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Finite-Element Design of Metamaterial Beams For FINITE-ELEMENT DESIGN OF METAMATERIAL BEAMS FOR BROADBAND WAVE ABSORPTION _______________________________________ A Thesis presented to the Faculty of the Graduate School at the University of Missouri-Columbia _______________________________________________________ In Partial Fulfillment of the Requirements for the Degree Master of Science _____________________________________________________ by SHUYI JIANG Dr. P. Frank Pai, Thesis Supervisor MAY 2015 The undersigned, appointed by the Dean of the Graduate School, have examined the thesis entitled FINITE-ELEMENT DESIGN OF METAMATERIAL BEAMS FOR BROADBAND WAVE ABSORPTION Presented by Shuyi Jiang A candidate for the degree of Master of Science And hereby certify that in their opinion it is worthy of acceptance. Professor P. Frank Pai Professor Steven Neal Professor Stephen Montgomery-Smith ACKNOWLEDGEMENTS I would like to express my deepest appreciation to my advisor Dr. P. Frank Pai. Without his patient guidance, I wouldn’t have grown as a good researcher. His continuous encouragement and valuable suggestions on my thesis work meant a lot to me. Also I would like to thank my committee members, Dr. Steven Neal and Dr. Stephen Montgomery-Smith, for serving on my thesis committee and providing me assistance when I have difficulties. I would also like to extend my thanks to Dr. Hao Peng, Xuewei Ruan, Haoguang Deng, Yiqing Wang, Jamie Lamont and all my labmates. They helped my study and gave me confidence to reach the goal. Thanks to all the staff in the Mechanical and Aerospace Engineering Department for their hard work for me during my study at the University of Missouri. Finally, special thanks to my family for their mental and financial support through my life. Without their love and support my study could not be so smooth and fruitful. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS .......................................................................................... ii LIST OF FIGURES .................................................................................................... v ABSTRACT ............................................................................................................ ix Chapter 1 Introduction to Metamaterials ............................................................ 1 1.1 Background ..................................................................................................... 1 1.2 Double Negative Permittivity and Permeability ................................................. 2 1.3 Methods of Construction .................................................................................. 6 1.4 Negative Refractive Index and Perfect Lens ..................................................... 10 1.5 Backward Waves and Reversed Doppler Effect ................................................ 16 1.6 Acoustic Metamaterials .................................................................................. 19 1.7 Objectives of Research ................................................................................... 20 Chapter 2 Basic Concepts of Acoustic Metamaterials ......................................... 22 2.1 Negative Effective Mass ................................................................................. 22 2.2 Negative Effective Stiffness ............................................................................ 25 2.3 Conventional Vibration Absorber .................................................................... 28 2.4 Broadband Vibration Absorbers ..................................................................... 32 2.5 Dispersive Material ........................................................................................ 35 2.5.1 Non-dispersive Wave .......................................................................................... 35 2.5.2 Dispersive Wave ................................................................................................. 37 2.6 Dispersive Metamaterial Bar .......................................................................... 38 iii 2.6.1 Governing Equation and Dispersion Relation ..................................................... 39 2.6.2 Finite Element Modeling .................................................................................... 45 Chapter 3 Acoustic Metamaterial Beams with Local Vibration Absorbers ........... 50 3.1 Governing Equations and Dispersion Relations ................................................ 50 3.2 Finite-Element Modeling ................................................................................ 55 3.2.1 Euler-Bernoulli Beam Element ........................................................................... 56 3.2.2 Timoshenko Beam Element ................................................................................ 58 3.2.3 Infinite Beams with Uniform Absorbers ............................................................. 63 3.3 Numerical Analysis ......................................................................................... 66 3.3.1 Supported Beam with Uniform Absorbers ......................................................... 66 3.3.2 Supported Beam with Varying Absorbers .......................................................... 70 3.4 Actual Working Mechanism ............................................................................ 77 Chapter 4 Multi-stopband Metamaterial Beams ................................................ 80 4.1 Basic Concept ................................................................................................ 80 4.2 Governing Equation and Finite-Element Modeling .......................................... 83 4.3 Numerical Analysis ......................................................................................... 88 Chapter 5 Conclusions and Recommendations for Future Work ......................... 94 5.1 Conclusions ................................................................................................... 94 5.2 Recommendations for Future Work ................................................................ 95 References: ......................................................................................................... 97 iv LIST OF FIGURES Figure 1.1 Three prototypes of magnetic microstructures. Model (a): an array of metallic cylinders designed to have magnetic properties along the axial direction of the cylinder. Model (b): an array of copper cylinders which consist of external and internal ‘split ring’ configurations. Model (c): ‘Swiss Roll’ Capacitor. ............................ 7 Figure 1.2 Dispersion relation for magnetic microstructure with ‘split ring’ configuration. ................................................................................................................. 9 Figure 1.3 (a) a ‘split ring’ resonator (SRR), (b) a square-shaped SRR, and (c)SRRs being attached to three perpendicular plans of a cube. ............................................. 10 Figure 1.4 Reflection and refraction at the interface of two media. ........................... 12 Figure 1.5 Optical characteristic of conventional lenses: (a) a convex lens is a converging lens, and (b) a concave lens is a diverging lens ......................................... 12 Figure 1.6 Passage of rays through a slab lens which has an ideal refractive index n=-1. ............................................................................................................................. 13 Figure 1.7 Optical characteristics of lenses made of left-handed material (LHM): ..... 15 Figure 1.8 (a) A silver slab lens placed at the center between an objective plane and image plane, and (b) electrostatic objective potential intensity, and (c) image potential intensity with/without silver slab lens. ........................................................ 16 Figure 1.9 The wave propagation direction where S represents the Poynting vector (group velocity) and k represents the wave vector (phase velocity): (a) right-handed material, and (b) left-handed material. ....................................................................... 17 Figure 2.1 A 2-DOF mass-in-mass system to illustrate negative effective mass. ......... 22 Figure 2.2 A 2-DOF mass-in-spring system to illustrate negative effective stiffness. .. 26 Figure 2.3 The Frahm dynamic vibration control device model. ................................. 29 Figure 2.4 The frequency response of the Frahm vibration absorber. ........................ 29 Figure 2.5 A 2-DOF damped vibration absorber model. .............................................. 31 Figure 2.6 The frequency response of a damped vibration absorber. ......................... 31 Figure 2.7 A 6-DOF system. .......................................................................................... 33 v Figure 2.8 The frequency response of a 6-DOF system with m1100 kg , m 2 m 6 , ki m i 20 , i 0.05 ...................................................... 34 Figure 2.9 The frequency response of a 6-DOF system with ....................................... 35 Figure 2.10 The free-body diagram of a differential string element under tension. ... 36 Figure 2.11 Wave propagation in a metamaterial bar: (a) a metamaterial bar configuration, and (b) the free-body diagram (FBD) of a unit cell. ............................. 40 Figure 2.12 Dispersion curves of a metamaterial bar: (a) curves, and (b) curves. .......................................................................................................................... 43 Figure 2.13 Dispersion curves ( ) of a
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