Analysis on nonlinear vibrations near internal resonances of a composite laminated piezoelectric rectangular plate

Engineering Structures 173 (2018) 89–106

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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Analysis on nonlinear vibrations near internal resonances of a composite laminated piezoelectric rectangular plate

T

⁎

Y.F. Zhanga, W. Zhangb, , Z.G. Yaoc a

Faculty of Aerospace Engineering, Shenyang Aerospace University, Liaoning 110136, PR China Beijing Key Laboratory on Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, PR China c College of Mechanical Engineering, Beijing University Technology, Beijing 100124, PR China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Composite laminated piezoelectric plate Modelling Nonlinear vibrations Bifurcations Chaotic motions Internal resonance

The nonlinear vibrations and chaotic motions of a simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate subjected to the transverse and in-plane excitations are analyzed in the case of primary parametric resonance and 1:3 internal resonance. It is assumed that different layers of the symmetric cross-ply composite laminated piezoelectric rectangular plate are perfectly bonded to each other and with piezoelectric actuator layers embedded in the plate. Based on the Reddy’s third-order shear deformation plate theory, the nonlinear governing equation of motion for the composite laminated piezoelectric rectangular plate is derived by using the Hamilton’s principle. The Galerkin’s approach is employed to discretize the partial differential governing equation to a two-degree-of-freedom nonlinear system under combined the parametric and external excitations. The method of multiple scales is utilized to obtain the four-dimensional averaged equation. Numerical method is used to find the bifurcation diagram, the periodic and chaotic motions of the composite laminated piezoelectric rectangular plate. The numerical results illustrate the existence of the periodic and chaotic motions in the averaged equation. It is found that the chaotic responses are especially sensitive to the forcing and the parametric excitations. The influences of the transverse, in-plane and piezoelectric excitations on the bifurcations and chaotic behaviors of the composite laminated piezoelectric rectangular plate are investigated numerically.

1. Introduction Piezoelectric materials, which include piezoelectric lead-zirconatetitanate (PZT) and piezoelectric polyvinylidene fluoride (PVDF), are new functional materials in engineering applications. Piezoelectric materials can be used as the actuators and sensors in engineering structures. Therefore, composite laminated piezoelectric plates have been widely applied to aircraft, large space station and shuttle in the two past decades [1,2]. With the increasing use of composite laminated piezoelectric plates in engineering fields, for example, morphing structures or morphing wings and piezoelectric harvesters, composite laminated plates with piezoelectric materials can undergo the large oscillating deformation which leads to the nonlinear oscillations, bifurcations and chaos of composite laminated piezoelectric plates. The nonlinear vibrations with the large amplitude can cause serious damage of engineering structures under the certain conditions. Research works on the nonlinear dynamics, bifurcations, and chaos of composite laminated piezoelectric plate are playing the significant role in

⁎

Corresponding author. E-mail addresses: [email protected] (W. Zhang), [email protected] (Z.G. Yao).

https://doi.org/10.1016/j.engstruct.2018.04.100 Received 20 November 2017; Received in revised form 1 April 2018; Accepted 29 April 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.

engineering applications. However, up to now, a few studies on the bifurcations and chaos of the composite laminated piezoelectric plate have been conducted. For high-dimensional nonlinear dynamic systems, because of the existence of the modal interactions, there exist the relationships on several types of internal resonant cases which can lead to different forms of the nonlinear vibrations. When there exists a special internal resonant relationship between two linear natural frequencies, the large amplitude nonlinear responses may suddenly happen in composite laminated piezoelectric structures. The internal resonance is one of the main reasons resulting in the transfer of energy among many modes to cause the serious damage to composite laminated piezoelectric structures. In general, there exist three typical types of the nonlinear vibrations for structures and systems, namely, periodic, quasi-periodic and chaotic motions. As the amplitudes of the transverse and parametric excitations change, the periodic, quasi-periodic and chaotic vibrations can alternately appear in composite laminated piezoelectric structures. Therefore, this paper mainly investigates the nonlinear

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buckling and postbuckling responses of composite laminated plates subjected to an in-plane electromechanical load by using a nonlinear finite element method. Topdar et al. [23] analyzed the dynamic responses and the shape control of a simply-supported smart composite laminated sandwich plates. Kapuria and Achary [24] investigated the thermo-electromechanical responses of hybrid piezoelectric laminated plates and considered the variation of the displacements. Jose et al. [25] developed a finite element model for active control of thin laminated structures with piezoelectric sensor and actuator layers and illustrated that the negative velocity feedback control algorithm used in their model was effective for an active damping control of vibrations. Heidary and Eslami [26] investigated the piezo-control of forced vibrations of a thermoelastic composite plate and compared controlled and uncontrolled responses of plate. Fernandes and Pouget [27] studied the static and dynamic responses of a composite structure made of an elastic thin plate and a piezoelectric element which is perfectly attached to the elastic thin plate. They proposed a quadratic distribution of the electric potential through the piezoelectric layer. Zhu et al. [28] investigated the effects of strain rate dependency and inelasticity on the transient responses of composite laminated plates by using the finite element technique. Heuer [29] analyzed thermo-piezoelectrically induced flexural vibrations of viscoelastic laminated panels with timeharmonic thermo-piezoelectric excitation and considered the influence of the interlayer slip on the dynamic responses of laminate. Kusculuoglu and Royston [30] presented a finite element formulation for the vibration of piezoelectric laminated plates taking hysteretic behavior into account and studied transient responses of piezoelectric laminated plates using Mindlin plate theory. Zhang et al. [31] investigated the periodic and chaotic dynamics of a composite laminated piezoelectric rectangular plate with one-to-two internal resonance. Zhang et al. [32] used the energy-phase method to analyze higher-dimensional chaotic dynamics of a composite laminated piezoelectric rectangular plate. Chen et al. [33] theoretically and experimentally studied the nonlinear oscillations of symmetric cross-ply composite laminated plates. Guo et al. [34] studied a new kind of energy transfer from high-frequency mode to low-frequency mode in a composite laminated plate. Zhang et al. [35] investigated the nonlinear dynamic responses of a truss core sandwich plate. Yao and Zhang [36] utilized the extended Melnikov method to study the multi-pulse homoclinic orbits and chaotic motions in the laminated composite piezoelectric rectangular plate. Rafiee et al. [37] investigated the nonlinear dynamic stability of piezoelectric functionally graded carbon nanotube-reinforced composite plates with initial geometric imperfection. Yao et al. [38] analyzed nonlinear vibrations and chaotic dynamics of the laminated composite piezoelectric beam. Saviz [39] used an optimal approach to study the active damping of the nonlinear vibrations for the composite laminated plate with surface-bonded piezoelectric layer. Guo and Zhang [40] investigated the nonlinear vibrations and chaotic motions of a reinforced composite plate with carbon nanotubes. Kolahchi et al. [41] investigated the nonlinear dynamic stability of embedded temperature-dependent viscoelastic plates reinforced by single-walled carbon nanotubes. Kolahchi et al. [42] used the visco-nonlocal-piezoelasticity theories and differential cubature and quadrature-Bolotin method to study the dynamic stability response of embedded piezoelectric nanoplates. Kolahchi et al. [43] utilized the refined piezoelasticity zig-zag theory to study the wave propagation of embedded viscoelastic FG-CNT-reinforced sandwich plates integrated with sensor and actuator. Arani et al. [44] used DQM to investigated the electro-thermo nonlocal nonlinear vibration of an embedded polymeric piezoelectric micro plate reinforced by DWBNNTs. Arani et al. [45] utilized the viscoelastic quasi-3D sinusoidal shear deformation theory to study the wave propagation of FG-CNT-reinforced piezoelectric composite micro plates This paper focuses on the nonlinear vibrations, bifurcations and chaotic dynamics of a four-edges simply supported composite laminated piezoelectric rectangular plate subjected to the transverse, in-

