A Simple and Systematic Approach for Implementing Boundary Conditions in the Differential Quadrature Free and Forced Vibration Analysis of Beams and Rectangular Plates

Abstract

This paper presents a simple and systematic way for imposing boundary conditions in the differential quadrature free and forced vibration analysis of beams and rectangular plates. First, the Dirichlet- and Neumann-type boundary conditions of the beam (or plate) are expressed as differential quadrature analog equations at the grid points on or near the boundaries. Then, similar to CBCGE (direct Coupling the Boundary Conditions with the discrete Governing Equations) approach, the resulting analog equations are used to replace the differential quadrature analog equations of the governing differential equations at these points in order to solve the problem. But, unlike the CBCGE approach, the grid points near the boundaries are not treated as boundary points in the proposed approach. In other words, the degrees of freedom related to Dirichlet-type boundary conditions are only eliminated from the original discrete equations. This simplifies significantly the solution procedure and its programming. A comparison of the proposed approach with other existing methodologies such as the CBCGE approach and MWCM (modifying weighting coefficient matrices) method is presented by their application to the vibration analysis of beams and rectangular plates with general boundary conditions to highlight the advantages of the new approach.