DSpace at VNU: Simultaneous resonances involving two mode shapes of parametrically-excited rectangular plates

Journal of Sound and Vibration 332 (2013) 5103–5114

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Journal of Sound and Vibration
journal homepage: www.elsevier.com/locate/jsvi

Simultaneous resonances involving two mode shapes of
parametrically-excited rectangular plates
Hai Nguyen n
Department of Engineering Mechanics, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet Street, District 10,
Hochiminh City, Vietnam

a r t i c l e i n f o

abstract

Article history:
Received 20 November 2012
Received in revised form
21 March 2013
Accepted 10 April 2013
Handling Editor: L.N. Virgin
Available online 21 May 2013

It is known that, for multi-degree-of-freedom systems under time-dependent excitation,
the combination of an internal resonance with an external resonance will give rise to
simultaneous resonances, and these resonances are characterized by the fact that the
system in question can resonate simultaneously in more than one normal mode while
only one resonant mode is directly excited by the excitation. This work deals with the
problem of the occurrence of simultaneous resonances in a parametrically-excited and

simply-supported rectangular plate. The analysis is based on the dynamic analog of von
Karman's large-deflection theory, and the governing equations are satisfied using the
orthogonality properties of the assumed functions. The nonlinear temporal response of
the damped system is determined by the first-order generalized asymptotic method. The
solution for simply supported plates indicates the possibility of principal parametric
resonances and simultaneous resonances. Simultaneous resonances involving two modes
of vibration are presented in this paper, and it is shown for the first time that only one
possibility out of two cases can really occur. This phenomenon is verified by experimental
results and observations.
& 2013 Elsevier Ltd. All rights reserved.

1. Introduction
A typical example in regard to the dynamic instability of structures is the case of a thin and flat rectangular plate acted
upon by an in-plane load of the form n(t)¼n0+nt cosλt, where cosλt is a harmonic function of time. Under such
circumstances, the plate may become laterally unstable over certain regions of the (n0, nt, λ) parameter space. Thus, apart
from the forced in-plane vibrations, transverse vibrations may be induced in the plate and the plate is said to be dynamically
(or parametrically) unstable.
It is well known that when the natural frequencies of this system are distinct, and in the absence of internal resonances
and combination resonances, the periodic in-plane load can excite only one normal mode at a time; and when the plate
executes lateral vibration at half the driving frequency, the corresponding resonance is called principal parametric resonance
[10–12,14–16]. In contrast with this case of simple parametric resonance, simultaneous resonances and combination
resonances may also occur in multi-degree-of-freedom system subjected to parametric excitation such as a plate.
It has been shown that when a parametric resonance is excited in the presence of an internal resonance, the coincidence
of these two types of resonances will give rise to simultaneous resonances [14–17,19]. These kinds of resonances are
characterized by the fact that all resonantly involved modes might exist in the response, even though only one mode is
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H. Nguyen / Journal of Sound and Vibration 332 (2013) 5103–5114

directly excited by the parametric excitation. It has been specified that internal resonance is responsible for this
phenomenon and, as a consequence, for a significant transfer of energy from the directly excited mode to other modes
of vibration through internal mechanisms.
Research in the field of parametric instability of plates has not been as extensive as for columns, but their instability
behavior has been considerably clarified. The first investigation on a rectangular plate subjected to periodic in-plane loads
was performed by Einaudi [1]. Subsequent works on this subject, however, have only established the boundaries of
instability regions associated mostly to principal parametric resonances and a few to combination resonances. Bolotin [2]
was the first to investigate the nonlinear problem of parametric response of a rectangular plate. Schmidt [3] presented only
a few qualitatively symbolic results in his study on the nonlinear parametric vibrations of sandwich plates. Recently, Sassi
and Ostiguy [4–6] studied the effects of initial geometric imperfections on the interaction between forced and parametric or
combination resonances. More recently, chaotic dynamic stability of plates in large deflection was investigated by Sun and
Zhang [7], and by Yeh et al. [8], while Wu and Shih [9] considered the dynamic instability of rectangular plate with an edge
crack. Works with significant results on nonlinear response corresponding to principal parametric resonances of
parametrically-excited rectangular plates can be found in Refs. [10–17]. Research on nonlinear response related to
combination resonances of parametrically-excited rectangular plates was performed by the present author and results
can only be found in Refs. [15–18]. Similarly, nonlinear response corresponding to simultaneous resonances involving two
mode shapes of parametrically-excited rectangular plates can only be found in [14,15,17,19,20]. Recently, theoretical results
concerning the occurrence of simultaneous resonances involving three mode shapes of parametrically-excited rectangular
plates was presented by the author for the first time [21]. The first experimental studies on plates were conducted by
Somerset and Evan-Iwanowski [22,23], and they pertained mainly to the large amplitude, nonlinear parametric response of
simply-supported square plates. The most significant experimental results on combination and simultaneous resonances of
parametrically-excited rectangular plates were obtained by Nguyen [15,17,19].
The occurrence of simultaneous resonances involving two spatial forms of vibration of a parametrically-excited

