Nonlinear vibration analysis of viscoelastic plates with fractional damping

Abstract

Fractional calculus has the history as long as the classical calculus, but up to recent years, less attention has been paid to this method due to its complexity and difficulties in dealing with fractional derivatives and fractional integrals. In the present study, the nonlinear vibration analysis of a rectangular composite plate with fractional viscoelastic properties has been analytically investigated. The plate is modeled based on the von Kármán plate theory, and the Caputo’s fractional derivative has been utilized to mathematically model the viscoelastic behavior of the plate with higher accuracy. Galerkin procedure has been implemented to derive the time-dependent ordinary differential equations of motion. The first two orthogonal mode shapes are considered, and variational iteration method is applied to analyze the nonlinear free vibration of the system. The method of multiple time-scales is utilized to investigate the nonlinear forced vibration of the plate when subjected to an external harmonic force. Primary resonances of the system have been studied and analytical expressions for the frequency-amplitude relationships of each mode are derived. Parametric studies have been performed to examine the effect of various parameters, specifically the fractional derivative order as a key factor, on the damped free vibration response and the forced vibration frequency response.