Amabili_Nonlinear Vibrations of Rectangular Plates With Different Boundary Conditions_Theory and Experiments

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Nonlinear vibrations of rectangular plates withdifferent boundary conditions: theory and experiments

M. Amabili *

Dipartimento di Ingegneria Industriale, Universita di Parma, Parco Area delle Scienze 181/A, Parma 43100, Italy

Received 31 October 2003; accepted 10 March 2004

Abstract

Large-amplitude (geometrically nonlinear) vibrations of rectangular plates subjected to radial harmonic excitation in

the spectral neighborhood of the lowest resonances are investigated. The von Karman nonlinear straindisplacement

relationships are used. The formulation is also valid for orthotropic and symmetric cross-ply laminated composite plate;

geometric imperfections are taken into account. The nonlinear equations of motion are studied by using a code based

on arclength continuation method that allows bifurcation analysis. Comparison of calculations to numerical results

available in the literature is performed for simply supported plates with immovable and movable edges. Three different

boundary conditions are considered and results are compared: (i) simply supported plates with immovable edges; (ii)

simply supported plates with movable edges; and (iii) fully clamped plates. An experiment has been specifically per-

formed in laboratory in order to very the accuracy of the present numerical model; a good agreement of theoretical

and experimental results has been found for large-amplitude vibrations around the fundamental resonance of the alum-inum plate tested.

2004 Elsevier Ltd. All rights reserved.

Keywords: Plate; Vibration; Nonlinear; Large-amplitude; Rectangular; Experiment

1. Introduction

A literature review of work on the nonlinear vibra-

tion of plates is given by Chia [1], Sathyamoorthy [2]

and Chia [3]; curved panels and shells were reviewedby Amabili and Padoussis [4]. The fundamental study

in the analysis of large-amplitude vibrations of rectangu-

lar plates is due to Chu and Herrmann [5], who were the

pioneers in the field. They studied simply supported rec-

tangular plates with immovable edges and obtained the

resonance curve (also known as the backbone curve),

which gives the maximum vibration amplitude versus

the excitation frequency for any excitation level, for

the fundamental mode of rectangular plates with differ-

ent aspect ratios. The solution was obtained by using aperturbation procedure and shows strong hardening

type nonlinearity.

A series of interesting papers [69] compared different

results for the backbone curve of the fundamental mode

of isotropic plates with those of Chu and Herrmann [5];

all of them are in good agreement with the original re-

sults of Chu and Herrmann. In particular, Leung and

Mao [8] also studied simply supported rectangular

plates with movable edges, which present reduced

0045-7949/$ - see front matter 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2004.03.077

* Tel.: +39 521 905896; fax: +39 521 905705.

E-mail address: [email protected]

URL: http://me.unipr.it/mam/amabili/amabili.html

Computers and Structures 82 (2004) 25872605

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mailto:[email protected]:[email protected]


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hardening-type nonlinearity with respect to simply sup-

ported plates with immovable edges.

The effect of geometric imperfection was investigated

by Hui [10]. More recent studies by Han and Petyt

[11,12] and Ribeiro and Petyt [1315] used the hierarchi-

cal finite element method to deeply investigate the non-

linear response of clamped rectangular plates. A similarapproach was used by Ribeiro [16] to investigate the

forced response of simply supported plates with immov-

able edges. A simplified analytical approach was devel-

oped by El Kadiri and Benamar [17] for the case

studied by Chu and Herrmann. A reduced basis tech-

nique for nonlinear free vibration analysis of composite

plates was developed by Noor et al. [18].

Even if a large number of theoretical studies on large-

amplitude vibrations of plates are available in the scien-

tific literature, experimental results are very scarce.

Experiments on the response of a clamped rectangular

plate to acoustic excitation were performed by Maest-rello et al. [19]. Chaotic response was detected for high

excitation level. However, the resonance curve as func-

tion of the vibration amplitude was not measured; for

this reason, these results cannot be easily used as bench-

mark for validation of numerical studies.

