Natural frequencies of rectangular plates (Q1369840)

In the absence of point supports at the corner of rectangular plates where two free edges meet, 21 combinations of boundary conditions arise, with additional 12 cases occurring if point supports at corners are included. In this note, a formulation treating these 33 cases in a unified manner is presented. The formulation is variationally based, which leads to symmetrical matrices, and enables a modified form of the sign counting algorithm to be employed. The governing differential equation is satisfied within the plate, reducing the two-dimensional problem to the determination of unknown coefficients defining boundary displacement and normal moments. With thin plate theory, two boundary conditions must be satisfied on any boundary. In the proposed method, transverse displacement and normal moment boundary conditions are satisfied exactly, while normal rotation and Kirchhoff shear boundary conditions are satisfied in an integral sense as a consequence of the variational procedure. It is observed that with all combinations of boundary conditions occurring with single plates, the two normal moments at plate corners are zero. This condition is satisfied exactly in the method described in this note.