Vibration of arbitrarily-shaped triangular plates with elastically restrained edges

Abstract

A solution method is proposed for the vibration analysis of arbitrarily-shaped triangular plates with elastically restrained edges. An arbitrarily-shaped triangular plate is first mapped into a right-angled isosceles triangular plate and subsequently “stretched” to form a square plate by padding to it another (right-angled isosceles triangular) plate of near-zero thickness. The displacement function is then expressed as a two-dimensional Fourier cosine series supplemented by several one-dimensional series introduced to improve the convergence and accuracy of the displacement solution. Regardless of the shapes and boundary conditions of the triangular plate, the final stiffness and mass matrices can always be calculated analytically which makes the current solution method highly efficient in implementations. Numerical examples are presented to verify the applicability of the proposed method to triangular plates of different shapes, and, perhaps more importantly, different boundary conditions. The current method can be easily extended to plates with heterogeneous material properties, variable thickness, non-uniform elastic restraints, etc.