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.
Piezoelectric materials can be used as the actuators and sensors in engineering structures [
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 [
However, in paper [
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 (
The model of plate subjected to its plane and transverse excitations.
We introduce the following transformations on (
Then, the following equivalent form of (
Comparing (
We consider the unperturbed system (
The frequencies
Consider the cross-section of the phase space [
Let
It is noticed that system (
Let
It is known from foregoing analysis that the condition
It is known from analysis in paper [
The
Based on (
Based on (
Equation (
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 (
Chaotic motions of the laminated composite piezoelectric plate.
The largest Lyapunov exponents are calculated for the laminated composite piezoelectric plate. Figure
The largest Lyapunov exponents via
The largest Lyapunov exponents via
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
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
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).