Dynamics of Axially Transporting Viscoelastic Plates with Tension-Dependent Speed

This paper introduces a new model of tension-dependent speed to study the dynamic stability of axially transporting viscoelastic plates. The relationship between tension and speed with periodic variations is considered. By introducing the Kelvin viscoelastic constitutive relationship and applying the energy method and generalized Hamilton principle, the vibration governing equation and matching non-homogeneous boundary conditions are derived. The solvability conditions of the vibration governing equation and the matching system instability boundary conditions through the direct multiscale method are obtained. The complex frequencies under this model are calculated. The effects of some system parameters, including viscoelastic coefficient, internal resonance, non-homogeneous boundary conditions on the stability boundary of the system are investigated. In addition, a comparison is made between the new model and the old model. What is more, the introduction of internal resonance leads to the emergence of new W-shaped unstable boundaries in the system. The reason for this phenomenon is due to energy exchange between modes. Finally, the approximate analytical results are verified by the differential quadrature method. The results indicate that the two outcomes have the same trend of change, but there are slight differences in numerical values.