vibrations, bifurcations and chaos of a simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate in the case of 1:3 internal resonance and primary parametric resonance. Several researchers have focused their attention on investigating the nonlinear dynamic responses of composite laminated plates. Pai and Nayfeh [3] presented a general nonlinear theory for the studies on the dynamics of elastic composite plates undergoing moderate-rotation oscillations by considering the geometric nonlinearities. Oh and Nayfeh [4] used the experimental method to study the nonlinear combination resonances in cantilever composite laminated plates with a harmonic transverse excitation. Bhimaraddi [5] studied large amplitude nonlinear vibrations of imperfect antisymmetric angle-ply laminated plates. Shukla et al. [6] gave an analytical approach to examine the nonlinear dynamic responses of a laminated composite plate with spatially oriented short fibers in each layer of the composite. Ye et al. [7] dealt with the nonlinear dynamic characteristics of a parametrically excited, simply supported symmetric cross-ply laminated rectangular thin plate. The geometric nonlinearity and nonlinear damping were included in the governing equations of motion. Ye et al. [8] investigated the nonlinear oscillations and chaotic dynamics of a simply supported antisymmetric cross-ply composite laminated rectangular thin plate under parametric excitation. Lee and Reddy [9] studied the nonlinear responses of composite laminated plates under thermomechanical loading using the third-order shear deformation plate theory. Zhang et al. [10] investigated the nonlinear oscillations and chaotic dynamics of a parametrically excited simply supported symmetric cross-ply composite laminated rectangular thin plate with the geometric nonlinearity and nonlinear damping. Abe et al. [11] analyzed the nonlinear dynamic responses of clamped laminated shallow shells with 1:1 internal resonance by using the combination of the Galerkin’s procedure and the shooting method. Hao et al. [12] analyzed the nonlinear oscillations, bifurcations and chaos of a functionally graded materials (FGM) plate and found that the periodic, quasi-periodic and chaotic motions exist for the FGM rectangular plate under certain conditions. The responses of composite laminated piezoelectric plates were also considered by investigators in the two past decades. Ye and Tzou [13] developed a new piezoelectric composite finite element method, compared finite element solutions of a piezoelectric composite laminated plate with experimental data and evaluated the control effectiveness. Later, Ye and Tzou [14] analyzed the responses and distributed control of a laminated piezoelectric semicircular shell in the changing temperature environment. Krommer and Irschik [15] studied the flexural vibrations of composite piezoelectric plates in which piezoelectric layers are used to generate distributed actuation or to perform distributed sensing of strains in plates. They demonstrated that coupling among the mechanical, electrical and thermal fields can be taken into account by means of effective stiffness parameters and an effective thermal loading. Shen [16] used a mixed Galerkin perturbation technique to determine thermal buckling temperature and postbuckling equilibrium paths. Lim et al. [17] gave a research on three-dimensional electromechanical responses of a parallel piezoelectric bimorph. Lim and He [18] analysed the exact solution of a compositionally graded piezoelectric layer under uniform stretch, bending and twisting. Correia et al. [19] developed a semi-analytical axisymmetric shell finite element model with embedded or surface bonded piezoelectric actuators or sensors to study active damping vibration control of structures. Donadon et al. [20] investigated the effect of the in-plane piezoelectric induced stresses on the natural frequencies of composite plates which are square and clamped along two opposing edges and free along other two edges. The natural frequencies and vibration modes were computed taking the stress stiffening effects of these piezoelectric stresses into account. Lim and Lau [21] investigated the electro-mechanical behaviors of a thick, laminated actuator with piezoelectric and isotropic lamina under externally applied electric loading by using a new twodimensional computational model. In recent years, Varelis and Saravanos [22] analyzed the pre90

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plane and piezoelectric excitations in the case of 1:3 internal resonance and primary parametric resonance. Based on the von Karman-type equation and the Reddy’s third-order shear deformation plate theory, we employ the Hamilton’s principle to obtain the governing equations of motion for the composite laminated piezoelectric rectangular plate. Because only transverse nonlinear oscillations of the composite laminated piezoelectric rectangular plate are considered, the governing equations of motion can be reduced to a two-degree-freedom nonlinear system under combined the parametric and external excitations by using the Galerkin’s method. The case of 1:3 internal resonance and primary parametric resonance is considered. Based on the equations obtained here, the method of multiple scales is used to obtain the averaged equation of the original non-autonomous system. Numerical method is utilized to investigate the bifurcations, periodic and chaotic motions of the composite laminated piezoelectric plate. The bifurcation diagrams are also obtained by using numerical simulation. It is found from the numerical results that there exist the periodic and chaotic motions of the composite laminated piezoelectric plate under certain conditions.

v ( x , y , z , t ) = v0 (x , y , t ) + z ϕy ( x , y , t )−z 3

4 ⎛ ∂w0 ⎞ ϕ + , 3h2 ⎝ y ∂y ⎠ ⎜

⎟

(1b)

w ( x , y , z , t ) = w0 ( x , y , t ),

(1c)

where (u , v , w ) are the displacement components along the ( x , y , z ) directions, (u 0 , v0, w0 ) is the deflection of a point on the middle plane (z = 0 ), ϕx and ϕy respectively represent the rotations of transverse normal of the mid-plane about the x and y axes. The nonlinear strain-displacement relations are given as follows

εxx = ε yy =

∂u ∂x ∂v ∂y

+ +

1 2

( ),

∂w 2 ∂x

εxz =

1 2

(

∂u ∂z

+

∂w ∂x

),

εxy =

1 2

1 2

∂w 2 , ∂y

ε yz =

1 2

(

∂v ∂z

+

∂w ∂y

),

εzz =

∂w . ∂z

( )

(

∂u ∂y

+

∂v ∂x

+

∂w ∂w ∂x ∂y

), (2)

Substituting Eq. (1) into Eq. (2) yields the strains (0) (0) (2) ⎧ εx ⎫ ⎧ κx ⎫ ⎧ κx ⎫ εx (0) (2) ⎫ ⎪ (0) ⎪ ⎪ ⎪ ⎪ ⎪ γyz ⎧ γyz ⎫ ⎧ γyz ⎫ (0) (2) εy = ε y + z κy + z3 κy , ⎧ γ ⎫ = + z2 , (0) ⎨ ⎨γ ⎬ ⎨ ⎬ ⎬ ⎨ ⎬ ⎨ ⎨γ ⎬ ⎨ γ (2) ⎬ zx ⎬ ⎭ ⎩ xy (0) zx zx (0) (2) ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎪ γxy ⎪ ⎪ κ xy ⎪ ⎪ κ xy ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭

