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Comparison of discrete singular convolution and generalized differential quadrature for the vibration analysis of rectangular plates. (English) Zbl 1067.74600
Summary: This paper presents a comprehensive comparison study between the discrete singular convolution (DSC) and the well-known global method of generalized differential quadrature (GDQ) for vibration analysis so as to enhance the understanding of the DSC algorithm. The DSC method is implemented through Lagrange’s delta sequence kernel (DSC-LK), which utilizes local Lagrange polynomials to calculate weighting coefficients, whereas, the GDQ requires global ones. Moreover, it is shown that the treatments of boundary conditions and the use of grid systems are different in the two methods. Comparison study is carried out on 21 rectangular plates of different combinations of simply supported, clamped and transversely supported with nonuniform elastic rotational restraint edges, and five rectangular plates of mixed supporting edges, some of which with a range of aspect ratios and rotational spring coefficients. All the results of the DSC-LK agree very well with both those in the literature and newly computed GDQ results. Furthermore, it is observed that the DSC-LK performs much better for plates vibrating at higher-order eigenfrequencies. Unlike the GDQ, the DSC-LK is numerically stable for problems which require a large number of grid points.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74H45 Vibrations in dynamical problems in solid mechanics
74K20 Plates
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
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References:
[1] Leissa, A.W., The free vibration of rectangular plates, J. sound vib., 31, 257-293, (1973) · Zbl 0268.73033
[2] Leissa, A.W.; Laura, P.A.A.; Gutierrez, R.H., Vibrations of rectangular plates with non – uniform elastic edge supports, J. appl. mech., 47, 891-895, (1980) · Zbl 0447.73052
[3] Leissa, A.W., Vibration of plates (NASA SP-160), (1969), US Government Printing Office Washington, DC
[4] Fan, S.C.; Cheung, Y.K., Flexural free vibrations of rectangular plates with complex support conditions, J. sound vib., 93, 81-94, (1984)
[5] Chia, C.Y., Non-linear vibration of anisotropic rectangular plates with non-uniform edge constraints, J. sound vib., 101, 539-550, (1985)
[6] Leipholz, H.H.E., On some developments in direct methods of the calculus of variations, Appl. mech. rev., 40, 10, 1379-1392, (1987) · Zbl 0633.73019
[7] Zitnan, P., Vibration analysis of membranes and plates by a discrete least squares technique, J. sound vib., 195, 4, 595-605, (1996)
[8] Donning, B.M.; Liu, W.K., Meshless methods for shear-deformable beams and plates, Comput. methods appl. mech. engrg., 152, 47-71, (1998) · Zbl 0959.74079
[9] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077
[10] Nowacki, W., Free vibrations and buckling of a rectangular plate with discontinuous boundary conditions, Bull. acad. Pol. sci. biol., 3, 159-167, (1955) · Zbl 0067.17701
[11] M.S. Cheung, Ph.D. thesis, University of Calgary, Finite strip analysis of structures, 1971
[12] Keer, L.M.; Stahl, B., Eigenvalue problems of rectangular plates with mixed edge conditions, J. appl. mech., 39, 513-520, (1972) · Zbl 0235.73025
[13] Rao, G.V.; Raju, I.S.; Murthy, T.V.G.K., Vibration of rectangular plates with mixed boundary conditions, J. sound vib., 30, 257-260, (1973)
[14] Narita, Y., Application of a series-type method to vibration of orthotropic rectangular plates with mixed boundary conditions, J. sound vib., 77, 3, 345-355, (1981) · Zbl 0477.73066
[15] Gorman, D.J., An exact analytical approach to the free vibration analysis of rectangular plates with mixed boundary conditions, J. sound vib., 93, 235-247, (1984) · Zbl 0538.73082
[16] Mizusawa, T.; Leonard, J.W., Vibration and buckling of plates with mixed boundary conditions, Engrg. struct., 12, 285-290, (1990)
[17] Liew, K.M.; Hung, K.C.; Lam, K.Y., On the use of the substructure method for vibration analysis of rectangular plates with discontinuous boundary conditions, J. sound vib., 163, 451-462, (1993) · Zbl 0925.73374
[18] Piskunov, V.H., Determination of the frequencies of the natural oscillations of rectangular plates with mixed boundary conditions, Prikl. mekh., 10, 72-76, (1964), (in Ukrainian)
[19] Ota, T.; Hamada, M., Fundamental frequencies of simply supported but partially clamped square plates, Bull. jpn. soc. mech. engrg., 6, 397-403, (1963)
[20] Young, D., Vibration of rectangular plates by the Ritz method, Trans. amer. soc. mech. engrs.: J. appl. mech., 17, 448-453, (1950) · Zbl 0039.20701
[21] S.P. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, Singapore, 1970 · Zbl 0114.40801
[22] Bassily, S.F.; Dickinson, S.M., On the use of beam functions for problems of plates involving free edges, Trans. amer. soc. mech. engrs.: J. appl. mech., 42, 858-864, (1975) · Zbl 0326.73055
[23] Mizusawa, T.; Kajita, T.; Naruoka, M., Vibration of skew plates by using B-spline functions, J. sound vib., 62, 301-308, (1979) · Zbl 0413.73069
[24] Bhat, R.B., Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method, J. sound vib., 102, 493-499, (1985)
