DSC-Ritz element method for vibration analysis of rectangular Mindlin plates with mixed edge supports. (English) Zbl 1475.74128
Summary: Based on Mindlin’s first-order shear deformable plate theory, a DSC-Ritz element method is developed for the free vibration analysis of moderately thick rectangular plates with mixed supporting edges. The rationale of the present approach is not only to apply the discrete singular convolution (DSC) delta type wavelet kernel as a trial function with the Ritz method, but also to incorporate the method in finite elements in order to handle the mixed boundary constraints. The approach is novel and flexible as it passes through a bottleneck of the global DSC-Ritz method in treating the kinematic supporting edges with assorted discontinuities. A series of numerical simulations for rectangular Mindlin plates with various edge support discontinuities, plate thicknesses and aspect ratios are presented. For verification, the vibration frequencies thus established are directly compared with those reported in the open literature. New sets of numerical results for several other cases of moderately thick plates with mixed simply supported, clamped and free edges are presented and discussed in detail.
MSC:
| 74S99 | Numerical and other methods in solid mechanics |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 74K20 | Plates |
| 74H15 | Numerical approximation of solutions of dynamical problems in solid mechanics |
Keywords:
discrete singular convolution; moderately thick plate; first-order shear deformation plate theory; Gauss wavelet kernelReferences:
| [1] | Al-Gwaiz, M. A., Theory of Distributions (1992), Marcel Dekker: Marcel Dekker New York · Zbl 0759.46033 |
| [2] | Bardell, N. S., The free vibration of skew plates using the hierarchical finite element method, Comput. Struct., 45, 841-874 (1992) |
| [3] | Civalek, Ö., Nonlinear analysis of thin rectangular plates on Winkler-Pasternak elastic foundations by DSC-HDQ methods, Appl. Math. Model., 31, 606-624 (2007) · Zbl 1197.74198 |
| [4] | Civalek, Ö., A parametric study of the free vibration analysis of rotating laminated cylindrical shells using the method of discrete singular convolution, Thin-Walled Struct., 45, 692-698 (2007) |
| [5] | Civalek, Ö., Vibration analysis of conical panels using the method of discrete singular convolution, Commun. Numer. Methods Eng., 24, 169-181 (2008) · Zbl 1151.74361 |
| [6] | Fan, S. C.; Cheung, Y. K., Flexural free vibrations of rectangular plates with complex support conditions, J. Sound Vib., 93, 81-94 (1984) |
| [7] | 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 |
| [8] | He, L. H.; Lim, C. W.; Kitipornchai, S., A non-discretized global method for free vibration of generally laminated fibre-reinforced pre-twisted cantilever plates, Comput. Mech., 26, 197-207 (2000) · Zbl 1162.74373 |
| [9] | Hou, Y.; Wei, G. W.; Xiang, Y., DSC-Ritz method for the free vibration analysis of Mindlin plates, Int. J. Numer. Methods Eng., 62, 262-288 (2005) · Zbl 1179.74175 |
| [10] | Keer, L. M.; Stahl, B., Eigenvalue problems of rectangular plates with mixed edge conditions, Trans. ASME J. Appl. Mech., 39, 513-520 (1972) · Zbl 0235.73025 |
| [11] | Lai, S. K.; Xiang, Y., DSC analysis for buckling and vibration of rectangular plates with elastically restrained edges and linearly varying in-plane loading, Int. J. Struct. Stab. Dyn., 9, 511-531 (2009) · Zbl 1271.74165 |
