本論文提出以值積元素法(Quadrature Element Method)來探討複材厚板之動態特性。複材厚板的運動方程式是以Mindlin一階剪力變形理論為基礎。首先應用值積法的轉換法則，將複材厚板元素之運動方程式離散化成為代數方程式；再經由元素的結合，得到整個系統的離散化代數方程式。本文研究的數值結果將先與有正確解或數值解的複材厚板問題做比較，以驗證值積元素法的準確性，再進一步探討不同邊界條件、疊層數、彈性模數比及寬厚比對複材厚板自然頻率的影響，並與文獻做比較。再討論複材厚板在不均勻厚度及邊界條件下對其自然頻率的影響。結果顯示值積元素法於複材厚板的分析非常方便及快速並且具有極佳的數值準確性。

　　In this thesis, the dynamic characteristic of thick composite laminated plates is investigated by using the Quadrature Element Method (QEM). Governing equations of thick laminated plates are based on Mindlin first-order shear deformation theory. First, we apply the formulation of differential quadrature to discretize the differential governing equations of laminated plate elements into algebraic ones, which are then assembled to get the discrete equations for the entire plate. To evaluate the accuracy of the QEM, numerical results obtained by the QEM are compared with exact or numerical results in the literature. Furthermore, effects of boundary condition, number of layers, modulus ratio and width-to-thickness ratio on the natural frequencies of thick composite laminated plates are studied and compared with those in the literature. Furthermore, natural frequencies of thick composite laminated plates with non-uniform boundaries and thickness are obtained. Numerical results show the high accuracy and efficiency of the QEM for vibration analysis of laminated plates.

1. E. Reissner, 1945, “The Effect of Transverse Shear Deformation on the Bending of Elastic Plates,” Journal of Applied Mechanics, Vol. 12, pp. A69-77.
2. R. D. Mindlin, 1951, “Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic‚ Elastic Plates,” Journal of Applied Mechanics, Vol. 18, pp. 31-38.
3. E. Reissner and Y. Stavsky, 1961, “Bending and Stretching of Certain Types of Heterogeneous Aeolotropic Elastic Plates,” Journal of Applied Mechanics, Vol. 28, pp. 402-408.
4. Y. Stavsky, 1961, “Bending and Stretching of Laminated Aeolotropic Plates,” ASCE Journal of Engineering Mechanics Division, Vol. 8, pp. 31-56.
5. P. C. Yang‚ C. H. Norris and Y. Stavsky, 1966, “Elastic Wave Propagation in Heterogeneous Plates,” International Journal of Solids and Structures‚ Vol. 2, pp. 665-684.
6. J. M. Whitney and N. J. Pagano, 1970, “Shear Deformation in Heterogeneous Anisotropic Plates,” Journal of Applied Mechanics, Vol. 37, pp. 1031-1036.
7. R. D. Mindlin, A. Schacknow and H. Deresiewicz, 1956, “Flexural Vibration of Rectangular Plates,” Journal of Applied Mechanics, Vol. 23, pp. 431-436.
8. J. N. Reddy and N. D. Phan, 1985, “Stability and Vibration of Isotropic, Orthotropic and Laminated Plates According to a Higher-order Shear Deformation Theory,” Journal of Sound and Vibration‚ Vol. 98‚ pp. 157-170.
9. N. D. Phan and J. N. Reddy, 1985, “Analysis of Laminated Composite Plates Using a Higher-Order Shear Deformation Theory,” International Journal for Numerical Methods in Engineering, Vol. 21, pp. 2201-2219.
10. A. A. Khdeir, 1988, “Free Vibration and Buckling of Symmetric Cross-Ply Laminated Plates by an Exact Method,” Journal of Sound and Vibration‚ Vol. 126‚ No. 3, pp. 447-461.
11. N. S. Putcha and J. N. Reddy, 1986, “Stability and Natural Vibration Analysis of Laminated Plates by Using a Mixed Element Based on Refined Plate Theory,” Journal of Sound and Vibration‚ Vol. 104‚ pp. 285-300.
