Acoustic Analysis and Design of Short Elliptical End-Chamber Mufflers (SpringerBriefs in Applied Sciences and Technology)

The automobile engine exhaust mufflers that are located under the vehicle are oftenelliptical in shape as a consequence of lowering the center of gravity of the vehicle inorder to ensure mechanical stability. The desirability of minimum volume, weight andcost of the muffler necessitates acoustically small end-chambers in the flow-reversaltype muffler configurations: Plane wave propagation breaks down and higher-ordermodes start propagating in these small end-chambers right from frequencies of theorder of the firing frequency of the high-speed multi-cylinder automotive engines.This calls for 3-D analysis of the elliptical exhaust mufflers. The numerical finiteelement method is too cumbersome and costly to be of any use to muffler designers.Dr. Akhilesh Mimani steps in here to offer this Springer Monograph on acousticanalysis and design of short elliptical end-chamber mufflers.Based on his doctoral work at the Facility for Research in Technical Acoustics,Indian Institute of Science, Bangalore, and post-doctoral work at leading AustralianUniversities during the last 14 years, this monograph makes use of 3-D Green’sfunction derived in terms of Mathieu functions, modal summation, impedance matrixand a uniform piston-driven model. Probably for the first time, the parametric zerosand non-dimensional resonance frequencies of the higher-order transverse modesof a rigid-wall elliptical waveguide for a complete range of aspect-ratio have beencalculated and tabulated in this monograph.An ingenious feature of Mimani’s monograph is formulation of a set of compre-hensive guidelines that suggest optimal inlet/outlet port locations which wouldsuppress the propagation of certain higher-order transverse modes, thereby yieldinga broadband attenuation despite a small expansion volume. This feature is very usefulfor designing short elliptical (and circular) mufflers of the flow-reversal type that areoften used as end-chambers in modern-day automotive exhaust systems. Foreword 5 Preface 7 Acknowledgements 8 Contents 9 About the Author 11 1 Introduction 12 1.1 Elliptical Mufflers for Automotive Exhaust Application: Motivation 12 1.2 State of the Art and Scope 14 1.2.1 Book Outline 17 References 17 2 Acoustic Wave Propagation in an Elliptical Cylindrical Waveguide 19 2.1 Introduction: A Brief Review on Mathieu Functions and Objectives 19 2.2 The Convective 3-D Helmholtz Equation in Elliptical Cylindrical Co-ordinates 23 2.3 Solution of Wave Equation in Angular and Radial Co-ordinates 27 2.3.1 Computation of Mathieu Functions 27 2.3.2 Computation of Modified Mathieu Functions 31 2.4 Acoustic Pressure Field in a Rigid-Wall Elliptical Cylindrical Waveguide 33 2.5 Numerical Computation of the q Parameters 38 2.6 Tabulation of Parametric Zeros and Non-Dimensional Resonance Frequencies 38 2.7 Resonance Frequency Variation with Aspect-Ratio, Interpolating Polynomials and Mode Shapes 54 2.7.1 Circumferential Modes of Even and Odd Type 54 2.7.2 Radial Modes and Cross-Modes of Even and Odd Type 59 2.8 Family of Ellipses Having Equal Cross-Sectional Area 64 Appendix: Resonance Frequencies of a Clamped Elliptical Membrane 69 References 75 3 Characterization of an Elliptical Chamber Muffler 78 3.1 Three-Dimensional Green’s Function 79 3.2 Acoustic Pressure Response Based upon the Uniform Piston-Driven Model 82 3.2.1 Excitation at an End Port 83 3.2.2 Homogeneous Wave Equation Subject to Inhomogeneous Conditions 85 3.2.3 Excitation at a Side Port 90 3.3 Integration of Green’s Function Over the Piston Area Due to End or Side Port 92 3.3.1 End Port 92 3.3.2 Side Port 92 3.4 Influence of Location of End and Side Ports on the Propagation of Even and Odd Modes 93 3.5 Evaluation of the Impedance [ Z ] Matrix Parameters 95 3.5.1 End-Inlet and End-Outlet Muffler 96 3.5.2 End-Inlet and Side-Outlet Configuration 98 3.6 Computation of Transmission Loss Performance in Terms of [ Z ] Matrix Parameters 100 3.6.1 Analysis/Prediction of the Observed Peaks and Troughs in the TL Spectrum 100 References 101 4 Double-Tuned Short End-Chamber Mufflers 104 4.1 Multi-pass Perforated Duct Muffler System 104 4.1.1 Short End-Chambers: Integral Component of a Multi-pass Perforated Muffler 105 4.2 End-Inlet and End-Outlet Configurations 106 4.2.1 Inlet on the Major-Axis and Outlet on the Minor-Axis: Optimal Port Location 108 4.2.2 Inlet on the Major-Axis and Outlet Port Centered 130 4.2.3 Inlet Port Centered and Outlet Located on the Minor-Axis 143 4.2.4 Ports Offset at Equal Distances on the Major-Axis or Minor-Axis: Non-tuned Configuration 144 4.3 End-Inlet and Side-Outlet Configurations 145 4.4 Design Guidelines 148 References 149 5 Summary of Contribution and Directions for Future Work 151 5.1 Contribution 151 5.2 Future Investigation: Analysis and Design of Perforated Elliptical Mufflers for Industrial and Automotive Applications 152 References 157 Index 159 This book presents a three-dimensional analysis of acoustic wave propagation in an elliptical waveguide, and applies the equations and concepts to design axially short elliptical end-chamber muffler configurations which are an important component of a complex multi-pass muffler used in a modern-day automotive exhaust system. A general solution of the Helmholtz equation in elliptical cylindrical co-ordinates is presented in terms of the Mathieu and modified Mathieu modal functions. This is followed by the tabulation and analysis, for the first time, of the non-dimensional resonance frequencies of the transverse modes of a rigid-wall elliptical waveguide for a complete range of aspect ratio. The modal shape patterns of the first few circumferential, radial and cross-modes are examined with particular attention to the pressure nodal ellipses and hyperbolae. An analytical formulation is then outlined for characterizing a single-inlet and single-outlet elliptical muffler with the inlet located on the end face and the outlet located either on the end face or side-surface. The ensuing chapter is devoted toward analyzing the Transmission Loss (TL) performance of different short end-chamber mufflers, namely (a) the straight-flow configuration having ports located on the opposite face, (b) the flow-reversal configuration with ports located on the same end face and (c) configuration with inlet port on the end face and outlet on the side surface. Design guidelines are formulated in terms of the optimal location of inlet and outlet ports which suppresses the deteriorating influence of certain higher-order modes, thereby delivering a broadband TL performance. Directions for future work are discussed toward the end. In summary, this book is a one-stop solution for a practicing automotive engineer designing mufflers, for an applied mathematician studying wave propagation in elliptical geometries, and also as a niche area within noise control engineering.