Closed form exact solutions of viscously damped free and forced vibrations of rectangular membranes

In the present study, viscously damped free and forced vibrations of a rectangular membrane are investigated using a closed form exact method. The method is exact and yields closed form expressions for the vibratory displacements. This is in contrast with the well known eigenfunction superposition method which requires expressing the distributed forcing functions and the displacement response functions as infinite sums of free vibration eigenfunctions. Instead of undamped natural frequencies which are typically computed and applied in the free and forced vibrations, damped natural frequencies are done. The damped natural frequency equation and the critical viscous damping equation are exactly derived. In the damped free vibration, effects of viscous damping on natural frequencies and mode shapes are studied. Some contour plots of damped mode shapes are given. Accurate displacement amplitude vs. forcing frequency curves showing the forced response due to distributed loading are displayed.