rectangular plate is dealt in this paper and covers an existing gap in our understanding of the dynamic buckling of
structures. The simply-supported rectangular plate under investigation is acted upon by periodic in-plane forces uniformly
distributed along two opposite edges; the other two edges are stress-free. The analysis is based on the dynamic analog of
von Karman's large-deflection theory and the governing equations are satisfied using the orthogonality properties of the
assumed functions. The first-order generalized asymptotic method is used to solve the temporal equations of motion, and
attention is focused on principal parametric resonances and an internal resonance involving two modes of vibration. The
coincidence of this type of internal resonance with corresponding principal parametric resonances will give rise to two cases
of simultaneous resonances, in which only one case is resonantly possible, the other case is impossible. The reason for the
latter case is explained for the first time. This explanation is also based on experimental results obtained [15].
2. Statement of the problem
The mechanical system under investigation is a rectangular plate, simply supported along its edges (in-plane movable
edges) and subjected to the combined action of both static and dynamic compressive forces uniformly distributed along two
opposite edges. The two vertical edges are stress free. The geometry of the plate, the load configuration and the coordinate
system are shown in Fig. 1. The x–y plane is selected in the middle plane of the undeformed plate. The plate is assumed to be
thin, initially flat, of uniform thickness, and the plate material is elastic, homogeneous and isotropic.
Restricting the problem to the relatively low frequency range where the plate oscillations are predominantly flexural, the
effect of transverse shear deformations as well as in-plane and rotatory inertia forces can be neglected. From these
restrictions, the plate theory used in the analysis may be described as the dynamic analog of von Karman's large-deflection
theory. The dimensionless differential equations governing the nonlinear flexural vibrations of the plate can be written as
R4 F ;XXXX þ 2R2 F ;XXYY þ F ;YYYY ¼ R2 ½W 2;XY −W ;XX W ;YY Š;

(1)

R4 W ;XXXX þ 2R2 W ;XXYY þ W ;YYYY ¼ ζ½R2 ðF ;YY W ;XX −2F ;XY W ;XY þ F ;XX W ;YY Þ−R4 W ;TT Š;

(2)

in which a comma denotes partial differentiation with respect to the corresponding coordinates, R¼b⧸a is the plate aspect
ratio, ζ ¼12(1−ν2) where ν is the Poisson ratio, and where
X ¼ x=a;

Y ¼ y=b;

W ¼ w=h;

2

F ¼ f =Eh ;

2

T ¼ t½Eh =ρa4 Š1=2 :

(3)

In Eq. (3), w(x, y, t) is the lateral displacement and f(x, y, t) the Airy stress function, h denotes the plate thickness, ρ the
density per unit volume, E the Young modulus, and t the time. The nonlinearity arising in the problem under consideration
is due to large amplitudes generating membrane forces.
The non-dimensional membrane forces NX, NY and NXY, arising from a combination of the dimensionless in-plane loading
3

NY ðTÞ ¼ ða2 =Eh Þny ðtÞ ¼ N Y0 þ NYT cos ΛT;

(4)

with the large amplitude lateral deflection, are related to the non-dimensional force function by
ðN X ; NY ; NXY Þ ¼ ðF ;YY =R2 ; F ;XX ; −F ;XY =RÞ;

(5a)

H. Nguyen / Journal of Sound and Vibration 332 (2013) 5103–5114

5105

Fig. 1. Plate and load configuration.

where
3

ðNX ; NY ; NXY Þ ¼ ða2 =Eh Þðnx ; ny ; nxy Þ;

(5b)

in which nx, ny and nxy are membrane forces. In Eq. (4), NY0 is the dimensionless constant component of the in-plane force,
NYT is the dimensionless amplitude of the harmonic in-plane loading, and Λ is the dimensionless excitation frequency.
The boundary conditions are related to both the stress function, F, and the lateral displacement, W. The stress conditions
may be expressed in the dimensionless form as
F ;YY ¼ F ;XY ¼ 0

at

F ;XX ¼ −NY ðTÞ; F ;XY ¼ 0

X ¼ 0; 1;
at

Y ¼ 0; 1:

(6a)

(6b)

The supporting conditions for a simply-supported rectangular plate are written as
W ¼ R2 W ;XX þ νW ;YY ¼ 0

at

X ¼ 0; 1;

(7a)

W ¼ W ;YY þ νR2 W ;XX ¼ 0

at

Y ¼ 0; 1:

(7b)

The problem consists in determining the functions F and W which satisfy the governing equations, together with the
boundary conditions.

