Solution of the Nonlinear Transverse Vibration

Problem of a Prestressed Membrane Using the

Adomian Decomposition Method

Mohammed R. Sunny∗ Rakesh K. Kapania†

Cornel Sultan ‡

Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

This work is a general investigation into using the Adomian decomposition method for

solving a system of coupled nonlinear differential equations governing the large amplitude

vibration of a prestressed membrane under transverse dynamic pressure. At first, the

equation of motion of the large amplitude vibration of the membrane has been presented.

Using the Galerkin technique, the equation of motion has been converted into a system of

coupled nonlinear ordinary differential equations. These equations were next solved using

the Adomian decomposition method to find out the contribution of different modes in the

vibration. Comparison of the results obtained from this method with the results obtained

from the solver ‘ode45’ in MATLAB and the finite element method using ABAQUS have

proved the accuracy of the method. The limitations of the method are also described.

Nomenclature

u Displacement along the X direction

v Displacement along the Y direction

w Displacement along the Z direction

ρ Density

E Modulus of elasticity

∗Post Doctoral Associate, Department of Aerospace and Ocean Engineering, Sensors and Structural Health MonitoringGroup, sunny@vt.edu

†Mitchell Professor, Department of Aerospace and Ocean Engineering, Sensors and Structural Health Monitoring Group,rkapania@vt.edu, Associate Fellow, AIAA

‡Assistant Professor, Department of Aerospace and Ocean Engineering, csultan@vt.edu

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52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference<BR> 19th4 - 7 April 2011, Denver, Colorado

AIAA 2011-1753

Copyright © 2011 by Mohammed Rabius Sunny. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

ν Poisson’s ratio

Lj Easily invertible operator in the jth ordinary differential equation

Nj Nonlinear operator in the jth ordinary differential equation

gj Part of the jth equation independent of the dependent variables

Rj Remaining linear part in the jth ordinary differential equation

Aj Adomian polynomial for the jth ordinary differential variable

I. Introduction

Prestressed membranes are an integral part of gossamer structures like space habitats, inflatable space

antennas, inflatable wings etc. We are interested in attaching a prestressed membrane to a tensegrity struc-

ture so that we can harvest energy from it’s large amplitude vibration using electro mechanical transducers.

This necessitates an approximate analytical method to solve the governing equations of the nonlinear vi-

bration of a prestressed membrane with a CPU time lower than the traditional numerical methods like the

finite element method.

Finding out the exact or approximate analytical solutions of nonlinear or stochastic differential equations

is one of the most challenging problems in mathematics. Most of the existing methods of solving such prob-

lems are based on linearization or perturbation techniques or restriction on the magnitude of the nonlinear

or stochastic process. Transformation based methods sometimes gives exact solution. But, they don’t lend

their application to all types of problems. Laplace transform method requires the numerical evaluation of

an improper integral and Fourier transform method also needs a numerical implementation, either via a

fast Fourier transform or by numerical integration. Adomian decomposition method, invented by George

Adomian,1 addresses these issues and solves the nonlinear or stochastic ordinary and partial differential

equations without changing the actual problem to a simpler one.

Hence, the Adomian decomposition method enables solutions of more realistic models. Other advantages

associated with this method are: (i) Calculations are relatively easy to follow and understand; (ii) Though

the solution is of the form of an infinite series in many cases, it can be written in a closed form in some

cases, otherwise, it can also be satisfactorily represented by proper truncations because the series converges

relatively rapidly. The convergence of this method was analyzed by Abbaoui and Cherrault2 and Lesnic.3

Biazar, Babolian and Islam4 showed the use of this method for solving a system of coupled nonlinear

ordinary differential equations. Rach5 showed a comparison of Adomian decomposition method with Picard’s

method. His conclusion was that these two methods are not the same, with Picard’s method being applicable

only if the vector field satisfies Lipschitz condition. Al-Khaled and Allan6 used Adomian decomposition

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to solve variable-depth shallow water equations and numerically showed its convergence. Ndour et al.7

used this method to solve the system of differential equations governing the interaction of two species.

