Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed

Stability in transverse parametric vibration of axially accelerating viscoelastic beams is investigated. The governing equation is derived from Newton's second law, the Kelvin constitution relation, and the geometrical relation. When the axial speed is a constant mean speed with small harmonic variations, the governing equation can be regarded as a continuous gyroscopic system under small periodically parametric excitations and a damping term. The method of multiple scales is applied directly to the governing equation without discretization. The stability conditions are obtained for combination and principal parametric resonance. Numerical examples are presented for beams with simple supports and fixed supports, respectively, to demonstrate the effect of viscoelasticity on the stability boundaries in both cases.