Relationship between nonlinear free vibration behavior and nonlinear forced vibration behavior of viscoelastic plates

Abstract

In this paper, the relationship between nonlinear free vibration behavior and nonlinear forced vibration behavior of viscoelastic plates is studied based on Reddy higher order shear deformation theory and von Kármán geometric nonlinearity assumption. Kelvin–Voigt viscoelastic model is adopted to describe the viscoelasticity of the plate. According to the geometric equations and constitutive equations, the elastic potential energy, kinetic energy and virtual work done by the external force and viscous damping force are obtained. Furthermore, the nonlinear governing differential equations of the system are achieved by using Hamilton’s principle. On the basis of the harmonic balance method and Runge–Kutta method, the nonlinear free vibration responses and nonlinear forced vibration responses of the system are presented, respectively. The major innovation of the manuscript is that the nonlinear frequency ratio derived from the nonlinear free vibration of the system, is in good agreement with the resonant frequency ratio obtained from the nonlinear forced vibration of the system. Also, the viscous damping coefficient has a little effect on the nonlinear frequency ratio.