Ghosh et al.8 proposed a numeric-analytic technique to solve the governing equations of strongly nonlinear

oscillators of engineering interest. The analytic part of their technique makes use of Adomian decomposition

method, but unlike other analytical solutions it does not rely on the functional form of the solution over

the whole domain of the independent variable. Instead it discretizes the domain and solves multiple initial

value problems recursively. Zu-feng and Xiao-yan9 used this method to derive the analytical solution of a

viscoelastic continuous beam whose damping characteristics are described in terms of a fractional derivative

of arbitrary order. Two simple cases, step and impulse function responses, were considered respectively.

Infinite summation expressions were obtained for a simply supported beam structure subjected to two loading

conditions, step and impulse functions.

Yaman10 derived an approximate analytical solution of the variable coefficient fourth order governing

differential equations governing the transverse vibrations of the cantilever beam of varying orientation with

tip mass using the Adomian decomposition method. However, literature dealing with the use of the Adomian

decomposition to solve the nonlinear vibration problems and their range of applicability has been rather lim-

ited to the best of the authors’ knowledge. In this paper, at first an overview of the Adomian decomposition

has been given. After that, the condition of convergence of the method has been described. Next, a set of

simultaneous coupled nonlinear ordinary differential equations governing the nonlinear transverse vibration

of a prestressed membrane has been derived. Then those equations have been solved using the Adomian

decomposition method. A comparison of the results obtained from the Adomian decomposition method with

the result obtained using the solver ’ode45’ available in MATLAB and the finite element package ABAQUS

has been shown. After that, the range of applicability of this method for our problem and it’s limitation

have been described.

II. Overview of the Adomian Decomposition Method

We begin with the following form of a differential equation:

LY +NY +RY = g(t) (1)

where, Y , LY , NY , RY , and g(t) take value in a Hilber space H.

Here, L is an invertible operator, N is a nonlinear operator, g(t) is the part of the equation independent

of Y and R is the remaining linear part. Eq. 1 can be written as:

L−1LY = L−1g(t) − L−1NY − L−1RY (2)

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In an initial value problem, the initial conditions are incorporated in L−1LY . For example, if L = ∂2

∂t2

then

L−1LY = Y − Y (0) − tY′

(0) (3)

where, Y′

= ∂Y∂t

Substituting Eq. 3 in Eq. 2, we get

Y = Y0 − L−1NY − L−1RY (4)

where Y0 = Y (0) + tY′

(0) + L−1g(t)

The general solution of Eq. 1 is assumed to be of the form Y =α∑

n=0

Yn. Assuming NY to be analytic, it

can be written as NY =α∑

n=0

An(Y0, Y1, .., Yn). Here, An are polynomials of Y0, Y1, .., Yn which depend on the

type of nonlinearity in NY . If NY is expressed as a function of Y by f(Y ), then coefficients An(n = 0, .., α)

can be written as:1

A0 = f (Y0)

A1 = Y1d

dY0

f(Y0)

A2 = Y2d

dY0

f(Y0) +Y 2

1

2d2

dY 2

0

f(Y0)

A3 = Y3d

dY0

f(Y0) + Y1Y2d2

dY 2

0

f(Y0) + 13!Y

31

d3

dY 3

0

f(Y0)

.

.

. (5)

Now, the general solution of Eq. 1 can be written as:

Y = Y0 − L−1R

∞∑

n=0

Yn − L−1N

∞∑

n=0

An (6)

where,

Yn+1 = −L−1R

∞∑

n=0

Yn − L−1N

∞∑

n=0

An (7)

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In this way, after obtaining Y0, all the successive components Yn(n = 1, ..,∞) are determined by using

Eqs. 5 and 7. By adding all the components the solution Y is obtained.

Now, let us consider a system of P coupled ordinary differential equations, where the jth equation is of

the following form.

