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The Pennsylvania State University The Pennsylvania State University The Graduate School Department of Mechanical and Nuclear Engineering ACOUSTIC CONTROL IN ENCLOSURES USING OPTIMALLY DESIGNED HELMHOLTZ RESONATORS A Thesis in Mechanical Engineering by Patricia L. Driesch 2002 Patricia L. Driesch Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December, 2002 We approve the thesis of Patricia L. Driesch. Date of Signature Gary H. Koopmann Distinguished Professor of Mechanical Engineering Chair of Committee Martin Trethewey Professor of Mechanical Engineering Victor Sparrow Associate Professor of Acoustics Mary Frecker Assistant Professor of Mechanical Engineering Richard C. Benson Professor of Mechanical Engineering Head of the Department of Mechanical Engineering ABSTRACT A virtual design methodology is developed to minimize the noise in enclosures with optimally designed, passive, acoustic absorbers (Helmholtz resonators). A series expansion of eigen functions is used to represent the acoustic absorbers as external volume velocities, eliminating the need for a solution of large matrix eigen value problems. A determination of this type (efficient model/reevaluation approach) significantly increases the design possibilities when optimization techniques are implemented. As a benchmarking exercise, this novel methodology was experimentally validated for a narrowband acoustic assessment of two optimally designed Helmholtz resonators coupled to a 2D enclosure. The resonators were tuned to the two lowest resonance frequencies of a 30.5 by 40.6 by 2.5 cm (12 x 16 x 1 inch) cavity with the resonator volume occupying only 2% of the enclosure volume. A maximum potential energy reduction of 12.4 dB was obtained at the second resonance of the cavity. As a full-scale demonstration of the efficacy of the proposed design method, the acoustic response from 90-190 Hz of a John Deere 7000 Ten series tractor cabin was investigated. The lowest cabin mode, referred to as a “boom” mode, proposes a significant challenge to a noise control engineer since its anti-node is located near the head of the operator and often generates unacceptable sound pressure levels. Exploiting the low frequency capability of Helmholtz resonators, lumped parameter models of these resonators were coupled to the enclosure via an experimentally determined acoustic model of the tractor cabin. The virtual design methodology uses gradient optimization iv techniques as a post processor for the modeling and analysis of the unmodified acoustic interior to determine optimal resonator characteristics. Using two optimally designed Helmholtz resonators; potential energy was experimentally reduced by 3.4 and 10.3 dB at 117 and 167 Hz, respectively. TABLE OF CONTENTS LIST OF FIGURES ..................................................................................................... viii LIST OF TABLES....................................................................................................... xiii LIST OF SYMBOLS ................................................................................................... xv ACKNOWLEDGMENTS ........................................................................................... xix Chapter 1 INTRODUCTION....................................................................................... 1 1.1 Background..................................................................................................... 1 1.2 Methodology Development ............................................................................ 2 1.3 History of Helmholtz Resonators ................................................................... 4 1.4 Advantages of Design Methodology .............................................................. 6 1.5 Thesis Organization........................................................................................ 7 Chapter 2 REVIEW OF PREVIOUS WORK.............................................................. 9 2.1 Design Methodology ...................................................................................... 11 2.2 Use of Helmholtz Resonators ......................................................................... 14 2.3 Conclusions..................................................................................................... 19 Chapter 3 RESONATOR MODELING....................................................................... 20 3.1 Lumped Parameter Model .............................................................................. 20 3.2 Empirical Adaptations .................................................................................... 25 3.2.1 Entrained mass...................................................................................... 25 3.2.2 Viscosity and heat conduction contributions........................................ 26 3.2.3 Damping treatment end correction ....................................................... 27 3.3 Experimental Verification .............................................................................. 28 3.4 Parameter Study.............................................................................................. 37 Chapter 4 ACOUSTIC ENCLOSURE MODEL ......................................................... 42 4.1 Governing Equations ...................................................................................... 42 4.2 Finite Representation...................................................................................... 44 vi Chapter 5 DESIGN METHODOLOGY ...................................................................... 47 5.1 Optimization Strategy..................................................................................... 47 5.2 Procedural Flowchart...................................................................................... 48 5.3 Efficient Model Representation...................................................................... 53 Chapter 6 PERFORMANCE EVALUATION USING A 2D ENCLOSURE............. 57 6.1 Experimental Setup......................................................................................... 57 6.2 Investigative Study: Fine-Tuning Helmholtz Resonators .............................. 60 6.3 Performance Evaluation.................................................................................. 62 6.4 Comparison with Numerical Results.............................................................. 66 6.5 Discussion of the Results................................................................................ 69 Chapter 7 OPTIMIZATION OF A 2D ENCLOSURE USING TWO HELMHOLTZ RESONATORS........................................................................... 71 7.1 Optimization Setup ......................................................................................... 71 7.2 Experimental Results...................................................................................... 74 7.3 Conclusions..................................................................................................... 77 7.4 Multiple Resonator Numerical Study............................................................. 77 7.5 Conclusions..................................................................................................... 82 Chapter 8 APPLICATION OF THE DESIGN METHOD TO A JOHN DEERE TRACTOR CABIN .............................................................................................. 83 8.1 Description of Tractor Cabin.......................................................................... 83 8.2 Experimental Setup......................................................................................... 85 8.2.1 Acoustic mode determination............................................................... 85 8.2.2 Boundary velocity measurement .......................................................... 88 8.2.3 Sound pressure measurements.............................................................. 93 8.3 Numerical Model Verification........................................................................ 94 8.3.1 Numerical model NM1......................................................................... 96 8.3.2 Numerical model NM2......................................................................... 97 8.3.3 Numerical model NM3......................................................................... 102 Chapter 9 OPTIMIZATION OF A TRACTOR CABIN USING HELMHOLTZ RESONATORS .................................................................................................... 117 9.1 Optimization Approach .................................................................................. 117 9.2 Single Helmholtz Resonator Gradient Optimization Trials ........................... 121 9.2.1 Acoustic Admittance Matching............................................................ 123 9.2.2 Case 2 optimization of NM3 ................................................................ 127 9.2.3 Fabrication concerns............................................................................. 135 9.3 Multiple Helmholtz Resonator Optimization ................................................. 136 9.4 Conclusions..................................................................................................... 143 vii Chapter 10 CONCLUSIONS....................................................................................... 146 10.1 Basic Conclusions of the Research............................................................... 146 10.2 Future Research and Practical Implementations .........................................
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