Dynamic Stability of Superelliptical Plates Resting on Elastic Foundations under Periodic In-Plane Loads

The dynamic instability of thin, clamped, flat isotropic elastic panels of superelliptical planform set on a two-parameter Winkler-Pasternak foundation and subjected to uniformly distributed pulsating in-plane loads is theoretically investigated. The plate perimeter is described by a superelliptic function with a power corresponding to shapes ranging from a circle (ellipse) to a square (rectangle). The classical Galerkin procedure is used to reduce the problem into a set of nondimensional coupled Mathieu-Hill equations, and the regions of parametric instability for principal and combination resonant frequencies are determined by applying Hsu's technique. The effects of the superellipticity parameter, plate aspect ratio, and foundation stiffness on the dynamic stability behavior (onset and width of the instability regions) are examined. Moreover, the transient flexural response of the panel under static in-plane load and subjected to a transverse uniform step force is obtained. The validity of the results is established by comparison with the available data as well as with the aid of a commercial finite-element package. The proposed model is believed to be the first rigorous attempt in dynamic stability and/or flexural dynamic response analysis of superelliptical plates.