3. Method of solution
An approximate solution of the governing Eqs. (1) and (2) is sought in the form of double series in terms of separate
space and time variables. The non-dimensional force function is expressed as
1
FðX; Y; TÞ ¼ ∑ ∑ F mn ðTÞX m ðXÞY n ðYÞ− X 2 NY ðTÞ
2
m n

(8)

and the dimensionless lateral displacement as
WðX; Y; TÞ ¼ ∑ ∑ W pq ðTÞΦp ðXÞΨ q ðYÞ
p

q

(9)


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H. Nguyen / Journal of Sound and Vibration 332 (2013) 5103–5114

where Fmn and Wpq are undetermined functions of the dimensionless time T, and where Xm, Yn, Φp, and Ψq are beam
eigenfunctions given by


coshαm −cosαm
X m ðXÞ ¼ coshαm X−cosαm X−
(10a)
ðsinhαm X−sinαm XÞ;
sinhαm −sinαm
Y n ðYÞ ¼ coshαn Y−cosαn Y−



coshαn −cosαn
ðsinhαn Y−sinαn YÞ;

sinhαn −sinαn

(10b)

Φp ðXÞ ¼ sin pπX;

(10c)

Ψ q ðYÞ ¼ sin qπY;

(10d)

in which the coefficients αi are obtained from the transcendental equations
1−cosαi coshαi ¼ 0:

(11)

These beam functions satisfy the relevant boundary conditions.
Applying the approach of generalized double Fourier series [24] to the governing equations, using the orthogonality
properties of the assumed functions, together with the indicial notation, leads to
pqrs
∑ ∑ Amn
ij F mn ðTÞ ¼ ∑ ∑ ∑ ∑ Bij W pq ðTÞW rs ðTÞ;

(12)

 2
€ uv ðTÞ þ ω2 W uv ðTÞ− πv NY ðTÞW uv ðTÞ þ ∑ ∑ ∑ ∑ Gmnrs F mn ðTÞW rs ðTÞ ¼ 0
W
uv

uv
R
m n r s

(13)

m n

p

q

r

s

pqrs
mnrs
in which Amn
ij ; Bij and Guv are coefficient matrices, and


π4
2u2 v2 v4
u4 þ
ω2uv ¼
þ
ζ
R2
R4

(14)

is the vibration frequency of the unloaded plate. In Eq. (13), the dots denote differentiation with respect to the nondimensional time T.
Generally, Eq. (12) can be solved for the time-dependent stress coefficients Fmn(T) in terms of Wuv(T) coefficients and
substituting into Eq. (13), leads to
 2
€ uv þ ω2 W uv − πv NY W uv þ ∑ ∑ ∑ ∑ ∑ ∑ M klpqrs W kl W pq W rs ¼ 0
W
(15)
uv
uv
R
k l p q r s
where
mn −1 pqrs
½M klpqrs
Š ¼ ½Gmnkl
uv
uv Š½Aij Š ½Bij Š

(16)

It is known that the rectangular plate buckles in such a way that there can be several half-waves in the direction of
compression but only one half-wave in the perpendicular direction. Hence, omitting all indices associated with the halfwave spatial mode in the unloaded direction, and introducing linear (viscous) damping lead to a system of nonlinear
ordinary differential equations for the time functions as follows:
€ m þ 2C m W
_ m þ Ω2 ð1−2μ cos θÞW m þ ∑ ∑ ∑ M ijk W i W j W k ¼ 0;
W
m

m
m
i

j

m ¼ 1; 2; 3; ⋯

(17)

k

where Cm represents the coefficient of viscous damping, Ωm ¼ωm [1−NY0/Nm]1/2 is the free vibration circular frequency of a
rectangular plate loaded by the constant component NY0 of the in-plane force while Nm represents the static critical load
_
according to linear theory, θðTÞ
¼ Λ is the instantaneous frequency of excitation, and μm ¼NYT/2(Nm−NY0) is the load
(excitation) parameter, in which NYT is the dimensionless amplitude of the harmonic in-plane loading as previously
mentioned.
By taking the first three terms in the expansion for the lateral displacement, the continuous system is reduced to a threedegree-of-freedom system and we get the following set of temporal equations of motion:
€ 1 þ Ω2 W 1 ¼ −2C 1 W
_ 1 þ 2μ Ω2 cos θW 1 −ðΓ 11 W 3 þ Γ 12 W 2 W 2 þ Γ 13 W 2 W 3 þ Γ 14 W 1 W 2
W
1 1
1
1
1
1
2
þΓ 15 W 1 W 23 þ Γ 16 W 32 þ Γ 17 W 22 W 3 þ Γ 18 W 2 W 23 þ Γ 19 W 33 þ Γ 110 W 1 W 2 W 3 Þ;

