本文以Soedel [18]之厚殼理論及Mead與Markus [3]之拘束阻尼層(Constrained Layer Damping, CLD)理論為本，導出黏彈夾心之三層厚殼能量式，應用漢彌爾敦定理(Hamilton’s principle)，推導出具九個位移變數之三層厚殼振動理論的廣義運動方程式。之後假設拘束層相對於主結構層可視為薄殼結構，採Donnell-Mushtari-Vlasov之假設簡化拘束層，應用平衡方程式，推導出僅含五個位移變數之黏彈夾心三層厚殼振動理論，此五個位移函數來自：一為主結構層之橫向(transverse)位移，二為主結構層之共平面(in-plane)方向位移，另二為主結構層之橫向剪切(transverse shear)方向位移。本理論除可直接應用於具拘束阻尼層殼結構之理論分析，亦可簡化成單層厚結構之振動理論。此廣義理論可特殊化成各種工程上常見的幾何結構，如厚樑、厚板、圓柱厚殼…等等。在數值分析方面，藉相關文獻與單層、三層之薄結構體作比較，可提供單層薄、厚結構體與黏彈三層薄、厚結構體之適用範圍；續變動三層結構厚度，探討參數對系統的固有頻率與模態阻尼之影響，提供改變拘束阻尼層時之參數考量。

General differential equations of motion for a three-layer sandwich thick shell with viscoelastic core are derived. First of all, based on Soedel consider shear deflection and rotatory inertia on the equations of motion of shell structure and by referring to the theory of Mead and Markus, the energy expressions for a three-layer thick shell with a constrained layer damping (CLD) are derived. Then application Hamilton’s principle and Donnell-Mushtari-Vlasov simplification are employed in the derivation. The differential equations, unlike the existing models with nine displacements, contain only five displacements, which are one transverse displacement, two in-plane displacements and two transverse shear displacements of the host structure. The proposed theory is generic and can be specialized to account for many other commonly occurring geometry, such as cylindrical thick shells, thick plates, thick beams, …etc. The derived theory can be directly applied to the studies of structures with constrained layer damping (CLD). The theory can also degenerate into one single layer structures. Specialization of the theory to a cylindrical shell, to a rectangular plate, and to a beam is demonstrated in a sequel as some of the possible applications. The influences of CLD treatment on the reduction of vibration induced by change structure thickness are discussed.

[1] Kerwin, E. M. Jr., “Damping of Flexural Waves by a Constrained Visco-Elastic Layer,” Journal of the Acoustical Society of America, 31, pp. 952-962(1959).
[2] DiTaranto, R. A., “Theory of Vibratory Bending for Elastic and Viscoelastic Layered Finite-Length Beams,” Journal of the Applied Mechanics, ASME, pp. 881-886(1965).
[3] Mead, D. J. and Markus, S., “The Forced Vibration of a Three-Layer Damping Sandwich Beam with Arbitrary Boundary Conditions,” AIAA Journal, 10(2), pp. 163-175(1969).
[4] Yan, M. J. and Dowell, E. H., “Governing Equations of Vibrating Constrained-Layer Damping Sandwich Plates and Beams,” Journal of the Applied Mechanics, ASME, pp. 1041-1046(1972).
[5] Douglas, B. E. and Yang, J. C. S., “Transverse Compressional Damping in the Vibratory Response of Elastic-Viscoelastic Beams,” AIAA Journal, 16(9), pp. 925-930(1978).
[6] Van Nostrand, W. C. and Inman, D. J., “Finite Element Model for Active Constrained Layer Damping,” Proceedings 31st Society of Engineering Science Meeting publish by SPIE, Active Materials and Smart Structures, ec. G. L. Anderson and D. C. Lagoucas, 2427, pp. 124-139(1994).
[7] Lam, M. J., Saunders, W. R. and Inman, D. J., “Modeling Active Constrained-Layer Damping Using Finite Element Analysis and GHM Damping Approach,” North American Conference on Smart Structures and Materials, pp. 124-139(1994).
[8] Liao, W. H. and Wang, K. W., “On the Active-Passive Hybrid Vibration Control Actions of Structures with Active Constrained Layer Treatments,” Proceeding ASME 15st Biennial Conference on Vibration and Noise(DE 84-3), 3 Part C, pp. 124-141(1995).
[9] Rao, D. K., “Frequency and loss factors of sandwich beams under various boundary conditions,” Journal of Mechanical Engineering Science, 20(5), pp. 271-282(1978).
[10]Lall, A. K., Asnani, N. T. and Nakra, B. C., “Damping analysis of partially covered sandwich beams,” Journal of Sound and Vibration, 123(2), pp. 247-259(1988).
[11]Mead, D. J. and Yaman, Y., “The Harmonic Response of Rectangular Sandwich Plates with Multiple Stiffening: A Flexural Wave Analysis,” Journal of Sound and Vibration, 145(3), pp. 409-428(1991).
[12]Rao, M. D. and He, S., “Dynamic Analysis and Desing of Laminated Composite Beams with Multiple Damping Layers,” AIAA Journal, 31(4), pp. 736-745(1993).
[13]Huang, S. C. and Chen, L. H., “Dynamic control of active and passive constrained layer damping treatments on a cylindrical shell,” Journal of the Chinese Society of Mechanical Engineers, 20(4), pp. 385-397(1999).
[14]Huang, S. C. and Chen, Y. C., “Parametric Effects on the Vibration of Plates with CLD treatment,” Journal of the Chinese Society of Mechanical Engineers, 20(2), pp. 159-167(1999).
[15]Huang, S. C. and Hu, Y. C., “The frequency response and damping effect of three layer thin shell with viscoelastic core,” Computers and Structures, 76, pp. 577-591(2000).
[16]Nayfeh, S. A. and Varanasi, K. K., “A model for the damping of torsional vibration in thin-walled tubes with constrained viscoelastic layers,” Journal of Sound and Vibration, 278, pp. 825-846(2004).
[17]Huang, S. C., Inman, D. J., and Austin, E. M., “Some Design Considerations for Active and Passive Constrained Lager Damping Treatments,” Journal of Smart Material and Structure, 5, pp. 301-313(1995).
[18]Soedel, Werner., Vibrations of Shells and Plates, Marcel Dekker, Inc., New York(1993).
[19]Sun, C. T. and Lu, Y. P., Vibration Damping of Structural Elements, Prentice-Hall, New Jersey(1995).
[20]Meirovitch, L., Principles and Techniques of Vibration, Prentice-Hall, Inc.,(1997).