ELASTIC WAVE ABSORPTION IN LASER-CUT ACOUSTIC METAMATERIAL PLATES A Dissertation Presented to the Faculty of the Graduate School University of Missouri In Partial Fulfillment Of the Requirements for the Degree Doctor of Philosophy by HAOGUANG DENG Dr. P. Frank Pai, Dissertation Supervisor Dr. Guoliang Huang, Dissertation Co-Supervisor December 2016 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled ELASTIC WAVE ABSORPTION IN LASER-CUT ACOUSTIC METAMATERIAL PLATES Presented by Haoguang Deng A candidate for the degree of Doctor of Philosophy in Mechanical & Aerospace Engineering And hereby certify that in their opinion it is worthy of acceptance. Professor P. Frank Pai Professor Guoliang Huang Professor Ming Xin Professor Steven Neal Professor Stephen Montgomery-Smith ACKNOWLEDGEMENTS I would like to thank the many people who contributed to this dissertation. Without their help, I would certainly not have been able to complete this work. First and foremost, I would like to express my sincere gratitude to my advisor Dr. P. Frank Pai. Without his academic insight and encouragement, this dissertation would not have been possible. I am very grateful for his guidance and for the many stimulating discussions on a variety of professional and personal topics. It was a really great pleasure working under his supervision. What I learned from him is a treasure for my future. I hope Dr. Pai will recover from illness and get back to work very soon! I also wish to thank my dissertation co-advisor, Dr. Guoliang Huang for his helpful suggestions. His enthusiastic encouragements and unwavering supports to this research are greatly appreciated. At the same time, many thanks to all the committee members, Dr. Ming Xin, Dr. Steven Neal and Dr. Stephen Montgomery-Smith, for their invaluable suggestions and helps. Special thanks to Dr. Hao Peng who helped me have a complete understanding of metamaterial theory and finite element analysis. Moreover, I would like to extend my thanks to Jerome Rivers, Miles Barnhart, Xuewei Ruan, Rumian Zhong and Tiancheng Xu for their helps and encouragements. Their supports make it possible to finish this dissertation on time. ii Finally, I wish to give my special thanks to my parents Mr. Xiaodong Deng, Mrs. Shihui Chen and my girlfriend Tong Li. Their company, encouragements, full support of my study and endless love make things around me easier during my hard times. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS ................................................................................................ ii LIST OF FIGURES ........................................................................................................... vi ABSTRACT ....................................................................................................................... x Chapter 1. INTRODUCTION ........................................................................................... 1 1.1 Introduction to Metamaterials .......................................................................... 1 1.2 Dissertation Organization ................................................................................ 6 Chapter 2. BASIC CONCEPTS OF ELECTROMAGNETIC METAMATERIALS ....... 8 2.1 Introduction to Electromagnetic Metamaterials .............................................. 8 2.2 Negative Permittivity and Permeability ........................................................... 9 2.3 Reversed Doppler Effect and Reversed Vavilov-Cerenkov Effect ............... 12 2.4 Negative Refraction and Super Lens ............................................................. 14 2.5 Metamaterial Cloaking .................................................................................. 18 Chapter 3. BASIC CONCEPTS OF ACOUSTIC METAMATERIALS ........................ 20 3.1 Introduction to Acoustic Metamaterials ........................................................ 20 3.2 Negative Effective Mass of Mass-On-Mass System ..................................... 24 3.3 Negative Effective Stiffness of Mass-On-Spring System ............................. 26 3.4 Stopband and Dispersion of Acoustic Metamaterials .................................... 29 3.5 Single-Frequency Vibration Absorber ........................................................... 29 3.6 Multi-Frequency Vibration Absorber ............................................................ 34 3.7 Acoustic Metamaterials Application: Seismic Waveguide ........................... 42 Chapter 4. CONVENTIONAL ACOUSTIC METAMATERIAL PLATES .................. 44 4.1 Prototype and Finite Element Analysis of Single-Stopband Acoustic Metamaterial Plates ....................................................................................................... 