LjYj +Nj(Y1, Y2, .., YP ) +RjYj = gj(t) (8)

To solve this system of equations, at first the first approximation for Yj(j = 1, .., P ) is obtained from the

jth equation as:

Yj,0 = Yj(0) + tY′

j (0) + L−1gj(t) (9)

Adomian polynomials Ak for the jthe equation can be calculated 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) +

Y 2

j,1

2d2

dY 2

i,0

f(Y1,0, Y2,0, .., YP,0)

Aj,3 = Yj,3d

dYj,0f(Y1,0, Y2,0, .., YP,0) + Yj,1Yj,2

d2

dY 2

0

f(Y1,0, Y2,0, .., YP,0) + 13!Y

3j,1

d3

dY 3

0

f(Y1,0, Y2,0, .., YP,0)

.

.

. (10)

Now, the general solution of Eq. 10 can be written as:

Yj = Yj,0 − L−1j Rj

∞∑

n=0

Yj,n − L−1j Nj

∞∑

n=0

Aj,n (11)

III. Convergence of the Solution

At first, let us consider the solution of Eq. 1 obtained using the Adomian decomposition method.

According to Abboui and Cherruault2 and Hossaini and Nasabzadeh,11∑N

n=0 Yn converges to Y , which is

the exact solution of Eq. 1, if ∃0 ≤ α ≤ 1, ||Yk+1|| ≤ ||Yk||,∀k ∈ N ∪ 0.

According to the Adomian decomposition method, the solution of Eq. 1 is approximated by the sequence

Sn∞

n=0, where Sn = Y0 + Y1 + Y2 + .. + Yn. To prove the convergence of the sequence Sn∞

n=0, we need to

prove that Sn∞

n=0 is a Cauchy sequence.

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Let us consider, ||Sn+1 − Sn|| = ||Yn+1|| ≤ α||Yn||... ≤ αn+1||Y0||.

Now, ∀n,m ∈ N,n ≥ m, we can write

||Sn − Sm|| = ||(Sn − Sn−1) + (Sn−1 − Sn−2) + ...+ (Sm+1 − Sm)||

≤ ||(Sn − Sn−1)|| + ||(Sn−1 − Sn−2)|| + ...+ ||(Sm+1 − Sm)||

= (αn + αn−1 + ...+ αm)||y0||

≤ (αm+1 + αm+2 + ...)||y0||

=αm+1

1 − α||y0||

(12)

So, limn,m→+∞ ||Sn − Sm|| = 0, i.e. Sn∞n=0 is a Cauchy sequence in H. Hence, solution of Eq. 1

converges in H, if ∃0 ≤ α ≤ 1, ||Yk+1|| ≤ ||Yk||,∀k ∈ N ∪ 0.

Similarly, in case of P simultaneous equations, the solution for the jth dependent variable Yj converges

if ∃0 ≤ α ≤ 1, ||Yj,k+1|| ≤ ||Yj,k||,∀k ∈ N ∪ 0.

IV. Description of the Problem

We are interested in solving the governing differential equation of the large amplitude vibration of a

prestressed membrane (Fig. 1) made of linearly isotropic material. Each of the four edges of the membrane

are stretched by an amount ∆ which keeps the membrane under uniform and equal prestress (tensile stress)

along the X and Y direction. The membrane is assumed to be under plane stress condition. The governing

equations of motion are shown in the next section.

V. Governing Equations of Motion

The amount of prestress along the X and Y directions is T = E∆L(1−ν) , where E is the modulus of elasticity

and ν is the Poisson’s ratio of the material of the membrane. According to Von Karman’s hypotheses12 for

very thin plate, the in-plane Green’s strains in the membrane are related to the displacements along the

X,Y and Z directions, u, v and w respectively, by the following equations:

εxx =1

2

(

∂w

∂x

)2

(13)

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Figure 1. Prestressed membrane under study

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εyy =1

2

(

∂w

∂y

)2

(14)

γxy =∂w

∂x

∂w

∂y(15)

The Kirchhoff’s stress components in the plane stress condition for linearly isotropic and homogeneous

materials are related to the Green’s strain components by the following equations given in Ref.12

σxx =E

1 − ν2(εxx + νεyy) (16)

σyy =E

1 − ν2(εyy + νεxx) (17)

σxy =E

2(1 + ν)γxy (18)

Here, E is the modulus of elasticity and ν is the Poisson’s ratio. The three dimensional set of equations

for a thin structure under the plane stress condition is given in Ref.:12

ρ∂2u

∂t2− ∂σxx

∂x− ∂σyx

∂y= fx (19)

ρ∂2v

∂t2− ∂σxy

∂x− ∂σyy

∂y= fy (20)

ρ∂2w

∂t2− ∂

∂x

(

σxx

∂w

∂x+ σxy

∂w

∂y

)

− ∂

∂y

(

σyx

∂w

∂x+ σyy

∂w

∂y

)

= fz (21)

Here, fx, fy and fz are the applied forces and u, v and w are the displacements along X,Y and Z

directions respectively and ρ is the density of the material.

