Natural frequency and critical speed determination of an axially moving viscoelastic beam

Abstract

In this paper, the natural frequency and critical speed of an axially moving viscoelastic beam with clamped and simple supports are calculated analytically based on the Euler–Bernoulli and Timoshenko theories. The beam is incompressible in bulk and viscoelastic in shear, which obeys the linear standard solid model with material time derivative. The axial speed is characterized by a simple harmonic variation about a constant mean speed. By defining some dimensionless parameters, the governing equations are derived from the Newton’s second law. They contain two coupled partial differential equations with time depended coefficients. The straightforward method in perturbation theory is used to solve these equations. By considering the homogeneous equation, the natural frequencies are calculated. The critical speed is determined by a constant speed assumption. By a parametric study, the effects of mechanical and geometrical parameters on the natural frequency and critical speed are investigated.