑚+2
(𝑚 + 1) (𝑚 + 2) (𝑚 + 3) (𝑚 + 4)𝑋𝑚+4
−
∞
∑
𝑚=0
(𝑚 + 1)2𝑈2𝑅𝐶𝑚+1
(𝑚 + 1) (𝑚 + 2) (𝑚 + 3) (𝑚 + 4)𝑋𝑚+4
4 Mathematical Problems in Engineering
−
∞
∑
𝑚=0
0.5𝑈2𝑚(𝑚 + 1)𝐶
𝑚
(𝑚 + 1) (𝑚 + 2) (𝑚 + 3) (𝑚 + 4)𝑋𝑚+4
+𝜆2𝐶𝑚
(𝑚 + 1) (𝑚 + 2) (𝑚 + 3) (𝑚 + 4)𝑋𝑚+4.
(10)
Finally, the coefficients 𝐶𝑚in (10) can be determined by
using the following recurrence relations:
𝐶0= Φ (0) , 𝐶
1=𝑑Φ (0)
𝑑𝑋
𝐶2=1
2
𝑑2Φ (0)
𝑑𝑋2, 𝐶
3=1
6
𝑑3Φ (0)
𝑑𝑋3
(11)
𝐶𝑚+4
=0.5𝑈2(1 + 2𝑅)𝐶
𝑚+2
(𝑚 + 3) (𝑚 + 4)−
(𝑚 + 1)𝑈2𝑅𝐶𝑚+1
(𝑚 + 2) (𝑚 + 3) (𝑚 + 4)
−0.5𝑈2𝑚𝐶𝑚
(𝑚 + 2) (𝑚 + 3) (𝑚 + 4)
+𝜆2𝐶𝑚
(𝑚 + 1) (𝑚 + 2) (𝑚 + 3) (𝑚 + 4), 𝑚 ≥ 0.
(12)
We may approximate the above solution by the𝑀-termtruncated series, and (6) can be rewritten as follows:
Φ (𝑋) =
𝑀
∑
𝑚=0
𝐶𝑚𝑋𝑚. (13)
Equation (13) implies that ∑∞𝑚=𝑀+1
𝐶𝑚𝑋𝑚 is negligibly
small. The number of the series summation limit 𝑀 isdetermined by convergence requirement in practice.
From the above analysis, it can be found that there arefive unknown parameters (𝐶
0, 𝐶1, 𝐶2, 𝐶3, and 𝜆) for the free
vibration analysis of the rotating beam. These unknownparameters can be determined by using the boundary condi-tion equations of the beam, and then the natural frequenciesand corresponding mode shapes for the rotating beams canbe obtained.
3. Natural Frequencies and Mode Shapes
The boundary conditions of the rotating beam shown inFigure 1 can be expressed into dimensionless form [1–4], andwe get the following:
𝑑2Φ (0)
𝑑𝑋2− 𝐾𝐿1
𝑑Φ (0)
𝑑𝑋= 0,
𝑑3Φ (0)
𝑑𝑋3+ 𝐾𝐿0Φ (0) = 0,
(14)
𝑑2Φ (1)
𝑑𝑋2+ 𝐾𝑅1
𝑑Φ (1)
𝑑𝑋= 0,
𝑑3Φ (1)
𝑑𝑋3− 𝐾𝑅0Φ (1) = 0,
(15)
where 𝐾𝐿1= 𝑘𝐿1𝐿/𝐸𝐼, 𝐾
𝐿0= 𝑘𝐿0𝐿3/𝐸𝐼, 𝐾
𝑅1= 𝑘𝑅1𝐿/𝐸𝐼,
𝐾𝑅0
= 𝑘𝑅0𝐿3/𝐸𝐼, 𝑘
𝐿0and 𝑘
𝑅0are the stiffness of the
0 1 2 3 4 51
1.5
2
2.5
3
3.5
5th mode
𝜆(𝑛)/𝜆0(𝑛)
𝑈
1st mode2nd mode3rd mode
4th mode
Figure 4: The first five dimensionless natural frequency ratios𝜆(𝑛)/𝜆
0(𝑛) for (a) the clamped-free beam with various dimension-
less rotating speeds (offset length 𝑅 = 3).
0 0.5 1 1.5 2 2.5 3 3.5 41
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
𝜆(𝑛)/𝜆0(𝑛)
𝑅
5th mode1st mode2nd mode3rd mode
4th mode
Figure 5: The first five dimensionless natural frequency ratios𝜆(𝑛)/𝜆
0(𝑛) for the clamped-free beam with various dimensionless
offset lengths (rotating speed 𝑈 = 2).
translational springs, and 𝑘𝐿1
and 𝑘𝑅1
are the stiffness of therotational springs at 𝑥 = 0 and 𝐿, respectively.
Substituting (11) into (14), we get the following:
𝐶2= 𝐾𝐿1𝐶1, 𝐶
3= −𝐾𝐿0𝐶0. (16)
Mathematical Problems in Engineering 5
Table 1: The first five dimensionless natural frequencies 𝜆(n) for a clamped-free beam with different dimensionless rotating speeds U andoffset lengths R.
