structural-acoustic coupling effect on the nonlinear natural frequency of a rectangular box with one...

Structural-acoustic coupling effect on thenonlinear natural frequency of a rectangular

box with one flexible plate

Y.Y. Lee

Department of Building and Construction, City University of Hong Kong,

Kowloon Tong, Kowloon, Hong Kong

Received 26 November 2001; received in revised form 18 April 2002; accepted 8 May 2002

Abstract

The nonlinear natural frequency of a rectangular box, which consists of one flexible plateand five rigid plates, is studied in this paper. The flexible plate is assumed to vibrate like a

simple piston. The behavior of the structural-acoustic coupling between the flexible plate andthe air cavity is analyzed by using the proposed finite element modal method. The systemfinite element equation is reduced and expressed in terms of the modal coordinates with small

degrees of freedom by using the proposed reduction method. The system nonlinear stiffnessmatrix representing the large amplitude vibration can be transformed to be a constant modalmatrix. The natural frequencies are determined by using the harmonic balance method tosolve the eigenvalue equations of the structural-acoustic system. The effect of the cavity depth

on the natural frequencies and convergence studies are discussed in detail. # 2002 Publishedby Elsevier Science Ltd.

1. Introduction

The interaction problems of structural-acoustic systems, or large amplitudevibration of a flexible plate has been of great interest to many investigators in thepast. For example, the authors of Refs. [1–7] assumed the structural vibration waslinear, and studied the structural-acoustic phenomena of a flexible plate backed by a

Applied Acoustics 63 (2002) 1157–1175

www.elsevier.com/locate/apacoust

0003-682X/02/$ - see front matter # 2002 Published by Elsevier Science Ltd.

PI I : S0003-682X(02 )00033 -6

E-mail address: [email protected] (Y.Y. Lee).

cavity using various classical approaches and finite element methods. On the otherhand, the authors of [8–22] studied and focused on the large amplitude vibration ofan elastic plate without any considerations of acoustic effect. However, few investi-gators have explored problems combining structural-acoustic coupling and largeamplitude vibration.Structural-acoustic and large amplitude vibrating systems are of considerable

importance, especially in aerospace structures, which are thin and light weight.Understanding the dynamic behaviors of a structural-acoustic system inherent withnonlinear phenomena due to large amplitude vibrations is essential in the designstage of an aerospace structure. In this paper, the author tries to combine the twodifferent problems together, cavity-plate interaction and large amplitude vibration,to make a new research problem, which has not been explored in detail. This paperpresents a finite element multi-mode approach for the large amplitude free structuralvibrations of a rectangular box with one flexible plate. The finite element equation isexpressed in terms of the modal coordinates with small degrees of freedom by usingthe proposed reduction method. The system nonlinear stiffness matrix representingthe large amplitude vibration can be transformed to be a constant modal matrix.The reasons why the finite element method developed by Lee et al. [14] is employedinclude (1) it is a powerful and versatile approach for structural problems of com-plex geometries, boundary conditions, and loadings; (2) no updating of the reducednonlinear stiffness matrices is required because it is a constant matrix. The acousticpressure with the rectangular box acting on the isotropic plate is obtained by usingthe one-dimension wave equation. The natural frequencies are determined by usingthe harmonic balance method to solve the eigenvalue equations of the structural-acoustic system. The convergence of the fundamental frequency ratio is investigatedfor a simply supported square plate with different numbers of finite elements, linearmodes, and harmonic terms. Accurate frequency-cavity depth relations for the fun-damental and higher modes at various cavity depths are obtained for a simply sup-ported plate and a clamped plate.

