A Differential Quadrature Procedure with Regularization of the Dirac-delta Function for Numerical Solution of Moving Load Problem

Abstract

The differential quadrature method (DQM) is one of the most elegant and efficient methods for the numerical solution of partial differential equations arising in engineering and applied sciences. It is simple to use and also straightforward to implement. However, the DQM is well-known to have some difficulty when applied to partial differential equations involving singular functions like the Dirac-delta function. This is caused by the fact that the Dirac-delta function cannot be directly discretized by the DQM. To overcome this difficulty, this paper presents a simple differential quadrature procedure in which the Dirac-delta function is replaced by regularized smooth functions. By regularizing the Dirac-delta function, such singular function is treated as non-singular functions and can be easily and directly discretized using the DQM. To demonstrate the applicability and reliability of the proposed method, it is applied here to solve some moving load problems of beams and rectangular plates, where the location of the moving load is described by a time-dependent Dirac-delta function. The results generated by the proposed method are compared with analytical and numerical results available in the literature. Numerical results reveal that the proposed method can be used as an efficient tool for dynamic analysis of beam-and plate-type structures traversed by moving dynamic loads.

A Differential Quadrature Procedure with Direct Projection of the Heaviside Function for Numerical Solution of Moving Load Problem

“…The DQMR results are shown for three different values of α. It is noted that in DQMR, the Dirac-delta function is approximated as (Eftekhari, 2015a)…”

“…The parameters of the problem are assumed to be (Eftekhari, 2015a) In Table 1, the numerical results are given for dynamic magnification factors (DMFs) (maximum value of normalized deflection at the beam center) of a simply supported beam for different values of moving speed. The analytical solution values are also shown for comparison.…”

“…However, this technique involves a regularization parameter that should be carefully adjusted before the problem being solved. This parameter has also been shown to be a problem-dependent parameter (Eftekhari, 2015a).…”

“…By regularizing the Dirac-delta function, such singular function is treated as non-singular functions and can be easily and directly discretized using the DQM. Jung (2009), Jung and Don (2009), , Wang et al (2014), Eftekhari (2015a), and Tornabene et al (2015b) have successfully applied this technique in conjunction with the point discretization methods to solve various partial differential equations involving the Dirac-delta function. However, this technique involves a regularization parameter that should be carefully adjusted before the problem being solved.…”

A dierential quadrature procedure for free vibration of rectangular plates involving free corners

“…Most of these applications are related to static and dynamic analyses of structural components like beams, plates, and shells. Newer applications include the use of DQM for solving moving load problems [46,47] and uid-structure interaction problems [48,49]. The results of many research works show that the DQM is computationally e cient and is applicable to a large class of initial and/or boundary value problems.…”