Response of plates to pulse excitation (Q580012)

The ultraspherical polynomial approximation technique is presented for large-amplitude vibrations of thin plates subjected to step function loading, neglecting the longitudinal and rotatory inertia forces. The equations are solved by a one-term solution in spatial coordinates which satisfies the boundary conditions. By proper time transformation of the Airy stress function, the time can be eliminated and the stress function can be found. Applying the Ritz-Galerkin method to the deflection equation yields an ordinary nonlinear differential equation in time. An additional transformation function is selected for the displacement variable to be the solution of the linear system subjected to the same pulse. This reduces the nonlinear differential equation of motion to a form where \textit{G. L. Anderson}'s [e.g. J. Sound Vib. 32, 101-108 (1974; Zbl 0283.70012)] ultraspherical polynomial approximation (UPA) technique can be applied to obtain the nonlinear response of plates. The nonlinear response predicted for the simply supported square and clamped-in circular plates and the boundaries for the edges considered are immovably constrained and stress-free conditions subjected to the step function excitation \((P_ 0)\) are obtained.