2. Theory and formulation

2.1. Acoustic pressure force generated by the piston motion of the plate

The plate-cavity model is shown in Fig. 1. The boundary at z=c is flexible thatthey can vibrate in typical mode shapes while the other walls are acoustically rigid.In this paper, the plate vibration amplitude ranges from 0 to 1.4 � plate thickness,(about 2.2 mm) which is much smaller than the cavity depth interested (about 25mm). As a result of pressure equalization (see reference [1]), the plate is assumed tobehave as a simple piston (a simplified representation of the flexible plate is alsoshown on Fig. 1, having a surface velocity W

:o (t). This assumption is valid for a low

frequency range or low structural/acoustic modal density. It is imaginable that theair cavity is like an elastic spring connecting to the flexible plate when the platevibrating frequency is lower than the fundamental acoustic resonant frequency. The

1158 Y.Y. Lee / Applied Acoustics 63 (2002) 1157–1175

surface velocity represents the average velocity amplitude of the flexible plate and isgiven by

W:o tð Þ ¼

Ðplate areaW

:x; y; tð Þdxdy

�oð1Þ

where W(x,y,t)=the displacement of the plate area which is expressed in terms ofthe finite element modal coordinates later in this paper, its velocity is marked with adot sign; and �o=plate area.The acoustic pressure within the rectangular cavity is given by the following

homogeneous wave equation (see Ref. [23]),

@2P

@ z2�

1

C2a

@2P

@ t2¼ 0 ð2Þ

where Ca is the speed of sound.The acceleration in z direction within the rectangular box can be derived from the

following equations,

��aZ€ ¼@P

@zð3Þ

where �a is the density of air, Z€ represents the acceleration in the z direction.The boundary conditions of the acoustic pressure with the rectangular box to be

satisfied are

@P

@ z¼ 0; at z ¼ 0;

@P

@ z¼ ��aW€ o tð Þ; at z ¼ c ð4a;bÞ

Here Wo(t) is the average displacement of the plate and obtained by Eq. (1), so itsacceleration is marked with a double dot sign. By applying the boundary conditions(1) and (4a,b), the solution of Eq. (2) for the total pressure force at z=c can be

Fig. 1. Rectangular box with one flexible plate.

Y.Y. Lee / Applied Acoustics 63 (2002) 1157–1175 1159

written as

Pc ¼ Lcos �cð Þ þ Nsin �cð Þ½ �T !tð Þ ð5Þ

where Pc is the total pressure at z=c; � ¼!

Ca; ! is the linear natural frequency to be

determined, T(!t) is a periodic time function (e.g. cosine or sine); L and N are coeffi-cients which depend on the boundary conditions at z=0 and z=c.

2.2. Structural vibration of the plate

In Fig. 1, the finite element system equations of motion for the large-amplitudevibration of the flexible plate can be expressed by (see Ref. [24] or Appendix A)

M½ � W€n o

þ K½ � þ K2 Wð Þ½ �ð Þ Wf g ¼ Pf g ð6Þ

where [M] and [K ] are constant matrices and represent the system mass and stiffnessrespectively; and [K2] is the second order nonlinear stiffness matrix and dependsquadratically on the unknown structural nodal displacements {W}; and {P} is theforce vector represents the sound pressure at z=c due to the plate motion. It shouldbe noted that the flexible plate is isotropic (in other words, it is a symmetric plate).That is why there is no first order nonlinear stiffness matrix in Eq. (6) (see Ref. [24]).Eq. (6) is then transformed into modal or generalized coordinates of much smaller

degrees of freedom In the present formulation, the general Duffing-type modalequations will have constant nonlinear modal stiffness matrices, therefore updatingof the base vectors is not needed. This is accomplished by the modal transformationand truncation

Wf g ¼Xn

r¼1

qr tð Þ �rf g ¼ �½ � q� �

ð7Þ

where �½ � ¼ �1f g � � � �rf g� �

, q� �

¼ q1ðtÞ � � � qrðtÞ cT

. qr(t) is the modal coor-

dinate of the rth linear mode. n is the number of modes considered. {�r} is the nat-ural mode from the eigen-solution

