Stability analysis and numerical confirmation in parametric resonance of axially moving viscoelastic plates with time-dependent speed

Abstract

In this paper, stability in parametric resonance of axially moving viscoelastic plates subjected to plane stresses is investigated. The plate material obeys the Kelvin–Voigt model in which the material time derivative is used. The generalized Hamilton principle is employed to obtain the governing equation. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The governing equation can be regarded as a continuous gyroscopic system with small periodically parametric excitations and a damping term. The method of multiple scales is applied to the governing equation to establish the solvability conditions in principal and summation parametric resonances. The natural frequencies and modes of linear generating equation are numerically calculated based on the given boundary conditions. The necessary and sufficient condition of the stability is derived from the Routh–Hurwitz criterion. Some numerical examples are presented to demonstrate the effects of related parameters on the frequencies and the stability boundaries. The differential quadrature scheme is developed to solve numerically the linear generating system and the primitive equation model. The numerical calculations confirm the analytical results.