Multi-scale analysis and Galerkin verification for dynamic stability of axially translating viscoelastic Timoshenko beams

Abstract

Axially moving structures are applied extensively in many engineering equipments. In this paper, the parametric stability of an axially accelerating viscoelastic Timoshenko beam is analytically and numerically investigated. On account of the axial tension fluctuation, the relationship between the time dependent tension and the axial speed is introduced emphatically. The axial tension of the system is assumed as a harmonic variation over a constant initial tension. On the basis of the generalized Hamilton principle, a novel coupled dynamic model with the linear partial-differential equations and the corresponding boundary conditions are established. The material time derivative is employed to reveal the viscoelastic characteristic by the Kelvin-Voigt energy dissipation mechanism. The method of multiple scale is applied to analyze the governing equations. The instability boundaries of the moving beam are obtained according to the solvability condition and the Routh-Hurwitz criterion. The display expression of the instability boundary is given. The effects of some system parameters on the resonance instability region of the first two harmonic parameters are displayed. The stability of The Timoshenko Beam is first compared by two different methods. The dependence of the stability on the truncation order of Galerkin method is highlighted.