⎧

2. Formulation

(3) Consider a simply supported at four-edges composite laminated piezoelectric rectangular plate, where the edge lengths are a and b and thickness is h, respectively, as shown in Fig. 1. The composite laminated piezoelectric rectangular plate is considered as regular symmetric crossply laminates with n layers. Some of these layers are made of the PVDF piezoelectric materials as actuators, while others are made of fiber-reinforced composite materials. A Cartesian coordinate Oxyz is located in the middle surface of the composite laminated piezoelectric rectangular plate. Assume that (u , v , w ) and (u 0 , v0, w0 ) represent the displacements of an arbitrary point and a point in the middle surface of the composite laminated piezoelectric rectangular plate in the x, y and z directions, respectively. It is also assumed that the in-plane excitations of the composite laminated piezoelectric rectangular plate are loaded along the y direction at x = 0 and the x direction at y = 0 with the form of q0 + qx cosΩ1 t and q1 + qy cosΩ2 t , respectively. The transverse excitation, which loads to the composite laminated piezoelectric rectangular plate, is represented by q = q3cosΩ3 t . The dynamic electrical loading is expressed as Ez = Ez cosΩ4 t . Considering the shear deformations of the composite plate, we use the Reddy’s third-order shear deformation plate theory, and the displacement field of the composite laminated piezoelectric rectangular plate is assumed to be of the following form [46,47]

u ( x , y , z , t ) = u 0 (x , y , t ) + z ϕx (x , y , t )−z 3

where

( )

2

⎧ ∂u0 + 1 ∂w0 ⎫ (0) 2 ∂x ⎧ ε x ⎫ ⎪ ∂x ⎪ ⎪ (0) ⎪ ⎪ ∂v ⎪ 2 1 ∂w 0 0 εy = , + 2 ∂y ⎨ ⎬ ⎨ ∂y ⎬ (0) ⎪ γxy ⎪ ⎪ ∂u0 ∂v 0 ∂w 0 ∂w 0 ⎪ ⎩ ⎭ ⎪ ∂y + ∂x + ∂x ∂y ⎪ ⎩ ⎭

( )

(0) ⎧ γyz ⎫ ⎧ φy + = ⎨ γ (0) ⎬ ⎨ φ + ⎩ zx ⎭ ⎩ x

∂w 0 ⎫ ∂y ∂w 0 ⎬ ∂x ⎭

∂φ

x ⎧ ⎫ (0) ∂x ⎧ κx ⎫ ⎪ ⎪ ⎪ (0) ⎪ ⎪ ∂φy ⎪ κy , = ∂ y ⎨ ⎬ ⎨ ⎬ (0) ⎪ ∂φy ⎪ ⎪ κ xy ⎪ ∂φx ⎩ ⎭ ⎪ ∂y + ∂x ⎪ ⎩ ⎭

,

2

∂φ

∂ w x ⎫ ⎧ + 20 (2) ∂x ∂x ⎧ κx ⎫ ⎪ γ (2) ⎪ ⎧ φy + ⎪ ⎧ yz ⎫ ⎪ (2) ⎪ ⎪ ∂φy ∂2w 0 κy + = −c2 , = −c1 2 y ∂ y ∂ (2) ⎨φ + ⎬ ⎨γ ⎬ ⎨ ⎬ ⎨ zx ⎭ (2) ⎪ ∂φy ⎪ κ xy ⎪ ∂φx ⎩ x ∂2w 0 ⎪ ⎩ ⎩ ⎭ ⎪ ∂y + ∂x + 2 ∂x ∂y ⎪ ⎭ ⎩

c1 =

4 , 3h2

∂w 0 ⎫ ∂y ∂w 0 ⎬ ∂x ⎭

,

c2 = 3c1.

(4)

It is known that the constitutive relations are of the form s σij = σ ijkl εkl−eijk Ek ,

(i, j, k ,

l = x , y, z ),

(5)

where Ek is the electric field and eijk is the piezoelectric moduli. s The stress-strain relationship σ ijkl without the piezoelectric effect

4 ⎛ ∂w0 ⎞ ϕ + , 3h2 ⎝ x ∂x ⎠

is

represented as follows

(1a)

s

0 0 ⎫ εxx ⎧ σ xx ⎫ ⎧Q11 Q12 0 ⎧ ⎫ ⎪ σ syy ⎪ ⎪Q Q 0 0 0 ⎪ ⎪ ε yy ⎪ 21 22 ⎪ s ⎪ ⎪ ⎪ ⎪γ ⎪ τ yx = 0 yz , 0 Q44 0 0 ⎬ ⎨γ ⎬ ⎨ s ⎬ ⎨ 0 0 Q55 0 ⎪ ⎪ zx ⎪ ⎪ τxz ⎪ ⎪ 0 ⎪ γxy ⎪ ⎪ τs ⎪ ⎪ 0 0 0 0 Q66 ⎪ ⎭⎩ ⎭ ⎩ xy ⎭ ⎩

q cos Ω 3t y b q0 + q x cos Ω1 t

(6)

where

Q11 = Q22 =

E , 1−ν 2

Q12 = Q21 =

νE , 1−ν 2

Q44 = Q55 = Q66 =

E . 2 (1−ν ) (7)

x

According to the Hamilton's principle, the nonlinear governing equations of motion for the composite laminated piezoelectric rectangular plate are given as follows

a q1 + q y cos Ω 2 t

∂Nxy ∂w¨ 0 ∂Nxx = I0 u¨ 0 + J1 ϕ¨x −c1 I3 + , ∂y ∂x ∂x

Fig. 1. The model of composite laminated piezoelectric rectangular plates is depicted. 91

(8a)

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Fig. 2. The bifurcation diagram of the composite laminated piezoelectric rectangular plates for x2 and x 4 via the forcing excitation f2 is given when x10 = −0.01, x20 = −0.09 , x30 = −0.05, x 40 = −0.05, μ1 = −0.2 , μ 2 = −0.2 , σ1 = 0.2 , σ2 = 0.7 , a2 = 3.0 , a3 = 0.01, a4 = −0.01, a5 = −0.01, a6 = −0.03, a7 = −0.05 , b6 = 0.07 , b7 = −0.08, b8 = 0.09 , f2 = 10 ∼ 14 .

∂Nxy

+

∂x

∂Nyy ∂y

= I0 v¨0 + J1 ϕ¨y −c1 I3

∂w¨ 0 , ∂y

γ (0) ⎧Q y ⎫ = ⎧ A 44 0 ⎫ ⎧ yz ⎫ , (0) ⎬ ⎨ ⎨ 0 A55 ⎬ ⎭ ⎨ γxz ⎩Q x ⎬ ⎭ ⎩ ⎩ ⎭

(8b)

∂Q y ∂ ⎛ ∂w ∂w ∂ ⎛ ∂w ∂w ∂Qx Nxx 0 + Nxy 0 ⎞ + Nxy 0 + Nyy 0 ⎞ + + ∂y ∂x ⎝ ∂x ∂y ⎠ ∂y ⎝ ∂x ∂y ⎠ ∂x ⎜

⎟

⎜

⎟

(0)

⎡ ∂u¨ ∂v¨ ∂w¨ ∂w¨ 0 ⎞ = I0 w¨ 0−c12 I6 ⎛ 20 + + c1 ⎢I3 ⎛ 0 + 0 ⎞ 2 x y x ∂ ∂y ⎠ ∂ ∂ ⎝ ⎠ ⎣ ⎝ ∂ϕ¨y ⎞ ⎤ ⎛ ∂ϕ¨ + I4 ⎜ x + ⎥, ∂y ⎟ ∂x ⎠⎦ ⎝ ⎟