_ 2 þ 2μ Ω2 cos θW 2 −ð⋯Γ 2 ⋯Þ;
€ 2 þ Ω2 W 2 ¼ −2C 2 W
W
2

2

2

_ 3 þ 2μ Ω2 cos θW 3 −ð⋯Γ 3 ⋯Þ
€ 3 þ Ω2 W 3 ¼ −2C 3 W
W
3 3
3
in which Γm1 through Γm10 are the coefficients of the nonlinear (cubic) terms, and are defined as follows:
Γ m1 ¼ M111
m ;
Γ m4 ¼
Γ m7 ¼

M122
m
M223
m

þ
þ

121
211

Γ m2 ¼ M 112
m þ Mm þ Mm ;

M 212
m
M 232
m

þ
þ

M 221
m ;
M 322
m ;

Γ m5 ¼
Γ m8 ¼

M 133
m
M 233
m

131
311
Γ m3 ¼ M 113
m þ Mm þ Mm ;

331

þ M 313
m þ Mm ;

Γ m6 ¼ M 222
m ;

332
þ M 323
m þ Mm ;

Γ m9 ¼ M 333
m ;

(18a–c)


H. Nguyen / Journal of Sound and Vibration 332 (2013) 5103–5114
132
231
213
312
321
Γ m10 ¼ M123
m þ Mm þ Mm þ Mm þ Mm þ Mm ;

m ¼ 1; 2; 3:

5107

(19a–j)

Eq. (18) constitutes the final form assumed by the equations of motion. They represent a system of second-order
nonlinear differential equations with periodic coefficients, which may be considered as extensions of the standard
Mathieu-Hill equation.
4. Solution of the temporal equations of motion
Mathematical techniques for solving nonlinear problems are relatively limited, and approximate methods are generally
used. The method of asymptotic expansion in powers of a small parameter, ε, developed by Mitropolskii [25] and
generalized by Agrawal and Evan-Iwanowski [26,10], is an effective tool for studying nonlinear vibrating systems with
slowly varying parameters. In the present analysis, this method is used to solve the equations of motion.
Assuming that the actual mechanical system is weakly nonlinear, the damping, the excitation, and the nonlinearity can
be expressed in terms of the above-mentioned small parameter, and that the instantaneous frequency of excitation and the
load parameter vary slowly with time. Then, the generalized system of temporal equations of motion (17) can be rewritten
in the following asymptotic form:
€ m þ Ω2 W m ¼ ε½2μ Ω2 cos θW m −2C m W
_ m −∑ ∑ ∑ Mijk W i W j W k Š; m ¼ 1; 2; 3:
W
m m
m
m
i

j

(20)

k

Confining ourselves to the first order of approximation in ε, we seek a solution for the system of Eq. (20) in the following
form:
W m ¼ am ðτÞcosφm ðτÞ; m ¼ 1; 2; 3

(21)

where τ¼ εT represents the “slowing” time, and where am and φm are functions of time defined by the system of differential
equations

m
A1 (τ,θ,

dam =dT ¼ a_ m ¼ εAm
1 ðτ; θ; am ; φm Þ;

(22)

dφm =dT ¼ φ_ m ¼ Ωm ðτÞ þ εBm
1 ðτ; θ; am ; φm Þ:

(23)

m
B1 (τ,θ,

am,φm) and
am,φm) are selected in such a way that Eq. (21) will, after replacing am and φm by the
Functions
functions defined in Eqs. (22) and (23), represent a solution of the set of Eq. (20).
Following the general scheme of constructing asymptotic solutions for Eq. (18) and performing numerous transformations and manipulations, we arrive finally at a system of equations describing the nonstationary response of the discretized
system. By integrating this system of equations, amplitudes am and phase angles φm can be obtained as functions of time.
5. Stationary response
For simply-supported rectangular plates excited parametrically, the results of the investigations conducted by the

present author [15,16] indicate, besides the possibility of principal parametric resonances, the presence of internal
resonances. The stationary response associated with the assumed spatial forms of vibration of our system may be calculated
as a special case of the nonstationary motions in the resonant regime described previously.
It is known that governing equations with cubic nonlinearities are associated with many physical systems. The presence
of these nonlinear terms has an important influence upon the behavior of the system, especially under a condition of
internal resonance. An internal resonance is possible when two or more natural frequencies are commensurable or almost
commensurable
∑ mi Ωi ≅0