| [12] | Laura, P. A.A.; Gutierrez, R. H., Analysis of vibrating rectangular plates with non-uniform boundary conditions by using the differential quadrature method, J. Sound Vib., 173, 702-706 (1994) · Zbl 0925.73486 |
| [13] | Liew, K. M.; Wang, C. M., Vibration analysis of plates by the pb-2 Rayleigh-Ritz method: mixed boundary conditions, re-entrant corners and internal curved supports, Mech. Struct. Mech., 20, 281-292 (1992) |
| [14] | Liew, K. M.; Hung, K. C.; Lim, M. K., Roles of domain decomposition method in plate vibrations: treatment of mixed discontinuous periphery boundaries, Int. J. Mech. Sci., 35, 615-632 (1993) · Zbl 0800.73528 |
| [15] | Liew, K. M.; Xiang, Y.; Kitipornchai, S., Transverse vibration of thick rectangular plates-I. Comprehensive sets of boundary conditions, Comput. Struct., 49, 1-29 (1993) |
| [16] | Liew, K. M.; Xiang, Y.; Kitipornchai, S., Research on thick plate vibration: a literature survey, J. Sound Vib., 180, 163-176 (1995) · Zbl 1237.74002 |
| [17] | Liew, K. M.; Hung, K. C.; Lim, M. K., Vibration of Mindlin plates using boundary characteristic orthogonal polynomials, J. Sound Vib., 182, 77-90 (1995) |
| [18] | Liew, K. M.; Wang, C. M.; Xiang, Y.; Kitipornchai, S., Vibration of Mindlin Plates, Programming the P-version Ritz Method (1998), Elsevier: Elsevier Amsterdam · Zbl 0940.74002 |
| [19] | Lim, C. W.; Li, Z. R.; Wei, G. W., DSC-Ritz method for high-mode frequency analysis of thick shallow shells, Int. J. Numer. Methods Eng., 62, 205-232 (2005) · Zbl 1179.74178 |
| [20] | Lim, C. W.; Li, Z. R.; Xiang, Y.; Wei, G. W.; Wang, C. M., On the missing modes when using the exact frequency relationship between Kirchhoff and Mindlin plates, Adv. Vib. Eng., 4, 221-248 (2005) |
| [21] | Liu, F. L.; Liew, K. M., Vibration analysis of discontinuous Mindlin plates by differential quadrature element method, Trans. ASME J. Vib. Acoust., 121, 204-208 (1999) |
| [22] | Liu, F. L.; Liew, K. M., Analysis of vibrating thick rectangular plates with mixed boundary constraints using differential quadrature element method, J. Sound Vib., 225, 915-934 (1999) · Zbl 1235.74364 |
| [23] | Lü, C. F.; Zhang, Z. C.; Chen, W. Q., Free vibration of generally supported rectangular Kirchhoff plates: state-space-based differential quadrature method, Int. J. Numer. Methods Eng., 70, 1430-1450 (2007) · Zbl 1194.74141 |
| [24] | Mindlin, R. D., Influence of rotary inertia and shear in flexural motion of isotropic, elastic plates, Trans. ASME J. Appl. Mech., 18, 31-38 (1951) · Zbl 0044.40101 |
| [25] | Mizusawa, T.; Leonard, J. W., Vibration and buckling of plates with mixed boundary conditions, Eng. Struct., 12, 285-290 (1990) |
| [26] | Narita, Y., Application of a series-type method to vibration of orthotropic rectangular plates with mixed boundary conditions, J. Sound Vib., 77, 345-355 (1981) · Zbl 0477.73066 |
| [27] | Ng, C. H.W.; Zhao, Y. B.; Wei, G. W., Comparison of discrete singular convolution and generalized differential quadrature for the vibration analysis of rectangular plates, Comput. Methods Appl. Mech. Eng., 193, 2483-2506 (2004) · Zbl 1067.74600 |
| [28] | Ota, T.; Hamada, M., Fundamental frequencies of simply supported but partially clamped square plates, Bull. Jpn. Soc. Mech. Eng., 6, 397-403 (1963) |
| [29] | 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) |