Substituting (12) and (16) into (15), then (15) can beexpressed as a linear function of 𝐶
0and 𝐶
1:
𝑀
∑
𝑚=0
(𝑚 + 1) (𝑚 + 2) 𝐶𝑚+2
+ 𝐾𝑅1
𝑀
∑
𝑚=0
(𝑚 + 1) 𝐶𝑚+1
= 𝑓11(𝜆) 𝐶0+ 𝑓12(𝜆) 𝐶1= 0,
(17)
𝑀
∑
𝑚=0
(𝑚 + 1) (𝑚 + 2) (𝑚 + 3) 𝐶𝑚+3
− 𝐾𝑅0
𝑀
∑
𝑚=0
𝐶𝑚
= 𝑓21(𝜆) 𝐶0+ 𝑓22(𝜆) 𝐶1= 0.
(18)
From (17) and (18), the nth dimensionless frequencyparameter 𝜆(𝑛) can be solved by the following:
𝑓11(𝜆) 𝑓22(𝜆) − 𝑓
12(𝜆) 𝑓21(𝜆) =
𝑁
∑
𝑛=0
𝑆𝑛𝜆𝑛= 0, (19)
where𝑁 is the greatest power of 𝜆.Notice that (19) is a polynomial of degree𝑁 evaluated at𝜆.
By using the functions sym2poly and roots in MATLAB Sym-bolic Math Toolbox, (19) can be directly solved.The next stepis to determine the nth mode shape function correspondingto nth dimensionless frequency 𝜆(𝑛). Substituting the solved
6 Mathematical Problems in Engineering
0 0.5 1
0
2
0 0.5 1
0
2
0 0.5 1
0
2
0 0.5 1
0
2
0 0.5 1
0
2
0 0.5 1
0
2
0 0.5 1
0
2
0 0.5 1
0
2
0 0.5 1
0
2
0 0.5 1
0
2
0 0.5 1
0
2
0 0.5 1
0
2
Nor
mal
ized
mod
e sha
peN
orm
aliz
ed m
ode s
hape
Nor
mal
ized
mod
e sha
peN
orm
aliz
ed m
ode s
hape
1st mode2nd mode
3rd mode4th mode
1st mode2nd mode
3rd mode4th mode
1st mode2nd mode
3rd mode4th mode
𝑋𝑋 𝑋
−2
−2
−2
−2
−2
−2
−2
−2
−2
−2
−2
−2
Figure 6:The first four normalized mode shapes for the rotating beams listed in Table 4. Columns 1, 2, and 3 are (𝐾𝐿1= 𝐾𝑅1= 0), (𝐾
𝐿1= 10;
𝐾𝑅1= 20), and (𝐾
𝐿1= 100; 𝐾
𝑅1= 200), respectively. Rows 1, 2, 3, and 4 are 𝑈 = 1, 2, 3, and 4, respectively.
𝜆(𝑛) into (17) or (18), 𝐶1can be expressed as the function of
𝐶0:
𝐶1= −
𝑓11(𝜆)
𝑓12(𝜆)𝐶0= −
𝑓21(𝜆)
𝑓22(𝜆)𝐶0. (20)
Substituting (11), (12), and (20) into (13), then the modeshape function can be obtained. By normalizing (13), thenormalized mode shape is defined as follows:
Φ (𝑋) =Φ (𝑋)
√∫1
0[Φ (𝑋)]
2𝑑𝑋
. (21)
It can be found that the mode shape function by usingAMDM is a continuous function (closed-form series solu-tion) and not discrete numerical values at knot point by finiteelement or finite difference methods.
4. Results and Discussion
In order to verify the proposed method to analyze the freevibration of the rotating beam shown in Figure 1, severalnumerical examples will be discussed in this section.
As mentioned earlier, the closed-form series solutionsof mode shape functions in (13) will have to be truncatedin numerical calculations. It is important to check howrapidly the dimensionless natural frequencies 𝜆(𝑛) computed
Mathematical Problems in Engineering 7
Table 2: The first five dimensionless natural frequencies 𝜆(n) for a clamped-clamped beam with different dimensionless rotating speeds Uand offset lengths R.
throughAMDMconverge toward the exact value as the seriessummation limit𝑀 is increased. To examine the convergenceof the solution, a clamped-free beam with dimensionlessrotating speed 𝑈 = 4 and dimensionless offset length𝑅 = 3 is considered. In this study, the classical boundaryconditions (such as clamped, simply supported, and free)can be considered as the special cases of (14) and (15). Forexample, the clamped boundary condition is obtained bysetting the stiffness of the translational and rotational springsto be extremely large (which is represented by a very largenumber, 1×109, in this paper). Similarly, for simply supportedboundary condition, the stiffness of the translational androtational springs is set to 1 × 109 and 0, respectively. Forfree boundary condition, the stiffness of the translational androtational springs is set to 0. Figure 2 shows the first fivedimensionless natural frequencies 𝜆(𝑛) as the function of theseries summation limit 𝑀. Clearly, the 𝜆(𝑛) converges veryquickly as the series summation limit 𝑀 is increased. Theexcellent numerical stability of the solution can also be foundin Figure 2.
