Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law

This paper deals with the transverse vibration of an initially stressed moving viscoelastic string obeying a fractional differentiation constitutive law. The governing equation is derived from Newtonian second law of motion, and reduced to a set of non-linear differential–integral equations based on Galerkinʹs truncation. A numerical approach is proposed to solve numerically the differential–integral equation through developing an approximate expression of the fractional derivatives involved. Some numerical examples are presented to highlight the effects of viscoelastic parameters and frequencies of parametric excitations on the transient responses of the axially moving string.