Multipulse Chaotic Dynamics for a Laminated Composite Piezoelectric Plate

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.

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]:𝑤̈₁+𝜇₁𝑤̇₁+𝜔²₁𝑤₁+𝑓₁cosΩ₁𝑡+𝑓₂cosΩ₂𝑡+𝑓₃cosΩ₄𝑡𝑤₁+𝛼₁𝑤₁𝑤²₂+𝛼₂𝑤²₁𝑤₂+𝛼₃𝑤³₁+𝛼₄𝑤³₂=𝐹₁cosΩ₃𝑤̈𝑡,₂+𝜇₂𝑤̇₂+𝜔²₂𝑤₂+𝑓₄cosΩ₁𝑡+𝑓₅cosΩ₂𝑡+𝑓₆cosΩ₄𝑡𝑤₂+𝛽₁𝑤²₁𝑤₂+𝛽₂𝑤₁𝑤²₂+𝛽₃𝑤³₂+𝛽₄𝑤³₁=𝐹₂cosΩ₃𝑡, (2.1) where 𝛼_𝑖 and 𝛽_𝑖(𝑖=1-4) are nondimensional coefficients. In (2.1), the coefficients for 𝜔₁, 𝜔₂, 𝐹₁, 𝐹₂, 𝛼_𝑖, 𝛽_𝑖(𝑖=1-4), and 𝑓_𝑗(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):𝑥₁=𝑤₁,𝑥₂=𝑤̇₁,𝑥₃=𝑤₂,𝑥₄=𝑤̇₂. (2.2)

Then, the following equivalent form of (2.1) is obtained:𝑥̇₁=𝑥₂,𝑥̇₂=−𝜔²₁𝑥₁−𝛼₁𝑥₁𝑥²₃−𝛼₂𝑥²₁𝑥₃−𝛼₃𝑥³₁−𝛼₄𝑥³₃−𝑐₁𝑥₂+𝐹₁cosΩ₃𝑡−𝑓₁cosΩ₁𝑡+𝑓₂cosΩ₂𝑡+𝑓₃cosΩ₄𝑡𝑥₁,𝑥̇₃=𝑥₄,𝑥̇₄=−𝜔²₂𝑥₃−𝛽₁𝑥²₁𝑥₃−𝛽₂𝑥₁𝑥²₃−𝛽₃𝑥³₃−𝛽₄𝑥³₁−𝑐₂𝑥₄+𝐹₂cosΩ₃𝑡−𝑓₄cosΩ₁𝑡+𝑓₅cosΩ₂𝑡+𝑓₆cosΩ₄𝑡𝑥₃. (2.3) The parameters of the damping and the forces in (2.3) are considered as perturbation parameters the unperturbed system is as follows:𝑥̇₁=𝑥₂,𝑥̇₂=−𝜔²₁𝑥₁−𝛼₁𝑥₁𝑥²₃−𝛼₂𝑥²₁𝑥₃−𝛼₃𝑥³₁−𝛼₄𝑥³₃,𝑥̇₃=𝑥₄,𝑥̇₄=−𝜔²₂𝑥₃−𝛽₁𝑥²₁𝑥₃−𝛽₂𝑥₁𝑥²₃−𝛽₃𝑥³₃−𝛽₄𝑥³₁. (2.4) 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:𝑥̇₁=3𝜔⁴₁𝛼₃𝑚₁𝑥³₂+3𝜔²₁𝛼₃𝑚₁𝑥²₁𝑥₂+𝜔²₁𝛼₁𝑛𝑥₂𝑥²₃+𝜔²₁𝜔²₂𝛼₁𝑛𝑥₂𝑥²₄,𝑥̇₂=−3𝛼₃𝑚₁𝑥³₁−3𝜔²₁𝛼₃𝑚₁𝑥₁𝑥²₂−𝛼₁𝑛𝑥₁𝑥²₃−𝜔²₂𝛼₁𝑛𝑥₁𝑥²₄,𝑥̇₃=3𝜔⁴₂𝛽₃𝑚₂𝑥³₄+𝜔²₂𝛽₁𝑛𝑥²₁𝑥₄+𝜔²₁𝜔²₂𝛽₁𝑛𝑥₂𝑥²₄+3𝜔²₂𝛽₃𝑚₂𝑥₄𝑥²₃,𝑥̇₄=−3𝛽₃𝑚₂𝑥³₃−𝛽₁𝑛𝑥²₁𝑥₃−𝜔²₁𝛽₁𝑛𝑥₃𝑥²₂−3𝜔²₂𝛽₃𝑚₂𝑥₃𝑥²₄, (2.5) where 𝑚₁=1/(3+2𝜔⁴₁+3𝜔⁸₁), 𝑚₂=1/(3+2𝜔⁴₂+3𝜔⁸₂), 𝑛=1/(1+𝜔⁴₁+𝜔⁴₁𝜔⁴₂+𝜔⁸₂).