Now, following the same procedure described in Ref.12 and neglecting the bending stiffness, the equation

of motion along the Z direction can be written as:

ρh∂2w

∂t2−Nx

∂2w

∂x2−Ny

∂2w

∂y2− 2Nxy

∂2w

∂x∂y− p = 0 (22)

Here, p = p(x, y, t) is the applied transverse pressure, Nx, Ny and Nxy are the stress resultants which are

given by Eqs. 23, 24 and 25.

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Nx = Th+Eh

2(1 − ν2)

(

(

∂w

∂x

)2

+

(

∂w

∂y

)2)

(23)

Ny = Th+Eh

2(1 − ν2)

(

(

∂w

∂y

)2

+

(

∂w

∂x

)2)

(24)

Nxy =Eh

2(1 + ν)

∂w

∂x

∂w

∂y(25)

Using the expressions for Nx, Ny and Nxy given above, Eq. 22 can be written as:

ρh∂2w

∂t2− hT

(

∂2w

∂x2+∂2w

∂y2

)

− hE

2(1 − ν2)

(

(

∂w

∂x

)2∂2w

∂x2+

(

∂w

∂y

)2∂2w

∂y2

)

− hEν

2(1 − ν2)

(

(

∂w

∂x

)2∂2w

∂y2+

(

∂w

∂y

)2∂2w

∂x2

)

− E

1 + ν

∂w

∂x

∂w

∂y

∂2w

∂x∂y= p (26)

By solving this nonlinear partial differential equation, transverse 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) =∑P

j=1 φj(x, y)ψj(t), where φj(x, y) is the jth

mode shape.

Here, φj(x, y)(j = 1, .., P ) are orthogonal mode shapes. They satisfy the essential boundary conditions.

According to the orthogonality of the modes,∫

Ωφi(x, y)φj(x, y)dydx = 0, when i 6= j.

Following the Galerkin technique, by multiplying both sides of Eq. 26 by φj(x, y)(j = 1, .., P ) and

integrating over the domain Ω, we get a set of P coupled nonlinear ordinary differential equations, where

the jth equation is:

Mjψj +Kjψj +

P∑

k=1

Bjkψ3k +

P∑

k=1

P∑

l=1

Hjklψ2kψl = Fj(t) (27)

where

Mj = ρh

∫

Ω

φ2jdΩ (28)

Kj = Th

∫

Ω

(φ2jx + φ2

jy)φjdΩ (29)

Bjk = − Eh

2(1 − ν2)

∫

Ω

((

φ2kxφkxx + φ2

kyφkyy

)

+ ν(

φ2kxφkyy + φ2

kyφkxx

))

φjdΩ − Eh

1 + ν

∫

Ω

φkxφkyφkxyφjdΩ

(30)

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Hjkl = − Eh

2(1 − ν2)

∫

Ω

((

φ2kxφlxx + φ2

kyφlyy + 2φkxφkxxφlx + 2φkyφkyyφly

)

+ν(

φ2kyφlxx + φ2

kxφlyy + 2φkyφkxxφly + 2φkxφkyyφlx

))

φjdΩ

Eh

1 + ν

∫

Ω

(φkxφkyφlxy + (φkxφly + φkyφlx)φkxy)φdΩ (31)

Fj =

∫

Ω

pφjdΩ (32)

By adding a damping term βjKjψj to the jth equation (j = 1, .., P ), the system of equations can be

written as:

M1ψ1 + β1K1ψ1 +K1ψ1 +P∑

k=1

B1kψ3k +

P∑

k=1

P∑

l=1

H1klψ2kψl = F1(t)

M2ψ2 + β2K2ψ2 +K2ψ2 +

P∑

k=1

B2kψ3k +

P∑

k=1

P∑

l=1

H2klψ2kψl = F2(t)

.