In the present study, large-amplitude (geometrically

nonlinear) vibrations of rectangular plates subjected to

radial harmonic excitation in the spectral neighborhood

of the lowest resonances are investigated. The von Kar-

man nonlinear straindisplacement relationships are

used. The formulation is also valid for orthotropic and

symmetric cross-ply laminated composite plates; geo-

metric imperfections are taken into account. Calcula-

tions for different boundary conditions are performed.

A comparison to results of an experiment specifically

performed in laboratory in order to very the accuracy

of the present numerical model is performed. The exper-

imental results given in the present study are among the

few available in the literature and can be used for further

validation of numerical models.

2. Elastic strain energy of the plate

A rectangular plate with coordinate system

(O; x, y, z), having the origin O at one corner is consid-

ered (see Fig. 1). The displacements of an arbitrary point

of coordinates (x, y) on the middle surface of the plate

are denoted by u, v and w, in the x, y and out-of-plane(z) directions, respectively. Initial geometric imperfec-

tions of the rectangular plate associated with zero initial

tension are denoted by out-of-plane displacement w0;

only out-of-plane initial imperfections are considered.

The von Karman nonlinear straindisplacement rela-

tionships are used. The strain components ex, ey and cxyat an arbitrary point of the plate are related to the mid-

dle surface strains ex,0, ey,0 and cxy,0 and to the changes

in the curvature and torsion of the middle surface kx, kyand kxy by the following three relationships:

ex ex;0 zkx; ey ey;0 zky; cxy cxy;0 zkxy; 1

where z is the distance of the arbitrary point of the plate

from the middle surface. According to von Karmans

theory, the middle surface straindisplacement relation-

ships and changes in the curvature and torsion are given

by [20]

ex;0 ou

ox

1

2

ow

ox

2ow

ox

ow0

ox; 2a

ey;0 ov

oy

1

2

ow

oy

2ow

oy

ow0

oy; 2b

cxy;0 ou

oyov

oxow

ox

ow

oyow

ox

ow0

oyow0

ox

ow

oy; 2c

kx o

2w

ox2; 2d

ky o

2w

oy2; 2e

Fig. 1. Plate and coordinate system.

2588 M. Amabili / Computers and Structures 82 (2004) 25872605


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kxy 2o

2w

oxoy: 2f

As a consequence that only isotropic and symmetric

laminated plates will be analyzed, there is no coupling

between in-plane stretching and transverse bending.

The elastic strain energy UP of a plate, neglecting rzunder Kirchhoffs hypotheses, is given by

UP 1

2

Za0

Zb0

Zh=2h=2

rxex ryey sxycxy

dxdydz; 3

where h is the plate thickness, a and b are the in-plane

dimensions in x and y directions, respectively, and the

stresses rx, ry and sxy are related to the strain for homo-

geneous and isotropic material (rz = 0, case of plane

stress) by

rx

E

1 m2ex

mey

;

ry E

1 m2ey mex

; sxy E

2 1 m cxy;

4

where E is Youngs modulus and m is Poissons ratio. By

using Eqs. (1), (3) and (4), the following expression is

obtained:

UP 1

2

Eh

1 m2

Za0

Zb0

e2x;0 e2y;0 2mex;0ey;0

1 m

2c2xy;0

dxdy

12

Eh

3

12 1 m2

Za0

Zb0

k2x k2y 2mkxky

1 m

2k2xy

dxdy

Oh4; 5

where O(h4) is a higher-order term in h; the first term is

the membrane (also referred to as stretching) energy and

the second one is the bending energy. An expression

analogous to Eq. (5) for composite laminated plates is

given in Appendix A.

3. Mode expansion, kinetic energy and external loads

The kinetic energy TP of a rectangular plate, by

neglecting rotary inertia, is given by

TP 1

2qPh

Za0

Zb0

_u2 _v2 _w2 dxdy; 6

where qP is the mass density of the plate. In Eq. (6) the

overdot denotes a time derivative.

