Parametric resonance of an axially accelerating viscoelastic membrane with a fractional model

In this paper, the parametric resonance of an axially accelerating viscoelastic membrane is examined with a fractional model. The membrane is transported with a time-dependent velocity. The viscoelastic model incorporating fractional derivative into the Kelvin-Voigt relation is considered. The governing equation is formulated from Newton's second law and von-Karman theory, and solved via Galerkin truncation and the multiple-scale method. Furthermore, the frequency-responses of the nonlinear resonance are estimated by eliminating the secular terms, and the Routh-Hurwitz criterion is employed to identify the stability and bifurcation of the steady-state responses. Finally, the numerical results are used to analyze the effects of the fractional order, viscoelastic coefficient and time-dependent velocity on the frequency- responses, solution stability and bifurcation points. The novelty of this study is to introduce a fractional viscoelastic model and time-dependent velocity to analyze the resonance stability of axially accelerating viscoelastic membranes in a roll-to-roll system. The results of this investigation are expected to be useful for quantifying the relative influence of the system parameters in the roll-toroll manufacturing process.