Technical NoteSolution of Nonlinear Vibration Problem

of a Prestressed Membrane by

Adomian Decomposition

Mohammed R. Sunny,∗ Rakesh K. Kapania,† and

Cornel Sultan‡

Virginia Polytechnic Institute and State University,

Blacksburg, Virginia, 24061

DOI: 10.2514/1.J051445

Nomenclature

Aj = Adomian polynomial for the jth ordinary differentialequation

E = modulus of elasticitygj = part of the jth equation independent of the dependent

variablesLj = easily invertible operator in the jth ordinary differential

equationNj = nonlinear operator in the jth ordinary differential equationRj = remaining linear part in the jth ordinary differential

equationu = displacement along the X directionv = displacement along the Y directionw = displacement along the Z direction� = Poisson’s ratio� = density

I. Introduction

P RESTRESSED membranes are an integral part of gossamerstructures like space habitats, inflatable space antennas,

inflatable wings, etc. [1]. We are interested in attaching a prestressedmembrane to a tensegrity structure [2,3] so that we can harvestenergy from its large amplitude vibration using electromechanicaltransducers. Analysis of flexible shell and membrane structuresusing the discrete and finite-element methods has been done byseveral researchers like Gallagher et al. [4], Mohan and Kapania [5],and Kapoor et al. [6]. Our interest is in an approximate analyticalmethod to solve the governing equations of the nonlinear vibration ofa prestressed membrane with a central processing unit (CPU) timelower than traditional numerical methods like the finite-elementmethod.

Finding exact or approximate analytical solutions of nonlinear orstochastic differential equations is one of the most challengingproblems in applied mathematics. Most of the existing methods of

solving such problems are based on linearization, perturbationtechniques, or restriction on the magnitude of the nonlinear orstochastic process [7]. Transformation-based methods sometimesgive exact solutions, but they cannot be applied to all types ofproblems. For example, the Laplace transform method requires thenumerical evaluation of an improper integral and the Fouriertransform method also needs a numerical implementation, either viaa fast Fourier transform or by numerical integration. The Adomiandecomposition method [7] addresses these issues and solves thenonlinear or stochastic ordinary and partial differential equationswithout changing the actual problem to a simpler one.

Hence, the Adomian decomposition method enables solutions ofmore realistic models. The convergence of this method was analyzedby Abbaoui and Cherrault [8] and Lesnic [9]. They showed that theAdomian decomposition method provides a powerful and easilycomputable solution for different kinds of nonlinear equations.Furthermore, the convergence of the truncated series is generallyvery fast. Biazar et al. [10] showed the use of this method for solvinga system of coupled nonlinear ordinary differential equations. Rach[11] showed a comparison of Adomian decomposition method withPicard’s method. His conclusion was that these two methods are notthe same, with Picard’s method being applicable only if the vectorfield satisfies the Lipschitz condition. Yaman [12] derived anapproximate analytical solution of the variable-coefficient fourth-order differential equations governing the transverse vibrations of thecantilever beam of varying orientation with tip mass using theAdomian decomposition method.

Despite several studies, the literature dealing with theuse of Adomian decomposition to solve nonlinear vibrationproblems and their range of applicability has been rather limited, tothe best of the authors’ knowledge. In this study, we investigatethe use of the Adomian decomposition method to determine thedynamic response of a membrane in the presence of geometricnonlinearities.

First, an overviewof theAdomian decompositionmethod is given.After that, the condition of convergence of the method is described.Next, a set of simultaneous coupled nonlinear ordinary differentialequations governing the nonlinear transverse vibration of aprestressed membrane are derived. Then, those equations are solvedusing the Adomian decomposition method. A comparison of theresults obtained from the Adomian decomposition method with theresult obtained using the solver “ode45” available in MATLAB andthe finite-element package ABAQUS is shown. Finally, the range ofapplicability of this method for our problem and its limitations aredescribed.

