# Application of Adomian Modified Decomposition Method to Free Vibration Analysis of Rotating Beams

Authors: Qibo Mao
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

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 beam becomes a recursive algebraic equation. By using the boundary condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions as well as different offset length and rotational speeds are presented. The accuracy is assured from the convergence and comparison published results. It is shown that the AMDM offers an accurate and effective method of free vibration analysis of rotating beams with arbitrary boundary conditions.


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## Body

## 1. Introduction

The rotating Euler-Bernoulli beams have been the subject of numerous investigations because they are widely used in various aeronautical, robotic, and helicopter blade and wind turbine engineering fields. The free vibration analysis of rotating beams has been extensively studied by many researchers [1–10] with great success. Different numerical or analysis 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 solving free vibration problems of such structures. References in [4, 5] give an exhaustive literature survey on the free vibration analysis of rotating beams. References in [11–13] discussed dynamic response of rotating beams with piezoceramic actuation and localized damages. No attempt will be made here to present a bibliographical account of previous work in this area. Few selective recent papers [1–10] which provide further references on the subject are quoted.Until now, most of the vibration analysis of rotating beams has been limited to classical boundary conditions (i.e., which are either clamped, free, simply supported, or sliding). In practice, however, the characteristics of a test structure may be very well depart from these classical boundary conditions. In this paper, a relatively new computed approach called Adomian modified decomposition method (AMDM) [14–21] is used to analyze the free vibration for the rotating Euler-Bernoulli beams under various boundary conditions, rotating speeds, and offset lengths. The AMDM is a useful and powerful method for solving linear and nonlinear differential equations. The goal of the AMDM is to find the solution of linear and nonlinear, ordinary, or partial differential equation without dependence on any small parameter like perturbation method. The main advantages of AMDM are computational simplicity and do not involve any linearization, discretization, perturbation, or unjustified assumptions which may alter the physics of the problems [14]. In AMDM, the solution is considered as a sum of an infinite series and rapid convergence to an accurate solution [15]. Recently, AMDM has been applied to the problem of vibration of structural and mechanical systems, and this method has shown reliable results in providing analytical approximation that converges rapidly [16–21].Using the AMDM, the governing differential equation for the rotating beam becomes a recursive algebraic equation [14–17]. The boundary conditions become simple algebraic frequency equations which are suitable for symbolic computation. Moreover, after some simple algebraic operations on these frequency equations, we can obtain the natural frequency and corresponding closed-form series solution of mode shape simultaneously. Finally, some numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method.