(24)

i

where mi are positive or negative integers. When an internal resonance coincides with a parametric resonance, the
combination of the two types gives rise to simultaneous resonances. This kind of resonances is characterized by the fact that
the system in question vibrates simultaneously in more than one normal mode and at different frequencies, although only
one of the modes is directly excited by the parametric excitation.
In the absence of internal resonances, the parametric excitation can excite only one mode at a time. In this case, principal
parametric resonance occurs when the excitation frequency is approximately equal to twice the natural frequency
associated with a particular mode of vibration, that is, Λ≈2Ωm. Stationary values for principal parametric response
associated with various spatial modes of vibration are given by
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3
2
u

2
u
u 4Ωm 4
2μm Ω2m

Λ−2Ωm 7
−ð2C m Þ2 5; m ¼ 1; 2; 3
am ¼ t
(25)
3M m
Λ
where only positive real values for the amplitude are admitted. The “7” sign upon the inner radical indicates the possibility
of two solutions; the larger solution is stable and attainable by the real system, while the lower is unstable and physically


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H. Nguyen / Journal of Sound and Vibration 332 (2013) 5103–5114

not realizable. Hence, the most important condition for this resonant case is that the stable solution must be higher than its
unstable solution. When the unstable solution coincides with – or is larger than – its stable solution, the parametrically
excited system becomes stable; this condition is then related to the non-resonant case.
In the presence of internal resonances, as mentioned above, the coincidence between an internal resonance and a
principal parametric resonance will give rise to simultaneous resonances. Of all possible internal resonances associated with
a flat rectangular plate we will, for convenience, consider only an internal resonance of the type 3Ω1 ≅Ω3 . Consequently, the
two following cases of simultaneous resonances will be investigated: (1) Λ ¼ 2Ω1 and 3Ω1 ≅Ω3 , and (2) Λ ¼ 2Ω3 and Ω3 ≅3Ω1 .
Case 1. Λ ¼ 2Ω1 AND 3Ω1 ≅Ω3
It is supposed that the principal parametric resonance Λ ¼ 2Ω1 and the internal resonance 3Ω1 ≅Ω3 occur simultaneously.
Then, performing numerous transformations and manipulations of the asymptotic solutions, we arrive finally at a system of
equations describing the stationary response for this case as follows:
1
Γ 12
μ Ω2 a1 sin ψ þ
a2 a3 sinψ′ ¼ 0;
Λ 1 1

4ðΩ3 −Ω1 Þ 1

(26a)

3Γ 11 2 Γ 14 2 2
Γ 12
a1 a3 cosψ′ ¼ 0;
a −
a þ μ Ω2 cos ψ−
4Ω1 1 2Ω1 3 Λ 1 1
2ðΩ3 −Ω1 Þ

(26b)

Γ 31
a3 sinψ′ ¼ 0;
4ð3Ω1 þ Ω3 Þ 1

(26c)

−C 1 a1 þ

Λ−2Ω1 −

−C 3 a3 −






9Γ 11 Γ 32 2
3Γ 14 3Γ 36 2 3
−
−
a1 þ
a − μ Ω2 cos ψ
8Ω1 4Ω3
4Ω1 8Ω3 3 Λ 1 1


a31
3Γ 12
Γ 31
a1 a3 −
þ
cosψ′ ¼ 0;
4ðΩ3 −Ω1 Þ
4ð3Ω1 þ Ω3 Þ a3

3Ω1 −Ω3 þ

(26d)

where ψ¼θ −2φ1 is the phase angle associated with the principal parametric resonance involving the first spatial form, and
ψ′ ¼3φ1−φ3 represents the phase angle corresponding to the specified internal resonance. The steady-state amplitudes,
a1 and a3, and the phase angles, ψ and ψ′, can be obtained by solving Eq. (26) by a numerical technique.
It appears from Eq. (26) that there are two possibilities for a nontrivial solution: either a1 is nonzero and a3 is zero, or
both are nonzero. The first possibility indicates that the specified internal resonance has no effect on the system response
and only the principal parametric resonance involving the first mode may occur. For the latter possibility, as the first mode is
the only one excited by the parametric excitation, the presence of the third mode in the response is possible only by the

transfer of energy from the excitedly first mode to the third mode through internal mechanism. As mentioned earlier,
the only condition for this resonant case possible is that the stable solution for each mode must be larger than the
corresponding unstable solution. When the stable solution of any mode coincides with its corresponding unstable solution,
the transfer of energy stops, and the first mode continues to be excited by the parametric excitation. If any mode showing
the stable solution lower than its corresponding unstable solution, then this is a non-resonant case and only the principal
parametric resonance of the first mode is possible.
Case 2. Λ ¼ 2Ω3 AND Ω3 ≅3Ω1
As in the previous case, the internal resonance has the same relationship, but this time the third mode is excited
parametrically. Using the same analysis as before and after a series of calculations, we obtain the following stationary
solutions:
1
Γ 31
μ Ω2 a3 sin ψ þ
a3 sinψ′ ¼ 0;
Λ 3 3
4ð3Ω1 þ Ω3 Þ 1

