I... a} UV.- cm. .5. 9... (.0. uk all lit mu: 4 4...: 3 5“,“ “Us“ 0.... MW”. “ma... «3. 1” Pt. PM... L v. -L- ....L 3 up “'3‘ a w .1.‘\\ ”O”: Q: .. . 3 WWW R 3 “0)..” 9.90.“. 1.3...” E...“— WQ .. x» . L. a..." in mm. 9.... am... . . a» a» .. .I‘ m nun. an 9 +2 «3. Mn. am. «a mm. mm 3 . rm;— Mwn 4.. ran. mun”. ”£4? ‘ w ‘ o ‘. a b . W“... :p! mm.“ x f -u 0 .. § t o «E .m a... :1 min I: 5.. «U MW! 3 JUL "‘5 ‘I n‘ t (91‘ h 2.”. f.v\ “ _..w M”. 0?... . S 4.. q I... 1:. .0”. . . . . -l- ”MB 7;"! “w... .rd r u ‘1'“ 1.. g... L L {v . um. Pm 14.x P \n‘k" ' 3. I | u: I 4 i 3‘33: ‘4 f. ‘3. 33;. " ~.\.! 3A ii if; . LLLLLLL LLLLLL LLLLLLLL LLLLL LLLLLLL LLLLLLLLLLLL LLL LLLLLLLLL LLLL 1&me l-‘__‘-n.‘ . AL LIBRARY ‘4 M i chiga n, 8 rate University ABSTRACT THEORETICAL AND EXPERIMENTAL ANALYSIS OF AUTOMOBILE MUFFLER SYSTEMS by Roger R. Regelbrugge For many years, automobile mufflers have been designed, developed and ultimately approved or rejected on the basis of trial and error type approaches carried over from the era when accurate and reliable test instrumentation was not readily available. The lengthy and sometimes wasteful development programs consisted of building a great number of handmade samples in the hope that ultimate- ly the customer would be satisfied. With the advent of electronic instrumentation, and loudspeakers and micrOphones with favorable frequency response, research work in the field of acoustics became more promising. In 1954 a report was published by NACA, illustrating a possible theoretical approach to the muffler design problem. Building on the foundations laid in that report, this thesis relates the details of a research program on automobile mufflers. A reliable test set up was develOped and a series of test mufflers was built. The test mufflers incorporated the basic circuit components frequently used in present day muffler designs. Analysis of these curcuitS'was done by comparing test results to analytically derived frequency response curves. Roger R. Regelbrugge The scope of the investigation ranged from.single chambers to the complex assemblies which constitute the modern mufflers. The program revealed that by eliminating from the considerations some of the variables caused by the relatively unknown sound source; which is the automobile engine, good agreement can be found between theory and experiment on the true acoustical behavior of mufflers. Reverse-flow features, cross-bleed chambers, and the use of louvers in mufflers were investigated. Systems of equations were developed for complete muffler systems. By the use of computers theoretical attenuation curves could be arrived at. Comparisons were made between analytical and experimental attenuation curves. From these comparisons it can be concluded that the test set up is adequate for the further exploration of muffler acoustics, and that it is possible to set up systems of equations which will predict muffler performance with substantial accuracy. Recommendations have been made for the expansion of the test program.to on-the-engine testing. It can be foreseen that muffler development will soon be able to benefit from the tools now readily available in laboratories and computer centers. COPYRIGHT BY RO GER RAFAEL REGELBRUGCE 196M THEORETICAL AND EXPERIMENTAL ANALYSIS OF AUTOMOBILE MUFFLER SYSTEMS BY (.3- “ Roger R? 'Re gelbrugge A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1964 , a" A 1 (if1 PREFACE AND.ACKNOWLEDGMENTS The combination of strictly technical development problems and the need for passing an exclusively subjective final approval test makes muffler design work a rather complex art. The additional complicar tions introduced by annual changes in system layout and/or characteristics of the sound source have resulted in lengthy and expensive programs of development. A concentrated effort in the direction of advancing the science of sound muffling is necessary. The program, which is dealt with in this thesis, illustrates one aspect of this effort. In the capacity of chief development engineer of Hayes Industries, Inc., of Jackson, Michigan, I had the opportunity to be exposed to the complexities of the problem and to the unexplored territory ahead. The experience gained during the execution of the daily assignments was a very tangible help in the preparation of this thesis. Further, the continued interest of top management, especially Mr. G. B. Vass, President, has been greatly appreciated. The very useful suggestions made by Dr. F. S. Tse, and Dr. G. Martin, and the ideal COOperation of the peOple at the Nflchigan State University Computer Laboratory, have greatly contributed to the final success of this effort. To my wife, Sandra, however, I must attribute the credit for the finalization of this program. Her constant interest and concern have provided the stimulus required to complete this thesis and to chart the course for our future programs. -11- TABLE OF CONTENTS Chapter I. II. III. INTRODUCTION . . . . . . . . . . . ACOUSTICAL VARIABLES AND RELATIONSHIPS FOR USE IN SOUND TRANSNESSION LINES 1. Definition and Symbols 0 . . . 2. Acoustic Plane Waves . . . . . 3. Acoustic Impedance . . . . . . a. Specific Acoustic Impedance b. Acoustic Impedance . . . . 4. Analysis of Acoustic Transmissi a. Reflection in Pipes . . . b. Resonance in Pipes . . . . c. Transmission From.One Pipe d. Theory of a Side Branch . TEST INSTRUMENTATION AND EQUIPMENT 1. Basic Test Circuit . . . . . . 2. Infinite Tailpipe . . . . . . 3. Physical Layout . . . . . . . 4. Calibration Details . . . . . DETAILS OF MUFFLER TEST PROGRAM 1. Method of Test . . . . - a . ~ 2. Analysis of Measurements . . . 3. Reiteration of Objectives A. Series of Test Units . . . . . -iii- on Lines . to Another Page \0 \o -q 4? 4r 11 12 13 15 17 20 2o 23 27 27 31 31 32 34 35 Chapter. IV. MATHEMATICAL ANALYSIS OF NWFFLER SYSTEMS l. 2. 3. HGMOltZ Resonator o o o o o o o .... EbCpanSiOnChamber.......... Combination of Chambers . . . . . . . V. DISCUSSION OF TEST AND COMPUTER RESULTS . l. 2. 