# Multipulse Chaotic Dynamics for a Laminated Composite Piezoelectric Plate

Authors: J. H. Zhang; W. Zhang
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

We investigate the global bifurcations and multipulse chaotic dynamics of a simply supported laminated composite piezoelectric rectangular thin plate under combined parametric and transverse excitations. We analyze directly the nonautonomous governing equations of motion for the laminated composite piezoelectric rectangular thin plate. The results obtained here indicate that the multipulse chaotic motions can occur in the laminated composite piezoelectric rectangular thin plate. Numerical simulations including the phase portraits and Lyapunov exponents are used to analyze the complex nonlinear dynamic behaviors of the laminated composite piezoelectric rectangular thin plate.


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## Body

## 1. Introduction

Piezoelectric materials can be used as the actuators and sensors in engineering structures [1]. With the increased use of composite laminated piezoelectric plates in engineering structures, for example, in airplane and launch vehicles, research works and development on the responses of laminated composite piezoelectric plates have experienced tremendous growth in the last two decades. For instance, Ye and Tzou [2] developed a new piezoelectric composite finite element and gave a comparison between finite element solutions of a laminated composite piezoelectric plate and experimental data. Shen [3] analyzed nonlinear bending of a simply supported, shear deformable cross-ply laminated plate with piezoelectric actuators and subjected to a transverse uniform or sinusoidal load combined with electrical loads and thermal environments based on higher-order shear deformation plate theory and perturbation technique. Recently, Zhang et al. [4] established the nonlinear governing equations of motion for a simply supported laminated composite piezoelectric rectangular plate under combined parametric and transverse excitations and studied the periodic and chaotic dynamics in the case of one-to-two internal resonance. The linear and geometrically nonlinear forced vibrations of laminated composite plates with piezoelectric materials were studied through numerical approach by Tanveer and Singh [5]. Torres and Mendonca [6] developed a formulation for laminated plates with extensional distributed piezoelectric sensors and the analytical solutions for simply supported square laminates with piezoelectric layers.Laminated composite plates with piezoelectric materials can undergo large oscillating deformation, which leads to nonlinear oscillations of plates. However, little research deals with the complex nonlinear dynamics of laminated composite piezoelectric plates, such as the bifurcations and multipulse chaotic dynamics. We investigate chaotic phenomena in such systems in order that we can control the system through the piezoelectric change. With the development of the theories of nonlinear dynamics and chaos, prediction, understanding, and control become possible for more complicated nonlinear phenomena in laminated composite piezoelectric plates. Kovacic and Wiggins [7] developed a new global perturbation method which may be used to detect the Shilnikov-type single-pulse homoclinic and heteroclinic orbits of four-dimensional autonomous ordinary differential equations. Camassa et al. [8] proposed an extended Melnikov method to study the multipulse jumping of homoclinic and heteroclinic orbits in a class of perturbed Hamiltonian systems. Zhang et al. [9] improved the extended Melnikov method given by Camassa et al. [8] and employed it to study the multipulse Shilnikov-type chaotic dynamics for a nonautonomous buckled rectangular thin plate. Zhang et al. [10] presented an extended Melnikov method to nonautonomous nonlinear dynamical systems in mixed coordinates. The multipulse chaotic dynamics of a simply supported laminated composite piezoelectric rectangular thin plate under the combination of the parametric and transverse excitations was investigated by the proposed method.However, in paper [10], the normal form theory is used to simplify the equations of the laminated composite piezoelectric plate. In this paper, the multipulse chaotic dynamics of the simply supported laminated composite piezoelectric rectangular plate under combined parametric and transverse excitations is investigated by using the extended Melnikov method improved in paper [9]. The nonlinear terms, which were missing through simplification by normal form theory in paper [10], are retained in this paper and added small positive parameter.