⎜

n

⎟

∂Mxy ∂w¨ 0 ∂Mxx −Qx = J1 u¨ 0 + k2 ϕ¨x −c1 J4 + , ∂y ∂x ∂x

(2)

γ γ ⎧ Ry ⎫ = ⎧ D44 0 ⎫ ⎧ yz ⎫ + ⎧ F44 0 ⎫ ⎧ yz ⎫. (0) 0 D 0 F ⎨ ⎨ ⎬ ⎨ γ (2) ⎬ ⎨ ⎬⎨γ ⎬ ⎩ 55 ⎭ 55 ⎭ ⎩ Rx ⎬ ⎭ ⎩ ⎩ xz ⎭ ⎩ xz ⎭

∂2Pyy ⎞ ∂2Pxy ∂ 2P + + c1 ⎛⎜ xx +2 ⎟ + q−γẇ 0 2 ∂y 2 ⎠ ∂x ∂y ⎝ ∂x ⎜

NxP =

zk+1

∑ ∫zk

n k

k Q11 e31 Ez dz , NyP =

k=1

∂x

(8d)

(Aij Bij Dij Eij Fij , Hij ) = (8e)

Ii =

∑ ∫zk

ρk z i dz ,

Ji = Ii−c1 Ii + 2,

K2 = I2−2 c1 I4.

k=1

n

(Aij Dij Fij ) =

A11

(0)

(10a)

(1) (2) Mxxp ⎧ε ⎫ ⎧ε ⎫ ⎧ Mx ⎫ ⎧ D11 D12 0 ⎫ ⎪ x ⎪ ⎧ F11 F12 0 ⎫ ⎪ x ⎪ ⎧ ⎪ p⎫ ⎪ (1) (2) My = D21 D22 0 εy ε y −− Myy , + F21 F22 0 ⎨ ⎬ ⎨ ⎬⎨ ⎬ ⎨ ⎬⎨ ⎬ ⎨ p ⎬ 0 D66 ⎭ ⎪ γ (1) ⎪ ⎩ 0 0 F66 ⎭ ⎪ γ (2) ⎪ ⎪ Mxy ⎪ ⎩ Mxy ⎭ ⎩ 0 ⎭ ⎩ xy ⎭ ⎩ xy ⎭ ⎩ (10b)

(1) (2) ⎧ εx ⎫ H H 0 ⎫⎧ εx ⎫ ⎧ Px ⎫ ⎧ F11 F12 0 ⎫ ⎪ (1) ⎪ ⎧ 11 12 ⎪ (2) ⎪ Py = F21 F22 0 εy εy , + H21 H22 0 ⎨ ⎬ ⎨ ⎬⎨ ⎬ ⎨ ⎬⎨ ⎬ (1) 0 H66 ⎭ ⎪ γ (2) ⎪ ⎩ Pxy ⎭ ⎩ 0 0 F66 ⎭ ⎪ γxy ⎪ ⎩ 0 ⎩ ⎭ ⎩ xy ⎭

z , (i, j

zk+1

∑ ∫zk

(11a)

k

Qij ( 1 , z 2 z 4 ) dz , (i, j = 4 , 5).

(11b)

Substituting Eq. (10) into Eq. (8), we obtain the governing equations of motion in terms of generalized displacements (u 0 , v0 , w0 , ϕx , ϕy ) for the composite laminated piezoelectric rectangular plate as

(9)

The force and moment resultants related to the strains and curvatures of the plate constitutive equations then become

⎧ ε x ⎫ ⎧ Nxxp ⎫ ⎧ Nx ⎫ ⎧ A11 A12 0 ⎫ ⎪ (0) ⎪ ⎪ ⎪ Ny = A21 A22 0 ε y − Nyyp , ⎨ ⎬ ⎨ ⎬⎨ ⎬ ⎨ p ⎬ 0 A66 ⎭ ⎪ γ (0) ⎪ ⎪ Nxy ⎪ ⎩ Nxy ⎭ ⎩ 0 ⎩ xy ⎭ ⎩ ⎭

z

∑ ∫z k+1 Qijk (1 z z 2 z 3 z 5 z 6) d k

= 1 , 2 , 6),

k=1

zk+1

(10f)

k=1

where a dot represents the partial differentiation with respect to time t , a comma denotes the partial differentiation with respect to a specified coordinate, γ is the damping coefficient, and all kinds of inertias in Eq. (8) are calculated by n

k

k Q22 e32 Ez dz ,

where NiP = NiP cosΩ4 t (i = x , y ) represents the piezoelectric stress resultants, and Aij , Bij , Dij , Eij , Fij , and Hij respectively are the stiffness elements of the composite laminated piezoelectric rectangular plate, which are denoted as [46]

(8c)

∂Myy

∂w¨ −Q y = J1 v¨0 + k2 ϕ¨y −c1 J4 0 , + ∂y ∂y

zk+1

∑ ∫zk

(10e)

k=1

n

∂Mxy

(10d)

A66

(10c) 92

∂ 2u 0 ∂ 2u ∂ 2v 0 ∂w ∂2w0 ∂w ∂2w0 + A66 20 + (A12 + A66 ) + A11 0 + A66 0 ∂x 2 ∂y ∂x ∂y ∂x ∂x 2 ∂x ∂y 2 2 ∂w ∂ w0 ∂w¨ + (A12 + A66 ) 0 = I0 u¨ 0 + J1 ϕ¨x −c1 I3 0 , ∂y ∂x ∂y ∂x (12a)

∂ 2v 0 ∂ 2v ∂ 2u 0 ∂w ∂2w0 ∂w ∂2w0 + A22 20 + (A21 + A66 ) + A66 0 + A22 0 ∂x 2 ∂y ∂x ∂y ∂y ∂x 2 ∂y ∂y 2 2 ∂w ∂ w0 ∂w¨ + (A21 + A66 ) 0 = I0 v¨0 + J1 ϕ¨y −c1 I3 0 , ∂x ∂x ∂y ∂y (12b)

Engineering Structures 173 (2018) 89–106

Y.F. Zhang et al.

Fig. 3. The periodic motion of the composite laminated piezoelectric rectangular plate exists when f2 = 10.0 .

93

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Y.F. Zhang et al.

Fig. 4. The quasi-periodic motion of the composite laminated piezoelectric rectangular plate exists when f2 = 10.3.

94

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Y.F. Zhang et al.

Fig. 5. The multiple periodic motion of the composite laminated piezoelectric rectangular plate exists when f2 = 11.7 .

95

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Y.F. Zhang et al.

Fig. 6. The chaotic periodic motion of the composite laminated piezoelectric rectangular plate exists when f2 = 12.7 .

96

Engineering Structures 173 (2018) 89–106

Y.F. Zhang et al.

Fig. 7. The bifurcation diagram of the composite laminated piezoelectric rectangular plates for x2 and x 4 via the parametric excitation a2 is given when x10 = −0.01, x20 = −0.09 , x30 = −0.05, x 40 = −0.05, μ1 = −0.2 , μ 2 = −0.2 , σ1 = 3.61, σ2 = 3.13, a2 = 50 ∼ 250 , a3 = −0.01, a4 = 0.01, a5 = −0.01, a6 = −0.03, a7 = −0.05, b6 = 0.07 , b7 = −0.08, b8 = 0.09, f2 = 211.38.