II. Overview of the Adomian Decomposition Method

We begin with a set of P coupled nonlinear ordinary differentialequations, where the jth equation is as follows:

LjYj � Nj�Y1; Y2; . . . ; YP� � RjYj � gj�t� (1)

where Yj, LjYj, Nj�Y1; Y2; . . . ; YP�, RjYj, and gj�t� take value in aHilbert space H.

Here, Lj is an invertible operator,Nj is a nonlinear operator, gj�t�is the part of the equation independent of Yj, and Rj is the remaininglinear part. Equation (1) can be written as

L�1j LjYj � L�1j Nj�Y1; Y2; . . . ; YP� � L�1j RjYj � L�1j gj�t� (2)

In an initial value problem, the initial conditions are incorporatedin L�1j LjYj. For example, if Lj � @2=@t2, then

Presented as Paper 2011-1753 at the 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Denver,CO, 4–7 April 2011; received 10 October 2011; revision received 12 January2012; accepted for publication 13 January 2012. Copyright © 2012 by theAmerican Institute of Aeronautics and Astronautics, Inc. All rights reserved.Copies of this paper may be made for personal or internal use, on conditionthat the copier pay the $10.00 per-copy fee to theCopyright Clearance Center,Inc., 222RosewoodDrive, Danvers,MA01923; include the code 0001-1452/12 and $10.00 in correspondence with the CCC.

∗Postdoctoral Associate, Department of Aerospace and Ocean Engineer-ing; sunny@vt.edu.

†Mitchell Professor, Department of Aerospace and Ocean Engineering;rkapania@vt.edu. Associate Fellow AIAA.

‡Assistant Professor, Department of Aerospace and Ocean Engineering;csultan@vt.edu. Senior Member AIAA.

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L�1j LjYj � Yj � Yj�0� � tY0j�0� (3)

where Y 0j � @Yj=@t.Substituting Eq. (3) in Eq. (2), we get

Yj � Yj;0 � L�1j Nj�Y1; Y2; . . . ; YP� � L�1j RjYj (4)

where Yj;0 � Yj�0� � tY0j�0� � L�1j gj�t�.The general solution of Eq. (1) is assumed to be of the form

Yj �P

�n�0 Yj;n. Assuming Nj�Y1; Y2; . . . ; YP� to be analytic, it

can be written as Nj�Y1; Y2; . . . ; YP� �P

�n�0 Aj;n�Y1;0::Y1;n; . . . ;

YP;0::YP;n�. Here, Aj;n are polynomials, known as the Adomianpolynomials, which depend on the type of nonlinearity inNj�Y1; Y2; . . . ; YP�. If Nj�Y1; Y2; . . . ; YP� is expressed as a functionof Y1; Y2; . . . ; YP by f�Y1; Y2; . . . ; YP�, then coefficientsAj;n; n� 0; . . . ; �, can be written [7] as

Aj;0 � f�Y1;0; Y2;0; . . . ; YP;0�

Aj;1 � Yj;1d

dYj;0f�Y1;0; Y2;0; . . . ; YP;0�

Aj;2 � Yj;2d

dYj;0f�Y1;0; Y2;0; . . . ; YP;0�

�Y2j;1

2

d2

dY2j;0

f�Y1;0; Y2;0; . . . ; YP;0�

Aj;3 � Yj;3d

dYj;0f�Y1;0; Y2;0; . . . ; YP;0�

� Yj;1Yj;2d2

dY2j;0

f�Y1;0; Y2;0; . . . ; YP;0�

� 1

3!Y3j;1

d3

dY3j;0

f�Y1;0; Y2;0; . . . ; YP;0� . . .

(5)

Using the expressions forYj andNj, the general solution of Eq. (1)can be written as

Yj � Yj;0 � L�1j RjX1n�0

Yj;n � L�1jX1n�0

Aj;n (6)

where

Yj;n�1 ��L�1j RjXnk�0

Yj;k � L�1jXnk�0

Aj;k (7)

In this way, after obtaining Yj;0, all the successive componentsYj;n; n� 1; . . . ;1 are determined by using Eqs. (5) and (7). Byadding all the components, Yj is obtained.