!2L;r M½ � �rf g ¼ K½ � �rf g ð8Þ

where !L,r is the linear natural frequency of the rth modeThe nonlinear stiffness matrix [K2] in Eq. (6) can now be expressed as the sum of

the products of modal coordinates and nonlinear modal stiffness matrices as

K2½ � ¼Xn

r¼1

Xn

s¼1

qr tð Þqs tð Þ K2 �r; �sð Þ½ � rsð Þ ð9Þ


The nonlinear modal stiffness matrix [K2](rs) is assembled from the element non-linear modal stiffness term [k2](rs) as

K2½ � rsð Þ¼X

all elementsþ boundary conditions:

k2½ � rsð Þ ð10Þ

whereP

all elements þ boundary conditions:=the finite element assembly procedure; theelement nonlinear modal stiffness matrices are evaluated with the known linearmode {�r}. Thus, the nonlinear modal stiffness matrix [K2](rs) is a CONSTANT

matrix.Eq. (6) is thus transformed to the general Duffing-type modal equations as

M�h i

q€� �

þ K�h i

þ K2qq

� � �q� �

¼ P�n o

ð11Þ

where the modal mass and linear stiffness matrices are diagonal and given by

M�h i

; K�h i �

¼ �½ �T M½ �; K½ �ð Þ �½ � ð12Þ

and the cubic term is

K2qq

� �q� �

¼ �½ �TXn

r¼1

Xn

s¼1

qrqs K2½ � rsð Þ

!�½ � q� �

ð13Þ

and the force term is

P�n o

¼ �½ �T Pf g ð14Þ

wherefP� g is the modal force vector which represents the sound pressure at z=c due

to the plate motion.

2.3. Harmonic balance method

Approximate solutions of Eq. (11) are obtained by applying the method of har-monic balance (see Ref. [25] or Appendix B for detail) and assuming

qr tð Þ ¼XH

h¼1;3;5:::

Ahr cos h!tð Þ ð15Þ

where Ahr =the hth-order superharmonic amplitude of the rth modal response; H is

the number of superharmonic terms considered.


Therefore the acoustic pressure can be expressed in terms of Ahr by

P�n o

¼ �XH

h;1;3;5:::

!½ � Ah� �

cos h!tð Þ ð16Þ

where Ah� �T

¼ Ah1 � � � Ah

n

�; [!] is the matrix which depends on the two coeffi-

cients L and N and obtained by using Eq. (5) (see Appendix B for detail). The matrix[!], which is similar to the structural acoustic term derived in Ref. [18], represents thestiffness coupling due to cavity between the motions of the various plate modes.Inserting Eqs. (15) and (16) into Eq. (11) gives

�!2 M�h i A1

1 � � � h2Ah1

..

. . .. ..

.

A1n � � � h2Ah

n

2664

3775þ K�

h i A11 � � � Ah

1

..

. . .. ..

.

A1n � � � Ah

n

2664

3775

0BB@

1CCA

cos !tð Þ

..

.

cos h!tð Þ

8>><>>:

9>>=>>;

þXH

h0¼1;3;5:::

XH

h00¼1;3;5:::

Xn

r¼1

Xn

s¼1

K2½ � rsð Þ

A11 � � � Ah

1

..

. . .. ..

.

A1n � � � Ah

n

2664

3775

cos !tð Þ

..

.

cos h!tð Þ

8>><>>:

9>>=>>;cos h0tð Þcos h00!tð ÞAh0

r Ah00

s

þXH

h;1;3;5:::

!½ � Ah� �

cos h!tð Þ ¼ 0 ð17Þ

By neglecting the superharmonic terms, which orders are higher than H in Eq.(16), and comparing the coefficients preceding the cosine terms, a set of cubic equa-tions in Ah

r can be obtained. By inputting an initial condition of Ahr , solving Eq. (16)

for the nontrivial solution of the natural frequency o becomes an eigenvalue prob-lem. Then, the Newton Raphson method is adopted for solving Eq. (16).Here, the frequency ratio, and vibration amplitude ratio are defined as

Frequency ratio ¼!N;r

!L;rð18Þ

Vibration amplitude ratio ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnr¼1

qrð Þamp

&&& &&&2s

Plate thicknessð19Þ

where (qr)amp is the displacement amplitude of the rth modal coordinate; !N,r is thenonlinear natural frequency of the rth mode the plate with the cavity effect.

