Parametric resonance of fractional viscoelastic webs under time-dependent tension

Abstract

The focus of this paper is on the parametric resonance of fractional viscoelastic webs under time-dependent tension. The novelty of this work lies in the introduction of a fractional order model for viscoelastic webs and the time-dependent tension in roll-to-roll manufacturing. The time-dependent tension is induced by the web treatment processes in roll-to-roll mass production. The rheological properties of viscoelastic webs are represented by the fractional Kelvin-Voigt relation. Considering the geometric nonlinearity, the governing equation is derived based on the Hamilton principle, and then the resulting equation is reduced to a fractional ordinary differential equation by using the Bubnov-Galerkin method. The multiple-scale method is applied to analyze the occurrence condition and dynamic responses of parametric resonance, and the solution stability is demarcated by the Routh-Hurwitz criterion. The effects of the tension variation coefficient, viscoelastic parameter and initial tension on the resonance responses are discussed. The results reveal that the increase in tension variation coefficient and initial tension not only widen the unstable zone of the response curve, but also enhance the resonance amplitude. Moreover, the resonance amplitude decreases with a larger fractional order. This investigation aims to explore the cause of the instability phenomena and thus avoid the unexpected failure of flexible fabrication.

Keywords

• Parametric resonance
• Fractional viscoelastic webs
• Time-dependent tension