The Helmholtz Equation Least Squares Method: For Reconstructing and Predicting Acoustic Radiation (Modern Acoustics and Signal Processing)

This book represents the HELS (Helmholtz equation least squares) theory and its applications for visualizing acoustic radiation from an arbitrarily shaped vibrating structure in free or confined space. It culminates the most updated research work of the author and his graduate students since 1997. The book contains six chapters. The first serves as a review of the fundamentals in acoustics and the rest cover five specific topics on the HELS theory. Preface 8 Contents 12 Chapter 1: Introduction 15 1.1 Conventional Noise and Vibration Diagnoses 16 1.2 Holography 17 1.3 Acoustical Holography 17 1.4 Near-Field Acoustical Holography 18 1.5 Fourier Transform-Based NAH 19 1.6 Boundary Element Method-Based NAH 21 1.7 Helmholtz Equation Least-Squares Method-Based NAH 23 Chapter 2: The Spherical Wave Functions 25 2.1 The Helmholtz Equation Under the Spherical Coordinates 25 2.2 Solution to R(kr) 26 2.3 Solution to Theta(theta) 30 2.4 Solution to Phi(phi) 32 2.5 Solution to (r, theta, phi; omega) 34 Chapter 3: The Helmholtz Equation Least-Squares Method 41 3.1 The HELS Formulations 42 3.2 Reconstructing the Radiated Acoustic Field 45 3.3 Predicting the Radiated Acoustic Field 51 3.4 Error Analyses 56 3.5 Regularization 60 3.6 Regularization Through Truncation of the Expansion Functions 61 3.7 Other Regularization Techniques 64 Chapter 4: Validity of the HELS Method 77 4.1 Rayleigh Hypothesis 77 4.2 The Rayleigh Series Versus HELS Formulations 81 4.3 Justification of the HELS Formulations 83 4.4 Significance of the Justification 91 Chapter 5: Implementation of the HELS-Based NAH 94 5.1 Guidelines for Implementing the HELS Method 95 5.2 Practical Considerations in Implementing the HELS Method 101 5.3 Test Configuration 102 5.4 Test Environment 105 5.5 Clarifications 109 Chapter 6: Combined Helmholtz Equation Least-Squares (CHELS) Method 113 6.1 The Helmholtz Integral Theory 114 6.2 Nonuniqueness Difficulties 121 6.3 Discrete Helmholtz Integral Formulations 126 6.4 The Combined Helmholtz Equation Least-Squares Method 128 6.5 Applications of the CHELS Method 130 Chapter 7: Hybrid NAH 141 7.1 Modified HELS 141 7.2 Hybrid NAH 143 7.3 Reconstructing Acoustic Fields Using the Hybrid NAH 146 Chapter 8: Equivalent Sources Using HELS 157 8.1 Localized Spherical Waves 158 8.2 Distributed Spherical Waves 159 8.3 Distributed Point Sources 161 8.4 Regularization for LSW, DSW, and DPS Expansions 162 8.5 Performances of LSW, DSW, and DPS Expansions 163 8.6 Locations of the Auxiliary Sources 166 8.7 Condition Number of the Transfer Matrices 167 8.8 Effect of Measurement Number 168 8.9 Choice of Regularization 171 Chapter 9: Transient HELS 175 9.1 Transient Acoustic Radiation 177 9.2 Residue Theorem 183 9.3 Extension to Arbitrary Time-Dependent Excitations 186 9.4 Transient NAH Formulations 188 9.4.1 Reconstruction Through BEM-Based NAH 188 9.4.2 Reconstruction Through HELS-Based NAH 190 9.4.3 Transient NAH Formulations 191 9.4.4 Applications of the Transient NAH Formulations 192 Chapter 10: Panel Acoustic Contribution Analysis Using HELS 206 10.1 The HELS-Based Panel Acoustic Contribution Analysis 207 10.2 Procedures for Conducting HELS-Based Panel Acoustic Contributions Analyses 211 10.3 Stories of Panel Acoustic Contributions Analyses 214 10.3.1 Story 1: Sound Transmission Paths 214 10.3.2 Story 2: Panel Acoustic Contribution Analyses 218 10.3.3 Story 3: Engine Block Noise Analyses 223 References 232 Index 239 This book gives a comprehensive introduction to the Helmholtz Equation Least Squares (HELS) method and its use in diagnosing noise and vibration problems. In contrast to the traditional NAH technologies, the HELS method does not seek an exact solution to the acoustic field produced by an arbitrarily shaped structure. Rather, it attempts to obtain the best approximation of an acoustic field through the expansion of certain basis functions. Therefore, it significantly simplifies the complexities of the reconstruction process, yet still enables one to acquire an understanding of the root causes of different noise and vibration problems that involve arbitrarily shaped surfaces in non-free space using far fewer measurement points than either Fourier acoustics or BEM based NAH. The examples given in this book illustrate that the HELS method may potentially become a practical and versatile tool for engineers to tackle a variety of complex noise and vibration issues in engineering applications This book is written for those who want to learn the state-of-the-art technology on visualizing acoustic radiation from vibrating objects, to understand difficulties involved in these inverse acoustic problems, and to acquire the information necessary to conduct successful vibro-acoustic diagnostics on their own.℗ Attention is focused on the new development on nearfield acoustical holography (NAH) beyond the traditional Fourier acoustics that are suitable for separable geometries only.℗ Specifically, it presents:-Helmholtz equation least squares (HELS) method-inverse Helmholtz integral equatio Front Matter....Pages i-xiii Introduction....Pages 1-10 The Spherical Wave Functions....Pages 11-26 The Helmholtz Equation Least-Squares Method....Pages 27-62 Validity of the HELS Method....Pages 63-79 Implementation of the HELS-Based NAH....Pages 81-99 Combined Helmholtz Equation Least-Squares (CHELS) Method....Pages 101-128 Hybrid NAH....Pages 129-144 Equivalent Sources Using HELS....Pages 145-162 Transient HELS....Pages 163-193 Panel Acoustic Contribution Analysis Using HELS....Pages 195-220 Back Matter....Pages 221-233