MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 284720 10.1155/2013/284720 284720 Research Article Application of Adomian Modified Decomposition Method to Free Vibration Analysis of Rotating Beams Mao Qibo Coskun Safa Bozkurt School of Aircraft Engineering Nanchang HangKong University 696 South Fenghe Avenue Nanchang 330063 China nchu.edu.cn 2013 24 3 2013 2013 07 01 2013 23 02 2013 23 02 2013 2013 Copyright © 2013 Qibo Mao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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  with great success. Different numerical or analysis methods such as differential transformation method [1, 2], the Frobenius method , finite element method [4, 5], and dynamic stiffness method  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  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  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)  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 . In AMDM, the solution is considered as a sum of an infinite series and rapid convergence to an accurate solution . 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 .

Using the AMDM, the governing differential equation for the rotating beam becomes a recursive algebraic equation . 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 length L, 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.

A rotating beam elastically restrained at both ends.

According to modal analysis approach (for harmonic free vibration), the w(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 , Φ(X) in (5) can be expressed as an infinite series: (6)Φ(X)=m=0CmXm, where the unknown coefficients Cm will be determined recurrently.

Impose a linear operator G=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=00.5U2(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=00.5U2m(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 coefficients Cm 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),m0.

We may approximate the above solution by the M-term truncated series, and (6) can be rewritten as follows: (13)Φ(X)=m=0MCmXm.

Equation (13) implies that m=M+1CmXm 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 Figure 1 can be expressed into dimensionless form , 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+KR1m=0M(m+1)Cm+1=f11(λ)C0+f12(λ)C1=0,(18)m=0M(m+1)(m+2)(m+3)Cm+3-KR0m=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 Figure 1, 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.

The first five dimensionless natural frequencies λ(n) as the function of the series summation limit M.

For brief, the series summation limit M 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.

The first five dimensionless natural frequencies λ(n) for a clamped-free beam with different dimensionless rotating speeds U and offset lengths R.

R U Methods Mode index n
1 2 3 4 5
0 0 Present 3.516015 22.034492 61.697214 120.901916 199.859530
 3.5160 22.0345 61.6972 120.902 199.860
1 Present 3.681647 22.181011 61.841763 121.050922 200.011574
 3.6816 22.1810 61.8418
2 Present 4.137320 22.614922 62.273184 121.496695 200.466923
 4.1373 22.6149 62.2732
3 Present 4.797279 23.320264 62.984967 122.235547 201.223245
 4.7973 23.3203 62.9850
12 Present 13.170150 37.603112 79.614478 140.534354 220.536322
 13.1702 37.6031 79.6145 140.534 220.536

1 1 Present 3.888824 22.375014 62.043053 121.263205 200.230870
 3.8888 22.3750 62.0431
2 Present 4.833689 23.366042 63.067548 122.339546 201.340072
 4.8337 23.3660 63.0675
4 Present 7.475048 26.957262 66.986772 126.537325 205.706683
 7.475 26.9573 66.9868
8 Present 13.507389 37.953793 80.529532 141.877971 222.165291
 13.5074 37.9538 80.5295
15 Present 24.409202 61.437052 113.488866 182.695785 268.831067
 24.4092 61.4371 113.4889

2 1 Present 4.085335 22.567257 62.243575 121.475038 200.449876
 4.0853 22.5673 62.2436
2 Present 5.439949 24.092373 63.850233 123.175402 202.208659
 5.4399 24.0924 63.8502
4 Present 8.966379 29.380507 69.852259 129.712377 209.068207
 8.9664 29.3805 69.8523

3 1 Present 4.272659 22.757784 62.443336 121.686425 200.668594
 4.2727 22.7578 62.4433
2 Present 5.983713 24.796124 64.621696 124.004411 203.072744
 5.9837 24.7961 64.6217
4 Present 10.236798 31.604901 72.583135 132.794372 212.364626
 10.2368 31.6049 72.5831

The first five dimensionless natural frequencies λ(n) for a clamped-clamped beam with different dimensionless rotating speeds U and offset lengths R.