A66

4 ∂3ϕy ∂w0 ∂2u 0 2 ∂ w0 2 ) (2 H c c F F H c H c − + + − − 22 1 66 12 66 1 12 1 1 ∂y ∂x 2 ∂x ∂y 2 ∂y 4 4 2 ∂3ϕy ∂ w0 ∂w ∂ u 0 + A11 0 + c1 (F22−H22 c1) 3 −H11 c12 ∂y ∂x 4 ∂x ∂x 2 ϕ ∂ ∂3ϕ y + c1 (F21 + 2F66−H21 c1−2H66 c1) 2 x + (F44 c22−2D44 c2 + A 44 ) ∂y ∂x ∂y ∂ 4w0 P 2 2 −c1 (H21 + 4H66 + H12) 2 2 + (A 44 −Ny cos(Ω4 t ) + F44 c2 ∂y ∂x ∂ 2w 0 ∂w ∂w ∂2w0 −2D44 c2) 2 + (A21 + 4A66 + A12 ) 0 0 ∂y ∂x ∂y ∂y ∂x ∂3ϕx ∂w0 ∂2u 0 ∂u ∂2w0 + A21 0 + c1 (F11−H11 c1) 3 + (A21 + A66 ) ∂x ∂y ∂y ∂x ∂x ∂y 2

(D66−2F66 c1 + H66 c12)

∂2ϕy ∂x 2

−c1 (F21 + 2F66−H21 c1−2H66 c1)

+ (H21 c12 + D66 + D21−2F21 c1 + H66 c12−2F66 c1) + (H22 c12 + D22−2F22 c1) −(F44 c22−2D44 c2 + A 44 ) = J1 v¨0 + K2 ϕ¨y −c1 J4

∂2ϕy ∂y 2

−c1 (F22−H22 c1)

∂3w0 ∂y ∂x 2

∂2ϕx ∂y ∂x

∂3w0 ∂y 3

∂w0 + (2D44 c2−F44 c22−A 44 )ϕy ∂y

∂w¨ 0 . ∂y

(12e)

The simply supported boundary conditions of the composite laminated piezoelectric rectangular plate can be represented as

2

+ A66 + A22

∂w ∂2v ∂w ∂ 2w 0 ∂w0 ∂2v0 1 + A22 0 20 + (A12 + 2A66 ) ⎛ 0 ⎞ 2 ∂y ∂y ∂ ∂y ∂x 2 2 y ⎝ ⎠ ∂x ⎜

x = 0:

⎟

v = w = ϕy = Nxy = Mxx = 0; x = a:

∂w ∂2v0 ∂2w0 ∂v0 + (A12 + A66 ) 0 ∂x ∂y ∂x ∂y 2 ∂y

y = 0:

∂2w ∂w 2 ∂w 2 ∂2w0 1 3 ∂2w0 ∂u 0 + (A21 + 2A66 ) 20 ⎛ 0 ⎞ + A11 ⎛ 0 ⎞ + A11 2 ∂y ⎝ ∂x ⎠ ∂ ∂ 2 2 ∂x 2 ∂x x x ⎝ ⎠ ∂2w0 ∂u 0 ∂2w0 ∂v0 ∂2w0 ∂v0 + 2A66 + A12 + 2A66 ∂y ∂x ∂y ∂x 2 ∂y ∂y ∂x ∂x

(13a)

u = w = ϕx = Nxy = Myy = 0; y = b:

∫0

3 ∂w ∂ 2w 0 + (A55 + qx cos(Ω1t )−NxP cos(Ω4 t ) + F55 c22 A22 ⎛ 0 ⎞ 2 2 ∂ y ⎝ ⎠ ∂y ⎜

h

(13b)

∂ϕ¨y ⎞ ⎛ ∂ϕ¨ , + c1 J4 ⎜ x + ∂y ⎟ ∂x ⎠ ⎝

⎜

(qx cosΩ1 t ) dz ,

∫0

h

Nyy |y = 0 dz

(qy cosΩ2 t ) dz.

(13c)

The boundary condition (13c) also includes the influence of the inplane load. We consider the complex nonlinear dynamics of the composite laminated piezoelectric rectangular plate in the first two modes of u 0 , v0 , w0 , ϕx and ϕy . In Refs. [33,34], the experimental results demonstrated that the mode shapes of symmetric composite cross-ply composite laminated piezoelectric rectangular plates are mainly due to the effects of the fiber orientation, the number of layers and the lay-up in composite laminated piezoelectric rectangular thin plates. Here, we write u 0 , v0 , w0 , ϕx and ϕy in the following forms [46]

∂ϕ ∂w ∂ 2w 0 + (F55 c22−2D55 c2 + A55 ) x −qcos(Ω3 t ) + μ 0 ∂x ∂t ∂x 2 ¨ ¨ u v ∂w¨ 0 2 ⎛ ∂2w¨ 0 ∂2w¨ 0 ⎞ ∂ ∂ = I0 2 −c1 I6 + + c1 I3 ⎛ 0 + 0 ⎞ 2 ∂t ∂y 2 ⎠ ∂y ⎠ ⎝ ∂x ⎝ ∂x −2D55 c2)

⎟

h

∫0 h = −∫ 0

Nxx |x = 0 dz = −

⎟

⎜

u = v = w = ϕx = Nxy

= Myy = 0,

2

+

u = v = w = ϕy = Nxy

= Mxx = 0,

⎟

(12c)

∂ 2ϕ ∂2ϕx + (D66−2F66 c1 + H66 c12) 2x ∂y ∂x 2 ∂3w0 ∂ w −c1 (F11−H11 c1) 3 −(F55 c22−2D55 c2 + A55 ) 0 ∂x ∂x ∂2ϕy + (D12 + D66 + H66 c12−2F66 c1 + H12 c12−2F12 c1) ∂y ∂x ∂3w0 −c1 (2F66 + F12−2H66 c1−H12 c1) 2 + (2D55 c2−A55 −F55 c22)ϕx ∂y ∂x ∂w¨ = J1 u¨ 0 + K2 ϕ¨x −c1 J4 0 , (12d) ∂x

πy πy πx 3 πx cos + u2 (t )cos cos , 2a 2b 2a 2b

(14a)

πy πy πx 3 πx cos + v2 (t )cos cos , 2b 2a 2b 2a

(14b)

u 0 = u1 (t )cos

(D11−2F11 c1 + H11 c12)

v0 = v1 (t )cos

πy πy πx 3π x sin + w2 (t )sin sin a b a b

(14c)

ϕx = ϕ1 (t )cos

πy πy πx 3π x sin + ϕ2 (t )cos sin , a b a b

(14d)

ϕy = ϕ3 (t )cos

πy πy πx 3π x sin + ϕ4 (t )cos sin . b a b a

(14e)

w0 = w1 (t )sin

97

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Fig. 8. The chaotic motion of the composite laminated piezoelectric rectangular plate exists when a2 = 88.0 .