The virtual work W done by the external forces is

written as

W

Za0

Zb0

qxu qyv qzw

dxdy; 7

where qx, qy and qz are the distributed forces per unit

area acting in x, y and z directions, respectively. Initially,

only a single harmonic force orthogonal to the plate is

considered; therefore qx = qy = 0. The external distrib-uted load qz applied to the plate, due to the out-of-plane

concentrated force ~f, is given by

qz ~fdy ~ydx ~x cosxt; 8

where x is the excitation frequency, t is the time, d is the

Dirac delta function, ~f gives the force magnitude posi-

tive in z direction, ~x and ~y give the position of the point

of application of the force; here, the point excitation is

located at the centre of plate, i.e. ~x a=2, ~y b=2. Eq.(7) can be rewritten in the following form:

W ~f cosxt w xa=2;yb=2: 9

Eq. (9), specialised for the expression of w used in the

present study, is given in Section 5, together with the vir-

tual work done by uniform pressure.

In order to reduce the system to finite dimensions, the

middle surface displacements u, v and w are expanded by

using approximate functions.

Three different boundary conditions are analyzed in

the present study: Case (a), simply supported plate with

immovable edges; Case (b), simply supported plate with

movable edges; Case (c), clamped plate.

The boundary conditions for the simply supported

plate with immovable edges (Case (a)) are:

u v w w0 Mx o2w0=ox

2 0 at x 0; a;

10af

u v w w0 My o2w0=oy

2 0 at y 0; b;

11af

where M is the bending moment per unit length.

The boundary conditions for the simply supported

plate with movable edges (Case (b)) are:

v w w0 Nx Mx o2w0=ox2 0 at x 0; a;

12af

u w w0 Ny My o2w0=oy

2 0 at y 0; b;

13af

where N is the normal force per unit length.

The boundary conditions for the clamped plate (Case

(c)) are:

u v w w0 ow=ox ow0=ox 0 at x 0; a;

14af

M. Amabili / Computers and Structures 82 (2004) 25872605 2589
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u v w w0 ow=oy ow0=oy 0; at y 0; b:

15af

Two bases of panel displacements are used to discre-

tise the system for different boundary conditions. For

Cases (a) and (c) the displacements u, v and w are ex-

panded by using the following expressions, which satisfyidentically the geometric boundary conditions (10ac)

and (11ac):

ux;y; t XMm1

XNn1

u2m;nt sin2mpx=a sinnpy=b;

16a

vx;y; t XMm1

XNn1

vm;2nt sinmpx=a sin2npy=b;

16b

wx;y; t X^Mm1

X^Nn1

wm;nt sinmpx=a sinnpy=b; 16c

where m and n are the numbers of half-waves in x and y

directions, respectively, and t is the time; um,n(t), vm,n(t)

and wm,n(t) are the generalized coordinates that are un-

known functions of t. Mand Nindicate the terms neces-

sary in the expansion of the in-plane displacements and

are generally larger than ^M and ^N, respectively, which

indicate the terms in the expansion of w.

For Case (b), the displacements u, v and w are ex-panded by using the following expressions, which satisfy

identically the geometric boundary conditions (12a,b)

and (13a,b):

ux;y; t XMm1

XNn1

um;nt cosmpx=a sinnpy=b; 17a

vx;y; t XMm1

XNn1

vm;nt sinmpx=a cosnpy=b; 17b

wx;y; t X^Mm1

X^Nn1

wm;nt sinmpx=a sinnpy=b: 17c

By using a different number of terms in the expansions,

it is possible to study the convergence and accuracy of

the solution. It will be shown in Section 6.3 that a suffi-

ciently accurate model for the fundamental mode of the

plate (Case b) has 9 degrees of freedom. In particular, it

is necessary to use the following terms: m, n = 1, 3 in

Eqs. (17a) and (17b) and m, n = 1 in Eq. (17c). For the

boundary conditions of Cases (a) and (c), more terms

are necessary in the expansion to achieve the same

accuracy.

