Nonlinear Parametric Resonance of a Fractional Damped Axially Moving String

Nonlinear parametric vibration of an axially accelerating viscoelastic string is investigated. The string is constituted by the fractional Kelvin model. The principal parametric resonance is analyzed by using an asymptotic approach. The modulation equation is derived from the solvability condition. Closed-form expressions of the amplitudes and the existence conditions of steady-state responses are obtained from the modulation equation. Numerical examples are presented to highlight the effects of fractional order and other system parameters on the responses.