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Acoustic GRIN Lens California Polytechnic State University San Luis Obispo EE 462 Senior Project Advisor: Dean Arakaki Acoustic GRIN Lens Author Matthew Luu Date: June 12, 2020 Abstract Museum displays with audio explanations typically use headphones. A speaker’s sound at one display interferes with patrons at other displays, thus museums use headphones. These devices are either integrated into the display itself, or attached to an mp3 player. When integrated into the display, headphones can be unsanitary; when attached to an mp3 player, these devices are inconvenient. If speakers can direct sound toward an intended area, headphones can be eliminated. Acoustic horns direct sound toward a specific point, but require a large spatial footprint. Acoustic spherical concave lenses also direct sound toward a point, but exhibit excessive acoustic attenuation. Current speakers on the market that focus sound toward a location are costly compared to headphones. An acoustic gradient index of refraction lens, GRIN lens, accomplishes sound wave focusing while minimizing attenuation and decreasing material costs compared to current directive speakers. The goal of this project is to design and optimize a GRIN lens using numerical simulations in COMSOL Multiphysics and scripts developed in MATLAB. This GRIN lens must direct sound for museum exhibits; therefore, must operate in the human speech frequency band (300Hz - 3400Hz). An acoustic test chamber was designed and built in order to test and validate the physical prototype. The prototype was fabricated using PETG on the MakerGearM2 3D printer. This chamber is 1m length x 1m height x 1m depth and characterizes acoustic radiation pattern vs. azimuth angle (300Hz - 3400Hz) and the frequency response - microphone directly in front of speaker (300Hz - 10kHz) - for 3” maximum diameter speakers. 1 Acknowledgements I would like to thank Dr. Arakaki for advising the project and giving me support to achieve results. He helped guide me in the right direction and helped me write reports for funding opportunities. I would like to thank CPConnect and the metamaterials team that helped me develop the equipment used throughout the project. The CPConnect funding award helped gather the equipment needed to measure acoustic devices. Furthermore, the team I assembled helped build the test equipment as well as research lens designs. Team members include: Baylor Whitehead, Scott Finfer, Christopher Chen, Christopher Lim, Jackie Kimoto, Alex Goldstein, and Alejandro Quintero. Finally, I would like to thank Stratedigm and Jabil, two companies in San Jose. Stratedigm helped manufacture the lens while Jabil let me use the simulation software needed for acoustics. I wanted to give a special thanks to Dave Logan at Jabil for helping me learn acoustics and the knowledge needed to succeed in this project. 2 Contents List of Figures 6 List of Tables 7 1 Introduction 8 1.1 Motivations ................................................. 8 1.2 Customer Needs, Specifications and Design Requirements ..................... 10 1.3 Project Scope ................................................ 13 2 Introduction to Acoustics 15 2.1 Fundamental Parameters of Acoustics ................................. 15 2.2 The Acoustic Wave ............................................ 17 2.3 Transmission and Reflection Coefficients ................................ 19 2.4 Acoustic Radiation/Directivity Index ................................. 20 3 Introduction to GRIN Theory 23 3.1 GRIN Theory and Optics Introduction ................................ 23 3.2 Discretized Rows and Focal Length ................................... 26 4 Design Process 30 4.1 Design Process Summary ......................................... 30 4.2 Terminology ................................................. 31 4.3 Focal Length and Outer Lens Dimensions ............................... 32 4.4 Index of Refraction Row Design ..................................... 36 4.5 Radiation Pattern Simulations ...................................... 43 4.6 Final Design Results ............................................ 48 5 GRIN Lens Testing 50 5.1 Manufacturing ............................................... 50 5.2 Testing Setup ................................................ 53 5.3 Radiation Patterns Measurement .................................... 57 5.4 Frequency Response Measurement ................................... 59 3 6 Further Simulation Work 67 6.1 Horn + Acoustic Lens ........................................... 67 6.2 Tabs and Diffraction Limiting ...................................... 69 7 Conclusion 71 7.1 Scope Review ................................................ 71 7.2 Results .................................................... 72 7.3 Future Work ................................................ 