Free in-plane vibration of an axially moving membrane

The free in-plane vibration of an axially moving membrane is studied considering the effects of both the translating speed and aspect ratio of the membrane. Two sets of boundary conditions, which are free and fixed constraints in the lateral direction at boundaries with mass transport, are discussed in this study. From the extended Hamilton principle, the coupled equations of the longitudinal and lateral motions are derived. These equations are then discretized by using the Galerkin method. From the discretized equations, the natural frequencies and mode shapes are obtained for the variations of the translating speed and aspect ratio. The results show that the translating speed, aspect ratio, and boundary conditions have significant effects on the in-plane vibrations of the moving membrane.