3. 1+. 5. 6. Reliability of the Test Setup . . . . Acoustical Value of Louvers . . . . . Cross-Bleed Chamber . . . . . . . . . TurnrAround Chambers . . . . . . . . . General Approach to Predicting Muffler Major Variables Not Accounted For . . CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . BIBLIOGRAPIIY O O O O O O O O O O O O O O O O O 0 Performance Page Ln Lt7 50 52 57 57 59 89 61 62 68 7o Figure 12. 13. 14. 15. 16. 17. 18. 19. 2o. 21. 22. LIST OF FIGURES Sound Transmission From One Pipe to Another . . . . Transmission From A Smaller Pipe to A Larger One Detail of Acoustical Side Branch . . . . . . . . . Schematic Diagram of Test System . . . . . . . . . MicrOphone and Pipe Assembly . . . . . . . . . . . Location of Measuring Stations in Exhaust Pipe . . Calibration of Infinite Tailpipe . . . . . . . . . Test Setup in Anechoic Room . . . . . . . . . . . . Calibration Curves of Condenser Microphones . . .'. Correction Factors for Measured Attenuation . . . . Simple Helmholtz Resonator . . . . . . . . . . . . Louver Fed Expansion Chamber . . . . . . . . . . . Equivalent Expansion Chamber with 1/8 Inch Holes . Equivalent Expansion Chamber with 1/4 Inch Holes . Conventional Expansion Chamber . . . . . . . . . . Double Expansion Chamber . . . . . . . . . . . . . Double Expansion Chamber . . . . . . . . . . . . . Expansion Chamber and Turn-Around Chamber . . . . . Triple Chamber Muffler . . . . . . . . . . . . . . Triple Chamber Muffler with Double Reversal of Flow Sketch of Resonator . . . . . . . . . . . . . . . . SketCh Of manSion Cllamber o o o o o o o o o o o o -v- Page 15 17 17 21 22 24 26 28 29 33 36 38 : 39 ii 1.2 43 1+4 45 46 L+7 5O Figure Page 23. Sketch of Double Expansion Chamber . . . . . . . . . 52 24. Equivalent to Double Expansion Chamber . . . . . . . 53 25. Conventional Muffler Circuit . . . . . . . . . . . . 55 26. Louver Design . . . . . . . . . . . . . . . . . . . 59 Appendix 1. LIST OF APPENDICES Sample Calculation of Tailpipe Resonance . . . . . . -vii- Page 71 INTRODUCTION In a report published in 1954, engineers of the Langley Aeronautical Laboratory of NACA summarized theory and experimental approaches to engine-exhaust muffler performance and design. Prior to that report very little information had been published on the specific subject of noise filtration in engine exhaust systems. In 1959, a series of papers on the subject of mufflers were presented at the annual passenger car convention of the SAE. These papers dealt with specific considerations of muffler design without generalizing theory for future reference. The NACA report prompts us to recognize several important conclur sions and/or limitations: -Laboratory test methods were developed for muffler testing under "true acoustical" conditions, eliminating some important variables of the engine test conditions. -Some correlation between "cold" and "hot" tests seemed possible. . -Substantial agreement was found between cold test results and attenuation values derived analytically. -thhematical analysis of common simple muffler chambers was achieved. -Only relatively simple sound filters were considered. -Only straight-through type silencers which permit through-flow of sound without turning around were considered. -1- -2- -The physical dimensions of the mufflers used in the NACA study were such that they far exceeded dimensions normally encountered in automobile mufflers. In an effort to determine to what extent the NACA experiments and theory had contributed to the understanding of the Operation of auto- mobile mufflers the author set up the following program; -Deve10p an infinite tailpipe capable of retaining its basic characteristics after repeated exposure to sound levels of up to 160 DB in a frequency range from 50 to 1000 Cps. This infinite tailpipe should have a diameter of not more than 2 inches. -Design an exhaust pipe (muffler inlet) of sufficient length to study standing waves in sound fields of 50 Cps. This exhaust pipe should further enable us to measure sound levels inside the pipe at random locations. -Construct a sound generating system of high quality and intensity as the sound source for the test program. -Design and build a series of mufflers incorporating parts and materials normally used in muffler design. This series of mufflers would become progressively more complicated as the units incorporate more and more features of the commercial mufflers. -Study, experimentally and theoretically the behavior of the various mufflers, limiting the study to "cold testing." -Evaluate test results against theoretical analysis. -Consider variables in hot systems not accounted for in this program. -3- As a result of this program discuss the merits of cold testing in muffler research and suggest a program for combination of hot and cold testing. It is this program which forms the basis for this thesis. The various formulae required to follow the mathematical analyses are de- rived in Chapter I. Chapter II is devoted to the test system and instrumentation. Armed with the mathematical background and equipped ‘with satisfactory test apparatus, a basic test approach is developed, and the sequence of designs to be tested is discussed. Actual calculations and test results are covered in a chapter where the contribution from computers is stressed. Finally, a correlation of results allows us to draw our conclusions and make recommendations for future programs. It should be recognized that no absolute and final answers could be obtained in this study. The objective was primarily that of gathering the theoretical background, the laboratory setup and the basic approach for a continuation of a research program. The author believes the results of this analytical and experimental investigation substantially increase the amount of technical data published on muffler research. All features of commercial mufflers were studied and their be- havior analytically predicted with good agreement between computed and actual performance. It is believed that this analysis will contribute substantially to the goal of introducing a higher degree of predictability in muffler development. Ultimate reductions in cost of develOpment programs and of product designs are sure to follow. CHAPTER I ACOUSTICAL VARIABLES AND RELATIONSHIPS FOR USE IN SOUND TRANSMISSION LINES The following definitions will be important in the interpretation and understanding of the acoustics theory as treated in this manuscript. Abbreviations and units of measurement are given in an effort to con- solidate this information in one section. 