73 A Appendix A: Acoustic Test Chamber Contributions 74 B Appendix B: General Derivations 75 B.1 Speed of Sound Derivation ........................................ 75 B.2 Focal Point Derivation .......................................... 77 C Appendix C: Developed Code 81 D Appendix D: ABET 82 D.1 Summary of Functional Requirements ................................. 82 D.2 Primary Constraints ............................................ 82 D.3 Economics .................................................. 82 D.4 Manufacturing ............................................... 86 D.5 Environmental ............................................... 86 D.6 Manufacturability ............................................. 87 D.7 Sustainability ................................................ 87 D.8 Ethical .................................................... 88 D.9 Health and Safety ............................................. 88 D.10 Social and Political ............................................ 89 D.11 Development ................................................ 89 4 List of Figures 1.1 Speaker Museum Example [4] ...................................... 8 1.2 Radiation Pattern ............................................. 9 1.3 SPL to Loudness Perception [9] ..................................... 11 1.4 Lens Dimensions .............................................. 11 1.5 Project Flowcharts ............................................. 14 2.1 Time Varying Sound Pressure Wave [10] ................................ 16 2.2 Transmitted and Reflected Wave for an Incident Plane Wave ................... 20 2.3 On-Axis Directivity Plot Example ................................... 21 2.4 Beranek Radiation Patterns [1] ..................................... 22 3.1 Ray and Wave Theory [14] ........................................ 23 3.2 Ray Theory Lenses ............................................. 24 3.3 Wave Theory Lenses ............................................ 25 3.4 Generalized GRIN Lens .......................................... 25 3.5 Discretized Secant Pattern ........................................ 27 3.6 Beam Trajectory .............................................. 29 4.1 Rows and Columns Definitions ..................................... 31 4.2 Unit Cell [7] ................................................. 32 4.3 Discretized Hyperbolic Secant Pattern ................................. 34 4.4 Focal Length vs. Number of Rows ................................... 34 4.5 Lens Index of Refraction Discretized Pattern ............................. 35 4.6 Transmission and Reflection Coefficient Simulation ......................... 36 4.7 Duke’s Unit Cell for Simulated Verification .............................. 37 4.8 Simulated Transmission and Reflection Coefficient .......................... 38 4.9 Simulated Effective Bulk Modulus and Mass Density ........................ 38 4.10 Simulated Pillar Dimensions and Lens Profile ............................ 39 4.11 Simulated Broadband Effective Parameters Middle Row ...................... 40 4.12 Simulated Broadband Effective Parameters Outer Row ....................... 40 4.13 Simulated Index of Refraction and Relative Acoustic Impedance Pillar Dimensions at 2.3kHz 41 4.14 Simulated Pillar Dimensions and Index of Refraction Profile at 2.3kHz ............. 42 4.15 Radiation Pattern Simulation ...................................... 44 5 4.16 Radiation Pattern Simulation using a 2.3kHz Plane Wave ..................... 44 4.17 Radiation Pattern Simulation using a 2.3kHz Plane Wave ..................... 46 4.18 Radiation Pattern Simulations Varying Distance from Lens using a 2.3kHz Plane Wave ... 47 4.19 CAD Model of Final Design ....................................... 49 5.1 Failed Print ................................................. 51 5.2 GRIN Lens Isometric view ........................................ 52 5.3 GRIN Lens Top Down .......................................... 52 5.4 Testing Block Diagram .......................................... 54 5.5 Test Chamber Side View ......................................... 56 5.6 Top Down of Test Chamber ....................................... 57 5.7 1kHz Radiation Pattern for GRS 3FR-4 Speaker ........................... 58 5.8 Speaker Test Configuration ........................................ 59 5.9 Speaker Input ................................................ 60 5.10 Speaker Time Domain Recording .................................... 61 5.11 Speaker Recording Max Amplitude A = 2 Spectrogram ...................... 62 5.12 Speaker Transfer Function Max Amplitude A = 2 Spectrogram .................. 62 5.13 Speaker Impulse Response 4 Different Amplitudes .......................... 63 5.14 SPL (GRS 3FR-4 Speaker) Frequency
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