Application of Differential Quadrature Method to Structural and Vibration Analysis

Abstract

As shown in the previous chapters, the differential quadrature method has a feature in that it can obtain very accurate numerical results by using just a few grid points. This feature has a particular merit in its application to structural and vibration analysis. For example, the vibration of a thin plate is governed by a fourth order partial differential equation. When a numerical method is applied to discretize the spatial derivatives, the partial differential equation can be reduced to a set of algebraic equations. The eigenvalues of the resultant algebraic equation system provide the vibrational frequencies of the problem. Usually, the number of interior grid points is equal to the dimension of the resultant algebraic equation system, thus providing the same number of eigenfrequencies. Among all the computed eigenfrequencies, only low frequencies are of practical interest. As we know, low order methods such as finite differences and finite elements are only capable of obtaining accurate numerical results by using a large number of grid points. So, when low order methods are applied, they need to use a large number of grid points to obtain highly accurate values for low frequencies. As a result, a lot of virtual storage and computational effort are required. On the other hand, when the DQ method is applied, the low frequencies can be obtained very accurately by using a considerably smaller number of grid points due to the feature of the method. As a consequence, very little computational effort and virtual storage are needed when the DQ method is used.