## 2. AMDM for the Rotating Beams

Consider the free vibration of a rotating Euler-Bernoulli beam with lengthL, constant thickness h, and width b, as shown in Figure 1. The partial differential equation describing the free vibration of a rotating beam is as follows [1, 2]:
(1)EId4w(x,t)dx4+ρAd2w(x,t)dt2-ddx[T(x)dw(x,t)dx]=0,
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 the axial force due to the centrifugal stiffening and is given by the following:
(2)T(x)=∫xL[ρAΩ2(r+x)]dx=0.5ρAΩ2(L2+2rL-2rx-x2),
where Ω is the angular rotating speed of the beam and r is offset length between beam and rotating hub.Figure 1
A rotating beam elastically restrained at both ends.According to modal analysis approach (for harmonic free vibration), thew(x,t) can be separable in space and time:
(3)w(x,t)=ϕ(x)eiωt,
where i=-1, ϕ(x) and ω are the structural mode shape and the natural frequency, respectively.Substituting (3) into (1), then separating variable for time t and space x, the ordinary differential equation for the rotating beam can be obtained:
(4)EId4ϕ(x)dx4-ddx[T(x)dϕ(x)dx]-ρAω2ϕ(x)=0.Substituting (2) into (4), then rewriting (4) in dimensionless form,
(5)d4Φ(X)dX4-0.5U2(1+2R)d2Φ(X)dX2+U2RddX[XdΦ(X)dX]+0.5U2ddX[X2dΦ(X)dX]-λ2Φ(X)=0,
where X=x/L, Φ(X)=ϕ(x)/L, R=r/L, U=ρAΩ2L4/EI 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:
(6)Φ(X)=∑m=0∞CmXm,
where the unknown coefficients Cm will be determined recurrently.Impose a linear operatorG=d4/dX4, then the inverse operator of G is therefore a 4-fold integral operator defined by the following:
(7)G-1=∫∫∫∫0x(⋯)dXdXdXdX,(8)G-1G[Φ(X)]=Φ(X)-C0-C1X-C2X2-C3X3.Applying both sides of (5) with G-1, we get the following:
(9)G-1G[Φ(X)]=-G-1{-0.5U2(1+2R)d2Φ(X)dX2+U2RddX[XdΦ(X)dX]+0.5U2×ddX[X2dΦ(X)dX]-λ2Φ(X)(d2Φ(X)dX2)}.Substituting (6) and (8) into (9), we get the following:
(10)Φ(X)=∑m=03CmXm+∑m=0∞0.5U2(1+2R)(m+1)(m+2)Cm+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-∑m=0∞0.5U2m(m+1)Cm(m+1)(m+2)(m+3)(m+4)Xm+4+λ2Cm(m+1)(m+2)(m+3)(m+4)Xm+4.Finally, the coefficientsCm in (10) can be determined by using the following recurrence relations:
(11)C0=Φ(0),C1=dΦ(0)dXC2=12d2Φ(0)dX2,C3=16d3Φ(0)dX3(12)Cm+4=0.5U2(1+2R)Cm+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.We may approximate the above solution by theM-term truncated series, and (6) can be rewritten as follows:
(13)Φ(X)=∑m=0MCmXm.Equation (13) implies that ∑m=M+1∞CmXm is negligibly small. The number of the series summation limit M is determined by convergence requirement in practice.From the above analysis, it can be found that there are five unknown parameters(C0,C1,C2,C3, and λ) for the free vibration analysis of the rotating beam. These unknown parameters can be determined by using the boundary condition equations of the beam, and then the natural frequencies and corresponding mode shapes for the rotating beams can be obtained.


## 3. Natural Frequencies and Mode Shapes

The boundary conditions of the rotating beam shown in Figure1 can be expressed into dimensionless form [1–4], and we get the following:
(14)d2Φ(0)dX2-KL1dΦ(0)dX=0,d3Φ(0)dX3+KL0Φ(0)=0,(15)d2Φ(1)dX2+KR1dΦ(1)dX=0,d3Φ(1)dX3-KR0Φ(1)=0,
where KL1=kL1L/EI, KL0=kL0L3/EI, KR1=kR1L/EI, KR0=kR0L3/EI, kL0 and kR0 are the stiffness of the translational springs, and kL1 and kR1 are the stiffness of the rotational springs at x=0 and L, respectively.Substituting (11) into (14), we get the following:
(16)C2=KL1C1,C3=-KL0C0.Substituting (12) and (16) into (15), then (15) can be expressed as a linear function of C0 and C1:
(17)∑m=0M(m+1)(m+2)Cm+2+KR1∑m=0M(m+1)Cm+1=f11(λ)C0+f12(λ)C1=0,(18)∑m=0M(m+1)(m+2)(m+3)Cm+3-KR0∑m=0MCm=f21(λ)C0+f22(λ)C1=0.From (17) and (18), the nth dimensionless frequency parameter λ(n) can be solved by the following:
(19)f11(λ)f22(λ)-f12(λ)f21(λ)=∑n=0NSnλn=0,
where N is the greatest power of λ.Notice that (19) is a polynomial of degree N evaluated at λ. By using the functions sym2poly and roots in MATLAB Symbolic Math Toolbox, (19) can be directly solved. The next step is to determine the nth mode shape function corresponding to nth dimensionless frequency λ(n). Substituting the solved λ(n) into (17) or (18), C1 can be expressed as the function of C0:
(20)C1=-f11(λ)f12(λ)C0=-f21(λ)f22(λ)C0.Substituting (11), (12), and (20) into (13), then the mode shape function can be obtained. By normalizing (13), the normalized mode shape is defined as follows:
(21)Φ-(X)=Φ(X)∫01[Φ(X)]2dX.It can be found that the mode shape function by using AMDM is a continuous function (closed-form series solution) and not discrete numerical values at knot point by finite element or finite difference methods.