According to Abboui and Cherruault [8] and Hossaini andNasabzadeh [13],

PNn�0 Yn converges to Y, which is the exact

solution of Eq. (1), if 9 0 � � � 1, jjYk�1jj � �jjYkjj, and 8 k 2 N.Here k � k denotes the norm used in Hilbert space.

III. Description of the Problem

We are interested in solving the governing differential equation ofthe large amplitude vibration of a prestressed membrane (Fig. 1)made of linearly isotropic material. In the unstretched configuration,the membrane has a square section of side length L � 2�. In thestretched configuration, the membrane has a thickness h. Each of thefour edges of the membrane is stretched by an amount �, whichkeeps the membrane under uniform and equal prestress (tensilestress) along theX and Y directions. The membrane is assumed to beunder plane stress condition. The governing equations of motion areshown next.

IV. Governing Equations of Motion

The amount of prestress along the X and Y directions isT � �E��=��L � 2���1 � ��, where E is the modulus ofelasticity, and � is the Poisson’s ratio of the material of themembrane.

Now, following the same procedure described byAmabili [14] andneglecting the bending stiffness, the equation of motion along the Zdirection can be written as

�h@2w

@t2� hT

�@2w

@x2� @

2w

@y2

�� hE

2�1 � �2�

��@w

@x

�2 @2w

@x2

��@w

@y

�2 @2w

@y2

�� hE�

2�1 � �2�

��@w

@x

�2 @2w

@y2��@w

@y

�2 @2w

@x2

�

� Eh

1� �@w

@x

@w

@y

@2w

@x@y� p (8)

By solving this nonlinear partial differential equation, thetransverse displacement w at any point at any time can be obtained.

Now, let us assume that w can be written as

w�x; y; t� �XPj�1

�j�x; y� j�t� (9)

Here,�j�x; y�, j� 1; . . . ; P are linear orthogonal mode shapes of therectangular membrane taken as the basis functions.

Following the application of the Galerkin method, i.e.,by multiplying both sides of Eq. (8) by �j�x; y�, j� 1; . . . ; Pand integrating over the domain �, we obtain a set of Pcoupled nonlinear ordinary differential equations, where the jthequation is

Mj� j � Kj j �

XPk�1

XPl�1

XPm�1

Hjklm k l m � Fj�t� (10)

where

Mj � �hZ�

�2j d� (11)

Kj � ThZ�

��2jx � �2jy��j d� (12)

Hjklm �Z�

�Eh

2�1 � �2� ��kx�lx��mxx � ��myy� � �ky�ly��myy

� ��mxx� � 2�1 � ���kx�ly�mxy�d� (13)

Fig. 1 Prestressed membrane under study.

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Fj �Z�

p�j d� (14)

Here, �jx � @�j=@x, �jy � @�j=@y, and �mxy � �@2�m�=�@x@y�.By adding a damping term �jKj _ j to the jth equation

(j� 1; . . . ; P), the jth equation can be written as

Mj� j � �jKj _ j � Kj j �

XPk�1

XPl�1

XPm�1

Hjklm k l m � Fj�t�

(15)

Next, we present an example problem that has been solved usingboth the Adomian decomposition method and the ordinarydifferential equation solver “ode45”. Values of different parametersused in our example problem are L� 0:6 m, h� 0:002,�� 0:06 m, E� 10 MPa, �� 0:33, p� 10 sin!t kPa, and!� 12:22 rad=s.

We assumed the vertical displacement w�t� to consist of nine

normal modes. Hence, we can write w�t� �PQ

j�1 �j�x; y� j�t�.Here, Q� 9 and �j�x; y� � sin��m�x�=L sin��n�y�=L, wherej� �p � 1� 3� q. The values of the integers range from 1 to 3.

Using the Galerkin technique and adding a damping term to eachequation as described already, we obtain a system of nine equations,where the jth equation is

Mj� j � �jKj _ j � Kj j �

XPk�1

XPl�1

XPm�1

Hjklm k l m

�Fj2i�ei!t � e�i!t� (16)

where i���������1p

. The function sin!t has been written as�1=2i��ei!t � e�i!t� because the exponential functions can beintegrated very easily.