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 𝛼₂𝑥²₁𝑥₃, 𝛼₄𝑥³₃, 𝛽₂𝑥₁𝑥²₃, and 𝛽₄𝑥³₁ 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):𝑥₁=𝑢₁,𝑥₂=𝑢₂,𝑥₃=𝑣₁,𝑥₄=𝑐₂𝑣₁+𝑐₂𝑣₂. (2.6) Then, the following equivalent form of (2.3) is obtained:𝑢̇₁=𝑢₂,𝑢̇₂=−𝜔²₁𝑢₁−𝛼₁𝑢₁𝑣²₁−𝛼₃𝑢³₁−𝛼₂𝑢²₁𝑣₁−𝛼₄𝑣³₁−𝑐₁𝑢₂+𝐹₁cosΩ₃𝑡−𝑓₁cosΩ₁𝑡+𝑓₂cosΩ₂𝑡+𝑓₃cosΩ₄𝑡𝑢₁,𝑣̇₁=𝑐₂𝑣₁+𝑐₂𝑣₂,𝑣̇₂=−𝜔²₂𝑣₁−𝛽₁𝑢²₁𝑣₁−𝛽₃𝑣³₁−𝛽₂𝑢₁𝑣²₁−𝛽₄𝑢³₁−2𝑐₂𝑣₂+𝐹₂cosΩ₃𝑡−𝑓₁cosΩ₁𝑡+𝑓₂cosΩ₂𝑡+𝑓₃cosΩ₄𝑡𝑣₁, (2.7) where 𝜔²₂=2𝑐₂+𝜔²₂/𝑐₂, 𝛽_𝑖=𝛽_𝑖/𝑐₂, 𝑖=1,2,3,4, 𝑓_𝑗=𝑓_𝑗+3/𝑐₂, 𝑗=1,2,3, 𝐹₂=𝐹₂/𝑐₂.

The frequencies Ω_𝑖  (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:𝑢̇₁=𝑢₂,𝑢̇₂=−𝜔²₁𝑢₁−𝛼₁𝑢₁𝑣²₁−𝛼₃𝑢³₁−𝜀𝛼₂𝑢²₁𝑣₁−𝜀𝛼₄𝑣³₁−𝜀𝑐₁𝑢₂+𝜀𝐹₁𝑓cos𝜙−𝜀₁cos𝑍₁𝜙+𝑓₂cos𝑍₂𝜙+𝑓₃cos𝑍₄𝜙𝑢₁,𝑣̇₁=𝜀𝑐₂𝑣₁+𝜀𝑐₂𝑣₂,𝑣̇₂=−𝜔²₂𝑣₁−𝛽₁𝑢²₁𝑣₁−𝛽₃𝑣³₁−𝜀𝛽₂𝑢₁𝑣²₁−𝜀𝛽₄𝑢³₁−2𝜀𝑐₂𝑣₂+𝜀𝐹₂cos𝜙−𝜀𝑓₁cos𝑍₁𝜙+𝑓₂cos𝑍₂𝜙+𝑓₃cos𝑍₄𝜙𝑣₁. (2.8)

Consider the cross-section of the phase space [9]Σ^𝜙0=𝑢₁,𝑢₂,𝑣₁,𝑣₂,𝜙∣𝜙=𝜙₀. (2.9) In the following analysis, we consider the system on the cross-section Σ^𝜙0 firstly. Then, let 𝜙 vary throughout the circle 𝐒¹, see [9] for details.

