A New Approach for Plate Vibrations: Combination of Transfer Matrix and Finite-Element Technique

Abstract

When the finite-element method is used in the vibration analysis of plates and shells, it results in large matrices requiring a large digital computer. A commonly used method of reducing the matrix size is to eliminate certain “slave” displacements by minimizing strain energy. The approach requires good judgement in the selection of the “master” displacements and involves additional approximations and some loss of accuracy. In the present method small matrices are obtained without any further approximations and without reducing the number of degrees of freedom. The transfer matrix technique, generally known as the Holzer-Myklestad method, is well known for beams and shafts. The present method is an extension of this idea to plates. The structure is divided into several strips, with a number of nodes on the left and right sections of each strip. Each strip is subdivided into elements and the stiffness and mass matrices are obtained for individual strips. The nodal equilibrium equations are rearranged to obtain a relation between the section variables of the left and the right sections. The section variables are the forces and the displacements of all the nodes on the section. Requirements of displacement continuity and force equilibrium at the nodes, on common sections of two adjacent strips, gives the transfer matrix relation. Successive matrix multiplication finally relates the variables of the left and right boundary of the structure. Boundary conditions require the determinant of a portion of the overall transfer matrix to vanish at the correct frequency. By calculating the determinant at various assumed values of frequency, the correct frequencies are obtained. The method also gives the corresponding mode shapes. The method as applied to several plate problems gives satisfactory results.