.

Mjψj + βjKjψj +Kjψj +P∑

k=1

Bjkψ3k +

P∑

k=1

P∑

l=1

Hjklψ2kψl = Fj(t)

.

.

MP ψP + βPKP ψP +KPψP +

P∑

k=1

BPkψ3k +

P∑

k=1

P∑

l=1

HPklψ2kψl = FP (t)

(33)

VI. Solution Using the Adomian Decomposition Method

We solved an example problem using both Adomian decomposition method and the ordinary differential

equation solver ‘ode45’. Values of different parameters used in our example problem are L = 0.6 m, h = 0.001,

∆ = 0.05 m, E = 10 MPa, ν = 0.33, p = 10Sinωt KPa, ω = 12.22 rad/sec.

We, assumed the vertical displacement w(t) to consist of 9 normal modes. Hence, we can write w(t) =

∑9j=1 φj(x, y)ψj(t), where

φ1(x, y) = Sin(πxL

)Sin(πyL

)

φ2(x, y) = Sin(πxL

)Sin( 2πyL

)

φ3(x, y) = Sin(πxL

)Sin( 3πyL

)

φ4(x, y) = Sin( 2πxL

)Sin(πyL

)

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φ5(x, y) = Sin( 2πxL

)Sin( 2πyL

)

φ6(x, y) = Sin( 2πxL

)Sin( 3πyL

)

φ7(x, y) = Sin( 3πxL

)Sin(πyL

)

φ8(x, y) = Sin( 3πxL

)Sin( 2πyL

) and

φ9(x, y) = Sin( 3πxL

)Sin( 3πyL

)

Using the Galerkin technique and adding a damping term to each equation as described in the previous

section, we got a system of nine equations, where jth equation is:

M1ψ1 + β1K1ψ1 +K1ψ1 +9∑

k=1

B1kψ3k +

9∑

k=1

9∑

l=1

H1klψ2kψl = F1Sinωt =

F1

2i

(

eiωt − e−iωt)

M2ψ2 + β2K2ψ2 +K2ψ2 +

9∑

k=1

B2kψ3k +

9∑

k=1

9∑

l=1

H2klψ2kψl = F2Sinωt =

F2

2i

(

eiωt − e−iωt)

.

.

Mjψj + βjKjψj +Kjψj +9∑

k=1

Bjkψ3k +

9∑

k=1

9∑

l=1

Hjklψ2kψl = FjSinωt =

Fj

2i

(

eiωt − e−iωt)

.

.

M9ψ9 + β9K9ψ9 +K9ψj +

9∑

k=1

BPkψ3k +

9∑

k=1

9∑

l=1

HPklψ2kψl = F9Sinωt =

F9

2i

(

eiωt − e−iωt)

(34)

Here, i =√−1. The function Sinωt has been written as 1

2i

(

eiωt − e−iωt)

, because the exponential

functions can be integrated very easily.

For the jth equation, we can write Ljψj = Mψj+αKψj+Kψj , Njψj =∑9

k=1Bjkψ3k+∑9

k=1

∑9l=1Hjklψ

2kψlkl,

and gj(t) = FjSin(ωt)

Using the procedure described in section II, the approximations for ψj and the Adomian polynomials

were obtained as:

ψj,0(t) = L−1j FjSin(ωt) =

Fj

2i

(

1

Dk1− 1

Dk1m

)

(35)

Aj0 =

9∑

k=1

Bjkψ3k,0 +

9∑

k=1

9∑

l=1

Hjklψ2k,0ψl (36)

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ψj,1 = −9∑

k=1

BjkF3k

8i3

(

− 3eiωt

D2k1Dk1mDj1

+3eiωt

D2k1mDk1Dj1m

+ei3ωt

D3k1Dj3

− ei3ωt

D3k1mDj3m

)