Let 𝜀=0 in (2.8); the new unperturbed system is of the form𝑢̇₁=𝑢₂,𝑢̇₂=−𝜔²₁𝑢₁−𝛼₁𝑢₁𝑣²₁−𝛼₃𝑢³₁,𝑣̇₁𝑣̇=0,₂=−𝜔²₂𝑣₁−𝛽₁𝑢²₁𝑣₁−𝛽₃𝑣³₁. (2.10) The Hamiltonian for space of 𝑢=(𝑢₁,𝑢₂) is of the form1𝐻=2𝜔²₁𝑢²₁+12𝛼₁𝑢²₁𝑣²₁+14𝛼₃𝑢⁴₁−12𝑢²₂. (2.11)

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=𝜔²₂/𝛽₃. Consider the first two decoupled equations𝑢̇₁=𝑢₂,𝑢̇₂=−𝜔²₁𝑢₁−𝛼₁𝑢₁𝑣²₁−𝛼₃𝑢³₁. (2.12)

Let 𝑅=−𝜔²₁−𝛼₁𝑣²₁, 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 as𝑢₁=±2𝑅𝛼₃sech√𝑢𝑅𝑡,₂=±2𝛼₃𝑅sech√𝑅𝑡tanh√𝑅𝑡. (2.13)

It is known from foregoing analysis that the condition 𝑅=−𝜔²₁−𝛼₁𝑣²₁>0 holds; in conclusion, the system considered here satisfies the following conditions:𝛽₃𝜔²₁<𝛼₁𝜔²₂<0,𝛼₃>0. (2.14)

It is known from analysis in paper [9] that the first pulse Melnikov function is given by the integral𝑀𝑣₀,𝜙₀,𝜇=^+∞_−∞𝐧𝑝^ℎ(𝑡),𝐠𝑝^ℎ(𝑡),𝜔𝑡+𝜙₀,𝜇,0𝑑𝑡. (2.15) It can be seen easily that (2.8) can be written as the following form:𝑢̇=𝐽𝐷_𝑢𝐻𝑢,𝑣₁+𝜀𝑔^𝑢𝑢,𝑣,𝜙₀,𝑣̇,𝜇,𝜀₁=𝜀𝑔^𝑣1𝑢,𝑣,𝜙₀,𝑣̇,𝜇,𝜀₂=Ω𝑢,𝑣₁+𝜀𝑔^𝑣2𝑢,𝑣,𝜙₀,𝜇,𝜀, (2.16) where 𝑢=(𝑢₁,𝑢₂)∈𝐑², 𝑣=(𝑣₁,𝑣₂)∈𝐑², 𝜇∈𝐑^𝑝 denotes the parameters of the system, and 𝐽 is a symplectic matrix𝐽=01−10. (2.17) Then, the vectors 𝐧 and 𝐠 in (2.15) are of the following form:𝐧=𝐷_𝑢𝐻𝑢,𝑣₁,𝐷_𝑣1𝐻𝑢,𝑣₁−𝐷_𝑣1𝐻𝑢₀𝑣₁,𝑣₁,𝑔,0𝐠=^𝑢𝑢,𝑣,𝜔𝑡,𝜇,0,𝑔^𝑣1𝑢,𝑣,𝜔𝑡,𝜇,0,𝑔^𝑣2,𝑝𝑢,𝑣,𝜔𝑡,𝜇,0_ℎ(𝑡)=𝑢^ℎ𝑡,𝑣₁,𝑣₁,𝑣^ℎ₂𝑡,𝑣₁+𝑣₂₀. (2.18)

The 𝑘-pulse Melnikov function 𝑀_𝑘(𝑘=1,2,…) is defined as𝑀_𝑘𝑣₀,𝜙₀,𝜇=^𝑘−1_𝑗=0𝑀𝑣₁₀,𝑣₂₀+𝑗Δ𝑣₂𝑣₁₀,𝜙₀,𝜇, (2.19) where Δ𝑣₂𝑣₁₀=^+∞_−∞Ω𝑢^ℎ𝜏,𝑣₁,𝑣₁₀𝑑𝜏. (2.20)

