Dynamic analysis of an axially translating plate with time-variant length

The linear dynamic analysis of a translating cantilever plate model characterized by time-variant length and axial velocity is investigated. A length-dependent governing partial differential equation (PDE) of motion is formulated by the extended Hamilton's principle based on Kirchhoff-Love plate theory. The tension in the system arising from the longitudinal accelerations and in-plane stresses are incorporated. Further, the extended Galerkin method along with the Newmark direct time integration scheme is employed to simulate the response of the system. Stability and vibration characteristics are studied according to the quadratic eigenvalue problem from the governing PDE, which demonstrates that the coupling effects between the axial translation motion and the flexural deformation stabilizes the system during the extension and destabilizes it in the retraction. The computation results show that the translating velocity and the aspect ratio affect the natural frequencies and stability for the out-of-plane vibration of the moving plate. A domain for the traveling velocity associated with the stable system is given, and the critical failure velocity is also predicted based on numerical simulations.