## 4. Results and Discussion

In order to verify the proposed method to analyze the free vibration of the rotating beam shown in Figure1, several numerical examples will be discussed in this section.As mentioned earlier, the closed-form series solutions of mode shape functions in (13) will have to be truncated in numerical calculations. It is important to check how rapidly the dimensionless natural frequencies λ(n) computed through AMDM converge toward the exact value as the series summation limit M is increased. To examine the convergence of the solution, a clamped-free beam with dimensionless rotating speed U=4 and dimensionless offset length R=3 is considered. In this study, the classical boundary conditions (such as clamped, simply supported, and free) can be considered as the special cases of (14) and (15). For example, the clamped boundary condition is obtained by setting the stiffness of the translational and rotational springs to be extremely large (which is represented by a very large number, 1×109, in this paper). Similarly, for simply supported boundary condition, the stiffness of the translational and rotational springs is set to 1×109 and 0, respectively. For free boundary condition, the stiffness of the translational and rotational springs is set to 0. Figure 2 shows the first five dimensionless natural frequencies λ(n) as the function of the series summation limit M. Clearly, the λ(n) converges very quickly as the series summation limit M is increased. The excellent numerical stability of the solution can also be found in Figure 2.Figure 2
The first five dimensionless natural frequenciesλ(n) as the function of the series summation limit M.For brief, the series summation limitM in (13) will be simply truncated to M=60 in all the subsequent calculations. 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 natural frequencies λ(n) of the beam under various dimensionless rotating speeds U and offset lengths R for clamped-free, clamped-clamped, and clamped-simply supported boundary conditions, respectively. Those calculated results are compared with those listed in [1, 3, 4], and excellent agreement is found. Figure 3 shows the first four normalized mode shapes for different boundary conditions when dimensionless rotating speed U=4 and offset length R=3.Table 1
The first five dimensionless natural frequenciesλ(n) for a clamped-free beam with different dimensionless rotating speeds U and offset lengths R.
R
U
Methods
Mode indexn
1
2
3
4
5
0
0
Present
3.516015
22.034492
61.697214
120.901916
199.859530
[4]
3.5160
22.0345
61.6972
120.902
199.860
1
Present
3.681647
22.181011
61.841763
121.050922
200.011574
[3]
3.6816
22.1810
61.8418
—
—
2
Present
4.137320
22.614922
62.273184
121.496695
200.466923
[3]
4.1373
22.6149
62.2732
—
—
3
Present
4.797279
23.320264
62.984967
122.235547
201.223245
[3]
4.7973
23.3203
62.9850
—
—
12
Present
13.170150
37.603112
79.614478
140.534354
220.536322
[4]
13.1702
37.6031
79.6145
140.534
220.536
1
1
Present
3.888824
22.375014
62.043053
121.263205
200.230870
[3]
3.8888
22.3750
62.0431
—
—
2
Present
4.833689
23.366042
63.067548
122.339546
201.340072
[3]
4.8337
23.3660
63.0675
—
—
4
Present
7.475048
26.957262
66.986772
126.537325
205.706683
[3]
7.475
26.9573
66.9868
—
—
8
Present
13.507389
37.953793
80.529532
141.877971
222.165291
[3]
13.5074
37.9538
80.5295
—
—
15
Present
24.409202
61.437052
113.488866
182.695785
268.831067
[1]
24.4092
61.4371
113.4889
—
—
2
1
Present
4.085335
22.567257
62.243575
121.475038
200.449876
[3]
4.0853
22.5673
62.2436
—
—
2
Present
5.439949
24.092373
63.850233
123.175402
202.208659
[3]
5.4399
24.0924
63.8502
—
—
4
Present
8.966379
29.380507
69.852259
129.712377
209.068207
[3]
8.9664
29.3805
69.8523
—
—
3
1
Present
4.272659
22.757784
62.443336
121.686425
200.668594
[3]
4.2727
22.7578
62.4433
—
—
2
Present
5.983713
24.796124
64.621696
124.004411
203.072744
[3]
5.9837
24.7961
64.6217
—
—
4
Present
10.236798
31.604901
72.583135
132.794372
212.364626
[1]
10.2368
31.6049
72.5831
—
—Table 2
The first five dimensionless natural frequenciesλ(n) for a clamped-clamped beam with different dimensionless rotating speeds U and offset lengths R.
R
U
Methods
Mode indexn
1
2
3
4
5
0
0
Present
22.373285
61.672823
