Nonlinear transversal vibration and stability of an axially moving viscoelastic string supported by a partial viscoelastic guide

In this paper, transversal nonlinear vibration of an axially moving viscoelastic string supported by a partial viscoelastic guide is analytically investigated. The string is traveling under time-variant velocity, which includes a mean velocity along with small harmonic fluctuations. The model of the viscoelastic guide is also a parallel combination of springs and viscous dampers. The governing partial-differential equation is derived from Hamilton's principle and geometrical relations. The method of multiple scales is applied to the governing partial-differential equation to obtain solvability conditions for both non-resonance and principal parametric resonance cases. Additionally, in the case of principal parametric resonance, the stability and bifurcation of trivial and non-trivial steady-state responses are analyzed through the Routh–Hurwitz criterion. Eventually, numerical simulations are presented to highlight the effects of mean velocity, guide length, stiffness and damping coefficient of the guide and viscosity coefficient of the string on the natural frequencies, stability, frequency-response curves and bifurcation points of the system.