[25] Liew, K.M.; Lam, K.Y., A set of orthogonal plate functions for vibration analysis of regular polygonal plates, Trans. amer. soc. mech. engrs.: J. vib. acoust., 113, 182-186, (1991)
[26] Lim, C.W.; Liew, K.M., Vibrations of perforated plates with rounded corners, J. engrg. mech., 121, 2, 203-213, (1995)
[27] Liew, K.M.; Wang, C.M.; Xiang, Y.; Kitipornchai, S., Vibration of Mindlin plates, programming the p-version Ritz method, (1998), Elsevier Amsterdam · Zbl 0940.74002
[28] Leung, A.Y.T.; Chan, J.K.W., Fourier p-element for analysis of beams and plates, J. sound vib., 212, 1, 179-185, (1998)
[29] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, vol. 1, New York, McGraw-Hill, 1989 · Zbl 0991.74002
[30] Bardell, N.S.; Dunsdon, J.M.; Langley, R.S., Free vibration of coplanar sandwich panels, Compos. struct., 38, 463-475, (1997)
[31] Cheung, Y.K.; Chen, W.J., Hybrid quadrilateral element based on Mindlin/Reissner plate theory, Comput. struct., 32, 327-339, (1989) · Zbl 0711.73241
[32] Bellman, R.; Kashef, B.G.; Casti, J., Differential quadrature: a technique for the rapid solution of non-linear partial differential equations, J. comput. phys., 10, 40-52, (1972) · Zbl 0247.65061
[33] Bert, C.W.; Jang, S.K.; Striz, A.G., Two new approximate methods for analyzing free vibration of structural components, Amer. inst. aeronaut. astronaut. J., 26, 612-618, (1988) · Zbl 0661.73063
[34] Jang, S.K.; Bert, C.W.; Striz, A.G., Application of differential quadrature to static analysis of structural components, Int. J. numer. methods engrg., 28, 561-577, (1989) · Zbl 0669.73064
[35] Quan, J.R.; Chang, C.T., New insights in solving distributed system of equations by the quadrature method–I. analysis, Comput. chem. engrg., 13, 779-788, (1989)
[36] Chen, W.; Zhong, T.X., A note on the DQ analysis of anisotropic plates, J. sound vib., 204, 1, 180-182, (1997)
[37] Shu, C.; Richards, B.E., Application of generalized differential quadrature to solve two – dimensional incompressible navier – stokes equations, Int. J. numer. methods fluids, 15, 791-798, (1992) · Zbl 0762.76085
[38] Du, H.; Lim, M.K.; Lin, R.M., Application of generalized differential quadrature method to structure problems, Int. J. numer. methods engrg., 37, 1881-1896, (1994) · Zbl 0804.73076
[39] Shu, C.; Du, H., A generalized approach for implementing general boundary conditions in the GDQ free vibration analysis of plates, Int. J. solids struct., 34, 837-846, (1997) · Zbl 0944.74644
[40] Shu, C.; Wang, C.M., Treatment of mixed and non-uniform boundary conditions in GDQ vibration analysis of rectangular plate, Engrg. struct., 21, 125-134, (1999)
[41] Wei, G.W., Discrete singular convolution for the solution of the fokker – planck equation, J. chem. phys., 110, 8930-8942, (1999)
[42] Wei, G.W., A unified approach for the solution of the fokker – planck equation, J. phys. A: math. gen., 33, 4935-4953, (2000) · Zbl 0988.82047
[43] Wei, G.W., Wavelets generated by using discrete singular convolution kernels, J. phys. A: math. gen., 33, 8577-8596, (2000) · Zbl 0961.42019
[44] Schwarz, L., Théore des distributions, (1951), Hermann Paris
[45] Wei, G.W., A new algorithm for solving some mechanical problems, Comput. methods appl. mech. engrg., 190, 2017-2030, (2001) · Zbl 1013.74081
[46] Wan, D.C.; Patnaik, B.S.V.; Wei, G.W., Discrete singular convolution-finite subdomain method for the solution of incompressible viscous flows, J. comput. phys., 180, 229-255, (2002) · Zbl 1130.76403
[47] Wei, G.W., Discrete singular convolution method for the sine-Gordon equation, Physica D, 137, 247-259, (2000) · Zbl 0944.35087
[48] Guan, S.; Lai, C.-H.; Wei, G.W., Fourier – bessel analysis of patterns in a circular domain, Physica D, 151, 83-98, (2001) · Zbl 1076.35535
[49] Wei, G.W., Synchronization of single-side locally averaged adaptive coupling and its application to shock capturing, Phys. rev. lett., 86, 3542-3545, (2001)
[50] Wei, G.W., Generalized perona – malik equation for image restoration, IEEE signal process. lett., 6, 165-168, (1999)
[51] Wei, G.W., Discrete singular convolution for beam analysis, Engrg. struct., 23, 1045-1053, (2001)
[52] Wei, G.W., Vibration analysis by discrete singular convolution, J. sound vib., 244, 535-553, (2001) · Zbl 1237.74095
[53] Zhao, Y.B.; Wei, G.W., DSC analysis of rectangular plates with nonuniform boundary conditions, J. sound vib., 255, 203-225, (2002)
[54] Wei, G.W.; Zhao, Y.B.; Xiang, Y., Discrete singular convolution and its application to the analysis of plates with internal supports–I. theory and algorithm, Int. J. numer. methods engrg., 55, 913-946, (2002) · Zbl 1058.74643
[55] Zhao, Y.B.; Wei, G.W.; Xiang, Y., Plate vibration under irregular internal supports, Int. J. solids struct., 39, 5, 1361-1383, (2002) · Zbl 1090.74603
[56] Zhao, Y.B.; Wei, G.W.; Xiang, Y., Discrete singular convolution for the prediction of high frequency vibration of plates, Int. J. solids struct., 39, 1, 65-88, (2002) · Zbl 1090.74604
[57] Wei, G.W.; Zhao, Y.B.; Xiang, Y., A novel approach for the analysis of high frequency vibrations, J. sound vib., 257, 207-246, (2002)
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