Parametrical Resonance of the Excited Axially Moving String with an Integral Constitutive Law

The two-to-one resonance in a parametrically excited axially moving viscoelastic string is investigated. The material of the viscoelastic string obeys an integral constitutive law. The method of multiple scales is applied directly the nonlinear partial-differential-integral equation that governs the transverse vibration of the string. Solvability conditions, which lead to the differential equations satisfied by the amplitude and the phase angle of the nontrivial response, are derived. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the two-to-one resonance are presented. The effects of the viscoelastic parameter, the amplitude of excitation, the frequency of excitation, and the transport speed on the nontrivial solutions are demonstrated in several numerical examples.