3. Numerical results

The fundamental frequency of a simply supported square plate of 12 � 12 � 0.048in (0.3048 m � 0.3048 m � 1.2192 mm) backed by a 2 in (0.0508 m) cavity at various


vibration amplitude ratios is obtained by using the harmonic balance method. Aquarter of the plate is modeled with 6 � 6, 7 � 7, 8 � 8 and 9 � 9 mesh sizes. The C1

conforming rectangular plate element with 24 (16 bending and 8 membrane) DOF isused. The in-plane boundary conditions are u=v=0 on all four edges. The first sixsymmetrical structural modes, and first four harmonic terms are used for the con-vergence studies. Tables 1–3 show that the 8 � 8 mesh and 4 structural mode and3-harmonic term model is good enough for a converged and accurate frequency

Table 1

Mesh convergence of the fundamental frequency ratios for the simply supported square plate backed by a

2 inch cavity (Poisson’s ratio=0.3)

Vibration amplitude ratio Frequency ratio, !N,11/!L,11

4 Structural modes, 3 harmonic terms

Mesh size

6 � 6 7 � 7 8 � 8 9 � 9

0.0 2.072 2.072 2.072 2.072

0.2 2.083 2.083 2.083 2.083

0.4 2.114 2.114 2.114 2.114

0.6 2.165 2.165 2.165 2.165

0.8 2.233 2.233 2.233 2.233

1.0 2.316 2.316 2.315 2.315

1.2 2.412 2.411 2.411 2.411

1.4 2.518 2.518 2.517 2.517

Table 2

Structural mode convergence of the fundamental frequency ratios for the simply supported square plate

backed by a 2 inch cavity (Poisson’s ratio=0.3)


3 Harmonic terms, 8 � 8 mesh size

Number of structural modes used

1 3 4 6

0.0 2.101 2.072 2.072 2.071

0.2 2.111 2.083 2.083 2.081

0.4 2.139 2.115 2.114 2.113

0.6 2.184 2.165 2.165 2.164

0.8 2.247 2.233 2.233 2.232

1.0 2.324 2.316 2.315 2.315

1.2 2.415 2.411 2.411 2.410

1.4 2.517 2.518 2.517 2.516


solution. Fig. 2 shows the variation of the fundamental frequency ratio with respectto the variation of the vibration amplitude ratio for the cavity depth=2 in.Table 4 shows the frequency ratios of the fundamental mode of a simply sup-

ported aluminum plate backed by a 6 in (0.1524 m) cavity. It is assumed that thevibration amplitude is small (in other words, it is a linear case). The width, length,and thickness of the plate are 6 in (0.1524 m), 12 in (0.3048 m), and 0.0064 in

Table 3

Harmonic term convergence of the fundamental frequency ratios for the simply supported square plate

backed by a 2 inch cavity (Poisson’s ratio=0.3)


4 Structural modes, 3 harmonic terms

Number of harmonic terms used

1 2 3 4

0.0 2.072 2.072 2.072 2.072

0.2 2.083 2.083 2.083 2.083

0.4 2.114 2.114 2.114 2.114

0.6 2.165 2.165 2.165 2.165

0.8 2.233 2.233 2.233 2.233

1.0 2.317 2.316 2.315 2.315

1.2 2.413 2.411 2.411 2.411

1.4 2.521 2.517 2.517 2.517

Fig. 2. Vibration amplitude ratio–frequency ratio relation for the fundamental mode of the simply sup-

ported aluminum square plate backed by a 2 in cavity.