1. Definitions and Symbols Acoustics: The Science of Sound Bel: A fundamental division of a logarithmic scale for expressing the ratio of two amounts of power. The number of bels denoting such a ratio is the logarithm to the base 10 of this ratio. Decibel: (db) One tenth of a bel., with w_.L and W2 designating two amounts of power, and n the number of decibels denoting their ratio n(in db) = 10 loglO-%l 2 When impedances are such that ratios of pressures are the square roots of the corresponding power ratios, the number of decibels by which the cor- responding powers differ is expressed -11.— Acoustic Impedance: Particle Velocity: Loudness: Pitch: Sound Pressure Level: Standing waves: -5- byn = 20 log-gidb See Chapter I Section 3 In a sound wave is the instantaneous velocity of a given infinitesimal part of the medium, relative to the medium as a whole, due to the passage of a sound wave. Units are cm/sec. That aspect of auditory sensation in terms of which sounds may be ordered on a scale running form "soft" to "loud". Loudness is chiefly a function of the intensity of a sound, but it is also dependent on the frequency and the composition. The unit is the sone. That aspect of auditory sensation in terms of which sounds may be ordered in a scale running from "low” to "high". Pitch is chiefly a function of the frequency of a sound, but it is also dependent on the intensity and composition. The unit is the mel. Is applied to data taken by a sound pressure meter with a "flat" response. The reference pressure is 0.0002 dyne/cme. Constitute the wave system resulting from the interference of progressive -6- waves of the same frequency and kind. They are characterized by the existence of nodes or partial nodes in the in- terference pattern. In order to obtain standing waves the interfering waves must have components traveling in opposite directions. The following symbols are of importance in the development and dis- cussion of the general wave equation. x,y,z coordinates of a particle of the medium. §,n,§ component particle displacements along the x, y, and z axes, respectively. u,v,w component particle velocities, i.e., p' instantaneous density at any point. p constant mean density at any point. s condensation at any point, as defined by S=2L:_2 p p' instantaneous pressure at any point. pO constant mean pressure at any point p excess pressure or acoustic pressure at any point, as defined by p = p' - pO ¢ velocity potential c velocity of propagation of the wave ;7- 2. Acoustic Plane Waves Acoustic waves are disturbances propagated in a compressible fluid. The frequency of these waves is in the so-called audible range, roughly between twenty cycles per second and twenty thousand cycles per second. The propagation of acoustiC'waves in a fluid medium is generally three dimensional. The general wave equation, applicable to both liquids and gases is of the form signer. St The solution of this equation represents a propagation of the velocity potential ¢, the velocity of propagation being c. The velocity potential ¢ = ¢(x,y,z,t) andl =22 =§2 LE u va Syw 52 L.52 8:42? P 95% pc Of particular importance in the study of transmission lines is the case in which the propagation of the wave is bounded in two dimensions. Such a wave is called a plane wave. When a plane acoustic wave travels in the x-direction, all particle motions are assumed to be in this direc- tion. Consequently -—— = v = o and -—— - w = 0 because the y and 2 components of velocity (v and w respectively) are zero. lNumbers refer to references in the Bibliography. -8- The velocity potential ¢ now becomes a function of x and t only. The general wave equation then becomes 82 = C2 52¢ '5t2 '5}? This equation has the general solution ¢ = fl(ct - x) + f2(ct + x) where c is the velocity of prOpagation. Expressing the motion as a function of harmonic waves with propagation in the positive and in the negative x direction, ¢=Aej(wt-kx)+BEj(th-ka) - (l) where A and B are complex amplitudes of waves traveling in the positive and in the negative x direction respectively, and where 9"! sf:- = propaga- tion velocity. 8 sume ¢+ = Aej(mt - kx) and ¢- = Bej(wt + kx) then1 P 5% = -J'<1>p(¢+ + ¢_) (2) k s=3§2—-Jg(¢++¢_) (3) u = $13- = -Jk(¢+ - ¢_) (1+) From these complex relationships we find that--acoustic pressure and condensation lag velocity potential by 90°. Also, E = fudt =-%5 = -'% (¢+ ' ¢_) (5) g5; = ._:. ('Jk¢+ - Jk¢_) = ch (A. + ¢_) (6) and = f§£ S 5x which indicates that a rare faction or a negative condensation is present in the medium whenever i 5.2. “5.x is positive i.e., whenever particle displacement is increasing as x increases. Particle velocity leads particle displacement by 90°. For waves traveling in the positive x, the particle velocity lags the velocity potential by 90°. For waves traveling in the negative x direction the particle velocity leads the velocity potential by 90°. The actual equations giving the various acoustic variables are the real parts of equations (1) through (6). Assuming the case where A and B are real constants A and B we find A cos (at - kx) + B cos (at + kx) (la) velocity potential- ¢ pm A sin (wt - kx) + pa) B sin (wt + kx) (2a) pressure (acoustic)- p condensation- s =-%.A sin.(dt - kx) +‘% B sin (at + kx) (3a) particle velocity- u = k A sin (wt - kx) -k B sin.(at + kx) (4a) particle displacement- §‘= :éicos (at - kx) +-§ cos (at + kx) (5a) strain— -§§ = 1% A sin (wt - kx) --g B sin.