## 2. Analysis on Multipulse Chaotic Dynamics of the Plate

In this section, we investigate the multipulse chaotic dynamics for the simply supported laminated composite piezoelectric rectangular thin plate under combined parametric and transverse excitations. The two-degree-of-freedom governing equation of motion for the plate in dimensionless nonautonomous nonlinear system is shown in (2.1), and the details about the equation can be seen in paper [4]:(2.1)ẅ1+μ1ẇ1+ω12w1+(f1cos⁡Ω1t+f2cos⁡Ω2t+f3cos⁡Ω4t)w1+α1w1w22+α2w12w2+α3w13+α4w23=F1cos⁡Ω3t,ẅ2+μ2ẇ2+ω22w2+(f4cos⁡Ω1t+f5cos⁡Ω2t+f6cos⁡Ω4t)w2+β1w12w2+β2w1w22+β3w23+β4w13=F2cos⁡Ω3t,
where αi and βi(i=1–4) are nondimensional coefficients. In (2.1), the coefficients for ω1, ω2, F1, F2, αi, βi(i=1–4), and fj(j=1–6) are presented in [4]. Figure 1 is a sketch on the model of the simply supported laminated composite piezoelectric rectangular thin plate under combined parametric and transverse excitations and the coordinate system.Figure 1
The model of plate subjected to its plane and transverse excitations.We introduce the following transformations on (2.1):(2.2)x1=w1,x2=ẇ1,x3=w2,x4=ẇ2.Then, the following equivalent form of (2.1) is obtained:(2.3)ẋ1=x2,ẋ2=-ω12x1-α1x1x32-α2x12x3-α3x13-α4x33-c1x2+F1cos⁡Ω3t-(f1cos⁡Ω1t+f2cos⁡Ω2t+f3cos⁡Ω4t)x1,ẋ3=x4,ẋ4=-ω22x3-β1x12x3-β2x1x32-β3x33-β4x13-c2x4+F2cos⁡Ω3t-(f4cos⁡Ω1t+f5cos⁡Ω2t+f6cos⁡Ω4t)x3.
The parameters of the damping and the forces in (2.3) are considered as perturbation parameters the unperturbed system is as follows:(2.4)ẋ1=x2,ẋ2=-ω12x1-α1x1x32-α2x12x3-α3x13-α4x33,ẋ3=x4,ẋ4=-ω22x3-β1x12x3-β2x1x32-β3x33-β4x13.
Executing the Maple program given by Zhang et al. [11], we can obtain the 3-order normal form of unperturbed system (2.4) as follows:(2.5)ẋ1=3ω14α3m1x23+3ω12α3m1x12x2+ω12α1nx2x32+ω12ω22α1nx2x42,ẋ2=-3α3m1x13-3ω12α3m1x1x22-α1nx1x32-ω22α1nx1x42,ẋ3=3ω24β3m2x43+ω22β1nx12x4+ω12ω22β1nx2x42+3ω22β3m2x4x32,ẋ4=-3β3m2x33-β1nx12x3-ω12β1nx3x22-3ω22β3m2x3x42,
where m1=1/(3+2ω14+3ω18), m2=1/(3+2ω24+3ω28), n=1/(1+ω14+ω14ω24+ω28).Comparing (2.4) with (2.5), we can conclude that the parameters of normal form (2.5) are independent of parameters α2, α4, β2, and β4; that is to say, the four terms α2x12x3, α4x33, β2x1x32, and β4x13 in (2.4) can only have influence on the terms which are higher than cubic of normal form (2.5). Therefore, the four terms forenamed have little contribution on 3-order normal form (2.5), which means that these four nonlinear terms affect unperturbed system (2.4) to a lesser extent compared with other terms of the system. On the other hand, the analytical solution can be obtained if there are not the four terms forenamed in (2.4). With these in mind and in order to obtain the homoclinic orbits of the unperturbed system, we add the small positive parameter ε on the four terms except the perturbed parameters aforementioned—the parameters of the damping and the forces.We consider the unperturbed system (2.3), firstly, we introduce the following transformations on (2.3):(2.6)x1=u1,x2=u2,x3=v1,x4=c2v1+c2v2.
Then, the following equivalent form of (2.3) is obtained:(2.7)u̇1=u2,u̇2=-ω12u1-α1u1v12-α3u13-α2u12v1-α4v13-c1u2+F1cos⁡Ω3t-(f1cos⁡Ω1t+f2cos⁡Ω2t+f3cos⁡Ω4t)u1,v̇1=c2v1+c2v2,v̇2=-ω̅22v1-β̅1u12v1-β̅3v13-β̅2u1v12-β̅4u13-2c2v2+F̅2cos⁡Ω3t-(f̅1cos⁡Ω1t+f̅2cos⁡Ω2t+f̅3cos⁡Ω4t)v1,
where ω̅22=2c2+ω22/c2, β̅i=βi/c2, i=1,2,3,4, f̅j=fj+3/c2, j=1,2,3, F̅2=F2/c2.The frequenciesΩi  (i=1,2,3,4) of the forces in (2.7) are rationally related, which have the following relations Z1ϕ=Ω1t, Z2ϕ=Ω2t, ϕ=Ω3t, Z4ϕ=Ω4t, where Z1, Z2, and Z4 are all nonnegative integers. Next, we add small parameter ε on the four terms aforementioned and the parameters of the damping and the forces in (2.7). Therefore, the following system can be obtained as follows:(2.8)u̇1=u2,u̇2=-ω12u1-α1u1v12-α3u13-εα2u12v1-εα4v13-εc1u2+εF1cos⁡ϕ-ε(f1cos⁡Z1ϕ+f2cos⁡Z2ϕ+f3cos⁡Z4ϕ)u1,v̇1=εc2v1+εc2v2,v̇2=-ω̅22v1-β̅1u12v1-β̅3v13-εβ̅2u1v12-εβ̅4u13-2εc2v2+εF̅2cos⁡ϕ-ε(f̅1cos⁡Z1ϕ+f̅2cos⁡Z2ϕ+f̅3cos⁡Z4ϕ)v1.Consider the cross-section of the phase space [9](2.9)Σϕ0={(u1,u2,v1,v2,ϕ)∣ϕ=ϕ0}.