(0.16256 mm), respectively. The frequency ratios obtained by the present method iswell agree with those from Pretlove [2].In Fig. 3, the fundamental frequency ratio of a simply supported plate is plotted

against the cavity depth for different vibration amplitude ratios. The dimensions andmaterial properties of the plate are the same as those of the one presented in theconvergence studies. Table 5 shows the frequency ratios of the fundamental mode ofthe simply supported aluminum plate without any cavity effect. The air stiffness ofthe 1 inch cavity (cavity depth/plate width=0.083) can increase the frequency ratio

Table 4

Comparison between the frequency ratios of the fundamental mode of the simply supported aluminum

plate obtained form Pretlove and present method

Frequency ratio, !N,11/!L,11

Pretlove[2] Present


Number of structural modes used

1 3 4 6

1.034 1.036 1.035 1.035 1.035

Fig. 3. Frequency–cavity depth relation for the fundamental mode of the simply supported aluminum

square plate backed by a cavity.


by about 170, 108, and 83% for vibration amplitude ratio=0, 1, and 1.4 respectivelywhile that of the 6 in cavity (cavity depth/plate width=1.67) can increased the fre-quency ratio by only about 4, 6, and 9% for vibration amplitude ratio=0, 1, and1.4, respectively. It can be seen that the air stiffness of a shallow cavity has a sig-nificant effect on the resonant frequencies of the plate. For the plate vibrating in alarger amplitude, the effect of the cavity has less effect. The frequency ratios of thethree curves in Fig. 2 generally decrease with increasing cavity depth because a dee-per cavity gives a smaller air stiffness when the air pressure acting on the plate in

Table 5

The fundamental frequency ratios for the simply supported square plate without any cavity effect (Pois-

son’s ratio=0.3)



Number of structural modes=4

0.0 1.000

0.2 1.020

0.4 1.076

0.6 1.164

0.8 1.275

1.0 1.404

1.2 1.547

1.4 1.699

Fig. 4. The (1,1) mode shape of a simply supported aluminum square plate backed by a 2 in cavity.


Fig. 5. Frequency–cavity depth relation for the fundamental mode of the clamped aluminum square plate

backed by a cavity.

Fig. 6. The (1,1) mode shape of a clamped aluminum square plate backed by a 2 in cavity.


phase to the plate motion gives a positive air stiffness to increase the frequency ratio.Negative air stiffness occurs when the air pressure acting on the plate is out of phaseto the plate motion so that the frequency ratios may be lower than those withoutcavity effect. There are two sharp jumps on each curve due to 180� phase changes ofthe acoustic pressure acting on the plate, turning the air stiffness from negative topositive. Under this situation, the air cavity acts like a solid backing the plate and itsimpedance becomes very high. Therefore, the structural resonant frequency of theplate is increased sharply. The linear mode shape and nonlinear mode shape of thesimply supported plate backed by a 2 in cavity at the vibration amplitude ratio=1.4are shown in Fig. 4. It can be seen that there is no big difference between the non-linear and linear mode shapes.In Fig. 5, the fundamental frequency ratio of a clamped aluminum square plate,

which is modeled with 8 � 8 mesh for a quarter of plate, is plotted against the cavitydepth for different vibration amplitude ratios. The dimensions and material prop-erties of the plate are the same as those of the one presented in the convergencestudies. Similar to those seen in the case of a simply supported plate, the frequencyratios generally decrease with increasing cavity depth and two sharp jumps arefound on each curve. Fig. 6 gives the linear mode shape and nonlinear modeshape of the clamped plate backed by a 2 in cavity at the vibration amplituderatio=1.4.