(d¢ + kx) (6a) 3- Acoustic Impedance Important relationships between acoustical variables can be ex- pressed using the concept of impedance. A. Specific Acoustic Impedance. The ratio of acoustic pressure in a medium to the associated particle velocity is defined as the specific acoustic impedance of the medium. For plane wave -10- LELMLQC + u -Jk¢+ + =E—- ¢‘— .— ug ‘ Jk¢_ '°C This impedance is also called the characteristic impedance of the medium. Acoustic Impedance. The use of electrical analogues has resulted in the defini- tion of acoustical variables facilitating the interpretation of the analogy. The Acoustical analogue of a voltage across part of an electric circuit would be the pressure difference across an acoustic element. Analogous of electric current at a point of the electric .‘ circuit would be the acoustic flux or volume velocity of the fluid. If we assume particle displacement through a surface to be normal to that surface and the same at all points, we find the volume displacement X = g S where S is the cross section of the element. The velocity then Eli-(=2 S 5t OD The acoustic impedance Z of a fluid medium acting on or through a surface of given area is the complex quotient of the acoustic pressure at the surface divided by the volume velocity at the surface. ‘ ax dt -11- In general, the acoustic impedance is equal to the mechanical impedance divided by the square of the area of the surface being considered. The acoustic ohm, unit of acoustic impedance, has dimensions of Pressure = DvnesZCm2 2 g mass Volume Velocity Cm Sec Cm Sec The acoustic resistance R is defined as the real component of the acoustic impedance. It is associated with the dissipation of energy. Acoustic Reactance X of a medium is the imaginary component of the acoustic impedance; it results from the effective mass and stiffness of the medium. The acoustic impedance is related to the specific acoustic impedance at a surface by Z = z/S. The acoustic compliance C of an element is defined as the volume displacement X that is produced by the application of unit pressure. The units of acoustic compliance are Cmu Sec2 / g. The acoustic inertance M of an element is defined as M = m/‘o2 where m is the effective mass of the element. Acoustic inertance has dimensions of g / Cm . 4. Analysis of Acoustic Transmission Lines Only the considerations which are of importance in the specific subjects handled in this'thesis are summarized. In order to properly interpret the results of our test programs, it will be necessary that we understand such phenomena as reflection in pipes, resonance in pipes, the effect of changes in pipe cross section, and the basic theory of a side branch. These subjects are now dealt with in preparation of the considera- tions of the specific problems in our test program. -12- Reflection in Pipes Assume that at some point x along a pipe the acoustic impedance changes from its characteristic value of -93 to z S x where Zx may be either real or complex. If an initial wave traveling in the positive x direction and represented by P1 = Aej(mt - kx) is incident at this point, a reflected wave p z leQRt + kx) r will in general be produced. The volume velocities of fluid flow corresponding to these two waves are given respectively by dXi Pi er _ Pr -——- = and .—__ ‘- - dt be S dt 0c 8 When both waves are present, the varying phase relationship between them causes the acoustic impedance to vary from point to point along the pipe, rather than remaining the same at all points as is true when only the incident wave is present. A general expression for acoustic impedance which includes the reflected wave is + Zzfl .7; 3:: u (7) dX dX S pi ‘ Pr __i. +. __£ dt dr or no Ae-ka + Be'jkx z = ——-.. 7 . (8) s Ae-jkx _ Bejkx -13- At the cross section of the pipe where the impedance changes, the usual conditions of continuity of pressure and volume Velocity may be replaced by a condition of continuity of their ratio, that is continuity of acoustic impedznice, hence the phases and amplitudes of the incident and reflected waves must be so related as to cause Z in equation (8) to be equal to f/JX. Resonance in Pipes f .- Assumc that the f .1 old in a pipe of length l and area :3 == ml. is driven by a vibrating piston located at the left-hand end where x = o and that the pipe is terminated in an acoustic impedance Z at the l-ight—hsuid end where x -= 1. Application 1. of equation (8) at x -= 1 results in ADI-J“ 4’ Bifjkl ()0 Zn " "'— -‘----.-~:'---~- L—-.— (9) .. _ , l l 5 Ac Jit- _ Bjkl This equation in effect; d<-'te.rmine.~; the reflected pressure amplitude B in terms of the incident amplitude A. The input impedance ZQ at x. =4 o is ccn'respomiingly given by 2:- iii: (10) [‘o ' s A - B Equations (9) and (10) may be combined to eliminate the complex pressure amplitudes A and B and then simplified to give of: Z + J 2'2 trail kl l s " ‘ Z0 "'3— n( (11) .L... I" Q .' t b + ‘,Lllv. k3 It is apparent that th: input impc-dzmce depends not only on the f,' 'b~mnlnating impedance ml but also on the JLtr‘rngjth of the pipe l and the wave length constant k. .1h. The resonant frequency of such a pipe may be defined as that at which the reactive component of the input impedance vanishes. At this frequency the input impedance is a minimum and the power radiated out of an Open-ended tube is a maximum. Assuming 21 = ($3) (a + 36) then equation (11) may be rewritten as __ c a+fiitanlk+fi) o ”'5‘ (1 - 5 tan k1) + 30 tan kl (12) Application of the condition that X0 = 0 gives s tan 2kl He2 + a2 - 1) tan lk - s = o (13) Two special cases are of particular interest. One is when the pipe is terminated at x = l in an infinite flange. Both a and B are small as compared to unity. In that case equation (13) is approximated by tan kl = -B (at low fre- quencies). _ * For this condition 5 has been determined to be 5 =-§§2 where a is the pipe radius. We then have tan kl = - 8ka / 3n satisfied by tan (nn - kl) = gfifi where n is an integer. Hence Ska nn - kl '3;- and n0 f: 8 2(1 + 