(27a)

a31
Γ 32 2 3Γ 36 2 2
Γ 31
a −
a þ μ Ω2 cos ψ−
cosψ′ ¼ 0;
2Ω3 1 4Ω3 3 Λ 3 3
2ð3Ω1 þ Ω3 Þ a3

(27b)

Γ 12
a2 a3 sinψ′ ¼ 0;
4ðΩ3 −Ω1 Þ 1

(27c)

−C 3 a3 þ

Λ−2Ω3 −

−C 1 a1 −
3Ω1 −Ω3 þ

h

9Γ 11
8Ω1

i
h
i
Γ 32
3Γ 36
2
2 1
14
a21 þ 3Γ
− 4Ω
4Ω1 − 8Ω3 a3 − Λ μ3 Ω3 cos ψ
3


þ


a31
3Γ 12
Γ 31
a1 a3 −
cosψ′ ¼ 0;
4ðΩ3 −Ω1 Þ
4ð3Ω1 þ Ω3 Þ a3

(27d)

where ψ¼θ−2φ3 is the phase angle corresponding to the principal parametric resonance involving the third mode, and as
before, ψ′¼3φ1−φ3 is the phase angle associated with the specified internal resonance. A numerical method is used to find
the solutions of Eq. (27).


H. Nguyen / Journal of Sound and Vibration 332 (2013) 5103–5114

5109

As in Case 1, we have two possibilities: either a1 is zero and a3 is nonzero, or neither is zero. The first possibility means
that only the principal parametric resonance involving the third mode shape may exist. The second possibility indicates the
presence of the two spatial forms in the system response and, hence, a transfer of energy from the directly-excited third
mode to the first mode through internal resonance. As before, the only condition for this resonant case is that the unstable
solution of each mode must be lower than its corresponding stable solution. When the unstable solution of any mode
coincides with its stable solution, the energy transfer stops and only the third mode continues to be excited by the
parametric excitation. If any of the involving modes having the unstable solution larger than its corresponding stable

solution, no transfer of energy is possible and only the third mode is parametrically excited.

6. Numerical results
In order to get more insight into the occurrence of simultaneous resonances, numerical evaluation was performed for a
thin rectangular plate of aspect ratio R¼ 1.73. The various values of the plate parameters and material constants used for the
numerical calculations are as follows: a ¼293 mm, b¼508 mm, h¼1 mm, E¼ 2.385 GPa, ν ¼0.45, and ρ¼ 1200 kg/m3. In the
analysis and numerical calculations, Dcr ( ¼NYT/Nn) denotes the dynamic component NYT of the periodic in-plane force
normalized to the lowest critical load Nn and is called the ratio of dynamic critical loading, Pcr (¼NY0/Nn) designates the
static in-plane load NY0 normalized to the lowest critical load Nn and is called the ratio of static critical loading, and
Δ (¼2πCm/Ωm) is the decrement of viscous damping. Typical results associated with the two cases of simultaneous
resonances analyzed above are shown in Figs. 2–6.
The stationary frequency–response curves are illustrated in Figs. 2–4. In these figures, the ordinate am represents the
steady-state amplitude, corresponding to the involved mode of vibration m, as a function of the plate thickness, while the
abscissa λ denotes the exciting frequency (in Hz). The bar over an amplitude means that the amplitude is associated with
simultaneous resonances, while the subscript i following the mth mode shape denotes the amplitude is possible due to
internal resonance. Solid and broken lines in the figures represent the stable and unstable solutions, respectively.
Figs. 2 and 3 show respectively the amplitudes of the first and third modes (a1 and a3) as functions of the excitation frequency
(λ, in Hz). In Fig. 2, the frequency–response curves are associated with the resonances Λ≅2Ω1 (principal parametric resonance
involving the first spatial form) and 3Ω1≅Ω3 (internal resonance involving the first and third mode shapes); hence, the first mode
is parametrically excited and the presence of the third mode is possible only by the transfer of energy from the first mode to the
third mode through internal mechanism. The result also shows that both stable branches of a1 and a3 are larger than the
corresponding unstable branches. This means that the energy transfer is possible, and this case of simultaneous resonances can
occur. The frequency–response curves in Fig. 3 are correspondent to the resonances Λ≅2Ω3 (principal parametric resonance
involving the third mode shape) and Ω3≅3Ω1 (the same internal resonance as specified). In this case, the third mode is
parametrically excited and its energy is transferred to the first mode through the specified internal resonance. However, it can be
seen that the unstable branch of a1 is over its stable solution; hence, this case of simultaneous resonances cannot occur and only
the principal parametric resonance involving the third spatial form is possible.