R U Methods Mode index n
1 2 3 4 5
0 0 Present 22.373285 61.672823 120.903392 1.998594481274627 2.985555379664196
 22.3733 61.6728 120.9034
1 Present 22.465244 61.801647 121.044116 200.006494 298.706437
 22.4652 61.8016 121.0441
2 Present 22.738323 62.186191 121.465166 200.446898 299.158620
 22.7383 62.1862 121.4652
4 Present 23.791502 63.696418 123.132818 202.197641 300.959712
 23.7915 63.6964 123.1328

1 1 Present 22.601469 61.987491 121.248139 200.220805 298.927273
 22.6015 61.9875 121.2481
2 Present 23.269013 62.919871 122.275462 201.300307 300.039256
 23.2690 62.9199 122.2755
4 Present 25.721997 66.488904 126.285440 205.551660 304.439765
 25.7220 66.4889 126.2854

2 1 Present 22.736642 62.172635 121.451744 200.434838 299.147913
 22.7366 62.1726 121.4517
2 Present 23.784414 63.642885 123.079231 202.149350 300.916791
 23.7844 63.6429 123.0792
4 Present 27.477304 69.138595 129.343258 208.839952 307.872284
 27.4773 69.1386 129.3433

3 1 Present 22.870783 62.3570858594266 121.6549319912815 200.6485958268648 299.3683574772145
 22.8708 62.3571 121.6549
2 Present 24.285626 64.355627 123.876600 202.994081 301.791251
 24.2856 64.3556 123.8766
4 Present 29.093946 71.663126 132.313123 212.065591 311.259430
 29.0939 71.6631 132.3131

The first five dimensionless natural frequencies λ(n) for a clamped-simply supported beam with different dimensionless rotating speeds U and offset lengths R.

R U Methods Mode index n
1 2 3 4 5
0 0 Present 15.418206 49.964862 104.247696 178.269729 272.030971
 15.4182 49.9649 104.2477
1 Present 15.512970 50.093465 104.388569 178.416902 272.181974
 15.5130 50.0935 104.3886
2 Present 15.793333 50.476967 104.809884 178.857595 272.634416
 15.7933 50.4770 104.8099
4 Present 16.861201 51.977798 106.475988 180.608162 274.435777
 16.8612 51.9778 106.4760

1 1 Present 15.650431 50.276757 104.591434 178.630496 272.402329
 15.6504 50.2768 104.5914
2 Present 16.324050 51.198754 105.614686 179.707690 273.512869
 16.3240 51.1988 105.6147
4 Present 18.739775 54.700871 109.594297 183.942227 277.903130
 18.7398 54.7009 109.5943

2 1 Present 15.786476 50.459230 104.793819 178.843781 272.622470
 15.7865 50.4592 104.7938
2 Present 16.834938 51.908185 106.412044 180.552930 274.387932
 16.8349 51.9082 106.4120
4 Present 20.412962 57.264050 112.606262 187.203776 281.318676
 20.4130 57.2641 112.6063

3 1 Present 15.921144 50.640895 104.995728 179.056759 272.842396
 15.9211 50.6409 104.9957
2 Present 17.327885 52.605804 107.202130 181.393382 275.259636
 17.3279 52.6058 107.2021
4 Present 21.932288 59.690141 115.520706 190.396605 284.684295
 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 speed U=4 and offset length R=3.

Figures 4 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.

The first five dimensionless natural frequency ratios λ(n)/λ0(n) for (a) the clamped-free beam with various dimensionless rotating speeds (offset length R=3).

The first five dimensionless natural frequency ratios λ(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.

Table 4 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.

The first five dimensionless natural frequencies λ(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 index n
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 303.679326

The first four normalized mode shapes for the rotating beams listed in Table 4. 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.

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.

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.

Acknowledgments

This work was partly sponsored by the National Natural Science Foundation of China (no. 51265037), Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (no. 2012-44), and Technology Foundation of Jiangxi Province, China (no. KJLD12075).