4. Conclusions

A model for predicting the nonlinear natural frequency of a rectangular box withone flexible plate has been presented. The flexible plate is assumed as a simplevibrating piston. The finite element modal formulation developed by Lee et al. [14] ismodified and developed for analyzing the behavior of the structural-acoustic cou-pling between the flexible plate and the cavity formed by the rectangular box. Theuse of the modal coordinate transformation enables to reduce the number of ordin-ary nonlinear differential modal equations to a much smaller one. The natural fre-quencies are determined by using the harmonic balance method to solve theeigenvalue equations of the structural-acoustic system. The convergence studies ofthe model have been discussed in detail. The comparison between the resultsobtained from the present method and Pretlove shows the good validity of themodel. From the numerical results, it is found that the air cavity formed by a rec-tangular box can have a negative stiffness effect on the flexible plate as well as apositive stiffness effect when the air pressure acting on plate is out of phase to theplate motion.

Acknowledgements

The research work was supported by the Strategic Research Grant (7001370) ofCity University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong.


Appendix A

Strain-displacement and constitutive relations

The von Karman strain-displacement relations are applied. The strains at anypoint z through the thickness are the sum of membrane and change of curvaturestrain components:

"f g¼

"x

"y

xy

8<:

9=;¼

u;xv;y

u;y þv;x

8<:

9=;þ

w2;x=2

w2;y=2

w;x w;y

8><>:

9>=>;þ z

�w;xx

�w;yy

�2w;xy

8<:

9=;¼ "om

� �þ "ob� �

þz �f g ðA1Þ

where "om� �

and "ob� �

are the membrane strain components due to in-plane dis-placements u and v and the transverse deflection w, respectively. The stress resul-tants, membrane force {N} and bending moment {M}, are related to the straincomponents as follows:

NM

( )¼

½A� ½B�½B� ½D�

* +"o

�

( )ðA2Þ

where [A] is the elastic extensional matrix, [D] is the flexural rigidity matrix, [B] is theextension coupling matrix of the laminated plate.

Element displacements, matrices and equations

Proceeding from this point, the displacements in Eq. (A1) are approximated overa typical plate element using the corresponding interpolation functions The C1 rec-tangular element, Bogner–Fox–Schmit element, which has the 16 bending and the 8in-plane nodal displacements at the four vertices is adopted. The in-plane displace-ments and the linear strains are interpolated from nodal values by

uv

( )¼

Hub c

Hvb c

* +Tm½ � wmf g; "0m

� �¼

Hub c;xHvb c;y

Hub c;yþ Hvb c;x

24

35 Tm½ � wmf g ðA3a;bÞ

where Hucb and Hvcb denote the in-plane displacement interpolation matrices,respectively, and {wm} is the in-plane nodal displacement vector. The subscripts ‘‘x’’and ‘‘y’’ represent the first derivatives in the x and y directions.

Hub c ¼ 1 x y xy 0 0 0 0 �

Hvb c ¼ 0 0 0 0 1 x y xy �


Tm½ � ¼

1 0 0 0 0 0 0 01 ae 0 0 0 0 0 01 ae be aebe 0 0 0 00 0 be 0 0 0 0 00 0 0 0 1 0 0 00 0 0 0 1 ae 0 00 0 0 0 1 ae aebe be

0 0 0 0 1 0 be 0

266666666664

377777777775

�1

ae and be are the finite element length and width, respectively.The transverse displacement, slopes and curvatures are interpolated from the

nodal values by

w ¼ Hbb c Tb½ � wbf g;w;x

w;y

( )¼

Hbb c;x

Hbb c;y

" #Tb½ � wbf g; �f g ¼

� Hbb c;xx

� Hbb c;yy

�2 Hbb c;xy

264

375 Tb½ � wbf g

ðA4a�cÞ

where Hbcb denotes the bending displacement interpolation matrix, {wb} denotes thenodal transverse displacements, the subscripts ‘‘xx’’, ‘‘yy’’, and ‘‘xy’’ represent thesecond derivatives in the x and y directions.