For brief, the series summation limit 𝑀 in (13) will besimply truncated to 𝑀 = 60 in all the subsequent calcu-lations. The dimensionless natural frequencies 𝜆(𝑛) are kept
accurate to the sixth decimal place for comparison purpose.Tables 1, 2, and 3 list the first five dimensionless naturalfrequencies 𝜆(𝑛) of the beam under various dimensionlessrotating speeds 𝑈 and offset lengths 𝑅 for clamped-free,clamped-clamped, and clamped-simply supported boundaryconditions, respectively. Those calculated results are com-pared with those listed in [1, 3, 4], and excellent agreementis found. Figure 3 shows the first four normalized modeshapes for different boundary conditionswhendimensionlessrotating speed 𝑈 = 4 and offset length 𝑅 = 3.
Figures 4 and 5 show the first five dimensionless naturalfrequency ratios 𝜆(𝑛)/𝜆
0(𝑛) for the clamped-free beam as the
functions of the dimensionless rotating speed 𝑈 and offsetlength 𝑅, where 𝜆
0(𝑛) is the corresponding dimensionless
natural frequencies when 𝑈 = 0 (nonrotating beam). FromFigures 4 and 5, it can be found that the natural frequencies’ratios increase when the rotating speed or offset lengthincreases for both beams. However, the variations on thenatural frequency ratios of the low order modes are moresensitive to the rotating speed or offset length.
Next, the beams with general boundary conditions arediscussed. Because the proposed method based on AMDMtechnique offers a unified and systematic procedure for
8 Mathematical Problems in Engineering
Table 3: The first five dimensionless natural frequencies 𝜆(n) for a clamped-simply supported beam with different dimensionless rotatingspeeds U and offset lengths R.
vibration analysis, the modification of boundary conditionsfrom one case to another is as simple as changing the valuesof the stiffness of translational and rotational springs. And itdoes not involve any changes to the solution procedures oralgorithms.
Table 4 lists the first five dimensionless natural frequency𝜆(𝑛) for the beam with different dimensionless rotatingspeeds 𝑈 and different rotational springs 𝐾
𝐿1and 𝐾
𝑅1when
the translational springs 𝐾𝐿0= 𝐾𝑅0= 1 × 10
9 and thedimensionless offset length 𝑅 = 3. From Table 4, it is foundthat the natural frequencies increase when the offset length orrotating speed increases, as expected. Figure 6 shows the firstfour normalized mode shapes of the rotating beam listed inTable 4. From Figure 6, it can be found that the discrepanciesof the mode shapes under different rotating speeds are verysmall. However, the natural frequencies are quite different, asshown in Table 4.
Based on the developments achieved and results obtainedin this paper, the following remarks can be made.
(1) The essential steps of the AMDMapplication includestransforming the governing differential equationfor the rotating beam into algebraic equation; byusing the boundary condition equations, any desired
dimensionless natural frequencies and correspondingmode shapes can be easily obtained simultaneously.
(2) All the steps of the AMDM are very straightforward,and the application of the AMDM to both equa-tions of motion and the boundary conditions seemsto be very involved computationally. However, allthe algebraic calculations are finished quickly usingsymbolic computational software (such asMATLAB).Besides all these, the analysis of the convergence ofthe results shows that AMDM solutions convergefast. The results of the AMDM are found in excellentagreement with available published results.
5. Conclusions
In this paper, free vibrations of the uniform rotatingEuler-Bernoulli beams under different boundary condi-tions are analyzed using Adomian modified decomposi-tion method (AMDM). The advantages of the AMDM areits fast convergence of the solution and its high degreeof accuracy. Natural frequencies and corresponding modeshapes with various boundary conditions, dimensionlessoffset length, and dimensionless rotating speed are presented.
Mathematical Problems in Engineering 9
Table 4:The first five dimensionless natural frequencies 𝜆(n) for the beamwith different dimensionless offset lengths R and rotational springstiffness 𝐾
Furthermore, the natural frequencies obtained by usingAMDM are in excellent agreement with published results.
It should be noted that the proposed method can be usedto analyze the vibration of the rotating beams under arbitraryboundary conditions. The vibration analysis for differentboundary conditions and/or rotating speed is as simple aschanging the value of corresponding parameters and does notinvolve any changes to the solution procedures or algorithms.
The results in this paper show that the AMDM techniqueis reliable, powerful, and promising for solving free vibrationproblems for rotating beams. The author believes that theAMDM can further be applied to the Timoshenko rotatingbeam problems and also it can be used as an alternativeto other solution techniques such as finite element method,differential quadrature method, and Frobenius method.
Acknowledgments
This work was partly sponsored by the National Natural Sci-ence Foundation of China (no. 51265037), Scientific ResearchFoundation for the Returned Overseas Chinese Scholars,State Education Ministry (no. 2012-44), and TechnologyFoundation of Jiangxi Province, China (no. KJLD12075).
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