三明治結構是一種常用的複合材料，由於其具有高比勁度、高比強度以及質量輕等優點，近年來已被大量地使用，特別是在航空、汽車、造船、國防工業方面。三明治結構的芯材多半是選擇質量輕的材料（發泡性高分子材料或者是蜂巢狀結構）；而面材通常是高強度以及高勁度的材料，例如纖維複合材料(碳纖維、玻璃纖維)或是金屬材料。本文所使用的三明治結構其芯材由環氧樹脂添加奈米碳管組成，而面材則為碳纖維疊層板的複合材料，並利用環氧樹脂做為面材和芯材的黏著劑。
本文以有限單元分析軟體ANSYS及實驗探討具有脫膠區之三明治結構之振動行為，並觀察芯材奈米碳管含量、面材碳纖維疊層板堆疊角度、以及脫膠區等三項因素對振動頻率及其模態的影響。
結果顯示芯材中添加奈米碳管會提高三明治結構的自然頻率，而不同面材堆疊方式也對自然頻率及模態有所影響；此外脫膠區越長時，自然頻率下降越多。

Because of its’ high specific strength and high specific stiffness, sandwich structure has been widely used in many applications, especially in automotive industry, aviation, shipbuilding and national defense industries. Usually, the core material of sandwich structure is light foam material or honeycomb structure, and high strength material, like fiber composites or metal material, is chosen for the facesheet. In this paper, we use MWNTs/epoxy for the core and carbon fiber laminate for the facesheet. The facesheet and core were glued together by epoxy.
The finite element analysis software ANSYS and experiment were used to study the vibration behavior of sandwich structure with or without debonding region. The influence of MWNTs content, facesheet stacking sequence and debonding length were discussed in this paper.
The results show that with the increase of MWNTs in core, the natural frequencies of the sandwich structure increased. Different CFRP stacking sequences influenced the natural frequencies and mode shapes. Besides, with the increase of debonding length, the natural frequencies of the sandwich structure decreased more.

1. N. J. Pagano and R. B. Pipes, “The Influence of Stacking Sequence on Laminate Strength,” Journal of Composite Materials, Vol. 5, pp.50-57, 1971.
2. A. Muc, “Optimal Fiber Orientation for Simply-Supported, Angle-Ply Plate Under Biaxial Compression,” Composite Structures, Vol. 9, pp.161-172, 1988
3. J. Brillaur and A. E. Mahi, “Numerical Simulation of the Influence of Stacking Sequence on Transverse Ply Cracking in Composite,” Composite Structures, Vol. 17, pp. 23-35, 1991.
4. T. C. Truong, M. Vettori, S. Lomov and I. Verpoest, “Carbon composites based on multi-axial multi-ply stitched preforms. Part 4. Mechanical properties of composites and damage observation,” Composites: Part A, Vol. 36, pp. 1207-1221, 2005.
5. L. D. Patrick and D. Laurent, “Numerical tools for advanced composite materials: a mean to reduce costs and improve performance,” The 24th international SAMPE Europe conference, Paris, France; 2003.
6. M. K. Yeh, N. H. Tai and J. H. Liu, “Mechanical behavior of phenolic-based composites reinforced with multiwalled carbon nanotubes,” Carbon, Vol. 44(1), pp. 1-9, 2006.
7. D. M. Luo, H. Yang, D. Yu and W. L. Zhu, “The Effects of Nano-Tube's Properties on the Mechanical Behaviors of Nano-Reinforced Composites,” Advanced Science Letters, Vol. 4, pp. 1819-1823, 2011.
8. X. Zhou, E. Shin, K. W. Wang and C. E. Bakis, “Interfacial damping characteristics of carbon nanotube-based composite,” Composites Science and Technology, Vol. 64, pp. 2425-37, 2004.
9. M. K. Yeh and T. H. Hsieh, “Bending property of sandwich beams with MWNT/polymer nanocomposites as core materials,” Key Engineering Materials, pp. 345-46, 1265-8, 2007.
10. N. Mathew, Z. Jiang and D. Wei, “Analysis of Multi-Layer Sandwich Structures by Finite Element Method,” Advanced Science Letters, Vol. 4, pp. 3243-3248, 2011.
11. J. A. Zapfe and G. A. Lesieutre, “A discrete layer beam finite element for the dynamic analysis of composite sandwich with integral damping layer,” Computers & Structures, Vol. 70, pp. 647-66, 1999.
12. C. A. Steevens and N. A. Flect, “Material Selection in Sandwich Beam Construction,” Scripta Materialia, Vol. 50, pp. 1335-1339, 2004.
13. J. Kim and S. R. Swanson, “Design of Sandwich Structures for Concentrated Loading,” Composite Structures, Vol. 52, pp. 365-373, 2001.
14. G. R. Monfortion and I. M. Ibrahim, “Analysis of Sandwich Plates with Unbalanced Cross-ply Faces,” International Journal of Mechanical Sciences, Vol. 17, pp. 227-238. 1975.