Hbc ¼ 1 x y x2 xy y 2 x3 x2y xy2 y3 x3y xy3 x2y2 x3y2 x2y3 x3y3 c

Tb½ � ¼

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 ae 0 a2e 0 0 a3e 0 0 0 0 0 0 0 0 01 ae be a2e aebe b2e a3e a2ebe aeb

2e b3e a3ebe a2eb2e aeb

3e a3eb2e a2eb3e a3eb3e

1 0 be 0 0 b2e 0 0 0 b3e 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 2ae 0 0 3a3e 0 0 0 0 0 0 0 0 00 1 0 2ae be 0 3a3e 2aeb b2e 0 3a2ebe 2aeb

2e b3e 3a2eb2e 2aeb

3e 3a2eb3e

0 1 0 0 be 0 0 0 b2e 0 0 0 b3e 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 ae 0 0 a2e 0 0 a2e 0 0 0 0 00 0 1 0 ae 2be 0 a2e 2aebe 3b2e a3e 2a2ebe 3aeb

2e 2a3ebe 3a2eb2e 3a3eb2e

0 0 1 0 0 2be 0 0 0 3b2e 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 1 0 0 2ae 0 0 3a2e 0 0 0 0 00 0 0 0 1 0 0 2ae 2be 0 3a2e 4aebe 3b2e 6a2ebe 6aeb

2e 9a2eb2e

0 0 0 0 1 0 0 0 2be 0 0 0 3b2e 0 0 0

2666666666666666666666666664

3777777777777777777777777775

�1

Through the use of Hamilton’s principle, the equations of motion for a plate ele-ment undergoing large amplitude vibration may be written in the form

m½ � w€f g þ k½ � þ k2½ ��

wf g ¼ p� �

ðA5Þ


where [m] and [k] are constant matrices representing the element mass and linearstiffness characteristics, respectively; [k2] is the second order non-linear stiffnessmatrix; and [k2] depends quadratically on the unknown plate slopes; {p} representsthe element air pressure force vector. The derivation of the element matrices andtheir explicit expressions are referred to [16].

System equations

After assembling the individual finite elements for the complete plate and applyingthe kinematic boundary conditions, the finite element system equations of motionfor the large-amplitude vibration of a thin laminated composite plate can beexpressed as

M½ � W€n o

þ K½ � þ K2 Wð Þ½ �ð Þ Wf g ¼ Pf g ðA6Þ

where [M] and [K] are constant matrices and represent the system mass and stiffnessrespectively; and [K2] is the second order nonlinear stiffness matrix; The {P} repre-sents the system air pressure force vector. The system equations presented in (A6)are not suitable for the harmonic balance method because: (a) the nonlinear stiffnessmatrix [K2] is a function of the unknown nodal displacements, and (b) the numberof degrees of freedom (DOF) of the system nodal displacements {W} is usually toolarge. Therefore, Eq. (A6) has to be transformed into modal or generalized coordi-nates of much smaller DOF.

Appendix B

The solution of Eq. (2) for the pressure at z is given by

Ph ¼ Lhcos �hz. /

þ Nhsin �hz. /� �

cos h!tð Þ ðB1Þ

where Lh and Nh are the coefficients to be determined for the hth-order super-harmonic solution, �h=h!/Ca

According to Eq. (4a), the Nh in Eq. (B1) can be given by

@Ph

@Z¼ 0 at z ¼ 0

) �hNhcos h!tð Þ ¼ ) Nh ¼ 0

ðB2Þ


Considering Eq. (4b) and the Lh in Eq. (B1) gives

@Ph

@Z¼ ��aW€ h

o at z ¼ c

) ��hLhsin �hc. /

cos h!tð Þ ¼ ��aW€ ho

ðB3Þ

where

W€ ho ¼ � h!ð Þ

2Xn

r¼1

Ahr

�r

�ocos h!tð Þ;

Ahr =the hth-order superharmonic amplitude of the rth modal response;