44 iv 4.2 Prototype and Finite Element Analysis of Multi-Stopband Acoustic Metamaterial Plates ....................................................................................................... 54 4.3 Design Guidelines for Vibration Absorbers .................................................. 59 Chapter 5. LASER-CUT ACOUSTIC METAMATERIAL PLATES ............................ 60 5.1 Prototype of Single-Stopband Laser-Cut Acoustic Metamaterial Plates ....... 60 5.2 Prototype of Multi-Stopband Laser-Cut Acoustic Metamaterial Plates ........ 63 Chapter 6. FINITE ELEMENT ANALYSIS OF RECTANGULAR PLATES .............. 66 6.1 Finite Element Analysis of Conforming Rectangular Plate Elements........... 66 6.2 Modal Analysis of Rectangular Plates ........................................................... 72 6.3 Dispersion Analysis: Bloch Wave Analysis of Rectangular Plates ............... 74 6.4 Frequency Response Analysis of Rectangular Plates .................................... 76 6.5 Transient Analysis of Rectangular Plates ...................................................... 77 Chapter 7. NUMERICAL RESULTS ............................................................................. 81 7.1 Isotropic Plate ................................................................................................ 81 7.2 Single-Stopband Laser-Cut Acoustic Metamaterial Plate ............................. 84 7.3 Multi-Stopband Laser-Cut Acoustic Metamaterial Plate ............................ 100 Chapter 8. CONCLUSIONS AND RECOMMENDATIONS ...................................... 121 8.1 Concluding Remarks ................................................................................... 121 8.2 Recommendations for Future Work ............................................................ 123 REFERENCES ............................................................................................................... 124 VITA ................................................................................................................... 129 v LIST OF FIGURES Figure Page Figure 2-1 Material types based on different signs of permittivity ( ) and permeability ( ). .................................................................................................................................... 9 Figure 2-2 Design for a cubic SRR, proposed by C. R. Simovski and S. He, 2003 [31]. 12 Figure 2-3 (a) Doppler effect in a right-handed medium; (b) Doppler effect in a left-handed medium. The letter A represents the source of radiation while the letter B represents the receiver. ............................................................................................................................. 13 Figure 2-4 Passage of a ray through the interface between two media. 1-incident ray; 2- reflected ray; 3-refracted ray for a left-handed material; 4-refracted ray for a right-handed material. ............................................................................................................................ 15 Figure 2-5 Passage of rays of light through a plate of thickness d made of a left-handed substance. A represents the source of radiation and B represents the detector of radiation. ........................................................................................................................................... 17 Figure 2-6 Paths of rays through lenses made of left-handed substances, situated in a vacuum. ............................................................................................................................. 17 Figure 2-7 ([36] From Pendry, “Controlling electromagnetic field,” Science 312, 1780- 1782 (2006), Reprinted with people from AAAS) a schematic that shows the principle on which coordinate-transformation cloaking schemes work. The electromagnetic rays are detoured away from the cloaked region. ........................................................................... 19 Figure 3-1 A 2-DOF mass-on-mass system. .................................................................... 26 Figure 3-2 A 2-DOF mass-on-spring system. .................................................................. 28 Figure 3-3 A Simplified 2-DOF EMM unit. .................................................................... 32 Figure 3-4 Absolute frequency response of m1 with (a) mm2/ 1 0,0.1,0.2,0.3,0.4, 1 2 0.01, (b) mm21/ 0.1, 2 0.01,0.03,0.05,0.07,0.1. 33 Figure 3-5 A Simplified 3-DOF EMM unit. .................................................................... 36 Figure 3-6 Absolute frequency response of with mm2 3 0, 1 0.01, 2 3 0 (black) and mm21/ 0.2, mm3/ 2 0.1, 1 2 3 0.01(red). ..................................................... 37 Figure 3-7 Average frequency response of m1 with mm2/