In order to obtain the dimensionless governing equation of motion for the composite laminated piezoelectric rectangular plate, we introduce the transformations of the variables and parameters

u0 v w , v = 0, w = 0, a b h y b2 y = , q = q , b Eh3

u=

qx =

b2 q , Eh3 x

Ωi =

1 ⎛ abρ ⎞1/2 Ωi π⎝ E ⎠

qy =

b2 q, Eh3 y

ϕx = ϕx ,

ϕy = ϕy ,

x =

Aij =

x , a

Fij =

⎜

⎟

(ab)1/2 Fij , Eh6

Bij =

Hij =

(ab)1/2 Bij , Eh3

(ab)1/2 Hij , Eh8

Dij =

Ii =

(ab)1/2 Dij , Eh4

1 Ii. (ab)(i + 1)/2ρ

Eij =

(ab)1/2 Eij, Eh5

(15)

For simplicity, we drop the overbar in the following analysis. We only consider the transverse nonlinear oscillations of the composite laminated piezoelectric rectangular plate. Therefore, we can neglect all inertia terms in Eqs. (12a), (12b), (12d) and (12e). Substituting Eqs. (13)–(15) into Eq. (12c) and applying the Galerkin procedure, we obtain the two-degree-of-freedom governing equations of motion for the composite laminated piezoelectric rectangular plate for the dimensionless as follows

1/2

E ⎞ t = π2 ⎛ ⎝ abρ ⎠

(ab)1/2 Aij , Eh2

t,

(i = 1 , 2),

98

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Fig. 9. The multiple periodic motion of the composite laminated piezoelectric rectangular plate exists when a2 = 207.0 .

w¨ 1 + μ 1ẇ 1 + ω12 w1 + (a2cosΩ1t + a3cosΩ2 t + a4 cosΩ4 t ) w1 + a5 w12 w2 + a6 w22 w1 + a7 w13 + a8 w23 = f1 cosΩ3 t ,

ω12 = π2

(16a)

w¨ 2 + μ 2ẇ 2 + ω22 w2 + (b2cosΩ1t + b3cosΩ2 t + b4 cosΩ4 t ) w2 + b5 w22 w1 +

b6 w12 w2

+

b7 w23

+

b8 w13

= f2 cosΩ3 t ,

(16b)

F44 c22−2D44 c2 + A 44 −2 D55 c2 + A55 + F55 c22 + π2 m1 b2 m1 a2

+π

A55 k 7 + F55 c22 k 7−2 D55 c2 k 7 m1 a

+π

A 44 k 9 + F44 c22 k 9−2 D44 c2 k 9 m1 b

+ π3

where ε is a small parameter, ai , bi (i = 2 , ⋯ , 8) are the non-dimensional coefficients,

+ π3

a4 = −

p π2Nxp π2Ny − 2 , a4 = G0, 2 a b

(17)

and

99

−c1 F11 k 7 + H11 c12 k 7 m1 a3

+ π4

H22 c12 m1 b4

H22 c12 k 9−c1 F22 k 9 H11 c12 H12 c12 +4 H66 c12+H21 c12 + π4 +π 4 3 4 m1 b m1 a m1 b2a2

+ π3

H12 c12 k 9−c1 F12 k 9−2 c1 F66 k 9 + 2 π3H66 c12 k 9 m1 b a2

+ π3

2 c1 F66 k 7 +2H66 c12 k 7−c1 F21 k 7+H21 c12 k 7 , m1 b2a

(18a)

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Fig. 10. The periodic motion of the composite laminated piezoelectric rectangular plate exists when a2 = 250.0 .

ω22 = π

A 44 k10−2D44 c2 k10 + F44 c22 k10 A + F44 c2 2−2D44 c2 + π2 44 m2 b m2 b2

+ π2 +π

+ +

−18 D55 c2 + 9 F55 c22 + 9 A55 m2 a2

3A55 k8−6D55 c2 k8 + 3 F55 c22 k8 π 4 H22 c12 + m2 a m2 b4

+ π3 +

To save space, the lengthy expressions of ai , bi (i = 4 , ⋯ , 8) are omitted herein. The aforementioned equation, which includes the cubic terms, parametric and transverse excitations, describes the transverse nonlinear oscillations of the composite laminated piezoelectric rectangular plate subjected to the in-plane and transverse excitations and with the PVDF piezoelectric materials in the first two modes.

36 H66 c12 + 9 H12 c12 + 9H21 c12 − c1 F22 k10 + H22 c12 k10 + π4 3 m2 b m2 b2a2

−3 c1 F21 k8 π3

3. Perturbation and stability analysis

81 π 4H11 c12 + 3 H21 c12 k8−6 c1 F66 k8 + 6 H66 c12 k8 + 2 m2 b a m2 a4

−9 c1 F12 k10 + 9 H12 c12 k10−18 c1 F66 k10 π3 m2 b a2 2 27 27 c F k H c − + 1 11 8 11 1 k8 π3 . m2 a3

+

We study the nonlinear vibrations, bifurcations and chaos of a simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate subjected to the transverse and in-plane excitations are analyzed in the case of primary parametric resonance and 1:3 internal resonance. In this case, there are the following resonant relations

18 H66 c12 k10

(18b)

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Fig. 11. The bifurcation diagram of the composite laminated piezoelectric rectangular plates for x2 and x 4 via the piezoelectric excitation G0 when x10 = −1.08 , x20 = 0.5 , x30 = −0.01, x 40 = 9.16 , μ1 = −0.2 , μ 2 = −0.2 , σ1 = 4.37 , σ2 = 1.42 , a2 = −300 , a3 = 300 , G0 = 0.0 ∼ 100.0 , a5 = −11.66, a6 = 11.27 , a7 = −20.68 , b6 = −2.2 , b7 = −19.69 , b8 = −22.32 , f2 = 22.7 .

ω12 =

ω2 2ω + εσ1, ω22 = ω2 + εσ12, Ω1 = ω, Ω2 = Ω3 = Ω4 = , ω2 ≈ 3ω1. 9 3 (19)

A = x1 + i x2 ,

x1̇ =

(20)

where T0 = t and T1 = εt The time derivative used in the method of multiple scales are given as

d ∂ ∂ ∂T1 = + + ⋯=D0 + εD1 + ⋯, dt ∂T0 ∂T1 ∂t

(21a)

d2 = (D0 + εD1 + ⋯)2 = D20 + 2εD0 D1 + ⋯, d t2

(21b)

∂

(23)

Introducing Eq. (23) into Eq. (22), the four-dimensional averaged equations in the Cartesian form are obtained as

The method of multiple scales [48] are used to find the uniform solutions of Eq. (16)

w (x , t , ε) = w0 (x , T0, T1) + εw1 (x , T0, T1),

B = x3 + i x 4.

x2̇ =

x3̇ =

∂

where D0 = ∂T and D1 = ∂T . 1 0 Introducing Eqs. (19)–(21) into Eq. (16) and eliminating secular terms, we obtain the following averaged equation in the complex form

x 4̇ =

1 1 1 1 D1 A = − μ 1 A + i σ1 A−i a5 A 2 B−i a6 A B B − i (a2 + a3 + a4 ) A 2 2 2 4 3 − i a7 A2 A , (22a) 2

1 1 1 1 μ x1− σ1x2 + (a2 + a3 + a4 ) x2 + a5 x 4 (x 32−x42) + a6 x2 (x 32 + x42) 2 1 2 4 2 3 2 2 + a7 x2 (x1 + x 2 ), (24a) 2

