Dynamic responses of the in-plane and out-of-plane vibrations for an axially moving membrane

The dynamic responses of both the in-plane and out-of-plane vibrations are investigated for an axially moving membrane. Using the extended Hamilton principle, equations of motion are derived for the moving membrane with no-slip boundary conditions at rollers. The equations of motion are discretized by using the Galerkin method and then the generalized-α time integration method is applied to compute the dynamic responses of the membrane. Based on the computed results, the responses are compared between the in-plane and out-of-plane vibrations. In addition, the effects of translating velocity/acceleration, critical speeds and velocity profiles on the dynamic behaviours for displacements and stresses are investigated and discussed.