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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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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.
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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
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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
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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