Bending vibration of simply supported rectangular plates with internal rigid support

Abstract

A previously derived invariant expression for the amplitude of the displacement of homogeneous, isotropic, harmonically vibrating plates with internal rigid supports or cracks is supplemented here by terms representing possible point discontinuities at the tips of the support or of the crack. This expression being given in tensor notation can be easily adapted to curvilinear plates with regular boundaries (separable coordinate systems) and arbitrary discontinuities and, in particular, to rectangular plates with curvilinear discontinuities. Vibration of a rectangular simply supported plate with arbitrarily located rectilinear rigid support is discussed as an example. The unknown discontinuity of the shear force across the support is described by a coupled system of integro-algebraic equations valid for any position and inclination of the rigid support. This system is subsequently reduced to an infinite system of linear-algebraic equations. Fundamental frequencies of natural vibration are obtained for a square plate with centrally located but arbitrarily inclined rigid support of any length. In the special cases—support parallel to one edge of the plate or diagonal support reaching the corners—the results show very good agreement with the ones known from previous analyses.