3-3) or for n = l the fundamental resonant frequency is * 3 Radiation reactance for ilanged open pipe. This indicates that the "effective length" of the pipe is l +-§% rather than 1. Experiments have shown the end correc- tion to be .82a. For unflanged pipes, this end correction is .6a. The other special case is that where the pipe is closed by a rigid cap. In that condition Zl = m and =&c__.l___=-9£ , Z0 8 J tan kl J 8 COt kl The reactance is zero when cot kl = 0 or for ki=(2n-1)-‘2‘- n equals any integer and f __ 2n - l o E . — H - 1 For __ C n - 1 fr - Ir]: 0 C. Transmission from one pipe to another. FIGUREl f 1 3, _..!L. s.2 1"? ~ xeo Let 81 and 82 be the cross sectional areas of two pipes, conductors of an acoustic wave. The incident pressure wave is p the reflected wave p and the transmitted wave}; . If i r - t = J(wt - Rx) -16- then we would have a reflected wave = B 63(am + kx) pr 1 (l5) and a transmitted wave _ 3(dt - kx pt — Age ) (16) Since we are not considering any changes in the medium or its characteristic impedance, the values of the wave length constant k on both sides of the junction are identical; Let the junction be at x = o . we have a condition of continuity of pressure which gives A1 + B1 = A2 (17) In as much as there is also continuity of flow (volume flow) we can establish the following condition: The total volume flow at any point in pipe 1 is + Sl (ui ur) That in pipe 2 is S2 (ut). - =Pi* =_Pr =P‘t ReplaCing ui ‘56 and ur .55 and 111} .55 we have at x = o to insure continuity of flow S iEi.:4Enl== 5 Pt 1 pc 2'56 81 (A1 ' Bl) — S2 A2 and _ Se A1 ' B1 "‘3; A2 Similarly for a'wave traveling from the smaller cross section to the larger one. -17.. _ S .__ pc 2 pc and S FIGURE 2 Theory of the Side Branch. Consider a pipe of uniform cross section S to which is attached a side branch of input acoustic impedance. Zb The pressure of such a branch causes the acoustic impedance at the junction to differ from the characteristic value-ES . Consequently a reflected wave is produced. It is further possible that a portion of the incident acoustic energy is transmitted into and dissipated in the branch. FIGURE 3 Zb I H— _______ ; Pz : i Pr 1 Pt I ~18- let = ej(am - kx) Pi A1 Then in general p = B ej(mi + kx) r l and P = A 8(Jam - kx) t 2 Let x = 0 at the junction then = jwt = jam = int pi Ale pr Be pt A2e . The pressure at the branch entrance may be similarly shown _ jwt wee Assuming pipe cross-sectional dimensions to be small in comparison with the wave length of the sound we have continuity of pressure and volume flow. Hence pi + pr = pt = pb (18) Let the volume velocities be .032: 10+ sis dt pc7s dt “ zb Then continuity of volume velocity means that dXi er dXi dXfi sr+sr=sr*sr (19) Dividing (19) by (18) we get dXi/dt + dXE/dt = dXi/dt + de/dt Pi + Pr pi Pb which by definition of acoustic impedance is the same as NI}-l I NIH + where Z:£S 8 (See equation 10). :29. Zt s It should be noted that if Zb = m which corresponds to no branch, all incoming acoustic power is transmitted past x = o . If however Zb;6 which implies both Rb~>o and Xbeo then we have the equivalent of Z = o and no power is transmitted.*' This does not mean that the branch Absorbs all energy. In fact for Rb = o it absorbs no energy at all but reflects 100 percent of the incoming energy back to the source. Also we recognize that if Rb has a finite value, some acoustic energy is dissipated in the branch; and some is transmitted. * We know that Al + B = 22 1 Z S Ai'Bl for Z O A1="Bi and the magnitude of the reflected wave is equal to that of the incident wave. CHAPTER II TEST INSTRUMENTATION AND EQUIPMENT The basic theory required to follow the reasoning of the following chapters was covered in Chapter I. Several details in connection with the test instrumentation are of very substantial importance. In fact, part of the original objective of this program was to develop a workable test setup, suitable for further research in this field. Consequently this chapter will be entirely devoted to details of instrumentation and construction of equipment. 1. Basic Test Circuit The basic test circuit consists of the following major components: a. Sound Source b. Muffler Inlet (Suitable for Measuring Standing waves) c. Test Mfoler d. Termination e. Measuring Circuit The schematic diagram is shown in Figure #. The sound source consists of the power supply, the oscillator, db attenuator, the amplifier, and the loud speaker with exponential horn, back enclosure and monitoring circuit. The exponential horn was made to reduce the cross section of the wave front from 15" loud speaker diameter, to a muffler inlet 01‘113/4" 1.1). .. 20- FIGURE A LOUOSPEAJKI R HORN YEASURING STAT/0N5 INFINITE AIL P/pg' . ““““““““ 1 \ T I l ' X XX I. - _ _ _. _ _ _ - MONITOR mm: "' \ l—BACK ENCLOSURE TEST HUFFL ER PICKUP MIKE TAIL PIPE ' L _j MEASURING .STA nous , vacuum TUBE AMPLIFIER VOLTHETER DB ATTENUAHHQ —— "—- RECORDER It was made of fiberglas with perfectly smooth inside surfaces. An average of four inches of concrete was poured around the fiberglas horn to reduce the possibility of setting up vibrations as a result of high intensity sound. A concrete back enclosure was also built to as much as possible avoid variation of the mechanical impedance of the source as a result of anything other than the conditions introduced by the load. (test muffler) A.monitor micrOphone was used to automatically reduce the oscillator output§;(through a compressor circuit) whenever sound levels in the system.exceeded 160 db. This way adequate protection was afforded for the loudspeaker. The loudspeaker itself was a 15" diameter, high-intensity, low frequency driver built by Transducers, Inc. The microphones were high intensity condenser microphones built by Altec Lansing. The db attenuator was used to enable us to turn down the output power by -22- predetermined steps and to subsequently switch back to the original output level with perfect accuracy. A high fidelity McIntosh amplifier was used in