3.1. Geometric imperfections

Initial geometric imperfections of the rectangular

plate are considered only in z direction. They are associ-

ated with zero initial stress. The imperfection w0 is ex-

panded in the same form of w, i.e. in a double Fourier

sine series satisfying the boundary conditions (10d,f)and (11d,f) at the plate edges

w0x;y X~Mm1

X~Nn1

Am;n sinmpx=a sinnpy=b; 18

where Am,n are the modal amplitudes of imperfections; ~N

and ~M are integers indicating the number of terms in the

expansion.

4. Satisfaction of boundary conditions

4.1. Case (a)

The geometric boundary conditions, Eqs. (10ad,f)

and (11ad,f), are exactly satisfied by the expansions

of u, v, w and w0. On the other hand, Eqs. (10e)

and (11e) can be rewritten in the following form

[20]:

Mx Eh3

121 m2kx mky

0 at x 0; a; 19

My Eh3

121 m2ky mkx

0 at y 0; b: 20

Eqs. (19) and (20) are identically satisfied for the expres-

sions ofkx and ky given in Eqs. (2d) and (2e). Therefore

all the boundary conditions are exactly satisfied in this

case.

4.2. Case (b)

In addition to the geometric boundary conditions

and constraints on bending moment (see Eqs. (19) and

(20)), the following constraints must be satisfied [20]:

Nx Eh

1 m2ex;0 mey;0

0 at x 0; a; 21

Ny Eh

1 m2ey;0 mex;0

0 at y 0; b: 22

Eqs. (21) and (22) are not identically satisfied. Elim-

inating null terms at the panel edges, Eqs. (21) and (22)

can be rewritten as

ou

ox

1

2

ow

ox 2

ow

ox

ow0

ox m

ov

oy

" #x0;a

0; 23

2590 M. Amabili / Computers and Structures 82 (2004) 25872605


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ov

oy

1

2

ow

oy

2ow

oy

ow0

oy m

ou

ox

" #y0;b

0; 24

where u and v are terms added to the expansion ofu and

v, given in Eqs. (17ac), in order to satisfy exactly the

boundary conditions Nx = 0 and Ny = 0. As a conse-

quence that u and v are second-order terms in the panel

displacement, they have not been inserted in the

second-order terms that involve u and v in Eqs. (23)

and (24).

Non-trivial calculations give

4.3. Case (c)

As discussed in Section 4.1, Eqs. (14ad,f) and (15a

d,f) are identically satisfied by the expansions of u, v, w

and w0. On the other hand, Eqs. (14e) and (15e) can be

rewritten in the following form:

Mx Eh3

121 m2kx mky kow=ox at x 0; a;

27

My Eh3

121 m2ky mkx

kow=oy at y 0; b;

28

where k is the stiffness per unit length of the elastic, dis-

tributed rotational springs placed at the four edges,

x = 0, a and y = 0, b. Eqs. (27) and (28) represent the

case of an elastic rotational constraint at the shell edges.

They give any rotational constraint from zero bending

moment (Mx = 0 and My = 0, unconstrained rotation,

obtained for k= 0) to perfectly clamped plate (ow/

ox = 0 and ow/oy = 0, obtained as limit for k ! 1),

according to the value of k. In case of k different from

zero, an additional potential energy stored by the elastic

rotational springs at the plate edges must be added. This

potential energy UR is given by

UR 1

2Zb

0

kow

ox

x0 !

2

ow

ox

xa !

2

( )dy

1

2

Za0

kow

oy

y0

" #2

ow

oy

yb

" #28

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Amabili_Nonlinear Vibrations of Rectangular Plates With Different Boundary Conditions_Theory and Experiments