In the following analysis, we consider the system on the cross-section Σϕ0 firstly. Then, let ϕ vary throughout the circle S1, see [9] for details.Letε=0 in (2.8); the new unperturbed system is of the form(2.10)u̇1=u2,u̇2=-ω12u1-α1u1v12-α3u13,v̇1=0,v̇2=-ω̅22v1-β̅1u12v1-β̅3v13.
The Hamiltonian for space of u=(u1,u2) is of the form(2.11)H=12ω12u12+12α1u12v12+14α3u14-12u22.It is noticed that system (2.10) is an uncoupled two-degree-of-freedom nonlinear system. The v1 variable appears in (u1,u2) components of system (2.10) as a parameter since v̇1=0. We set v1=ω̅22/β̅3. Consider the first two decoupled equations(2.12)u̇1=u2,u̇2=-ω12u1-α1u1v12-α3u13.LetR=-ω12-α1v12, and consider R>0, α3>0. Therefore, (2.12) has three singular points: one is (u1,u2)=(0,0), it is a saddle point, and the other two are centers: (u1,u2)=(±R/α3,0); therefore, (2.12) can exhibit homoclinic bifurcations. The homoclinic orbits which connect the saddle point (u10,u20)=(0,0) are obtained as(2.13)u1=±2Rα3sechRt,u2=±2α3RsechRttanhRt.It is known from foregoing analysis that the conditionR=-ω12-α1v12>0 holds; in conclusion, the system considered here satisfies the following conditions:(2.14)β̅3ω12<α1ω̅22<0,α3>0.It is known from analysis in paper [9] that the first pulse Melnikov function is given by the integral(2.15)M(v0,ϕ0,μ)=∫-∞+∞〈n(ph(t)),g(ph(t),ωt+ϕ0,μ,0)〉dt.
It can be seen easily that (2.8) can be written as the following form:(2.16)u̇=JDuH(u,v1)+εgu(u,v,ϕ0,μ,ε),v̇1=εgv1(u,v,ϕ0,μ,ε),v̇2=Ω(u,v1)+εgv2(u,v,ϕ0,μ,ε),
where u=(u1,u2)∈R2, v=(v1,v2)∈R2, μ∈Rp denotes the parameters of the system, and  J is a symplectic matrix(2.17)J=(01-10).
Then, the vectors n and g in (2.15) are of the following form:(2.18)n=(DuH(u,v1),Dv1H(u,v1)-Dv1H(u0(v1),v1),0),g=(gu(u,v,ωt,μ,0),gv1(u,v,ωt,μ,0),gv2(u,v,ωt,μ,0)),ph(t)=(uh(t,v1),v1,v2h(t,v1)+v20).Thek-pulse Melnikov function Mk(k=1,2,…) is defined as(2.19)Mk(v0,ϕ0,μ)=∑j=0k-1M(v10,v20+jΔv2(v10),ϕ0,μ),
where (2.20)Δv2(v10)=∫-∞+∞Ω(uh(τ,v1),v10)dτ.Based on (2.15), the 1-pulse Melnikov function is computed as follows:(2.21)M=∫-∞+∞u2[F1cos⁡ϕ-α2u12v1-α4v13-c1u2-(f1cos⁡Z1ϕ+f2cos⁡Z2ϕ+f3cos⁡Z4ϕ)u1]dt-c2(v1+v2)(-ω̅22v1-β̅1u12v1-β̅3v13)dt=∑i=1,2,4fiΩi2πα3sin(Ziϕ0)cschΩiπ2R-4c1R3/23α3-F1π2α3sin⁡ϕ0sech⁡π2R-c2Δv2(v10+v20),
where (2.22)Δv2=∫-∞+∞(-ω̅22v1-β̅1u12v1-β̅3v13)dt=-α1α3ω̅22R-β̅3.Based on (2.19) and the foregoing analysis, the k-pulse Melnikov function is obtained as follows:(2.23)Mk=∑i=1,2,4fiΩi2πα3ksin(Ziϕ0)cschΩiπ2R-4c1R3/23α3k-F1πk2α3sin⁡ϕ0sech⁡π2R-c2Δv2v10k-c2Δv2v20k-k(k-1)2c2Δv22.
If the k-pulse Melnikov function Mk has simple zero points, then we obtain(2.24)∑i=1,2,4fiΩi2πα3sin(Ziϕ0)cschΩiπ2R-4c1R3/23α3-F1π2α3sinϕ0sechπ2R-c2Δv2v10-c2Δv2v20-(k-1)2c2Δv22=0.
It is known that the following holds:(2.25)Dv10Mk=-c2Δv2k=c2kα1α3ω̅22R-β̅3≠0.Equation (2.24) can be rewritten as follows:(2.26)k=∑i=1,2,42fiΩi2πα3c2Δv22sin(Ziϕ0)cschΩiπ2R-8c1R3/23α3c2Δv22-2F1πc2Δv222α3sinϕ0sechπ2R-2c2(v10+v20)Δv2+1.
For k∈N, we may choose proper parameters in (2.26) so that the value of the following formula: (2.27)∑i=1,2,42fiΩi2πα3c2Δv22sin(Ziϕ0)cschΩiπ2R-8c1R3/23α3c2Δv22-2F1πc2Δv222α3sinϕ0sechπ2R-2c2(v10+v20)Δv2
is a nonnegative integer. Based on analysis in [9], the stable manifold Ws(Mεϕ0) and unstable manifold Wu(Mεϕ0) of system (2.8) intersect transversely, which means that the multipulse chaotic motions exist in the simply supported laminated composite piezoelectric rectangular thin plate. To verify the analytical results obtained above, numerical method will be utilized to indicate that the multipulse chaotic motions occur for the laminated composite piezoelectric plate.


