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Differential Quadrature Analysis of Moving Load Problems

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical methods

Published online by Cambridge University Press:  27 May 2016

Xinwei Wang  and
Chunhua Jin
Xinwei Wang*
Affiliation:
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Chunhua Jin*
Affiliation:
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China School of Civil Engineering and Architecture, Nantong University, Nantong 224019, China
*
*Corresponding author. Email:[email protected] (X. W. Wang), [email protected] (C. H. Jin)
*Corresponding author. Email:[email protected] (X. W. Wang), [email protected] (C. H. Jin)
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Abstract

The 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.

Keywords

Differential quadrature methodDirac-delta functionmoving point loaddynamic responsework-equivalent load

MSC classification

Secondary: 74S30: Other numerical methods 65M99: None of the above, but in this section
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 4 , August 2016 , pp. 536 - 555
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Bellman, R. and Casti, J., Differential quadrature and long term integration, J. Math. Anal. Appl., 34(2) (1971), pp. 235–238.CrossRefGoogle Scholar
[2]Bellman, R., Kashef, B. G. and Casti, J., Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, J. Comput. Phys., 10 (1972), pp. 40–52.CrossRefGoogle Scholar
[3]Bert, C. W., Jang, S. K. and Striz, A. G., TWO new approximate methods for analyzing free vibration of structural components, AIAA J., 26 (1988), pp. 612–618.CrossRefGoogle Scholar
[4]Fung, T. C., Solving initial value problems by differential quadrature method: Part 1: first order equations, Int. J. Numer. Methods Eng., 50(6) (2001), pp. 1411–1427.3.0.CO;2-O>CrossRefGoogle Scholar
[5]Fung, T. C., Solving initial value problems by differential quadrature method: Part 2: second and higher order equations, Int. J. Numer. Methods Eng., 50(6) (2001), pp. 1429–1454.3.0.CO;2-A>CrossRefGoogle Scholar
[6]Shu, C., Yao, Q. and Yeo, K. S., Block-marching in time with DQ discretization: an efficient method for time-dependent problems, Comput. Methods Appl. Mech. Eng., 191 (2001), pp. 4587–4597.CrossRefGoogle Scholar
[7]Chen, W. and Tanaka, M., A study on time schemes for DRBEM analysis of elastic impact wave, Comput. Mech., 28 (2002), pp. 331–338.CrossRefGoogle Scholar
[8]Chen, W., Shu, C., He, W. and Zhong, T., The application of special matrix product to differential quadrature solution of geometrically nonlinear bending of orthotropic rectangular plates, Comput. Struct., 74(1) (2000), pp. 65–76.CrossRefGoogle Scholar
[9]Bert, C. W. and Malik, M., Differential quadrature method in computational mechanics: A review, Appl. Mech. Rev., 49 (1996), pp. 1–28.CrossRefGoogle Scholar
[10]Shu, C., Differential Quadrature and Its Application in Engineering, London, Springer-Verlag, 2000.CrossRefGoogle Scholar
[11]Zong, Z. and Zhang, Y., Advanced Differential Quadrature Methods, CRC Press, 2009.CrossRefGoogle Scholar
[12]Wang, X., Differential Quadrature and Differential Quadrature Based Element Methods: Theory and Applications, Elsevier Inc., 2015.Google Scholar
[13]Engquist, B., Tornberg, A. K. and Tsai, R., Discretization of Dirac delta functions in level set methods, J. Comput. Phys., 207 (2005), pp. 28–51.CrossRefGoogle Scholar
[14]Zhang, C., Hu, Z. and Zhong, Z., Wavelet-based differential quadrature method for the analysis of graded composite beam under concentrated loading, Acta. Mechnica Solida Sinica, 27 (2006), pp. 38–42.Google Scholar
[15]Han, H. T., Zhang, Z. and Lu, Z. X., Differential quadrature (DQ) method of discontinous problem, J. Mech. Strength, 32(2) (2010), pp. 333–337.Google Scholar
[16]Wang, X. and Gu, H., Static analysis of frame structures by the differential quadrature element method, Int. J. Numer. Methods Eng., 40 (1997), pp. 759–772.3.0.CO;2-9>CrossRefGoogle Scholar
[17]Khalili, S. M. R., Jafari, A. A. and Eftekhari, S. A., A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads, Compos. Struct., 92(10) (2010), pp. 2497–2511.CrossRefGoogle Scholar
[18]Jafari, A. A. and Eftekhari, S. A., A new mixed finite element-differential quadrature formulation for forced vibration of beams carrying moving loads, ASME J. Appl. Mech., 2011, 78(1) (2011), pp. 011–020.CrossRefGoogle Scholar
[19]Wang, X. and Bert, C. W., A new approach in applying of differential quadrature to static and free vibrational analyses of beams and plates, J. Sound Vib., 162(3) (1993), pp. 566–572.CrossRefGoogle Scholar
[20]Wang, X., Liu, F., Wang, X. and Gan, L., New approaches in application of differential quadrature method for fourth-order differential equations, Commun. Numer. Methods Eng., 21(2) (2005), pp. 61–71.CrossRefGoogle Scholar
[21]Leissa, A. W., The free vibration of rectangular plates, J. Sound Vib., 31(3) (1973), pp. 257–293.CrossRefGoogle Scholar
[22]Olsson, M., On the fundamental moving load problem, J. Sound Vib., 145(2) (1991), pp. 299–307.CrossRefGoogle Scholar
[23]Rieker, J. R., Lin, Y. H. and Trethewey, M. W., Discretization considerations in moving load finite element beam models, Finite Elem. Anal. Des., 21 (1996), pp. 129–144.CrossRefGoogle Scholar