15. G. R. Monfortion and I. M. Ibrahim, “Modified Stiffness Formulation of Unbalanced Anisotropic Sandwich Plates,” International Journal of Mechanical Sciences, Vol. 19, pp. 335-343 1977.
16. I. M. Ibrahim, A. Farah and M. N. F. Rizk, “Dynamic Analysis of Unbalanced Anisotropic Sandwich Plates,” Journal of Engineering Mechanics ASCE, Vol. 107, pp. 405-418, 1981.
17. C. M. Wang, K. K. Ang and L. Yang, “Free Vibration of Skew Sandwich Plates with Laminated Facings,” Journal of Sound and Vibration, Vol. 235(2), pp. 317-340, 2000.
18. Y. Y. Yu, “Flexural Vibrations of Elastic Sandwich Plates,” Journal of Aerospace Sciences, Vol. 27, pp. 272-282, 1960.
19. Y. Y. Yu, “Simplified Vibration Analysis of Elastic Sandwich Plates,” Journal of Aerospace Sciences, Vol. 27, pp. 894–900, 1960.
20. J. S. Chang and H. C. Chen, “Vibration of Sandwich Plates of Variable Thickness,” Journal of Sound and Vibration, Vol. 155, pp. 195-208, 1992.
21. S. S. F. Ng and B. Das, “Free Vibration and Buckling Analysis of Clamped Skew Sandwich Plates by the Galerkin Method,” Journal of Sound and Vibration, Vol. 107, pp. 97-106, 1986.
22. U. K. Vaidya, S. Pillay, S. Bartus, C. A. Ulven, D. T. Grow and B. Mathew, “Impact and Post-Impact Vibration Response of Protective Metal form Composite Sandwich Plates,” Materials Science and Engineering A, Vol. 428, pp. 59-66, 2006.
23. 謝宗翰，多壁奈米碳管補強高分子樹脂之複材樑與三明治結構動態特性分析，國立清華大學動力機械工程研究所碩士論文，2006。
24. H. H. Kanematsu and Y. Hirano, “Bending and Vibration of CFRP-Faced Rectangular Sandwich Plates,” Composite Structures, Vol. 10, pp. 145-163, 1988.
25. W. X. Yuan and D. J. Dawe, “Free Vibration of Sandwich Plates with Laminated Faces,” International Journal For Numerical Methods in Engineering, Vol. 54, pp. 195-217, 2002.
26. M. E. Raville and C. E. S. Veng, “Determination of Natural Frequencies of Vibration of Sandwich Plates,” Experimental Mechanics, Vol. 7, pp. 490-493, 1967.
27. L. Jing, Q. Yan, Z. Wang and L. Zhao, “The Dynamic Mechanical Behavior of Sandwich Beams with Aluminum Honeycomb Cores,” Advanced Science Letters, Vol. 4, pp. 731-735, 2011.
28. N. S. Bardell and J. M. Dunsdon, R. S. Langley, “Free Vibration Analysis of Coplanar Sandwich Panels,” Composite Structures, Vol. 38, pp. 463-475, 1997.
29. L. J. Lee and Y. J. Fan, “Bending and Vibration Analysis of Composite Sandwich Plates,” Computers and Structures, Vol. 60, pp. 103-112, 1996.
30. T. P. Khatua and Y. K. Cheung, “Bending and Vibration of Multi-layer Sandwich Beams and Plates,” International Journal for Numerical Methods in Engineering, Vol. 6, pp. 11-24, 1973.
31. J. T. S. Wang, Y. Y. Liu and J. A. Gibby, “Vibrations of Split Beams,” Journal of Sound and Vibration, Vol. 84(4), pp. 491-502, 1982.
32. P. M. Mujumdar and S. Suryanarayan, “Flexural Vibrations of Beams with Delaminations,” Journal of Sound and Vibration, Vol. 125(3), pp. 441-461, 1988.
33. J. J. Tracy and G. C. Pardoen, “Effect of delamination on the natural frequencies of composite laminates,” Journal of Composite Materials, Vol. 23(12), pp. 1200-1215, 1989.
34. P. Cawley and R. D. Adams, “The Location of Defects in Structures from Measurements of Natural Frequencies from Measurements of Natural Frequencies,” Journal of Strain Analysis for Engineering Design, Vol. 14(2), pp. 49-57, 1979.
35. B. Crema, L., A. Castellani and I. Pcroni, “Modal Tests on Composite Material Structures. Application in Damage Detection,” in Proceedings of the 3rd International Modal Analysis Conference (IMAC), pp. 708-713, 1985.
36. D. C. Zimmerman, T. Simmermacher and M. Kauok, “Structural Damage Detection Using Frequency Response Functions,” in Proceedings of the 13th International Modal Analysis Conference (IMAC), Nashville TN, pp. 179-184, 1995.
