Free-Vibration of an L-Shaped Plate: The General Solution and an Example of a Simply-Supported Plate With a Clamped Cutout

This study develops a new accurate method for finding the natural vibration frequencies of plates with cutouts. The method is based on replacing the plate with a cutout by a rectangular plate. This is achieved by filling the cutout with a “dummy” plate made of the same material, and of the same thickness as the original from which it is separated by an infinitesimal gap. Thanks to this device it is possible to apply finite Fourier transformation of discontinuous functions in a rectangular domain. The expression for the deflection now depends on the unknown quantities along the boundary and across the gap. Subsequent application of the available boundary conditions leads to a system of boundary integral equations. An L-shaped plate simply supported along the perimeter, and fixed along the cutout, is analyzed as an example. The frequencies of natural vibration are calculated and compared with the results obtained using the finite element method. The method presented here is also applicable to two- and three-dimensional problems of solids with holes or cavities and to similar thermoelastic problems. Application to plates with curved boundaries is also possible.