Primary and Secondary Resonance Analyses of Viscoelastic Nanoplates Based on Strain Gradient Theory

Abstract

In this study, the nonstationary oscillation, secondary resonance and nonlinear dynamic behavior of viscoelastic nanoplates with linear damping are investigated based on the modified strain gradient theory extended for viscoelastic materials. The viscous component of the nonclassical and classical stress tensors are evaluated on the basis of the Leaderman viscoelastic model. Then, incorporating the size-dependent potential energy, kinetic energy and an external excitation force work, the governing equations of the oscillations are obtained based on the Hamilton’s principle. The governing formula is obtained as a nonlinear second-order integro-differential partial equation. This size-dependent viscoelastic formula is solved using analytical Harmonic balance method (HBM) and the fourth-order Runge–Kutta technique after applying the expansion theory. Additionally, the stability of the steady-state response is examined by means of HBM. Then, the secondary resonance conditions due to the super-harmonic motion are determined by performing frequency response, force response, Poincare map and phase portrait analyses. In addition, the nonstationary transient vibration of viscoelastic nanosystem is analyzed by performing Hilbert–Huang transform.