For the jth equation, we can write Lj j �Mj� j � �jKj _ j�

Kj j, Nj� 1; 2; . . . j:; ; P��P

Pk�1P

Pl�1P

Pm�1Hjklm k l m,

and gj�t� � Fj sin!t. Using the procedure described in Sec. II,the approximations for j and the Adomian polynomials areobtained as

j;0�t� � L�1j Fj sin!t� Cj01ei!t � Cj01ae�i!t (17)

Aj0 �XPk�1

XPl�1

XPm�1

Hjklm k;0 l;0 m;0 (18)

j;1 � Cj03ei3!t � Cj03ae�i3!t � Cj01ei!t � Cj01ae�i!t (19)

Aj1 � j;1XPl�1

XPm�1

Hjjlm l;0 m;0 � j;1XPl�1

XPm�1

Hjkjm k;0 m;0

� j;1XPl�1

XPm�1

Hjklj k;0 l;0 (20)

j;2 � Cj05ei5!t � Cj05ae�i5!t � Cj03ei3!t � Cj03ae�i3!t

� Cj01ei!t � Cj01ae�i!t (21)

Here,

Cj01 � Fj=�2iDj10�; Cj01a ���Fj=�2iDj1a�;

Cj13 �XPk�1

XPl�1

XPm�1

�1Dj3

Ck01Cl01Cm01;

Cj13a �XPk�1

XPl�1

XPm�1

�1Dj3a

Ck01aCl01aCm01a;

Cj11 �XPk�1

XPl�1

XPm�1��1=Dj1��Ck01Cl01Cm01a � Ck01Cl01aCm01

� Ck01aCl01Cm01�;

Cj11a �XPk�1

XPl�1

XPm�1��1=Dj1a��Ck01aCl01aCm01 � Ck01aCl01Cm01a

� Ck01Cl01aCm01a�;

Cj25 �XPk�1

XPl�1

XPm�1��Cj13=Dj5��HjjlmCl01Cm01 �HjkjmCk01Cm01

�HjkljCk01Cl01�;

Cj25a �XPk�1

XPl�1

XPm�1��Cj13a=Dj5a��HjjlmCl01aCm01a

�HjkjmCk01aCm01a �HjkljCk01aCl01a�;

Cj23 �XPk�1

XPl�1

XPm�1�1=Dj3�fCj11�HjjlmCl01Cm01 �HjkjmCk01Cm01

�HjkljCk01Cl01� � Cj13�Hjjlm�Cl01Cm01a � Cl01aCm01��Hjkjm�Ck01Cm01a � Ck1aCm01� �Hjklj�Ck01Cl01a� Ck01aCl01�gCj23a

�XPk�1

XPl�1

XPm�1�1=Dj3a�fCj11a�HjjlmCl01aCm01a

�HjkjmCk01aCm01a �HjkljCk01aCl01a� � Cj13a�Hjjlm�Cl01aCm01� Cl01Cm01a� �Hjkjm�Ck01aCm01 � Ck01Cm01a��Hjklj�Ck01aCl01 � Ck01Cl01a�g;

Cj21 �XPk�1

XPl�1

XPm�1�1=Dj1�fCj13�HjjlmCl01aCm01a

�HjkjmCk01aCm01a �HjkljCk01aCl01a� � Cj11�Hjjlm�Cl01Cm01a� Cl01aCm01� �Hjkjm�Ck01Cm01a � Ck01aCm01��Hjklj�Ck01Cl01a � Ck01aCl01� � Cj1a�HjjlmCl01Cm01

�HjkjmCk01Cm01 �HjkljCk01Cl01�g;and

Cj21a �XPk�1

XPl�1

XPm�1�1=Dj1a�fCj13a�HjjlmCl01Cm01

�HjkjmCk01Cm01 �HjkljCk01Cl01� � Cj11a�Hjjlm�Cl01Cm01a� Cl01aCm01� �Hjkjm�Ck01Cm01a � Ck01aCm01��Hjklj�Ck01Cl01a � Ck01aCl01� � Cj1�HjjlmCl01aCm01a

�HjkjmCk01aCm01a �HjkljCk01aCl01a�g;

whereDjk � Kj � k2!2Mj � ik!�jKj, andDjka � Kj � k2!2Mj�ik!�jKj. These expressions have been obtained usingMathematica.