12. A. K. Noor, 1973, “Free Vibrations of Multilayered Composite Plates,” AIAA Journal, Vol. 11‚ pp. 1038-1039.
13. J. N. Reddy, 1997, Mechanics of Laminated Composite Plates .Theory and Analysis, CRC Press, Boca Raton, FL.
14. L. X. Luccioni and S. B. Dong, 1998, “Levy-type Finite Element Analysis of Vibration and Stability of Thin and Thick Laminated Composite Rectangular Plates,” Composites Part B , Vol. 29, pp. 459-475.
15. C. C. Chao and Y. C. Chern, 2000, “Comparison of Natural Frequencies of Laminated by 3-D Theory,” Journal of Sound and Vibration‚ Vol. 230, No. 5‚ pp. 985-1007.
16. M. Aydogdu and T. Timarci, 2003, “Vibration Analysis of Cross-ply Laminated Square Plates with General Boundary Condition,” Composites Science and Technology, Vol. 63, pp. 1061-1070.
17. K. M. Liew, K. C. Hung and M. K. Lim, 1993, “Method of Domain Decomposition in Vibration of Mixed Edge Anisotropic Plates,” International Journal of Solids and Structures, Vol. 30, No. 23, pp. 3281-3301.
18. T.-P. Chang and M.-H. Wu, 1997, “On the Use of Characteristic Orthogonal Polynomials in the Free Vibration Analysis of Rectangular Anisotropic Plates with Mixed Boundaries and Concentrated Masses,” Computers and Structures, Vol. 62, No. 4, pp. 699-713.
19. K. M. Liew, K. C. Hung and M. K. Lim, 1993, “Method of Domain Decomposition in Vibration of Mixed Edge Anisotropic Plates,” International Journal of Solids and Structures, Vol. 30, No. 23, pp. 3281-3301.
20. D. J. Dawe and D. Tan, 1999, “Finite Strip Buckling and Free Vibration Analysis of Stepped Rectangular Composite Laminated Plates,” International Journal for Numerical Methods in Engineering, Vol. 46, pp. 1313-1334.
21. R. E. Bellman and J. Casti, 1971, “Differential Quadrature and Long-Term Integration,” Journal of Mathematical Analysis and Application, Vol. 34, pp. 235-238.
22. F. Civan and C. M. Sliepcevich, 1984, “Differential Quadrature for Multi-Dimensional Problems,” Journal of Mathematical Analysis and Application, Vol. 101, pp. 423-443.
23. C. W. Bert, S. K. Jang and A. G. Striz, 1988, “Two New Approximate Methods for Analyzing Free Vibration of Structural Components,” AIAA Journal, Vol. 26, pp. 612-618.
24. C. Shu and B. E. Richards, 1992, “Application of Generalized Differential Quadrature to Solve Two-dimensional Incompressible Navier-Stokes Equations,” International Journal of Numerical Methods for Fluids, Vol. 15, pp. 791-798.
25. X. Wang and C. W. Bert, 1993, “A New Approach in Applying Differential Quadrature to Static and Free Vibrational Analyses of Beams and Plates,” Journal of Sound and Vibration, Vol. 162, No. 3, pp. 566-572.
26. C. W. Bert and M. Malik, 1996, “Differential Quadrature Method in Computational Mechanics: A Review,” ASME Applied Mechanics Review, Vol. 49, No. 1, pp. 1-28.
27. X. Wang, C. W. Bert and A. G. Striz, 1993, “Differential Quadrature Analysis of Deflection, Buckling, and Free Vibration of Beams and Rectanglar Plates,” Computers and Structures, Vol. 48, pp. 473-479.
28. C. W. Bert, X. Wang and A. G. Striz, 1994, “Static and Free Vibrational Analysis of Beams and Plates by Differential Quadrature Method,” Acta Mechanics, Vol. 102, No. 1, pp. 11-24.
29. J. Farsa, A. R. Kukreti and C. W. Bert, 1993, “Fundamental Frequency Analysis of Laminated Rectangular Plates by Differential Quadrature Method,” International Journal for Numerical Methods in Engineering, Vol. 36, pp. 2341-2356.