−P∑

k=1

P∑

l=1

F 2kFlHjkl

8i3

(

ei3ωt

D2k1Dl1Dj3

− e−i3ωt

D2k1mDl1mDj3m

− eiωt

Dj1

(

1

D2k1Dl1m

2

Dk1Dk1mDl1

)

+e−iωt

Dj1m

(

1

D2k1mDl1

+2

Dk1Dk1mDl1m

))

(37)

Aj1 = 3Bjjψ2j,0ψj,1 +

9∑

l=1

Hjjl

(

2ψj,0ψl,0 + ψ2j,0

)

ψj,1 (38)

ψj,2(t) =

P∑

k=1

3BjjBjkF3kF

2j

32i5

(

− ei5ωt

Dj5D3k1Dk3D

2j1

+e−i5ωt

Dj5mD3k1mDk3mD

2j1m

−ei3ωt

Dj3

(

1

D3j1D

2k1Dj1m

− 2

Dj1Dj1mDk3D3k1

)

+e−i3ωt

Dj3m

(

1

D3j1mD

2k1mDj1

− 2

Dj1mDj1Dk3mD3k1m

)

− 2eiωt

D3j1Dj1D

2k1Dk1m

+2e−iωt

D3j1mDj1D

2k1mDk1

)

+

P∑

k=1

P∑

l=1

3BjjHjklF2j F

2kFl

32i5

(

ei5ωt

D2j1D

2k1Dl1Dj3Dj5

− e−i5ωt

D2j1mD

2k1mDl1mDj3mDj5m

−ei3ωt

Dj3

(

1

D2k1Dl1mD

2j1

+2

Dk1Dk1mDl1D2j1

+2

Dj1Dj1mD2k1Dl1Dj3

)

+e−i3ωt

Dj3m

(

1

D2k1mDl1D

2j1m

+2

Dk1mDk1Dl1mD2j1m

+2

Dj1mDj1D2k1mDl1mDj3m

)

− 2eiωt

D3j1Dj1m

(

1

D2k1Dl1m

+2

Dk1Dk1mDl1

)

+2e−iωt

D3j1mDj1

(

1

D2k1mDl1

+2

Dk1mDk1Dl1m

)

)

(39)

Here, Djk = Kj − k2ω2Mj + ikωβjKj and Djkm = Kj − k2ω2Mj − ikωβjKj

VII. Results

At first, we assumed p = 10 KN and solved the system of equations using the Adomian decomposition

method and the function ‘ode45’ available in MATLAB. The calculations were done using a personal com-

puter (Dell Precision PWS690) with 2.66GHz processor speed and 2GB RAM. CPU times required by the

Adomian decomposition method and ode45 were 0.5 Seconds and 6.2 Seconds respectively. The first and

ninth modes (φ1 and φ5) were observed to be dominant.

Figure 2 shows the variation of ψ1 and ψ5 with time. The results obtained from the Adomian decom-

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position method using 2 terms (ψj,0 and ψj,1) and 3 terms (ψj,0, ψj,1 and ψj,2) approximations have been

compared with those obtained from ‘ode45’. After that, the problem was solved by the finite element method

ABAQUS. We used membrane element ’M3D4’ available in the ABAQUS element library. CPU time re-

quired by ABAQUS was 18.4 seconds. Figure 3 shows the variation of the total displacement at the center

(0.3m, 0.3m) of the membrane with time obtained using the Adomian decomposition method, ode45 and

ABAQUS. In the figures, the error of Sj(j = 1, .., Q) with respect to Uj(j = 1, .., Q) has been calculated

using the relation∑

j=1Q(Sj−Uj)

2

U2

j

, where Sj and Uj are the value of the data at the jth data point and Q

is the total number of data points. In Figs. 2 and 3, the errors have been calculated with respect to the

ode45 and ABAQUS results respectively.

Figure 2 shows that with increase in the number of terms in the Adomian decomposition solution, the

solution converges to the solution obtained by ‘ode45’.

