Nonlinear and chaotic vibration and stability analysis of an aero-elastic piezoelectric FG plate under parametric and primary excitations

Abstract

In this study, in the presence of supersonic aerodynamic loading, the nonlinear and chaotic vibrations and stability of a simply supported Functionally Graded Piezoelectric (FGP) rectangular plate with bonded piezoelectric layer have been investigated. It is assumed that the plate is simultaneously exposed to the effects of harmonic uniaxial in-plane force and transverse piezoelectric excitations and aerodynamic loading. It is considered that the potential distribution varies linearly through the piezoelectric layer thickness, and the aerodynamic load is modeled by the first order piston theory. The von-Karman nonlinear strain–displacement relations are used to consider the geometrical nonlinearity. Based on the Classical Plate Theory (CPT) and applying the Hamilton׳s principle, the nonlinear coupled partial differential equations of motion are derived. The Galerkin׳s procedure is used to reduce the equations of motion to nonlinear ordinary differential Mathieu equations. The validity of the formulation for analyzing the Limit Cycle Oscillation (LCO), aero-elastic stability boundaries is accomplished by comparing the results with those of the literature, and the convergence study of the FGP plate is performed. By applying the Multiple Scales Method, the case of 1:2 internal resonance and primary parametric resonance are taken into account and the corresponding averaged equations are derived and analyzed numerically. The results are provided to investigate the effects of the forcing/piezoelectric detuning parameter, amplitude of forcing/piezoelectric excitation and dynamic pressure, on the nonlinear dynamics and chaotic behavior of the FGP plate. It is revealed that under the certain conditions, due to the existence of bi-stable region of non-trivial solutions, system shows the hysteretic behavior. Moreover, in absence of airflow, it is observed that variation of control parameters leads to the multi periodic and chaotic motions.