Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model

The weakly forced vibration of an axially moving viscoelastic beam is investigated. The viscoelastic material of the beam is constituted by the standard linear solid model with the material time derivative involved. The nonlinear equations governing the transverse vibration are derived from the dynamical, constitutive, and geometrical relations. The method of multiple scales is used to determine the steady-state response. The modulation equation is derived from the solvability condition of eliminating secular terms. Closed-form expressions of the amplitude and existence condition of nontrivial steady-state response are derived from the modulation equation. The stability of nontrivial steady-state response is examined via the Routh-Hurwitz criterion.