Fig. 2. Frequency–response curves associated with the simultaneous resonances Λ≅2Ω1 (principal parametric) and 3Ω1≅Ω3 (internal). Pcr ¼ 0.5, Dcr ¼0.2,
Δ ¼0.11.

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Fig. 3. Frequency–response curves associated with the simultaneous resonances Λ≅2Ω3 (principal parametric) and 3Ω1≅Ω3 (internal). Pcr ¼ 0.5, Dcr ¼ 0.2,
Δ ¼ 0.11.

Fig. 4. Effect of the internal resonance 3Ω1−Ω3≅0 on the frequency–response curves corresponding to principal parametric resonances involving the first
and third mode shapes. Pcr ¼0.5, Dcr ¼0.2, Δ ¼ 0.11.

The interaction between an internal resonance and a principal parametric resonance on the frequency–response curves
is illustrated in Fig. 4. As can be seen, the parametric response of the first mode occurs when Λ≅2Ω1. At a certain frequency,
however, a small part of energy from the first mode is transferred to the third mode, due to modal coupling between these
two modes. Consequently, the amplitude of the first mode slightly decreases but remains larger than the one of the third
mode. After a certain range when both stable and unstable solutions coincide, the energy transfer vanishes; the amplitude of
the third mode decays and the steady-state amplitude of the first mode regains its full strength.
As explained previously, only the principal parametric resonance involving the third mode is possible, as shown in Fig. 4.
For reference, the theoretical interaction between the internal resonance Ω3≅3Ω1 and the principal parametric resonance
Λ≅2Ω3 is also illustrated in this figure by dotted curves. If this case of simultaneous resonances is possible, i.e., both stable
solutions are larger than their unstable solutions, we can see that the system response will be particularly interesting. It can
be observed that when the excitation frequency reaches the point where the first mode can be excited through internal
resonance, the amplitude of the third mode, which is directly excited by the parametric excitation, drops drastically and
becomes less than the amplitude of the first mode which is due to internal resonance. This implies that there is a significant
transfer of energy from the third mode to the first one. As before, when the transfer of energy stops, the amplitude of the
first mode disappears and the third mode continues to be excited by the parametric excitation.


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Fig. 5. (a) Load amplitude–response curves associated with the simultaneous resonances Λ≅2Ω1 and 3Ω1≅Ω3. (b) Effect of the specified internal resonance
on the amplitude–response curves associated with the principal parametric resonance involving the first spatial form of vibration. λ ¼20 Hz, Pcr ¼0.5,
Δ ¼0.11.

The load amplitude–response curves corresponding to the two cases of simultaneous resonances are shown in Figs. 5 and
6. In the figures, the ordinate am represents again the steady-state amplitude, corresponding to the involved mode of
vibration m, as a function of the plate thickness, while the abscissa μm denotes the load (or excitation) parameter. The bar
over an amplitude signifies once more that the amplitude is associated with simultaneous resonances, while the subscript i
following the mth mode shape denotes the amplitude is possible due to internal resonance. Solid and broken lines represent
respectively, as before, the stable and unstable solutions.
Fig. 5(a) shows the amplitude–response curves associated with the resonances Λ≅2Ω1 and 3Ω1≅Ω3. Since both unstable
solutions are lower than their corresponding stable solutions, this case of simultaneous resonances is totally possible. The
results illustrate again the domination of the first mode over the third mode. The interaction between these two resonances
is presented in Fig. 5(b). The transfer of energy from the first mode to the third mode closely resembles to the case of
frequency response as explained previously.
The amplitude–response curves corresponding to the simultaneous resonances Λ≅2Ω3 and Ω3≅3Ω1 are illustrated in
Fig. 6(a). As shown before, the deflection of the motion is dominated by the first mode, though the third mode is the only
one directly excited by the parametric excitation. The result shows, however, that the unstable solution of the first mode is
much more higher than its corresponding stable solution. In practice, this situation signifies a non-resonant case, and only
the load–response curves associated with the principal parametric resonance involving the third mode can occur, as shown
in Fig. 6(b). In the latter, theoretical results showing the effect of the specified internal resonance on the amplitude–
response curves associated with the principal parametric resonance involving the third spatial form of vibration are
represented by dotted lines. If this case of simultaneous resonances is possible, a very significant transfer of energy from the
third mode to the first mode through internal mechanism can be observed.
7. Comparison with experimental results
In order to gain further insight into the occurrence of simultaneous resonances, an experimental program [15,17] was
undertaken to investigate the effects of internal resonances on the responses of parametrically-excited rectangular plates.

