Principal and second instability regions of shear-deformable polygonal plates

The dynamic stability of plates subject to periodic in-plane forces is analyzed and the corresponding stability regions of the first and second order are calculated. Moderately thick plates are considered where the influence of shear and rotatory inertia is taken into account according to the Reissner–Mindlin theory. Plates of an arbitrary polygonal shape and simply supported boundaries are studied in detail. A two-parameter foundation of the plate is included. In-plane forces are assumed isotropic. This class of plates is characterized by the decoupling of the free flexural, thickness-shear and thickness-twist plate motions. The present contribution extends recent formulation derived by the authors, where the Reissner–Mindlin plate is modeled by a time-variant dynamic system of ODE's. The polygonal shape of the plate enters these Mathieu-type equations by means of the eigenvalues of second-order Helmholtz boundary value-problems via a proper membrane-analogy. Influence of shear and rotatory inertia is taken into account by tracers. In the present contribution, special emphasis is given to the calculation of the limits of the regions of instability in the p-Pt plane (stability chart), where p defines the exciting frequency and Pt the exciting force. In order to find the limits of the first and second order, suitable ansatz-functions are introduced. These ansatz-functions depend on the order of the regions, which are to be analyzed. The results are derived in a non-dimensional form and the two regions are graphically presented, compared and some parametric investigations are performed. In particular, the influence of the Helmholtz-eigenvalues, characterizing the specific shape of the plate, is studied. It is shown that an increase in the eigenvalue increases the domains of instability for each order and leads to a shift of the domains of instability towards higher frequencies.