Coupling Ritz Method and Triangular Quadrature Rule for Moving Mass Problem

Abstract

In this paper, a mixed method that combines the Ritz method and the triangular quadrature rule (TQR) is presented for solving time-dependent problems. In this study, the Ritz method is first used to discretize the spatial partial derivatives. The TQR is then employed to analogize the temporal derivatives. The resulting algebraic formulation is a triangular matrix equation, which reduces to the solution of a system of algebraic equations of the size of the problem for each time step. This requires less computational effort compared to the differential quadrature method (DQM) where a larger system of the size of the problem should be solved within each time element. The mixed formulation combines the simplicity of the Ritz method and accuracy and computational efficiency of the TQR. The stability property and computational efficiency of the scheme are discussed in detail. Numerical results show that the proposed mixed methodology can be used as an efficient tool for handling the time-dependent problems.

An Analytical Study on Dynamic Response of Multiple Simply Supported Beam System Subjected to Moving Loads

“…For decades, the dynamic response of a single beam has been extensively investigated. Many previous studies have provided better solutions to this problem [2][3][4][5][6], and there are also many papers on beam vibration excited by a moving mass [7][8][9][10]. In the recent decades, much effort has been devoted to the analysis of dynamic response of a double-beam system, and the results are significant [11][12][13][14][15][16].…”

A differential quadrature procedure for free vibration of circular membranes backed by a cylindrical fluid-filled cavity

“…Most of these applications are concerned with static and dynamic analysis of structural components like beams, plates, and shells. Newer applications include the use of the DQM for approximation of time-derivative terms [28][29][30][31]. It has been shown that the DQM is computationally efficient, and is applicable to a large class of initial and/or boundary value problems.…”

A modified differential quadrature procedure for numerical solution of moving load problem

“…[6][7][8][9][10][11][12] The DQM has also been successfully applied to initial value problems to discretize the time-derivative terms. [13][14][15][16] It has been found that the DQM time integration scheme is reliable, computationally efficient, and also suitable for time integration over long time duration. More recently, the DQM has been combined with other techniques like the Ritz and finite element (FE) methods and applied to the vibration problem of rectangular and skew plates.…”