微分值積元素法是一種新發展的數值方法，其特性除了延續微分值積法所具有的數值快速收斂性及準確性外，還可以處理結構具幾何外型、材料性質及邊界負載有不連續的問題。本論文的主旨在於利用微分值積元素法(Differential Quadrature Element Method, DQEM)來分析非均勻厚度之矩形Mindlin平板的動態行為。首先應用微分值積轉化規則，推導Mindlin平板元素的離散化代數方程式；再經由元素的接合，得到系統的離散化代數方程式。本文求解均勻或不均勻厚度的平板在不同邊界條件下的自然頻率，與文獻上的結果做比較。由分析結果顯示，應用微分值積元素法在Mindlin平板的振動分析上，可以得到準確的結果。
將微分值積元素法應用於非均勻厚度或階梯平板的振動分析，將可有效地節省計算時間，並快速地且準確地計算出平板振動的自然頻率。

In this thesis, the dynamic characteristics of rectangular Mindlin plates with non-uniform thickness is investigated by using the differential quadrature element method (DQEM). First, we apply the formulation of differential quadrature to obtain the discrete algebraic governing equations of Mindlin plate elements, which are then assembled to get the discrete equations for the entire plate. Natural frequencies of uniform, stepped and tapered plates with different boundary conditions are obtained, and compared with those in the literature. Numeral results show the high accuracy and efficiency of the DQEM for vibration analysis of rectangular Mindlin plates with non-uniform thickness.

參考文獻
Bellman, R. E., and Casti, J., “Differential Quadrature and Long-term Integration,” Journal of Mathematical Analysis and Application, Vol. 34, pp. 235-238, 1971.
Bellman, R. E., Kashef B. G., and Casti J., “Differential Quadrature: A Technique for the Rapid Solution of Nonlinear Partial Differential Equations,”Journal of Computational Physics, Vol. 10, pp. 40-52, 1972.
Bert, C. W., Jang, S. K., and Striz, A. G., “Two New Approximate Methods for Analyzing Free Vibration of Structural Components,”AIAA Journal, Vol. 26, pp. 612-618, 1988.
Bert, C. W., Wang, X., and Striz, A. G., “Static and Free Vibrational Analysis of Beams and Plates by Differential Quadrature Method,”Acta Mechanics, Vol. 102, pp. 11-24, 1994.
Bert, C. W., and Malik, M., “Differential Quadrature Method in Computational Mechanics: A Review,”ASME Applied Mechanics Review, Vol. 49, pp. 1-28, 1996.
Bert, C. W., and Malik, M., “Free Vibration Analysis of Tapered Rectangular Plates by Differential Quadrature Method: A Semi-analytical Approach,”Journal of Sound and Vibration, Vol. 190, pp. 41-63, 1996.
Bhat, R.B., Laura, P. A. A., Gutierrez, R. G., Cortines, V. H. and Sanzi, H. C. “Numerical Experiments on the Determination of Natural Frequencies of Transverse Vibrations of Rectangular Plates of Non-uniform Thickness,” Journal of Sound and Vibration, Vol. 138, pp. 205-219, 1990.
Choi, S.-T., and Chou, Y.-T., “Vibration Analysis of Elastically Supported Turbomachinery Blades by the Modified Differential Quadrature Method,” Journal of Sound and Vibration, Vol. 240, pp. 937-953, 2001.
Choi, S.-T., Chou, Y.-T. and Huang, C.-J., “Buckling and Vibration Analyses of Rectangular Plates by the Differential Quadrature Method,” Journal of the Chinese Society of Mechanical Engineers, Vol. 23, No. 1, pp. 1-9, 2002.
Choi, S. T., Wu, J. D., and Chou, Y. T., “Dynamic Analysis of a Spinning Timoshenko Beam by the Differential Quadrature Method,” AIAA Journal, Vol. 38, pp. 851-856, 2000.
Du, H., Lim, M. K., and Lin, R. M., “Application of Generalized Differential Quadrature Method to Structural Problems,” International Journal for Numerical Methods in Engineering, Vol. 37, pp. 1881-1896, 1994.
Gutierrez, R. H., and Laura, P. A. A., “Vibrations of Rectangular Plates with Linearly Varying Thickness and Non-uniform Boundary Conditions,” Journal of Sound and Vibration, Vol. 178, pp. 563-566, 1994.