A. Limitation of the Adomian Decoposition Method

Now, we need to verify the limitation of the Adomian decomposition method by an example. From Eqs. 35, 37

and 39, it can be observed that, when P = 30KN , ψ1,0(t) < ψ1,1(t), but ψ5,0(t) > ψ5,1(t) > ψ5,2(t). Hence,

in Fig. 4 it can be observed that the Adomian decomposition solution for ψ1(t) diverges from the solution

obtained using ‘ode45’ and that for ψ5(t) converges to the solution obtained using ’ode45’ with increase in

number of terms. This example verifies the condition of convergence of the Adomian decomposition method

described in section III.

VIII. Summary and Conclusions

Adomian decomposition method has been used to obtain an approximate analytical solution of a system

of coupled nonlinear ordinary differential equations. This method allows us to solve the nonlinear differential

equations without applying any simplification to them and the method requires significantly less CPU time

than the traditional numerical methods. A comparison of the solution obtained using this method with the

solutions obtained using the function ode45 in MATLAB and finite element method has been given to verify

its accuracy. A limitation of this solution method has been explained. For our problem, solution does not

converge if the magnitude of the applied pressure exceeds a certain limit depending on the structural and

geometric properties of the membrane. However, this upper limit of the magnitude of the applied pressure is

higher than the pressure this membrane is supposed to experience under normal operating conditions. Our

future work involves modification of the coupled nonlinear differential equations by incorporating the effect

of piezoelectric transducers in the membrane and solution of the modified equations using the Adomian

decomposition method to find out the contributions of different modes in energy harvesting.

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0 0.5 1 1.5 2 2.5 3 3.5−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time (Seconds)

ψ1(t

)

ode45 solutionAdomian decomposition solution with 2 terms (Error=1.4%)Adomian decomposition solution with 3 terms (Error=0.6%)

(a) ψ1 Vs. Time

0 0.5 1 1.5 2 2.5 3 3.5−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time (Seconds)

ψ9 (

t)

ode45 solutionAdomian decomposition solution with 2 terms(Error=0.3%)Adomian decomposition solution with 3 terms (Error=0.1%)

(b) ψ9 Vs. Time

Figure 2. Modal responses for P=10 KN/m2

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Acknowledgments

The research was performed under project grant from Institute for Critical Technologies and Sciences

(ICTAS) at Virginia Polytechnic Institute and State University, Blacksburg, VA.

References

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8Ghosh, S., Roy, A., and Roy, D., “An adaption of Adomian decomposition for numeric-analytic integration of strongly

nonlinear and chaotic oscillators,” Computer Methods in Applied Mechanics and Engineering, 196, 2007, pp. 1133–1153.

9Zu-feng, L., and Xiao-yan, T., “Analytical solution of fractionally damped beam by Adomian decomposition method,”

Applied Mathematics and Mechanics (English Edition), 28, 2007, pp. 219–228.

10Yaman, M., “Adomian Decomposition Method for Solving a Cantilever Beam of Varying Orientation with Tip Mass,”

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11Hosseini, M. M., Nasabzadeh, H., “On the Convergence of Adomian Decomposition Method ,” Applied Mathematics and

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12Amabili, M., “Nonlinear Vibrations and Stability of Shells and Plates,” Cambridge University Press, 2008.

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0 0.5 1 1.5 2 2.5 3 3.5−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time (Seconds)

Dis

plac

emen

t w (

m)

ode45 solution (Error=0.76%)Adomian decomposition solution with 2 terms (Error=1.72%)Adomian decomposition solution with 3 terms (Error=0.89%)ABAQUS Solution

Figure 3. Variation of the transverse displacement at the center (0.3m, 0.3m) with time

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0 0.5 1 1.5 2 2.5 3 3.5−8

−6

−4

−2

0

2

4

6

8

Time (Seconds)

ψ1(t

)

Adomian decomposition solution with 3 terms (Error=34961.38%)Adomian decomposition solution with 2 terms (Error=2320.67%)ode45 solution

(a) ψ1 Vs. Time

0 0.5 1 1.5 2 2.5 3 3.5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time (Seconds)

ψ9(t

)

ode45 solutionAdomian decomposition solution with 2 terms (Error=1.25%)Adomian decomposition solution with 3 terms (Error=0.27%)

(b) ψ9 Vs. Time

Figure 4. Modal responses for P=30 KN/m2

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