37. H. Baru and S. Ratan, “Damage Detection in Flexible Structures,” Journal of Sound and Vibration, Vol. 166(1), pp. 21-30, 1993.
38. B. T. Lee, C. T. Sun and D. Liu, “An Assessment of Damping Measurement in the Evaluation of Integrity of Composite Beams,” Journal of Reinforced Plastics and Composites, Vol. 6, pp. 114-125, 1987.
39. A. F. Jensen and F. Irgens, “Thickness Vibrations of Sandwich Plates and Beams and Delamination Detection,” Journal of Intelligent Material Systems and Structures, Vol. 10(1), pp. 46-55, 1999.
40. H. Y. Kim and W. Hwang, “Effect of Debonding on Natural Frequencies and Frequency Response Functions of Honeycomb Sandwich Beams,” Composite Structure, Vol. 55, pp. 51-62, 2002.
41. D. Shu, “Vibration of Sandwich Beams with Double Delaminations,” Composites Science and Technology, Vol. 54, pp. 101-109, 1995.
42. C. N. Della and D. Shu, “Free Vibration Analysis of Composite Beams with Overlapping Delaminations,” European Journal of Mechanics A/Solids, Vol. 24, pp. 491-503, 2005.
43. S. Lee, T. Park and G. Z. Voyiadjis, “Vibration Analysis of Multi-delaminated Beams,” Composites Part B-engineering, Vol. 34, pp. 647-659, 2003.
44. L. H, Tenek, E. G. Henneke II and M. D. Gunzburger, “Vibration of Delaminated Composite Plates and Some Applications to Non-destructive Testing,” Composite Structures, Vol. 23, pp. 253-262, 1993.
45. S. Yang, R. F. Gibson, L. Gu and W. H. Chen, “Modal Parameter Evaluation of Degraded Adhesively Bonded Composite Beams,” Composite Structures, Vol. 43, pp. 79-91, 1998.
46. B. O. Baba, S. Thoppul, “Experimental Evaluation of the Vibration Behavior of Fat and Curved Sandwich Composite Beams with Face/Core Debond,” Composite Structures, Vol. 91, pp. 110-119, 2009.
47. H. P. Chen, “Free Vibration of Prebuckled and Postbuckled Plates with Delamination,” Composites Science and Technology, Vol. 51(3), pp. 451-462, 1994.
48. A. Paolozzi and I. Peroni, “Detection of Debonding Damage in a Composite Plate through Natural Frequency Variations,” Journal of Reinforced Plastics and Composites, Vol. 9, pp. 369-389, 1990.
49. V. N. Burlayenko and T. Sadowski, “Influence of Skin/Core Debonding on Free Vibration Behavior of Foam and Honeycomb Cored Sandwich Plates,” International Journal of Non-Linear Mechanics, Vol. 45, pp. 959-968, 2010.
50. ANSYS Release 12.1, ANSYS, Inc., PA, 2010.
51. R. F. Gibson, Principles of Composite Material Mechanices, 2nd ed., McGraw-Hill, New York, 2007.
52. P. Qiao, W. Lestari, M. G. Shah and J. Wang, “Dynamics-based Damage Detection of Composite Laminated Beams using Contact and Noncontact Measurement Systems,” Journal of Composite Materials, Vol. 42, No. 10, pp. 1217-1252, 2007.
53. M. Mehdizadeh, K. Oruganti, M. Bannister, I. Herzberg and S. John, “Vibration Based Analysis of an Increasing Delamination in a Carbon/Epoxy Composite Structure,” The 3rd International Conference on Integrity, Reliability and Failure, Porto, Portugal, pp. 1-17, 2009.
54. J. W. Dally and W. F. Riley, Experimental Stress Analysis, McGraw-Hill, New York, 1991.
55. ASTM D638-10, “Standard Test Method for Tensile Properties of Plastics,” Annual Book of ASTM Standards, Vol. 8.1, 2010.
56. ASTM D3039-08, “Standard Test Method for Tensile Properties of Fiber-Resin Composites,” Annual Book of ASTM Standards, Vol. 15.03, 2008.
57. ASTM D3518-01, “Standard Practice for Inplane Shear Stress-strain Response of Unidirectional Reinforced Plastics,” Annual Book of ASTM Standards, Vol. 15.03, 2007.
58. L. A. Carlsson and R. B. Pipes, Experimental Characterization of Advanced Composite Materials, Prentic-Hall, New Jersey, 1987.
59. ASTM E756-05, “Standard Test Method for Measuring Vibration-Damping Properties of Materials,” Annual Book of ASTM Standards, Vol. 4.06, 2010.
60. M. K. Yeh and T. H. Hsieh, “Dynamic properties of sandwich beams with MWNT/polymer nanocomposites as core materials,” Composites Science and Technology, Vol. 68, pp. 2930-36, 2008.
61. 邱郁文，含脫層奈米複材三明治結構之彎曲特性與局部挫曲分析與實驗，國立清華大學動力機械工程研究所碩士論文，2011。