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V. Results

At first, we assumed p�x; y; t� � 10 sin!t kN=m2 and solved thesystem of equations using the Adomian decomposition method and“ode45” available in MATLAB. The CPU times required by theAdomian decomposition method and “ode45”were 1.10 s and 6.2 s,respectively. The first, fifth, and ninth modes (�1, �5, and �9) wereobserved to be dominant. Figure 2 shows the variation of 1, 5, and 9 with time when p�x; y; t� � 10 sin!t kN=m2. The resultsobtained from the Adomian decomposition method using twoterms ( j;0 and j;1), three terms ( j;0, j;1, and j;2), and four terms( j;0, j;1, j;2, and j;3) have been compared with those obtainedfrom “ode45”.

The problem was also solved by the finite-element method usingABAQUS using membrane element “M3D4”. The membrane wasdiscretized into 144 elements. The dynamic implicit scheme wasused for the solution of the nonlinear ordinary differential equations.

The CPU time required by ABAQUS was 18.4 s. Figure 3 shows thevariation of the total displacement at the center (0.3 m, 0.3 m) of themembrane with time obtained using the Adomian decompositionmethod, “ode45”, and ABAQUS. However, with 144 elements, thesize of the system is 144 when using the finite-element method. So,for a better comparison, we also analyzed the reduced systemobtained by projecting the system into the reduced basis formed bythe first nine modes using ABAQUS. The modes were found usingthe linear perturbation step, and the analysis of the reduced systemwas done by the “dynamic,susbapce” step in ABAQUS. CPU timeused by this analysis was 7.3 s. Comparison of the variation of thetotal displacement at the center (0.3 m, 0.3 m) of the membraneobtained by the Adomian decomposition method and “ode45” withthat obtained from the reduced system analysis by ABAQUSis shown in Fig. 4. In the figures, the error of Sj;j� 1; . . . ; G,with respect to Uj;j� 1; . . . ; G, has been calculated usingP

Gj�1��Sj � Uj�2=U2

j , where Sj and Uj are the values of the data

at the jth data point, and G is the total number of data points. InFig. 2, the errors have been calculated with respect to the “ode45”results, whereas Figs. 3 and 4 shows the errors with respect to theABAQUS results.

We need to verify the limitation of the Adomian decompositionmethod by an example. From Eqs. (17), (19), and (21), it can beobserved that, when P�x; y; t� � 30 sin!tk N=m2, 1;0�t�< 1;1�t�, but 5;0�t�> 5;1�t�> 5;2�t� and 9;0�t�> 9;1�t�> 9;2�t�. Hence, in Fig. 5, it can be observed that the Adomiandecomposition solution for 1�t� diverges from the solution obtainedusing “ode45” and that, for 5�t� and 9�t�, the Adomian convergesto the solution obtained using “ode45” with an increase in thenumber of terms.

0 0.5 1 1.5 2 2.5 3−0.1

−0.05

0

0.05

0.1

Time, s

ψ1(t

)

Adomian decomposition solution with 2 terms (Error=1.37%)Adomian decomposition solution with 3 terms (Error=0.94%)Adomian decomposition solution with 4 terms (Error=0.62%)ODE45 solution

a) 1 Vs. Time

b) 5 Vs. Time

c) 9 Vs. Time

0 0.5 1 1.5 2 2.5 3−5

0

5x 10

−3

Time, s

ψ3(t

),ψ

7(t)

Adomian decomposition solution with 2 terms (Error=1.03%)Adomian decomposition solution with 3 terms (Error=0.91%)Adomian decomposition solution with 4 terms (Error=0.73%)ODE45 solution

0 0.5 1 1.5 2 2.5 3−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

Time, s

ψ9(t

)

Adomian decmposition solution with 2 terms (1.86%)Adomian decmposition solution with 3 terms (0.95%)Adomian decmposition solution with 4 terms (0.72%)ODE45 solution

Fig. 2 Comparison of themodal responses obtained using theAdomian

decomposition method and “ode45” for P� 10 kN=m2.

