Multi-pulse orbits dynamics of composite laminated piezoelectric rectangular plate

Abstract

The multi-pulse orbits and chaotic dynamics of a simply supported laminated composite piezoelectric rectangular plate under combined parametric excitation and transverse excitation are studied in detail. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded. The nonlinear equations of motion for the laminated composite piezoelectric rectangular plate are derived from von Karman-type equation and third-order shear deformation plate theory of Reddy. The two-degree-of-freedom dimensionless equations of motion are obtained by using the Galerkin approach to the partial differential governing equation of motion for the laminated composite piezoelectric rectangular plate. The four-dimensional averaged equation in the case of primary parametric resonance and 1:3 internal resonances is obtained by using the method of multiple scales. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on the normal form obtained, the energy phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the laminated composite piezoelectric rectangular plate. The analysis of the global dynamics indicates that there exist multi-pulse jumping orbits in the perturbed phase space of the averaged equation. Based on the averaged equation obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the laminated composite piezoelectric rectangular plate are also found by numerical simulation. The results obtained above mean the existence of the chaos in the Smale horseshoe sense for the simply supported laminated composite piezoelectric rectangular plate. laminated composite piezoelectric rectangular plates, multi-pulse orbit, chaotic dynamics, energy phase method Citation:

Nonlinear stability analysis of a composite laminated piezoelectric rectangular plate with multi-parametric and external excitations

“…The Fig. 1 The model of a laminated composite piezoelectric rectangular plate dynamic electrical loading is expressed as E z = E z cos 4 t. The nonlinear governing equations of motion for the composite laminated piezoelectric rectangular plate in a dimensionless form can be written as follows [16]:…”

“…In the previous work [16], The Shilnikov type multi-pulse orbits and chaotic dynamics of a four-edge simply supported laminated composite piezoelectric rectangular plate subjected to the in-plane, transverse and piezoelectric excitations are investigated by using the energy phase method. The four-dimensional averaged equation in the case of primary parametric resonance and 1:3 internal resonances is obtained by using the method of multiple scales.…”

“…Yeh et al [15] presents a new approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection rectangular plate of thermo-mechanical coupling by utilizing the criteria of the fractal dimension and the maximum Lyapunov exponent. Yao et al [16] investigated the Shilnikov type multi-pulse orbits and chaotic dynamics of a four-edge simply supported laminated composite piezoelectric rectangular plate subjected to the in-plane, transverse and piezoelectric excitations by using the energy phase method. The method of multiple scales is utilized to obtain the four-dimensional averaged equation in the case of primary parametric resonance and 1:3 internal resonances Zhang et al [17] analyzed the Shilnikov type multi-pulse chaotic dynamics of a six-dimensional nonlinear system, which represents the averaged equation of a composite laminated piezoelectric rectangular plate subjected to the transverse, in-plane excitations and the excitation loaded by piezoelectric layers.…”

Multipulse Heteroclinic Orbits and Chaotic Dynamics of the Laminated Composite Piezoelectric Rectangular Plate

“…According to Hamilton's principle, the nonlinear governing equations of motion for the composite laminated piezoelectric rectangular plate are given in the previou studies as follows [10,35]:…”

“…McDonald and Namachchivaya [33] applied the Melnikov method, the Shilnikov method, and the energy-phase method to detect the presence of chaotic dynamics of parametrically excited pipes conveying fluid near a zero-to-one resonance. Yao and Zhang [34,35] utilized the energy-phase method to analyze the Shilnikov type multipulse heteroclinic or homoclinic orbits and chaotic dynamics in a parametrically and externally excited rectangular thin plate and a laminated composite piezoelectric rectangular plate. Yu and Chen [36,37] made use of the energy-phase method to examine the Shilnikov type multipulse homoclinic orbits of a nonlinear cyclic system and a harmonically excited circular plate.…”