Nonlinear Vibrations of an Axially Moving Beam with Fractional Viscoelastic Damping

The nonlinear vibrations of an axially moving viscoelastic beam under the transverse harmonic excitation are examined. The governing equation of motion of the viscoelastic beam is discretized into a Duffing system with nonlinear fractional derivative using Galerkin’s method. The viscoelasticity of the moving beam is described by the fractional Kelvin–Voigt model based on the Caputo definition. The primary resonance is analytically investigated by the averaging method. With the aid of response curves, a parametric study is conducted to display the influences of the fractional order and the viscosity coefficient on steady-state responses. The validations of this study are given through comparisons between the analytical solutions and numerical ones, where the stability of the solutions is determined by the Routh–Hurwitz criterion. It is found that suppression of undesirable responses can be achieved via changing the viscosity of the system.

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