30. C. W. Bert and M. Malik, 1997, “Differential Quadrature: A Powerful New Technique for Analysis of Composite Stuctures,” Composite Structures, Vol. 39, pp. 179-189.
31. S. T. Choi and Y. T. Chou, 2001, “Vibration Analysis of Elastically Supported Turbomachinery Blades by the Modified Differential Quadrature Method,” Journal of Sound and Vibration, Vol. 240, No. 5, pp. 937-953.
32. J. B. Han and K. M. Liew, 1997, “Numerical Differential Quadrature Method for Reissner/Mindlin Plates on Two-Parameter Foundations,” International Journal of Mechanical Science, Vol. 38, No. 9, pp. 977-989.
33. K. M. Liew and T. M. Teo, 1999, “Three-Dimensional Vibration Analysis of Rectangular Plates Based on Differential Quadrature Method,” Journal of Sound and Vibration‚ Vol. 220‚ No. 4, pp. 577-599.
34. K. M. Liew and F. L. Liu, 2000, “Differential Quadrature Method for Vibration Analysis of Shear Deformable Annular Sector Plates,” Journal of Sound and Vibration‚ Vol. 230‚ No. 2, pp. 335-356.
35. H. Zeng and C. W. Bert, 2001, “A Differential Quadrature Analysis of Vibration for Rectangular Stiffened Plates,” Journal of Sound and Vibration‚ Vol. 241‚ No. 2, pp. 247-252.
36. T. C. Fung, 2002, “Stability and Accuracy of Differential Quadrature Method in Solving Dynamic Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, pp. 1311-1331.
37. A. G. Striz, W. Chen and C. W. Bert, 1994, “Static Analysis of Structures by the Quadrature Element Method (QEM),” International Journal of Solids and Structures, Vol. 31, pp. 2807-2818.
38. K. M. Liew, J.-B. Han and Z. M. Xiao, 1997, “Vibration Analysis of Circular Mindlin Plates Using the Differential Quadrature Method,” Journal of Sound and Vibration, Vol. 205, pp. 617-630.
39. F.-L. Liu and K. M. Liew, 1998, “Static Analysis of Reissner-Mindlin Plates by Differential Quadrature Element Method,” Journal of Applied Mechanics, Vol. 65, pp. 705-710.
40. F.-L. Liu and K. M. Liew, 1999, “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.
41. 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, 台南市.
42. F.-L. Liu, 2000, “Static Analysis of Thick Rectangular Laminated Plates: Three-Dimensional Elasticity Solutions via Differential Quadrature Element Method,” International Journal of Solids and Structures, Vol. 37, pp. 7671-7688.
43. C.-N. Chen, 2001, “Vibration of Nonuniform Shear Deformable Axisymmetric Orthotropic Circular Plates Solved by DQEM,” Composite Structures, Vol. 53, pp. 257-264.
44. 黃俊智, 民國九十一年六月, 微分值積元素法於Mindlin平板之振動分析, 國立成功大學碩士論文, 台南市.
45. X. Wang , Y. Wang and Y. Zhou, 2004, “Application of a New Differential Quadrature Element Method to Free Vibration Analysis of Beams and Frame Structures,” Journal of Sound and Vibration, Vol. 269, pp. 1133-1141.
46. S. Srinivas and A. K. Rao, 1970, “Bending, Vibration and Buckling of Simply Supported Thick Orthotropic Rectangular Plates,” International Journal of Solids and Structures, Vol. 6, pp. 1463-1481.
47. J. N. Reddy, 1979, “Free Vibration of Antisymmetric Angle-ply Laminated Plates Including Transverse Shear Deformation by the Finite Element Method,” Journal of Sound and Vibration, Vol. 66, pp. 565-576.
48. C. W. Bert and T.-L. C. Chen, 1978, “Effect of Shear Deformation on Vibration of Antisymmetric Angle-ply Laminated Rectangular Plates,” International Journal of Solids and Structures, Vol. 4, pp. 465-473.