the input circuit. The measuring circuit consisted of a vacuum tube volt meter (with DB scale)’an.oscilloscope to examine the degree of distortion and a re- corder, automatically recording the sound levels as the microphone was moved from one station to another. Because of the small pipe diameters, it became impossible to have the microphone move inside the pipe without having an influence on the propagated wave. Instead, a substantial number of holes were drilled at random locations in the exhaust pipe. Small rubber grommets were inserted in all of those holes and closely fitting plugs closed the holes off entirely, whenever they were not in use, Probe tube micro- phones inserted through the hole in the grommet were used to pick up sound levels inside the pipe. See Figure 5 FIGURE 5 GROMMET PROBE TUBE PIPE WALL It was assumed that a sufficient number of sound readings at the randomly located measuring stations would produce at least one reading prOperly indicating the maximum sound level in the pipe. Upon completion -23- of a great number of experiments this assumption is considered to be fully justified. A detail of location of measuring stations is shown in Figure 6. With no reflection present in the tailpipe (See Section 2 on infinite tailpipe) one reading in back of the muffler is adequate. Three measuring stations were provided to continuously verify the prOper operation of the infinite tailpipe. Heavy walled tubing was used for both tailpipe and exhaust pipe (muffler inlet). This would reduce the amount of transmission and dissipa- tion through the walls. It is recognized however that under the cir- cumstances transmission through the walls could not be entirely avoided. 2. Infinite Tailpipe As we discussed in Chapter I a sudden change in the cross section of a pipe transmitting a sound wave will cause a reflected'wave to be set up. Any measurement of sound level in that pipe would be influenced by both the incident and the reflected waves. In order to determine the true incident sound level a detailed study of maximum and minimum sound levels measured in the standing wave as well as the distance from the cross sectional change would have to be made. An analysis of the effect of an Open ended tailpipe on the acoustical transmission characteristics of a muffler system is made in reference No. 2. In studying the perfor- mance of an acoustical filter in detail, it becomes advantageous to eliminate the effect of an open ended tailpipe. In reference No. 2 an infinite tailpipe was constructed to this end. The objective is of course, to construct a termination to the system in such a way that no reflected wave is set up at any point in the tailpipe. In trying to FIGURE 6 srA TION N0. DIMENSION l .STA TION N0 DIMENSION z I 8 5/; ' II 72/2 " 2 I974 ' I2 75" 3 30 %" I3 7874' 4 4/ 3A" / 4 8172" 5 47 Z,” I5 847’.” 6 5%" I6 as " 7 so " [7‘ .91“ 8 ea 94" /8 95%" 9 5772" /.9 , 99" IO 70 " 20 I03 72" I . DISTANCE or MEASURING 57.4 nm FR on END or HORN ——-I£ _— HORN MONITOR MICROPHONE J LOCATION OF MEASURING STATIONS IN EXHAUST PIPE -gk- -25- construct such a termination cotton was first used in the pipe to absorb a maximum of sound energy without reflection. It was soon found that, after exposure to high intensity sound levels the cotton would pack to varying degrees of density, completely modifying the percentage of absorbed and the percentage of reflected power. The cotton was then separated into small sections or compartments. Even then, the perfor- mance of the infinite tailpipe was very erratic and to tally unsatis- factory. Finally after many approaches were tried, and many more investigated, a totally satisfactory solution was arrived at. A.mesh of flat, thin stainless steel wire was inserted into the pipe. The material was heavy and resilient to the extent that high in- tensity sound levels did not alter the density with which the material had originally been packed. It was soon found that, even though reason- able absorption was attained, a denser material should be used in a portion of the pipe, to effect total absorption. The final construction consisted of 12 feet of packed mesh followed by 15 feet of cotton pre- packed and in fixed sections. The drop of intensity across the mesh was sufficient to reduce pressure levels to a point where the cotton portion of the tailpipe performed in a consistent manner. The ultimate reflection set up by this termination was always withing: .75 DB. In the frequency range from 50 to 1000 cps. A.typical calibration curve for the infinite tailpipe is shown in Figure 7. DEC/EELS +I FIGURE 7 \ \ \] FREQUENCY X [00 CPS CALIBRATION OF INFINITE TAILPIPE -26- -27- 3. Physical Layout A substantial portion of the physical layout has now been described. It will suffice to mention that all sound tests were run in the acoustical laboratory of Hayes Industries, Inc. in Jackson, Michigan. A.layout of the test set up in the anechoic room, with physical dimensions of the system is shown in Figure 8. It should be noted that all sound measurements were made inside the pipes, and the use of an anechoic room would not in itself have been required. A. Calibration Details Calibration of all electronic instrumentation was performed in accordance with the recommendations of the manufacturers. The calibra- tion curves of the condenser microphones are shown in Figure 9 and the calibration of the infinite tailpipe in Figure 7. The sound source calibration was not deemed important. Indeed inasmuch as the muffler evaluation was based upon an analysis of the muffler performance at one frequency point at the time, the particular loudspeaker efficiency at that frequency was not important. With all measurements both in front and in back of the test muffler taken through the same micrOphone, the micrOphone calibration was also of no major importance. Readings were important as they related to other readings. The absolute value of these readings was only of interest to determine approximate sound pressure levels at