## 3. Numerical Simulations

In this section, the nonlinear dynamic behaviors of the simply supported laminated composite piezoelectric rectangular thin plate under combined parametric and transverse excitations are presented by using Matlab programs.We choose parameters of (2.3) under condition (2.14) to do numerical simulations. The parameters are chosen as follows: ω1=5.6,  α1=2.8, α2=-2.5, α3=10.2, α4=1.8, μ1=0.18, f1=10, f2=12, f3=20, F1=10, Ω1=1, Ω2=2, Ω4=1.2, ω2=6.1,  β1=2.2, β2=8.3, β3=-3, β4=2.4, μ2=0.19, f4=12, f5=11, f6=28, F2=2, Ω3=ϕ=1. The initial values are given by Matlab stochastically as follows: x10=-0.1465, x20=0.1909, x30=0.1892, x40=-0.0376. Figure 2 demonstrates chaotic motions of the laminated composite piezoelectric plate. Figures 2(a) and 2(c) present the waveform on the planes (t,x1) and (t,x3). Figures 2(b) and 2(d) give the phase portraits on the planes (x1,x2) and (x3,x4). Multipulse jumping motions of system (2.3) can be obtained apparently from phrase portraits of Figures 2(e) and 2(f) in three-dimensional spaces (x1,x2,x3) and (x3,x4,x1).Chaotic motions of the laminated composite piezoelectric plate.
(a)(b)(c)(d)(e)(f)The largest Lyapunov exponents are calculated for the laminated composite piezoelectric plate. Figure3 represents the largest Lyapunov exponents of (2.3) with variations of parametric excitation f3 which dependent on piezoelectric excitation.  Figure 4 is the largest Lyapunov exponents via frequency Ω3=ϕ of external force.Figure 3
The largest Lyapunov exponents viaf3.Figure 4
The largest Lyapunov exponents viaϕ.


## 4. Conclusions

New and valuable results of analysis and computation have been achieved during the course of present study. According to our analysis, the extended Melnikov method is attributed to the nonautonomous ordinary differential equations of motion for the laminated composite piezoelectric rectangular plate by introducing the cross-sectionΣϕ0.We minimize the simplification processes on the system at the best possibility so that a better understanding of the nature and behavior of high-dimensional nonlinear systems can be acquired. By virtue of the theory of normal form, some nonlinear terms in the governing equations of the laminated composite piezoelectric rectangular plate have less effect than other terms on the system. Therefore, these nonlinear terms are retained herein and added with small positive parameters to analyze the complex nonlinear dynamics of the laminated composite piezoelectric rectangular plate. We obtain the simple zero point of thek-pulse Melnikov function, which means that there exist k-pulse chaotic motions for the laminated composite piezoelectric rectangular plate. The chaotic dynamics are also numerically investigated by means of the phase portraits and the largest Lyapunov exponents.



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