Experiments were conducted for four different sets of boundary conditions, namely: (1) all edges simply supported;
(2) loaded edges simply supported, the others loosely clamped; (3) loaded edges loosely clamped, the others simply
supported; (4) all edges loosely clamped. The laboratory apparatus, plate specimens, boundary conditions of the specimens,
test procedures and recorded data were described in detail in [17], and therefore are not repeated in this paper.


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Fig. 6. (a) Load amplitude–response curves associated with the simultaneous resonances Λ≅2Ω3 and 3Ω1≅Ω3. (b) Only the principal parametric resonance
involving the third spatial form of vibration is possible. λ¼ 60 Hz, Pcr ¼ 0.5, Δ¼ 0.11.

For the four test specimens under four different boundary conditions, it was shown in Ref. [17] that various internal
resonances were observed. Most of the types of internal resonance exhibited by real systems, however, could differ from
those generally assumed in the analytical investigations. In this case, the response curve associated with the internal
resonance mode shape was highly unpredictable and the frequency range of the response was not large.
Particularly important are the experimental results, shown in Fig. 7 for the third set of boundary conditions, involving the
first and third mode shapes, in which the first mode of vibration is directly excited by the parametric excitation.
The frequency spectra and mode shape records are also plotted for reference. An examination of the frequency spectra
reveals that the internal resonance relationship appears exactly as predicted by the theory, and that the natural frequencies
associated with the first and third mode are completely commensurable (that is, 3Ω1 ¼Ω3). The response shape of the third
mode appears in the same form as theoretically predicted, and it can be seen that the first mode dominates the response.
These experimental results show that the theoretical analysis gives an excellent prediction for this particular case of
simultaneous resonances. In the figure, the dotted curve represents the estimated response curve associated with the
principal parametric resonance of the first spatial mode in the absence of internal resonance.
For the same plate specimen under the third boundary conditions, experimental results recorded only the frequency–
response curves associated with the principal parametric resonance of the third mode, as shown in Fig. 8 by bold lines. This
means that the specified internal resonance, 3Ω1 ¼Ω3, has no effect on this principal parametric resonance and, as a
consequence, no significant transfer of energy from the directly-excited third mode to the first mode can occur. As explained

previously, the energy transfer cannot happen since the unstable solution of the first mode is higher than its stable solution.
Once again, the experimental results ascertain the validity of the analytical results thus obtained.
8. Concluding remarks
The present work deals primarily with the problem of the occurrence of simultaneous resonances involving two spatial
forms of vibration in a parametrically-excited rectangular plate and covers an existing gap in our understanding of the
parametric resonance and dynamic buckling of structures. Experimental results are also presented to verify the analytical
predictions.
The analysis shows that the presence of cubic nonlinearities has an important influence upon the behavior of the system,
especially under a condition of internal resonance involving two or more spatial forms of vibration. When this type of


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Fig. 7. Experimental results concerning a particular case of simultaneous resonances, λ¼ 2Ω1 and 3Ω1 ¼ Ω3. (a) Frequency–response curves: -−, increasing
sweep; −←, decreasing sweep. (b) Frequency spectra of the excitation and response. (c) Mode shape records for λ ¼31.5 Hz.

Fig. 8. Frequency–response curves associated with various parametric principal and simultaneous resonances. Theory: —, stable solution; − −, unstable
solution; Pcr ¼ 0.3; Dcr ¼ 0.13; Δ ¼0.1. Experiment: -−, increasing sweep; −←, decreasing sweep; n0 ¼31 N; nt ¼7.7 N.

internal resonance coincides with a principal parametric resonance, the combination of the two types gives rise to
simultaneous resonances. These kinds of resonances are characterized by the fact that the system in question resonates
simultaneously in more than one normal mode, although only one of the modes is directly excited by a single harmonic
excitation. Internal resonance is thus responsible for strong modal coupling and, as a consequence, for a significant transfer
of energy from the parametrically excited mode into other non-excited modes. Because of this modal interaction, modes
other than the one directly excited can dominate the response and, in general, the lowest mode dominates higher modes. In
the present investigation, based on the resonant condition, that is, the transfer of energy can only occur if the unstable
amplitude of each involved mode is lower than its corresponding stable amplitude, only one case of simultaneous
resonances out of two is possible. This conclusion is confirmed by experimental results.

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Acknowledgments
All theoretical and experimental results presented in this paper were obtained while the author was at Ecole
Polytechnique de Montréal 25 years ago. During that time, the research project was sponsored in part by the National
Sciences and Engineering Research Council of Canada through Grant A-4207, for which the author expresses gratitude.
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