which tests could proceed. These levels were between 130 and 160 DB. re .0002 dyne/em2 Calibration checks were made frequently however in order to make sure the micrOphones were functioning normally. The one very important ‘\ FIGURE 8 _ [2' :r‘ l5, WIRE HESH PACKED COTTON TEST MUFFLER NONI TOR NIKE. 26 FEET _ TEST SETUP IN THE ANECHOIC ROOM. ~28- FIGURE 9 ‘7 DYNE/cmz RATED SENSITIVITY e -55 / 2i BR -I50 / ALTEC LANSING / CONDENSER MICROPHONE -65 MICROPHONE SENSITIVITY- DB R. I I40 AI I35 I30 ACOUSTIC CALIBRATOR p’25 SIGNAL ( I VOLT INPUT) I20 0/23‘4.5 SOUND PRESSURE LEVEL " DEC/EELS. FREQUENCY X IOOi CBS CALIBRATION CURVES OF CONDENSER MICROPHONES -30- calibration, namely that of the infinite tailpipe was a source of extreme satisfaction. The consistency of the operation of the infinite tailpipe prompted us to disregard correcting tailpipe sound pressure levels. It was felt that the magnitude of the correction factor was ‘well'within acceptable limits for this program. CHAPTER III DETAILS OF MUFFLER TEST PROGRAM More information must be given.on how the tests were actually run, and how the measured values were interpreted. In addition it seems opportune to reiterate in detail the scope of this program and its true objective. This information then will set the stage for the analysis of the test results in Chapter IV and V. Correlation of experimental data with computer results will then enable us to arrive at our conclusions and recommendations covered in Chapter VI. 1. Method of Test 'Upon calibration of the equipment and after setting the compressor circuit controls in such a way that sound levels at the monitor micro~ phone never exceed 160 db over the entire frequency spectrum (from 50 to 650 Cpfi the recorder is started up and the tests can begin. The test microphone is moved from one measuring station to another. The sound level drOps by more than.UO db as the microphone is pulled from one of the measuring points in.the pipe. This gives us very clear distinction between the sound levels at each station. The recording of successive sound levels creates a good pattern of the standing wave system in the exhaust pipe. Continuous checks are made of the operating level of the power supply to the condenser microphones, calibration of the oscillator, -31- -32- and calibration of the recorder. At the same time the wave form at the measuring station gives an indication of the presence of harmonics of distortion of the wave. Even though in most cases distortion was negligible it seems as though a study of all distortions to the original sine wave would be of interest. Vibration of baffles inside the muffler as well as so-called "shell noise" vibration of the muffler shell seem to contribute to modifications of the sine wave signal. Upon measuring all sound levels in front of the muffler, the sound levels in the three stations in the tailpipe are also recorded. 2. Analysis of Measurements The maximum measured attenuation is the difference between the maximum sound level measured in the standing wave, and the sound level in the tailpipe. It is obvious that the maximum.sound level in the stand- ing wave is not representative of the magnitude of the incident wave. A rather complicated procedure of analysis is required to determine the true value of the incident wave. we would have to go through this proce- dure for every muffler tested, as well as for every frequency. To eliminate this tedious repetitive procedure, Reference No. 2 illustrates a."short cut" which is of sufficient accuracy in all cases and which allows for immediate correction of measured maximum attenuation to a reliable value of actual attenuation. The reasoning on this short procedure is as follows: Assume that all sound reflection takes place from a single point, and that the incident sound pressure is unity. Assume 20 percent of the incident wave is reflected; then 80 percent of the incident wave is TRUE ATTENUATION DB (.u 0 N (.71 N 0 \ Ch 0. FIGURE 10 / / / / IO I5 20 25 50 MEASURED ATTENUATION DB CORRECTION FACTORS FOR MEASURED ATTENUATION -33.. 55 -3A- transmitted. The maximum sound level measured in the standing wave is then 120 percent of the incident wave or 1.2 . The true attenuation would be 10 loglo (5513)2 = 20 loglo i8 = 1.9382 db The maximum.measured attenuation: 2O log«£f§% = 20 log 1.5 = 20 x .1761 = 3.522 db The correction required therefore would be: 3.522 - 1.938 = 1.58 db or 20 log-iigr= 1.58 db This implies that the correction factor is in fact a function of the measured attenuation. This being the case, a curve, shown in Figure 10, was plotted and for all determinations of attenuation, the procedure of applying a correction factor to the maximum measured attenuation was used. \ 3. Reiteration of Objectives To better illustrate the reasons for the choice of test units we repeat here in detail the objectives of the program. a. DevelOp a test system.capable of reliably "cold testing" muffler circuits of importance to automobile muffler design. b. Restricting the test program to cold testing thereby eliminating the variables of gas flow and temperatures in the exhaust system determine the acoustical value of so called louvers as compared to holes of various sizes, in quantities however to produce the same total outlet area over equal length. c. Extend the acoustical theory to the cross-bleed chamber and determine its' true nature as an acoustical filter. -35- d. Extend the theory and investigation tO'uunlaround chambers which are present in virtually every original equipment muffler. e. Through correlation of experimental and theoretical analysis establish an approach for predicting the attenuation of a muffler circuit. f. In gaining sufficient understanding of the important variables involved, discuss the anticipated influence of gas flow and elevated temperatures on the acoustical behavior of mufflers. Briefly discuss the subjective nature of all final judgment on sound. Make recommendations for future projects. A. Series of Test Units The