120.903392
1.998594481274627
2.985555379664196
[3]
22.3733
61.6728
120.9034
—
—
1
Present
22.465244
61.801647
121.044116
200.006494
298.706437
[3]
22.4652
61.8016
121.0441
—
—
2
Present
22.738323
62.186191
121.465166
200.446898
299.158620
[3]
22.7383
62.1862
121.4652
—
—
4
Present
23.791502
63.696418
123.132818
202.197641
300.959712
[3]
23.7915
63.6964
123.1328
—
—
1
1
Present
22.601469
61.987491
121.248139
200.220805
298.927273
[3]
22.6015
61.9875
121.2481
—
—
2
Present
23.269013
62.919871
122.275462
201.300307
300.039256
[3]
23.2690
62.9199
122.2755
—
—
4
Present
25.721997
66.488904
126.285440
205.551660
304.439765
[3]
25.7220
66.4889
126.2854
—
—
2
1
Present
22.736642
62.172635
121.451744
200.434838
299.147913
[3]
22.7366
62.1726
121.4517
—
—
2
Present
23.784414
63.642885
123.079231
202.149350
300.916791
[3]
23.7844
63.6429
123.0792
—
—
4
Present
27.477304
69.138595
129.343258
208.839952
307.872284
[3]
27.4773
69.1386
129.3433
—
—
3
1
Present
22.870783
62.3570858594266
121.6549319912815
200.6485958268648
299.3683574772145
[3]
22.8708
62.3571
121.6549
—
—
2
Present
24.285626
64.355627
123.876600
202.994081
301.791251
[3]
24.2856
64.3556
123.8766
—
—
4
Present
29.093946
71.663126
132.313123
212.065591
311.259430
[3]
29.0939
71.6631
132.3131
—
—Table 3
The first five dimensionless natural frequenciesλ(n) for a clamped-simply supported beam with different dimensionless rotating speeds U and offset lengths R.
R
U
Methods
Mode indexn
1
2
3
4
5
0
0
Present
15.418206
49.964862
104.247696
178.269729
272.030971
[3]
15.4182
49.9649
104.2477
—
—
1
Present
15.512970
50.093465
104.388569
178.416902
272.181974
[3]
15.5130
50.0935
104.3886
—
—
2
Present
15.793333
50.476967
104.809884
178.857595
272.634416
[3]
15.7933
50.4770
104.8099
—
—
4
Present
16.861201
51.977798
106.475988
180.608162
274.435777
[3]
16.8612
51.9778
106.4760
—
—
1
1
Present
15.650431
50.276757
104.591434
178.630496
272.402329
[3]
15.6504
50.2768
104.5914
—
—
2
Present
16.324050
51.198754
105.614686
179.707690
273.512869
[3]
16.3240
51.1988
105.6147
—
—
4
Present
18.739775
54.700871
109.594297
183.942227
277.903130
[3]
18.7398
54.7009
109.5943
—
—
2
1
Present
15.786476
50.459230
104.793819
178.843781
272.622470
[3]
15.7865
50.4592
104.7938
—
—
2
Present
16.834938
51.908185
106.412044
180.552930
274.387932
[3]
16.8349
51.9082
106.4120
—
—
4
Present
20.412962
57.264050
112.606262
187.203776
281.318676
[3]
20.4130
57.2641
112.6063
—
—
3
1
Present
15.921144
50.640895
104.995728
179.056759
272.842396
[3]
15.9211
50.6409
104.9957
—
—
2
Present
17.327885
52.605804
107.202130
181.393382
275.259636
[3]
17.3279
52.6058
107.2021
—
—
4
Present
21.932288
59.690141
115.520706
190.396605
284.684295
[3]
21.9323
59.6901
115.5207
—
—The first four normalized mode shapes for the (a) clamped-free beam, (b) clamped-clamped beam, and (c) clamped-simply supported beam when dimensionless rotating speedU=4 and offset length R=3.
(a)
(b)
(c)Figures4 and 5 show the first five dimensionless natural frequency ratios λ(n)/λ0(n) for the clamped-free beam as the functions of the dimensionless rotating speed U and offset length R, where λ0(n) is the corresponding dimensionless natural frequencies when U=0 (nonrotating beam). From Figures 4 and 5, it can be found that the natural frequencies’ ratios increase when the rotating speed or offset length increases for both beams. However, the variations on the natural frequency ratios of the low order modes are more sensitive to the rotating speed or offset length.Figure 4
The first five dimensionless natural frequency ratiosλ(n)/λ0(n) for (a) the clamped-free beam with various dimensionless rotating speeds (offset length R=3).Figure 5