0 0.5 1 1.5 2 2.5 3−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time, s

Dis

plac

emen

t (w

), m

Adomian decomposition solution with 4 terms (Error=1.86%)ODE45 solution (Error=1.38%)ABAQUS solution

Fig. 3 Comparison of the transverse displacement at the center (0.3 m,0.3 m) of the membrane after an analysis is run in ABAQUSwithout any

modal approximation for P� 10 kN=m2.

0 0.5 1 1.5 2 2.5 3−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time, s

Dis

plac

emen

t, m

Adomian decomposition solution with 4 terms (0.63%)ODE45 solution (0.32%)ABAQUS solution

Fig. 4 Comparison of the transverse displacement at the center (0.3 m,0.3 m) after a reduced system analysis is run in ABAQUS for

P� 10 kN=m2.

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VI. Conclusions

The Adomian decomposition method has been used to obtain anapproximate analytical solution of a system of coupled nonlinearordinary differential equations. This method allows us to solve thenonlinear differential equations without applying any simplificationto them, and themethod requires significantly less central processingunit time than the traditional numerical methods. A comparison ofthe solution obtained using this method with the solutions obtainedusing the function “ode45” in MATLAB and finite-element methodhas been given to verify its accuracy. A limitation of this solutionmethod has been explained. For this problem, the Adomian

decomposition solution does not converge if the magnitude of theapplied pressure exceeds a certain limit depending on the structuraland geometric properties of themembrane. However, this upper limitof the magnitude of the applied pressure is higher than the pressurethis membrane is supposed to experience under normal operatingconditions. Future work involves modification of the couplednonlinear differential equations by incorporating the effect ofpiezoelectric transducers in the membrane and solution of themodified equations using the Adomian decomposition method tofind out the contributions of different modes in energy harvesting.

Acknowledgment

The authors are grateful for support by the Institute for CriticalTechnology and Applied Science and the National ScienceFoundation under grant no. CMMI-0952558.

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[13] Hosseini, M. M., and Nasabzadeh, H.,“ On the Convergence ofAdomian Decomposition Method,” Applied Mathematics and

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[14] Amabili, M., Nonlinear Vibrations and Stability of Shells and Plates,CambridgeUniv. Press, NewYork/London/Cambridge, England, U.K.,2008.

0 0.5 1 1.5 2 2.5 3−4

−3

−2

−1

0

1

2

3

4

Time, s

ψ1(t

)Adomian decomposition solution with 2 terms (Error=561.72%)Adomian decomposition solution with 3 terms (Error=6839.85%)Adomian decomposition solution with 4 terms (Erorr=6842.53%)ODE45 solution

0 0.5 1 1.5 2 2.5 3−0.01

−0.005

0

0.005

0.01

Time, s

ψ3(t

),ψ

7(t)

Adomian decomposition solution with 2 terms (Error=1.98%)Adomian decomposition solution with 3 terms (Error=1.36%)Adomian decomposition solution with 4 terms (Error=0.96%)ODE45 solution

0 0.5 1 1.5 2 2.5 3−5

0

5x 10

−3

Time, s

ψ9(t

)

Adomian decomposition solution with 2 terms (Error=1.17%)Adomian decomposition solution with 3 terms (Error=1.04%)Adomian decomposition solution with 4 terms (Error=0.73%)ODE45 solution

a) 1 Vs. Time

b) 5 Vs. Time

c) 9 Vs. Time

Fig. 5 Comparison of themodal responses obtained using theAdomian

decomposition method and “ode45” for P� 30 kN=m2.

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