first portion of the test program.was planned in an effort to determine the reliability of the test procedure and the test apparatus. To this end three simple Helmholtz resonators were constructed. Their attenuation characteristics are easy to calculate, and they show a very sharp peak which would establish the degree of accuracy as a func- tion of frequency. In an effort to determine the reliability of the computer programs, the perfOrmance curve of one of the NACA.mufflers was computed and compared to the original NACA results. The next step was to study the effect of louvers as compared to holes of various sizes. Figures 12 through 1h show mufflers which were built and tested. In addition to the study of the louver effect it was of interest to determine whether or not such a chamber would behave as a true expansion chamber. The true equivalent expansion chamber is shown in Figure 15. Test results and computer data are shown in Figure 12 through 15. It should be noted that the blind tubes a and b in Figure 15 ATTENUATION - 06 FIGURE 11 t.— 4.50.... I I 7. 347 I l 1 [.50 DIA. _.. .— 2.00 ' i L75 IA 35 30 25 20 I5 I0 / K’\g__ I 2 3 Y«4 5 e FREQUENCY XIOO CPS. SIMPLE HELMHOLTZ RESONATOR ‘0-36- 6.5 -37- were installed only to maintain perfect similarity of construction with the other units and to keep the cross sections equal to those of the other test mufflers. In the next series of mufflers we proceed from the single expansion chamber and the Helmholtz chamber to combinations of chambers. The entire series was built with the purpose of ultimately combining cross bleed and turn around chambers as shown in Figures 16 through 19 and as we normally find in commercial automobile mufflers. The double expansion chamber shown in Figure 16 is fed through one single pipe. The next muffler shown in Figure 17 feeds the second chamber through two pipes. In the following circuit, ”true reversal of flow is accomplished. In Figure 19 we progress to a triple chamber design and further to an additional reversal of flow in Figure 20, to our final muffler of the program. In conclusion, a typical commercial muffler is sketched (Figure 25). This muffler was not actually constructed and tested. The purpose of showing it in this report is to illustrate which portions of a conventional muffler circuit were dealt with in detail. For the purposes of extending theory to this commerical muffler version, however, we have set up the applicable system of equations. AT TENUATION. DB 3O 25 20 I5 IO FIGURE 12 . PLUG THIS END I I" I I. 751.0. PLUG THIS END I [2 GROUPS QFBO COARSE EXTRUSIONS = 160 EX TRUSIQNS -PERJHAHBEL--_ o/ I ,/ \ I 2 3 4 5 6 6.5 FREQUENCY X I00 CPS LOU'VER FED EXPANSION CHAMBER -38- PLUG THIS END ATTE NUATI ON ' DB FIGURE 13 F— 8.00 r PLUG THIS I a. ' ' £7510 30 25 20 I5 fly/ ,0 e ///// Y 0 I 2 3 4 5'? 6 61? FREQUENCY X/OO CPS EQUIVALENT EXPANSION CHAMBER WITH l/8 INCH HOLES -39- FIGURE 11+ ' PLUG THIS I END r ------ ”1 ' ’ : : I.751.D. I“ """" ‘"” I 1 Z ’ u PLUG THIS END 48 HOLES /4 DIA. 30 25 a, 20 Q ' o o In 5% I5 .,/f E / < £2 I0 I :2 / h a 0/ 5 O I 2 3 4 5' 6 Cd? FREQUENCY X IOO CPS. EQUIVALENT EXPANSION CHAMBER WITH l/LI INCH HOLES -I+Q- ATTENUATION ‘ DB 30 25 20 I5 IO FIGURE 15 PLUG BOTH "-————- 8.00 --—————--~ ENDS / N / I 2 3 4 5 6 6.5 FREQUENCY XIOO CPS CONVENTIONAL EXPANSION CHAMBER -41- FIGURE 16 PLUG THIS END ~——————8.oo ———|I.8/ r— L. ___________ i I I“ """"""""" I I I I I /L, —————————— -_1 PLUG ms mo 45mm or so cams: urn/was 35 f , - T dd so / I y 25 / ¢fl 0 3 20 j 2 «r / 2 I5 5 u: ' e o I: ’0 §,5—“\ < 7 \ 5 A v O I 2 3 4 5 6’ Gli FREQUENCY X100 CPS DOUBLE EXPANSION CHAMBER -h2- FIGURE 17 PL us rm: mo 2 GROUPS or so coma: 'ExrR/Tua: 36 - ‘ 30 f v—J ATTENUATION - DB Br 3... 0 :F I \ 0 I 2 3 4 5 6 6.5 FREQUENCY X IOO CPS DOUBLE EXPANSION CHAMBER -43- ATIENUATION - 05 FIGURE 18 35 30 25 20 I5 I0 r— 8.00 ———- I8! I" ““““““““ “I I I I I I. ___________ I I" -------- '1 | I . I 5 ___________ J / ‘V 0/ 0 ° 9 4" O / I ‘2 3 4 ‘ 5 ,. L6 6.5 FREQUENCY “a" X I00 CPS EXPANSION CHAMBER AND TURN-AROUND CHAMBER -I-ILI- ATTEALUAZIQN - DB .35 30 25 20 [5 IO. FIGURE 19 LBI \; 9 9;. H ° h\\/r If I 2 3 4 5 a FREQUENCY xmo CPS TRIPLE CHAMBER MUFFLER -hs- 6:3 35 30 25 8 z 20 o E 3 I5 2 Lu .2 < IO 5 FIGURE 20 VT ----—----—-- I 2 GROUPS or so cams: (um/rues I \ «If \0 «r X 2’ X f V V [”2 3 4 5 FREQUENCY XIOO CPS 6.5 TRIPLE CHAMBER MUFFLER WITH DOUBLE REVEIBAL OF FLOW -45- up I. '~ ‘71 u. .. MATHEMATICAIIANAIISIS CI'TKHBINUTKLKR SYSTEMS I‘M Utilizing the background gained in Chapter I've will now proceed to deve10p the mathematical solutions to the variouszmuffler circuits considered. In most cases it will suffice to arrive ”J d- . a system of equations for the units, leaving the solution of that system of equations to the computers. we will stazt with the simplest coni’gurations, gradually moving into some of the more elaborate circuits. 1. Helmholtz P = m (A2 - B2) and s2 (Aee'Jkl - BeeJkl) = sl A3 (28) or m (Age-Jkl - Baejkl) = A3 (29) The simultaneous solution of equations 26 - 27 - 28 - 29 for'fig lead to the attenuation = 10 loglo '2: = 10 loglo [l +-% (m‘-'H)2 sin 2kl] -52. 3. Combinations of Chambers Various combinations of chambers make up the average automobile muffler. we are giving here the systems of equations for some of the mufflers in the test series. The mmffler as shown in Figure 16 consists essentially of two expansion chambers. An equivalent circuit (for low frequencies for which side inlet presents no noticeable difference from center inlet) is shown in Figure 23. FIGURE 23 1 1.: 2r i I ‘3! :‘As ______i£%L Iléi"i L____. A,|A2 A4 .A‘ 5’ 61:52 82 :54‘SSIBGS4IA755 In: I . : --.- ISL.— -"l [C butt-DI The conditions of continuity of pressure and volume flow give the followa ing system of equations: (30) A1 + B1 = A2 + B2 2 A3 + B3 (31) B3 = A3eJ2le (32) 81 (A1 ‘ B1) = (A2 ‘ Be) S2 ' (32 ‘ SI) (A3 ‘ B3) (33) AEe'3k(le ' ls ' 1c) + Bee3k(le ' ls ' 1c) = AA + Bu = A5 + B5 (3”) B = A.e-J2klc 5 5 (35) SHEAQe-Jk(le - ls _ le) - B ejk