Jang, S. K., Bert, C. W., and Striz, A. G., “Application of Differential Quadrature to Static Analysis of Structural Components,” International Journal for Numerical Methods in Engineering, Vol. 28, pp. 561-577, 1989.
Kanaka Raju, K., and Hinton, E., “Natural Frequencies and Modes of Rhombic Mindlin Plates,” Earthquake Engineering and Structural Dynamics, Vol. 8, pp. 55-62, 1980.
Kobayashi, H., and Sonoda, K., “Vibration and Buckling of Tapered Rectangular Plates with Two Opposite Edges Simply Supported and the Other Two Edges Elastically Restrained against Rotation,” Journal of Sound and Vibration, Vol. 146, pp. 323-337, 1991.
Kukreti, A. R., and Farsa, J., “Differential Quadrature and Rayleigh-Ritz Methods to Determine the Fundamental Frequencies of Simply Supported Rectangular Plates with Linearly Varying Thickness,” Journal of Sound and Vibration, Vol. 189, pp. 103-122, 1996.
Kukreti, A. R., and Farsa, J., “Fundamental Frequency of Tapered Plates by Differential Quadrature,” Journal of Engineering Mechanics, Vol. 118, pp. 1221-1238, 1992.
Leissa, A. W., “The Free Vibration of Rectangular Plates,” Journal of Sound and Vibration, Vol. 31, pp. 257-293, 1973.
Liew, K.M., Hung, K. C., and Lim, M. K., “Roles of Domain Decomposition Method in Plate Vibrations: Treatment of Mixed Discontinuous Periphery Boundaries ,” International Journal of Mechanical Science, Vol. 35, pp. 615-632, 1993a.
Liew, K. M., Xiang, Y., and Kitipornchai, S., “Transverse Vibration of Thick Rectangular Plates―I. Comprehensive Sets of Boundary Conditions,” Computers and Structures, Vol. 49, pp. 1-29, 1993b.
Liu, F.-L., and Liew, K. M., “Static Analysis of Reissner-Mindlin Plates by Differential Quadrature Element Method,” Journal of Applied Mechanics, Vol. 65, pp. 705-710, 1998.
Malik, M., and Bert, C. W., “Implementing Multiple Boundary Conditions in the DQ Solution of Higher Order PDE’s: Application to Free Vibration of Plates,” International Journal for Numerical Methods in Engineering, Vol. 39, pp. 1237-1258, 1996.
Liu, F.-L., and Liew, K. M., “Analysis of Vibrating Thick Rectangular Plates with Mixed Boundary Constraints Using Differential Quadrature Element Method,” Journal of Sound and Vibration, Vol. 225, pp. 915-934, 1999.
Mindlin, R. D., “Influence of Rotary Inertia and Shear on Flexural Motion of Isotropic, Elastic Plates,” Journal of Applied Mechanics, Vol. 18, pp. 31-38, 1951.
Shu, C., and Richards, B. E., “Application of Generalized Differential Quadrature to Solve Two-dimensional Incompressible Navier-Stokes Equations,” International Journal for Numerical Methods in Fluids, Vol. 15, pp. 791-798, 1992.
Shu, C., and Wang, C. M., “Treatment of Mixed and Nonuniform Boundary Conditions in GDQ Vibration Analysis of Rectangular Plates,” Enginnering Structures, Vol. 21, pp. 125-134, 1999.
Wang, X., and Bert, C. W., and Striz, A. G., “Differential Quadrature Analysis of Deflection, Buckling, and Free Vibration of Beams and Rectanglar Plates,” Computers and Structures, Vol. 48, pp. 473-479, 1993.
Wang, X., and Bert, C. W., “A New Approach in Applying Differential Quadrature to Static and Free Vibrational Analyses of Beams and Plates,” Journal of Sound and Vibration, Vol. 162, pp. 566-572, 1993.
Yuan, J. and Dickinson, S. M., “The Flexural Vibration of Rectangular Plate Systems Approached by Using Artificial Springs in the Rayleigh-Ritz Method,” Journal of Sound and Vibration, Vol. 159, pp. 39-55, 1992.
周玉端, 民國八十九年六月 , “改良型微分值積法及其元素法於結構力學之應用,” 國立成功大學博士論文。