1 1 1 1 μ x2 + σ1x1 + (a2 + a3 + a4 ) x1 + a5 x3 (x 22−x12)−a5 x1 x2 x 4 2 1 2 4 2 3 2 2 2 2 −a6 x1 (x 3 + x4 )− a7 x1 (x1 + x 2 ), (24b) 2 1 1 1 1 μ x3− σ2x 4 + b6 x 4 (x12 + x 22) + b7 x 4 (x 32 + x42) 2 2 6 3 2 1 + b8 x2 (x12−x 22), 6

(24c)

1 1 1 1 μ x 4 + σ2 x3− b6 x3 (x12 + x 22)− b7 x3 (x 32 + x42) 2 2 4 3 2 1 1 + b8 x1 (x 22 + x12)− f2 . 2 12

(24d)

The constant solution of the average Eq. (24) corresponds to the periodic solution of system (16). The stability of the solutions of the average Eq. (24) can be determined by investigating the characteristic equation. The stability of the solution for Eq. (24) is determined by the properties of its characteristic equation

1 1 1 1 1 1 D1 B = − μ 2 B + i σ2 B−i b6 A A B−i b7 B2 B −i b8 A3 −i f. 2 6 3 2 6 12 2 (22b) Let 101

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Fig. 12. The multiple periodic motion of the composite laminated piezoelectric rectangular plate exists when G0 = 0.0 .

dx1 dx dx dx 4 = 2 = 3 = = 0. dt dt dt dt

(25)

In order to determine the stability of the steady-state responses of the system, let

x1 = x10 + δ1, x2 = x20 + δ2 , x3 = x30 + δ3, x 4 = x 40 + δ4 ,

d δ3 = f31 δ1 + f32 δ2 + f33 δ3 + f34 δ4 , dt

(27c)

d δ4 = f41 δ1 + f42 δ2 + f43 δ3 + f44 δ4. dt

(27d)

(26) The coefficient matrices of Eq. (27) is of the form

where (x10 , x20 , x30 , x 40 ) is the solution of Eq. (25), δ1, δ2 , δ3 and δ4 are the small perturbations to the steady-state responses. Substituting Eq. (26) into Eq. (25), we have the variational equation

d δ1 = f11 δ1 + f12 δ2 + f13 δ3 + f14 δ4 , dt

(27a)

d δ2 = f21 δ1 + f22 δ2 + f23 δ3 + f24 δ4 , dt

(27b)

⎡ f11 ⎢ f21 A=⎢ ⎢ f31 ⎢f ⎣ 41 where

102

f12 f22 f32 f42

f13 f23 f33 f43

f14 ⎤ f24 ⎥ ⎥, f34 ⎥ f44 ⎥ ⎦

(28)

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Fig. 13. The multiple periodic motion of the composite laminated piezoelectric rectangular plate exists when G0 = 60.0 .

f11 =

1 1 1 2 2 μ + 3a7 x20 x10 , f12 = − σ1 + (a2 + a3 + a4 ) + a6 (x 30 + x40 ) 2 1 2 4 3 9 2 2 , + a7 x10 + a7 x 20 2 2

f13 = a5 x 40 x30 + 2a6 x20 x30 , f14 =

f21 =

f22 =

f24 = −a5 x10 x20−2a6 x10 x 40 , f31 = =

f33 =

1 3 2 2 a5 x 30 − a5 x40 + 2a6 x20 x 40 , 2 2

2 1 b6 x 4 x10 + b8 x20 x10 , f32 3 3

2 1 2 b6 x 40 x20− b8 x 20 , 3 2 1 1 1 1 3 2 2 2 2 μ + b7 x 40 x30 , f34 = − σ2 + b6 (x10 + x 20 ) + b7 x 30 + b7 x40 , 2 2 6 3 2 2

2 1 3 2 2 2 f41 = − b6 x30 x10 + b8 x 20 + b8 x10 , f42 = − b6 x30 x20 + b8 x10 x20 , 3 2 2 3

1 1 9 2 2 2 σ1 + (a2 + a3 + a4 )−a5 x30 x10−a5 x20 x 40−a6 (x 30 + x40 )− a7 x10 2 4 2 3 2 − a7 x 20 , 2

1 1 3 1 1 2 2 2 2 f43 = + σ2 − b6 (x10 + x 20 )− b7 x 30 − b7 x40 , f44 = μ 2−b7 x30 x 40. 4 3 2 2 2 (29) The characteristic equation of the non-trivial steady-state solution is of the form

1 1 2 2 μ + a5 x30 x20−a5 x10 x 40−3a7 x10 x20 , f23 = a5 x (x 20 −x10 )−2a6 x10 x30 , 2 1 2

(D (λ ) = |λI −A| = 0. 103

(30)

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Fig. 14. The chaotic motion of the composite laminated piezoelectric rectangular plate exists when G0 = 72.0 .

controlling parameters when the periodic and chaotic responses of the composite laminated piezoelectric rectangular plate are investigated. Through analyzing the bifurcation diagrams, the complicated nonlinear dynamics, including periodic and chaotic motions, may be observed globally from a range of parameter values. The two-dimensional phase portrait, waveform, three-dimensional phase portrait and Poincare map are depicted to illustrate the nonlinear dynamic behaviors of the composite laminated piezoelectric rectangular plate. It can be clearly found from the numerical results and the bifurcation diagrams that the periodic and chaotic motions occur for the composite laminated piezoelectric rectangular plate. Fig. 2 illustrates the bifurcation diagram of the composite laminated piezoelectric rectangular plate when the forcing excitation f2 is located in the interval 10 ∼ 14 . The other parameters and the initial conditions are respectively chosen as μ 1 = −0.2 , μ 2 = −0.2 , σ1 = 0.2, σ2 = 0.7 , a2 = 3.0 , a3 = 0.01, a4 = −0.01, a5 = −0.01, a6 = −0.03, a7 = −0.05,

If the real parts of all eigenvalues of Eq. (30) are negative, the corresponding solutions are stable. If one or more than one real parts of all eigenvalues of Eq. (28) are positive, the corresponding solutions are unstable. Furthermore, the Hopf bifurcation can cause the instability of the steady-state solutions, if a pair of pure imaginary eigenvalues exists in Eq. (30). 4. Numerical simulations In the following research, the fourth-order Runge-Kutta algorithm is utilized to numerically analyze the periodic and chaotic motions of the simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate subjected to the electric and mechanical loads for the case of 1:3 internal resonance and primary parametric resonance. We choose the averaged Eq. (24) to do numerical simulation. We use the parametric excitation a2 and forcing excitation f2 as the 104

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piezoelectric excitation amplitude G0 . When the piezoelectric excitation is G0 = 0.01, there exists the multiple periodic motion of the composite laminated piezoelectric rectangular plate, as shown in Fig. 12. When the piezoelectric excitation is G0 = 60.0 , the multiple periodic motion of the composite laminated piezoelectric rectangular plates also occurs, as shown in Fig. 13. When the piezoelectric excitation changes to G0 = 72.0 , the chaotic motion of the composite laminated piezoelectric rectangular plates exists, as shown in Fig. 14. In Figs. 11–14, the other parameters are the same as those given in Fig. 11. The aforementioned results are similar to the results obtained in Refs. [12,31,33].