�r ¼

ðplate area

�rðx; yÞdxdy ¼ �rb cTX

all elementsþ bdy: conds:

ðArea

½Tb�T½Hbðx; yÞ�Tdxdy;

�r(x,y) is the rth mode shape in the analytic form;[�] is the mode shape matrix.It is noted that W x; y; tð Þ ¼

Ph¼1;3;5:::W

h x; y; tð Þ=the plate displacement of thehth superharmonic response, which can be expressed in terms of the modal coordi-nates =

Pnr¼1qr tð Þ�r x; yð Þ, and qr tð Þ ¼

PHh¼1;3;5:::A

hr cos h!tð Þ. Finally the Lh is

expressed by

Lh ¼ ��a

h!ð Þ2Pn

r¼1

Ahr �r

�hsin �hcð ÞðB4Þ

The pressure force of the hth superharmonic component at z=c is given by

Pch ¼ ��a h!ð Þ2cot �

hc. /�h

Xn

r¼1

Ahr �rcos h!tð Þ ðB5Þ

Consider the modal force vector in Eq. (14)

P�n o

¼ �½ �T Pf g ¼

XH

h¼1;3;5:::

P� h1

..

.

P� hn

8>><>>:

9>>=>>; ðB6Þ

whereP� h

r ¼ fr½ �TPNEÐ

Area Tb½ �T Hb x; yð Þ½ �

TPchdxdy=the hth superharmonic componentof the rth modal force.


Express the modal force, P� hr in (B6) in terms of Ah

s

P� hr ¼ � fr½ �

TXNE ð

Area

½Tb�T Hb x; yð Þ½ �

T�a h!ð Þ2cot �

hc. /�h

Xn

s¼1

Ahs �scos h!tð Þdxdy

)

P� hr ¼ ��a h!ð Þ

2cot �hc

. /�h

Xn

s¼1

Ahs

�s�r

�0cos h!tð Þ ðB7Þ

Then the modal force vector P�n o

can be given by

P�n o

¼ ��a!2XH

h¼1;3;5:::

h2cot �hc. /

�h�ocos h!tð Þ ��½ � Ah

� �¼ �

XH

h;1;3;5:::

!½ � Ah� �

cos h!tð Þ

ðB8Þ

where

��½ � ¼

�1�1 � � � �n�1

..

. . .. ..

.

�1�n � � � �n�n

264

375

!½ � ¼ �a!2XH

h¼1;3;5:::

h2cot �hc. /

�h�o��½ �

Ah� �

¼

Ah1

..

.

Ahn

8><>:

9>=>;

Inserting Eq. (B8) into the equation of motion of the plate gives

M�h i

q€� �

þ K�h i

þ K2qq

� � �q� �

¼ �XH

h;1;3;5:::

!½ � Ah� �

cos h!tð Þ ðB9Þ

Since {q} can be expressed in terms of Ahr using Eq. (19), Eq. (B9) can be further

rewritten and given by


�!2 M�h i A1

1 � � � h2Ah1

..

. . .. ..

.

A1n � � � h2Ah

n

2664

3775þ K�

h i A11 � � � Ah

1

..

. . .. ..

.

A1n � � � Ah

n

2664

3775

0BB@

1CCA

cos !tð Þ

..

.

cos h!tð Þ

8>><>>:

9>>=>>;

þXH

h0¼1;3;5:::

XH

h00¼1;3;5:::

Xn

r¼1

Xn

s¼1

K2½ � rsð Þ

A11 � � � Ah

1

..

. . .. ..

.

A1n � � � Ah

n

2664

3775

cos !tð Þ

..

.

cos h!tð Þ

8>><>>:

9>>=>>;cos h0tð Þcos h00!tð ÞAh0

r Ah00

s

þXH

h;1;3;5:::

!½ � Ah� �

cos h!tð Þ ¼ 0:ðB10Þ

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