Based on (2.15), the 1-pulse Melnikov function is computed as follows:𝑀=^+∞_−∞𝑢₂𝐹₁cos𝜙−𝛼₂𝑢²₁𝑣₁−𝛼₄𝑣³₁−𝑐₁𝑢₂−𝑓₁cos𝑍₁𝜙+𝑓₂cos𝑍₂𝜙+𝑓₃cos𝑍₄𝜙𝑢₁𝑑𝑡−𝑐₂𝑣₁+𝑣₂−𝜔²₂𝑣₁−𝛽₁𝑢²₁𝑣₁−𝛽₃𝑣³₁=𝑑𝑡_𝑖=1,2,4𝑓_𝑖Ω²_𝑖𝜋𝛼₃sin𝑍_𝑖𝜙₀Ωcsch^𝑖𝜋2√𝑅−4𝑐₁𝑅^3/23𝛼₃−𝐹₁𝜋2𝛼₃sin𝜙₀𝜋sech2√𝑅−𝑐₂Δ𝑣₂𝑣₁₀+𝑣₂₀, (2.21) where Δ𝑣₂=^+∞_−∞−𝜔²₂𝑣₁−𝛽₁𝑢²₁𝑣₁−𝛽₃𝑣³₁𝛼𝑑𝑡=−¹𝛼₃𝜔²₂𝑅−𝛽₃. (2.22)

Based on (2.19) and the foregoing analysis, the 𝑘-pulse Melnikov function is obtained as follows:𝑀_𝑘=_𝑖=1,2,4𝑓_𝑖Ω²_𝑖𝜋𝛼₃𝑘sin𝑍_𝑖𝜙₀Ωcsch^𝑖𝜋2√𝑅−4𝑐₁𝑅^3/23𝛼₃𝑘−𝐹₁𝜋𝑘2𝛼₃sin𝜙₀𝜋sech2√𝑅−𝑐₂Δ𝑣₂𝑣₁₀𝑘−𝑐₂Δ𝑣₂𝑣₂₀𝑘()𝑘−𝑘−12𝑐₂Δ𝑣²₂. (2.23) If the 𝑘-pulse Melnikov function 𝑀_𝑘 has simple zero points, then we obtain_𝑖=1,2,4𝑓_𝑖Ω²_𝑖𝜋𝛼₃sin𝑍_𝑖𝜙₀Ωcsch^𝑖𝜋2√𝑅−4𝑐₁𝑅^3/23𝛼₃−𝐹₁𝜋2𝛼₃sin𝜙₀𝜋sech2√𝑅−𝑐₂Δ𝑣₂𝑣₁₀−𝑐₂Δ𝑣₂𝑣₂₀−(𝑘−1)2𝑐₂Δ𝑣²₂=0. (2.24) It is known that the following holds:𝐷_𝑣10𝑀_𝑘=−𝑐₂Δ𝑣₂𝑐𝑘=²𝑘𝛼₁𝛼₃𝜔²₂𝑅−𝛽₃≠0. (2.25)

Equation (2.24) can be rewritten as follows:𝑘=_𝑖=1,2,42𝑓_𝑖Ω²_𝑖𝜋𝛼₃𝑐₂Δ𝑣²₂sin𝑍_𝑖𝜙₀Ωcsch^𝑖𝜋2√𝑅−8𝑐₁𝑅^3/23𝛼₃𝑐₂Δ𝑣²₂−2𝐹₁𝜋𝑐₂Δ𝑣²₂2𝛼₃sin𝜙₀𝜋sech2√𝑅−2𝑐₂𝑣₁₀+𝑣₂₀Δ𝑣₂+1. (2.26) For 𝑘∈𝐍, we may choose proper parameters in (2.26) so that the value of the following formula: _𝑖=1,2,42𝑓_𝑖Ω²_𝑖𝜋𝛼₃𝑐₂Δ𝑣²₂sin𝑍_𝑖𝜙₀Ωcsch^𝑖𝜋2√𝑅−8𝑐₁𝑅^3/23𝛼₃𝑐₂Δ𝑣²₂−2𝐹₁𝜋𝑐₂Δ𝑣²₂2𝛼₃sin𝜙₀𝜋sech2√𝑅−2𝑐₂𝑣₁₀+𝑣₂₀Δ𝑣₂ (2.27) is a nonnegative integer. Based on analysis in [9], the stable manifold 𝑊^𝑠(𝑀^𝜙0_𝜀) and unstable manifold 𝑊^𝑢(𝑀^𝜙0_𝜀) 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).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

Chaotic motions of the laminated composite piezoelectric plate.

The largest Lyapunov exponents are calculated for the laminated composite piezoelectric plate. Figure 3 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 via 𝑓3.

Figure 4 

The largest Lyapunov exponents via 𝜙.

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-section Σ^𝜙0.

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 the 𝑘-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.

Acknowledgments

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant no. 10732020, 11072008, and 11002005 and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).