Extensions of the Ritz-Galerkin Method for the Forced, Damped Vibrations of Structural Elements,

The classical method for analyzing the forced vibrations of structural elements such as beams, plates and shells is to express the displacements as superpositions of the responses of the free vibration modes. This is only possible for those relatively few problems where exact eigenfunction solutions exist, and often only with considerable difficulty. Ritz-Galerkin methods are widely used to obtain approximate solutions for free undamped, vibration problems. The present paper demonstrates how these same methods may be used straightforwardly to analyze forced vibrations with damping. This is done directly without requiring the free vibration eigenfunctions. Two types of damping--viscous and material hysteretic are considered. Both distributed and concentrated exciting forces are treated. Numerical results are obtained for cantilevered beams and rectangular plates. Studies showing the rates of convergence of the solutions are made. In the case of the cantilever beam, approximate solutions from the present methods are compared with the exact solutions.