| [30] | Ramachandran, J., Large amplitude natural frequencies of rectangular plates with mixed boundary conditions, J. Sound Vib., 45, 295-297 (1976) |
| [31] | Reissner, E., The effect of transverse shear deformation on the bending of elastic plates, Trans. ASME J. Appl. Mech., 12, 69-76 (1945) · Zbl 0063.06470 |
| [32] | Schwartz, L., Théore des Distributions (1951), Hermann: Hermann Paris · Zbl 0042.11405 |
| [33] | Shu, C.; Wang, C. M., Treatment of mixed and nonuniform boundary conditions in GDQ vibration analysis of rectangular plates, Eng. Struct., 21, 125-134 (1999) |
| [34] | Su, G. H.; Xiang, Y., A non-discrete approach for analysis of plates with multiple subdomains, Eng. Struct., 24, 563-575 (2002) |
| [35] | Timoshenko, S. P.; Woinowsky-Krieger, S., Theory of Plates and Shells (1959), McGraw-Hill: McGraw-Hill New York |
| [36] | Venkateswara Rao, G.; Raju, I. S.; Murthy, T. V.G. K., Vibration of rectangular plates with mixed boundary conditions, J. Sound Vib., 30, 257-260 (1973) |
| [37] | Wei, G. W.; Zhang, D. S.; Kouri, D. J.; Hoffman, D. K., Lagrange distributed approximating functional, Phys. Rev. Lett., 79, 775-779 (1997) |
| [38] | Wei, G. W., Discrete singular convolution for the solution of the Fokker-Planck equation, J. Chem. Phys., 110, 8930-8942 (1999) |
| [39] | Wei, G. W., Discrete singular convolution for the sine-Gordon equation, Physica D, 137, 247-259 (2000) · Zbl 0944.35087 |
| [40] | Wei, G. W., A new algorithm for solving some mechanical problems, Comput. Methods Appl. Mech. Eng., 190, 2017-2030 (2001) · Zbl 1013.74081 |
| [41] | Wei, G. W., Vibration analysis by discrete singular convolution, J. Sound Vib., 244, 535-553 (2001) · Zbl 1237.74095 |
| [42] | Wei, G. W.; Zhao, Y. B.; Xiang, Y., The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution, Int. J. Mech. Sci., 43, 1731-1746 (2001) · Zbl 1018.74017 |
| [43] | Wei, G. W.; Zhao, Y. B.; Xiang, Y., Discrete singular convolution and its application to the analysis of plates with internal supports. Part 1: theory and algorithm, Int. J. Numer. Methods Eng., 55, 913-946 (2002) · Zbl 1058.74643 |
| [44] | 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) |
| [45] | Xiang, Y.; Liew, K. M.; Kitipornchai, S., Transverse vibration of thick annular sector plates, ASCE J. Eng. Mech., 119, 1579-1599 (1993) |
| [46] | Xiang, Y.; Zhao, Y. B.; Wei, G. W., Discrete singular convolution and its application to the analysis of plates with internal supports. Part 2: applications, Int. J. Numer. Methods Eng., 55, 947-971 (2002) · Zbl 1058.74644 |
| [47] | Xiang, Y., Lai, S.K., Zhou, L. DSC-element method for free vibration analysis of rectangular Mindlin plates. Int. J. Mech. Sci., in Press, doi:10.1016/j.ijmecsci.2009.12.001; Xiang, Y., Lai, S.K., Zhou, L. DSC-element method for free vibration analysis of rectangular Mindlin plates. Int. J. Mech. Sci., in Press, doi:10.1016/j.ijmecsci.2009.12.001 |
| [48] | Zhao, S.; Wei, G. W.; Xiang, Y., DSC analysis of free-edged beams by an iteratively matched boundary method, J. Sound Vib., 284, 487-493 (2005) · Zbl 1237.74198 |
| [49] | 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, 65-88 (2002) · Zbl 1090.74604 |
| [50] | Zhou, D.; Cheung, Y. K.; Lo, S. H.; Au, F. T.K., Three-dimensional vibration analysis of rectangular plates with mixed boundary conditions, Trans. ASME J. Appl. Mech., 72, 227-236 (2005) · Zbl 1111.74746 |
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