Differential Quadrature Analysis of Moving Load Problems

2016 ◽  
Vol 8 (4) ◽  
pp. 536-555 ◽  
Author(s):  
Xinwei Wang ◽  
Chunhua Jin
Keyword(s):  
Differential Equations ◽  
Differential Quadrature Method ◽  
Moving Load ◽  
Delta Function ◽  
Point Load ◽  
Differential Quadrature ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Singular Functions ◽  
Grid Points

AbstractThe differential quadrature method (DQM) has been successfully used in a variety of fields. Similar to the conventional point discrete methods such as the collocation method and finite difference method, however, the DQM has some difficulty in dealing with singular functions like the Dirac-delta function. In this paper, two modifications are introduced to overcome the difficulty encountered in solving differential equations with Dirac-delta functions by using the DQM. The moving point load is work-equivalent to loads applied at all grid points and the governing equation is numerically integrated before it is discretized in terms of the differential quadrature. With these modifications, static behavior and forced vibration of beams under a stationary or a moving point load are successfully analyzed by directly using the DQM. It is demonstrated that the modified DQM can yield very accurate solutions. The compactness and computational efficiency of the DQM are retained in solving the partial differential equations with a time dependent Dirac-delta function.

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

2015 ◽  
Vol 230 (5) ◽  
pp. 715-731 ◽  
Author(s):  
SA Eftekhari
Keyword(s):  
Differential Equations ◽  
Differential Quadrature Method ◽  
Moving Load ◽  
Finite Difference Methods ◽  
Delta Function ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Dirac Delta ◽  
Singular Functions ◽  
Grid Points

The differential quadrature method is a powerful numerical method for the solution of partial differential equations that arise in various fields of engineering, mathematics, and physics. It is easy to use and also straightforward to implement. However, similar to the conventional point discretization methods like the collocation and finite difference methods, the differential quadrature method has some difficulty in solving differential equations involving singular functions like the Dirac-delta function. This is due to complexities introduced by the singular functions to the discretization process of the problem region. To overcome this difficulty, this paper presents a combined differential quadrature–integral quadrature procedure in which such singular functions are simply handled. The mixed scheme can be easily applied to the problems in which the location of the singular point coincides with one of the differential quadrature grid points. However, for problems in which such condition is not fulfilled (i.e. for the case of arbitrary arranged grid points), especially for moving load class of problems, the coupled approach may fail to produce accurate solutions. To solve this drawback, we also introduce two simple approximations and show that they can yield accurate results. The reliability and applicability of the proposed method are demonstrated herein through the solution of some illustrative problems, including the moving load problems of Euler–Bernoulli and Timoshenko beams. The results generated by the proposed method are compared with analytical and numerical results available in the literature and excellent agreement is achieved.

Response of a Semi-Infinite Elastic Solid to an Arbitrary Line Load Along the Axis

1971 ◽  
Vol 38 (4) ◽  
pp. 906-910 ◽  
Author(s):  
G. L. Agrawal ◽  
W. G. Gottenberg
Keyword(s):  
Half Space ◽  
Axisymmetric Problem ◽  
Body Force ◽  
Elastic Solid ◽  
Delta Function ◽  
Point Load ◽  
Dirac Delta Function ◽  
Line Load ◽  
Dirac Delta ◽  
Arbitrary Line

The axisymmetric problem of a line load acting along the axis of a semi-infinite elastic solid is solved using Hankel transforms. In this solution the line load is interpreted as a body force loading and by assuming the line load to be of the form of a Dirac delta function the solution of Mindlin’s problem of a point load within the interior of the half space is obtained. Solutions of this problem presented in the literature have been obtained using semi-inverse techniques whereas the solution given here is obtained in a systematic step-by-step manner.

scholarly journals Finite Series of Distributional Solutions for Certain Linear Differential Equations

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 116
Author(s):  
Nipon Waiyaworn ◽  
Kamsing Nonlaopon ◽  
Somsak Orankitjaroen
Keyword(s):  
Differential Equations ◽  
Delta Function ◽  
Dirac Delta Function ◽  
Linear Differential Equations ◽  
Dirac Delta ◽  
Finite Series ◽  
Distributional Solutions ◽  
Linear Differential

In this paper, we present the distributional solutions of the modified spherical Bessel differential equations t2y″(t)+2ty′(t)−[t2+ν(ν+1)]y(t)=0 and the linear differential equations of the forms t2y″(t)+3ty′(t)−(t2+ν2−1)y(t)=0, where ν∈N∪{0} and t∈R. We find that the distributional solutions, in the form of a finite series of the Dirac delta function and its derivatives, depend on the values of ν. The results of several examples are also presented.

A numerical study of two-dimensional coupled systems and higher order partial differential equations

2019 ◽  
Vol 12 (05) ◽  
pp. 1950071
Author(s):  
R. Rohila ◽  
R. C. Mittal
Keyword(s):  
Differential Equations ◽  
Numerical Study ◽  
Differential Quadrature Method ◽  
Bernstein Polynomials ◽  
Differential Quadrature ◽  
Coupled Systems ◽  
Two Dimensional ◽  
Quadrature Method ◽  
Grid Points ◽  
The Stability

In this paper, a new approach and methodology is developed by incorporating differential quadrature technique with Bernstein polynomials. In differential quadrature method, approximations are done in a way that the derivatives of the function are replaced by a linear sum of functional values at the grid points of the given domain. In Bernstein differential quadrature method (BDQM), Bernstein polynomials are employed for spatial discretization so that a system of ordinary differential equations (ODE’s) is obtained which is solved by SSPRK-43 method. The stability of the method is also studied. The accuracy of the present method is checked by performing numerical experiments on two-dimensional coupled Burgers’ and Brusselator systems and fourth-order extended Fisher Kolmogorov (EFK) equation. Implementation of the method is very easy, efficient and capable of reducing the size of computational efforts.

Dirac delta regularization

2020 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Finite Result

I present a finite result for the Dirac delta "function."

scholarly journals Electromagnetic Waves Through Disordered Systems: Comparison of Intensity, Transmission and Conductance

2001 ◽  
Vol 694 ◽  
Author(s):  
Fredy R Zypman ◽  
Gabriel Cwilich
Keyword(s):  
Probability Distribution ◽  
Disordered Systems ◽  
Electromagnetic Waves ◽  
Delta Function ◽  
Rayleigh Distribution ◽  
Dirac Delta Function ◽  
Universal Form ◽  
Dirac Delta ◽  
Diffusive Regime ◽  
Analytical Expressions

AbstractWe obtain the statistics of the intensity, transmission and conductance for scalar electromagnetic waves propagating through a disordered collection of scatterers. Our results show that the probability distribution for these quantities x, follow a universal form, YU(x) = xne−xμ. This family of functions includes the Rayleigh distribution (when α=0, μ=1) and the Dirac delta function (α →+ ∞), which are the expressions for intensity and transmission in the diffusive regime neglecting correlations. Finally, we find simple analytical expressions for the nth moment of the distributions and for to the ratio of the moments of the intensity and transmission, which generalizes the n! result valid in the previous case.

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