Dynamic Stability of Axially Moving Plates With Periodically Varying Speeds Under Partially Distributed Edge Tensions

This paper investigates the dynamic stability of an axially moving web which translates with periodically varying speeds and is subjected to partially distributed tensions on two opposite edges. The web is modeled as a rectangular plate simply supported at two opposite edges where the tension is applied, and free at the other two edges. The plate is assumed to possess internal damping, which obeys the Kelvin-Voigt model. The moving speed of the plate is expressed as the sum of a constant speed and a periodical perturbation with a zero mean. Due to the periodically varying speed of the moving plate, terms with time-dependent coefficients appear in the equations of motion, which may bring about parametric instability under certain situations. First, the in-plane stresses of the plate due to the partially distributed edge tensions is determined exactly by the theory of plane elasticity. Then, the dependence on the spatial coordinates in the equations of motion is eliminated by the Galerkin method, which results in a set of discretized system equations in time. Finally, the method of multiple scales is utilized to solve this set of system equations analytically if the periodical perturbation of the moving speed is much smaller as compared with the average speed of the plate, from which the stability boundaries of the moving plate are obtained. Numerical results reveal that only combination resonances of the sum-type appear between modes having the same symmetry class in the transverse direction. Unstable regions of main resonances are generally larger than those of sum-type resonances.

Equilibrium Displacement and Stress Distribution in a Two-Dimensional, Axially Moving Web Under Transverse Loading

Von Karman nonlinear plate equations are modified to describe the motion of a wide, axially moving web with small flexural stiffness under transverse loading. The model can represent a paper web or plastic sheet under some conditions. Closed-form solutions to two nonlinear, coupled equations governing the transverse displacement and stress function probably do not exist. The transverse forces arising from the bending stiffness are much smaller than those arising from the applied axial tension except near the edges of the web. This opens the possibility that boundary layer and singular perturbation theories can be used to model the bending forces near the edges of the web when determining the equilibrium solution and stress distribution. The present analysis is applied to two examples: (I) a web deflecting under its own uniformly distributed weight; (II) a web deflecting under a transverse load whose distribution is described by the product of sine functions in the axial and width directions. Membrane theory and linear plate theory solutions are used to characterize the importance of the web deformation solutions.