The first five dimensionless natural frequency ratiosλ(n)/λ0(n) for the clamped-free beam with various dimensionless offset lengths (rotating speed U=2).Next, the beams with general boundary conditions are discussed. Because the proposed method based on AMDM technique offers a unified and systematic procedure for vibration analysis, the modification of boundary conditions from one case to another is as simple as changing the values of the stiffness of translational and rotational springs. And it does not involve any changes to the solution procedures or algorithms.Table4 lists the first five dimensionless natural frequency λ(n) for the beam with different dimensionless rotating speeds U and different rotational springs KL1 and KR1 when the translational springs KL0=KR0=1×109 and the dimensionless offset length R=3. From Table 4, it is found that the natural frequencies increase when the offset length or rotating speed increases, as expected. Figure 6 shows the first four normalized mode shapes of the rotating beam listed in Table 4. From Figure 6, it can be found that the discrepancies of the mode shapes under different rotating speeds are very small. However, the natural frequencies are quite different, as shown in Table 4.Table 4
The first five dimensionless natural frequenciesλ(n) for the beam with different dimensionless offset lengths R and rotational spring stiffness KL1 and KR1 when the translational spring stiffness KL0=KR0=1×109 and the dimensionless offset length R = 3.
U
K
L
1
K
R
1
Mode indexn
1
2
3
4
5
1
0
0
10.728125
40.379864
89.736290
158.826493
247.654314
10
20
18.722928
52.557379
104.894264
176.148714
266.614189
100
200
22.240871
60.660903
118.406571
195.403654
291.714050
2
0
0
12.901808
42.950832
92.403230
161.529064
250.373747
10
20
20.251238
54.708735
107.278904
178.653087
269.187647
100
200
23.660610
62.669418
120.641412
197.764434
294.153770
3
0
0
15.741169
46.861509
96.655335
165.918965
254.831087
10
20
22.506559
58.074886
111.115686
182.735812
273.412896
100
200
25.806448
65.847359
124.257999
201.625455
298.167186
4
0
0
18.856444
51.730723
102.251676
171.841633
260.921080
10
20
25.233188
62.406770
116.227213
188.272249
279.199963
100
200
28.463584
69.993005
129.113530
206.884873
303.679326Figure 6
The first four normalized mode shapes for the rotating beams listed in Table4. Columns 1, 2, and 3 are (KL1=KR1=0), (KL1=10; KR1=20), and (KL1=100; KR1=200), respectively. Rows 1, 2, 3, and 4 are U=1,2,3, and 4, respectively.Based on the developments achieved and results obtained in this paper, the following remarks can be made.(1)
The essential steps of the AMDM application includes transforming the governing differential equation for the rotating beam into algebraic equation; by using the boundary condition equations, any desired dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously.(2)
All the steps of the AMDM are very straightforward, and the application of the AMDM to both equations of motion and the boundary conditions seems to be very involved computationally. However, all the algebraic calculations are finished quickly using symbolic computational software (such as MATLAB). Besides all these, the analysis of the convergence of the results shows that AMDM solutions converge fast. The results of the AMDM are found in excellent agreement with available published results.


## 5. Conclusions

In this paper, free vibrations of the uniform rotating Euler-Bernoulli beams under different boundary conditions are analyzed using Adomian modified decomposition method (AMDM). The advantages of the AMDM are its fast convergence of the solution and its high degree of accuracy. Natural frequencies and corresponding mode shapes with various boundary conditions, dimensionless offset length, and dimensionless rotating speed are presented. Furthermore, the natural frequencies obtained by using AMDM are in excellent agreement with published results.It should be noted that the proposed method can be used to analyze the vibration of the rotating beams under arbitrary boundary conditions. The vibration analysis for different boundary conditions and/or rotating speed is as simple as changing the value of corresponding parameters and does not involve any changes to the solution procedures or algorithms.The results in this paper show that the AMDM technique is reliable, powerful, and promising for solving free vibration problems for rotating beams. The author believes that the AMDM can further be applied to the Timoshenko rotating beam problems and also it can be used as an alternative to other solution techniques such as finite element method, differential quadrature method, and Frobenius method.



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*Source: 284720-2013-03-24.xml*