b6 = 0.07 , b7 = −0.08, b8 = 0.09 x10 = −0.01, x20 = −0.09, x30 = −0.05, x 40 = −0.05. It is observed from Fig. 2 that the external excitation f2 has significant effect on the nonlinear dynamic responses of the composite laminated piezoelectric rectangular plate. In Fig. 2, the longitudinal coordinate denotes the deflection of the plate, while the abscissa denotes the external excitation f2 . There are the following physical explanations of motions for the simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate. From Fig. 2, it is seen that the motions of the composite laminated piezoelectric rectangular plate change from one period motion to quasi-period motion, and then from the multiple period motion to chaotic motions with the increase of the external excitation amplitude f2 . In the following investigation, we may change the external excitation amplitude f2 to find the periodic and chaotic motions of the composite laminated piezoelectric rectangular plate base on Fig. 2. Fig. 3 indicates existence of the periodic motion for the composite laminated piezoelectric rectangular plate when the forcing excitation f2 is 10.0. Here the other parameters are the same as those in Fig. 2. Fig. 3(a) and (c) represent the phase portraits on the planes (x1, x2) and (x3 , x 4 ) , respectively. Fig. 3(b) and (d) give the waveforms on the planes (t , x1) and (t , x3) , respectively. Fig. 3(e) and (f) represent the Poincare map on the plane (x1 , x2) and three-dimensional phase portrait in the space (x1, x2 , x3) , respectively. When the external excitation changes to f2 = 10.3, the quasi-period motion of the composite laminated piezoelectric rectangular plates is observed, as shown in Fig. 4. Fig. 5 illustrates that the multiple periodic motion of the composite laminated piezoelectric rectangular plate occurs when the external excitation changes to f2 = 11.7 . When the external excitation changes to f2 = 12.7 , the chaotic motion of the composite laminated piezoelectric rectangular plates occurs, as shown in Fig. 6. In Fig. 7, we depict the bifurcation diagram for x2 and x 4 via the parametric excitation a2 . In this case, the other parameters and the initial conditions are respectively chosen as μ 1 = −0.2 , μ 2 = −0.2 , σ1 = 3.61, σ2 = 3.13, a2 = 50.0 ∼ 250 , a3 = 0.01, a4 = −0.01, a5 = −0.01, a6 = −0.03, a7 = −0.05, b6 = 0.07 , b7 = −0.08, b8 = 0.09, f2 = 211.38, x10 = −0.01, x20 = −0.09, x30 = −0.05, x 40 = −0.05. Fig. 7 demonstrates the effect of the in-plane excitation on the nonlinear dynamic responses of the composite laminated piezoelectric rectangular plate. The longitudinal coordinate denotes the deflection of the plate and the abscissa denotes the in-plane excitation. It can be seen from Fig. 7 that the motions of the composite laminated piezoelectric rectangular plate change from the chaotic motion to the multi-periodic motion, and then from the multiple periodic motion to the periodic motion with the increase of the in-plane excitation a2 . Fig. 8 illustrates the existence of the chaotic motion for the composite laminated piezoelectric rectangular plate when the in-plane excitation is a2 = 88.0 , where the other parameters are the same as those in Fig. 7. When the in-plane excitation changes to a2 = 207.0 , the multiple periodic motion of the composite laminated piezoelectric rectangular plate occurs, as shown in Fig. 9. Fig. 10 demonstrates that there exists the periodic motion of the composite laminated piezoelectric rectangular plate when the in-plane excitation is a2 = 250.0 . In the last, we give the bifurcation diagram for x2 and x 4 via the piezoelectric excitation G0 , as shown in Fig. 11. In this situation, the other parameters and the initial conditions are respectively chosen as μ 1 = −0.2 , μ 2 = −0.2 , σ1 = 3.61, σ2 = 3.13, a2 = 300.0 , a3 = 300.0 , G0 = 0.0 ∼ 100.0 , a5 = −0.01, a6 = −0.03, a7 = −0.05, b6 = 0.07, b7 = −0.08, b8 = 0.09, f2 = 211.38, x10 = −0.01, x20 = −0.09, x30 = −0.05, x 40 = −0.05. Fig. 11(b) and (d) are local enlarged diagrams of Fig. 11(a) and (c), respectively. Fig. 11 demonstrates the influence of the piezoelectric excitation on the nonlinear dynamic response of the composite laminated piezoelectric rectangular plate. It is observed from Fig. 11 that the motions of the composite laminated piezoelectric rectangular plate change from the periodic motion to the chaotic motion, and then from the chaotic motion to the periodic motion with the increase of the

5. Conclusions We study the bifurcations, periodic and chaotic dynamics of a fouredges simply supported composite laminated piezoelectric rectangular plate subjected to the in-plane, transverse and piezoelectric excitations. Based on the von Karman-type equations and the Reddy’s third-order shear deformation plate theory, we utilize the Hamilton’s principle to establish the governing equations of motion for the composite laminated piezoelectric rectangular plate. Because the transverse nonlinear oscillations of the composite laminated piezoelectric rectangular plate are only considered, the governing equation of motion can be reduced to a two-degree-of-freedom nonlinear system under combined the parametric and external excitations by using the Galerkin’s method. The case of 1:3 internal resonance and primary parametric resonance is considered. Based on the equation obtained here, the method of multiple scales is used to obtain the averaged equation of the original nonautonomous system. Numerical method is used to investigate the bifurcations, periodic and chaotic motions of the composite laminated piezoelectric rectangular plate. The bifurcation diagrams are also obtained by using numerical simulation. It is found from the numerical results that there exist the periodic, quasi-periodic and chaotic motions of the composite laminated piezoelectric rectangular plate under certain conditions. We obtain three type bifurcation diagrams of the composite laminated piezoelectric rectangular plate, including the bifurcation diagram for x2 and x 4 via the forcing excitation f2 , the bifurcation diagram for x2 and x 4 via the parametric excitation a2 and the bifurcation diagram for x2 and x 4 via the piezoelectric excitation G0 , respectively. From Fig. 2, it is found that the varying procedure for the motions of the composite laminated piezoelectric rectangular plate is as follows: the periodic motion → quasi-periodic motion → the multiple periodic motion → the chaotic motion with the increase of the external excitation f2 . Figs. 3–6 present the corresponding phase portraits and waveforms for four types of motions for the composite laminated piezoelectric rectangular plate, respectively. It is observed from Fig. 7 that the varying procedure for the vibrations of the composite laminated piezoelectric rectangular plate is as follows: the chaotic motion → the multi-periodic motion → the periodic motion with the increase of the in-plane excitation a2 . Figs. 8–10 give the corresponding phase portraits and waveforms for three type motions of the composite laminated piezoelectric rectangular plate, respectively. It is also illustrated from Fig. 11 that varying procedure for the motions of the composite laminated piezoelectric rectangular plate is as follows: the multiple periodic motion → the chaotic motion → the periodic motion with the increase of the piezoelectric excitation G0 . Figs. 12–14 demonstrate the corresponding phase portraits and waveforms for three type motions of the composite laminated piezoelectric rectangular plate, respectively. The influences of the piezoelectric excitation on the nonlinear oscillations, bifurcations and chaos of the composite laminated piezoelectric rectangular plates are considered in this paper. The piezoelectric excitation can be considered to be a controlling parameter, which may control the nonlinear dynamic responses of the composite laminated piezoelectric rectangular plate. It is found that the forcing and the parametric excitations have significant influences on the 105

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