此篇論文中，我主要探討內含週期性薄膜型共振器之超穎材料結構樑的波傳行為，結構中包含了可以視為局部共振器之薄膜質量系統，從超穎結構樑之色散關係圖可以發現頻率能隙出現在低頻範圍中，此能隙之位置取決於薄膜質量系統的局部共振頻率，藉由調整質量大小與薄膜張力，就能夠控制頻率能隙之位置，我們利用薄板理論與能量守恆定律可以計算薄膜質量系統的基本頻率，並使用有限元素分析軟體(COMSOL Multiphysics)模擬超穎材料結構樑之波傳行為，而在有效質量密度之分析中，可以發現負質量密度的特性出現於能隙之頻率範圍下，運用多重質量共振器與雙層共振器都能夠達到拓寬頻率能隙寬度之目的。

In this thesis, I studied the wave motion in a metamaterial beam with circular membrane-mass structures that can be considered as local resonators. Due to the local resonance of the membrane-mass resonator, locally resonant band gaps can be created in the dispersion relation of metamaterial beam. The frequency range of a locally resonant band gap can be tuned by adjusting the mass magnitude and membrane tension. A thin plate theory and the principle of conservation of energy were used to calculate the fundamental frequency of a membrane-mass resonator. In order to simulate the propagation behavior of the metamaterial beam, I also used finite element software (COMSOL Multiphysics). The effective mass density analysis demonstrated that effective mass density becomes negative within the locally resonant band gap frequencies. Multiple kinds of cells and double-layer resonators were used for broadening the band gap width.

[1] M. M. Sigalas and E. N. Economou, Elastic and acoustic wave band structure, Journal of Sound and Vibration 158(2), 377-382 (1992).
[2] M. S. Kushwaha, P. Halevi, G. Martínez, L. Dobrzynski, and B. Djafari-Rouhani, Theory of acoustic band structure of periodic elastic composites, Physical Review B 49(4), 2313-2322 (1994).
[3] M. Torres, F. R. Montero de Espinosa, D. García-Pablos, and N. García, Sonic Band Gaps in Finite Elastic Media: Surface States and Localization Phenomena in Linear and Point Defects, Physical Review Letters 82(15), 3054-3057 (1999).
[4] Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng, Locally Resonant Sonic Materials, Science 289(5485), 1734-1736 (2000).
[5] K. M. Ho, C. K. Cheng, Z. Yang, X. X. Zhang, and P. Sheng, Broadband locally resonant sonic shields, Applied Physics Letters 83(26), 5566-5568 (2003).
[6] H. H. Huang and C. T. Sun, Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density, New Journal of Physics 11(1), 013003 (2009).
[7] Y. Xiao, J. Wen, D. Yu, and X. Wen, Flexural wave propagation in beams with periodically attached vibration absorbers: Band-gap behavior and band formation mechanisms, Journal of Sound and Vibration 332(4), 867-893 (2013).
[8] X. Yong, W. Jihong, and W. Xisen, Flexural wave band gaps in locally resonant thin plates with periodically attached spring–mass resonators, Journal of Physics D: Applied Physics 45(19), 195401 (2012).
[9] J. S. Chen and Y. J. Huang, Wave Propagation in Sandwich Structures With Multiresonators, Journal of Vibration and Acoustics 138(4), 041009 (2016).
[10] C. J. Naify, C.-M. Chang, G. McKnight, and S. Nutt, Transmission loss and dynamic response of membrane-type locally resonant acoustic metamaterials, Journal of Applied Physics 108(11), 114905 (2010).
[11] C. J. Naify, C.-M. Chang, G. McKnight, and S. Nutt, Transmission loss of membrane-type acoustic metamaterials with coaxial ring masses, Journal of Applied Physics 110(12), 124903 (2011).
[12] J. S. Chen and D. W. Kao, Sound Attenuation of Membranes Loaded with Square Frame-Shaped Masses, Mathematical Problems in Engineering 2016, 11 (2016).
[13] M. Nouh, O. Aldraihem, and A. Baz, Vibration Characteristics of Metamaterial Beams With Periodic Local Resonances, Journal of Vibration and Acoustics 136(6), 061012 (2014).
[14] M. Nouh, O. Aldraihem, and A. Baz. Metamaterial structures with periodic local resonances, Proceedings of SPIE 90641, (2014).
[15] M. Nouh, O. Aldraihem, and A. Baz, Wave propagation in metamaterial plates with periodic local resonances, Journal of Sound and Vibration 341, 53-73 (2015).
[16] M. A. Nouh, O. J. Aldraihem, and A. Baz, Periodic metamaterial plates with smart tunable local resonators, Journal of Intelligent Material Systems and Structures 27(13), 1829-1845 (2015).
[17] P. Hagedorn and A. DasGupta, Vibrations and Waves in Continuous Mechanical Systems, Wiley UK, 231-234 (2007).
[18] W. Soedel, Vibrations of Shells and Plates, Third ed, Marcel Dekker New York, 105 (2005).
[19] Structure Mechanics Module Model Library Manual, COMSOL version 5.0
[20] L. Y. Wu, The Foci and Dispersions of Negative Reaction Sonic Crystals with Circular and Elliptic Rods, Department of Mechanical Engineering, National Cheng Kung University, Tainan, (2007).
[21] C. S. Yu, C. Y. Hsiao, and H. H. Huang. Study of Wave Propagation Behavior of Elastic Metabeam, Proceedings of 36th Ocean Engineering Conference in Taiwan, National Chiao Tung University, (2014).
[22] Y. J. Huang, Flexural Wave Propagation in a Metamaterial Beam with Membrane-Mass Structures, Department of Engineering Science, National Cheng Kung University, Tainan, (2016).