A. G. Falkowski Chrysler Corporation, Detroit, Michigan 48227

Transcription

1 Three-ass mufflers with uniform erforations A. Selamet V. Easwaran The Ohio State University, Deartment of Mechanical Engineering Center for Automotive Research, 06 West 18th Avenue, Columbus, Ohio 4310 A. G. Falkowski Chrysler Cororation, Detroit, Michigan 487 Received 1 June 1998; revised 1 October 1998; acceted 17 November 1998 A quasi-one-dimensional aroach is resented to analyze three-ass mufflers with erforated elements using numerical decouling. The aroach is further develoed to include mufflers with ducts extended into the end cavities. Theoretical redictions are comared with exeriments for three different muffler configurations, one fabricated two commercially available mufflers, shown to agree reasonably well. The effect of orosity, length of the end cavities, exansion chamber diameter are studied. Also, the effect of ducts extending into the end cavities are investigated Acoustical Society of America. S PACS numbers: 43.0.Mv, Gf ANN LIST OF SYMBOLS A cross-sectional area c 0 seed of sound d duct diameter D d/dx f frequency; as defined in Eq. 4 i imaginary unit, 1 k 0 wave number, /c 0 M Mach number n unit vector normal to the control surface ressure t time; erforate thickness T transfer matrix TR rank reduced transfer matrix as defined by Eq. 63 u axial velocity u velocity vector U velocity of moving medium v radial velocity x axial coordinate Greek symbols control surface end correction length secific imedance of erforate eigenvalue density diagonal matrix comosed of eigenvalues orosity/oen-area ratio vector of generalized coordinates/transformation vector matrix of eigenvectors angular frequency, f control volume Subscrits 0 mean value LEC left end chamber REC right end chamber Suerscrits erturbed quantity d/dx h erforate T transose INTRODUCTION Perforated tube elements are widely used in resonators mufflers to suress engine exhaust noise. Tyical examles include concentric tube resonators cross-flow elements. The flow through the tubes can be either straightthrough or reversed. Aero-acoustic analysis of erforated elements in the form of a series exansion for the straightthrough silencer elements was first resented by Sullivan Crocker. 1 A segmentation method develoed by Sullivan,3 later lumed the effect of erforations into a number of discrete lanes with solid ies resent in between. This was followed u by Jayaraman Yam 4 who develoed a decouling method to obtain a closed form analytical solution. Thawani Jayaraman demonstrated the use of this method 5 by analyzing straight-through resonators in the absence of flow. Rao Munjal 6 extended the analysis of Jayaraman Yam by allowing the Mach numbers in the inner erforated duct the outer casing to be different. Munjal et al. 7 then develoed a decouling rocedure by extracting the roots of the characteristic olynomial of the system numerically. Later, Peat 8 resented another numerical decouling aroach where the eigenvalues the eigenvectors were obtained rather than the roots of the characteristic olynomial, thereby removing the numerical instability roblems associated with other methods described thus far. Gogate Munjal 9 extended the analysis of Munjal et al. 7 to include oen-ended two-ass erforated mufflers. Re J. Acoust. Soc. Am. 105 (3), March /99/105(3)/1548/15/$ Acoustical Society of America 1548

2 Three-duct erforated ele- FIG. 1. ment. cently, Dokumaci 10 resented a matrizant aroach for the analysis of erforated duct mufflers included the gradient of the mean flow in the analysis which was later extended to include multile duct erforated ie arrangement with identical erforated ies. 11 All of these studies, excet Ref. 10, are limited to concentric tube resonators, lug mufflers, or two-ass mufflers two erforated ducts lus an exansion chamber. As many of the commercially available mufflers imlement three asses three erforated ducts lus an exansion chamber, such configurations have been of more recent interest. Ross 1 attemted the analysis of the three-ass muffler using the finite element method, but only demonstrated the results for the case where the middle ie was just ass through not erforated. Recently, Dickey et al. 13 resented a time-domain comutational analysis of a multi-ass erforated muffler, illustrated the results for a three-ass muffler. The corresonding frequency domain analysis of this muffler was later resented by Munjal, 14 via a decouling method, which was also refined to include the extended-tube three-ass erforated element muffler. 15 This decouling aroach was shown by Peat 8 to lead to numerical instability roblems, esecially near transmission loss eaks. The resent study has develoed a quasi-onedimensional theoretical model for the analysis of a three-ass erforated duct muffler based on the transfer matrix method the numerical decouling of Peat. 8 The method is then generalized to include the analysis of three-ass mufflers where the ducts extend into the end cavities. A rototye three-ass muffler was built the transmission loss was measured in an imedance tube facility for this rototye muffler, as well as for two roduction mufflers. Exerimental results are shown to be in reasonable agreement with the theoretical redictions. Following this introduction, the general theoretical aroach is discussed in Sec. I for the analysis of the three-ass muffler. The numerical decouling of the differential equations is described in Sec. II, the boundary conditions for the exansion chamber the end cavities in Sec. III, a transfer matrix solution in Sec. IV. The results are resented in Sec. V, followed by concluding remarks in Sec. VI. I. THEORY The derivation of governing equations follows, in general, that of Sullivan,3 Peat. 8 The mean flow in the chamber designated by 4 in Fig. 1, however, is assumed to be zero. In addition, the thicknesses of the walls of the erforated tubes designated by 1 3 in Fig. 1 are assumed to be negligible comared to the diameter of the chamber, the gradient of the mean flow is neglected. Consider four control volumes, each of length dx, as shown in Fig. 1. For simlicity, the tube the chamber are assumed to be circular. Integrating the continuity equation, t u0, over a finite control volume alying the divergence theorem yields 1 t d u j n d0, j1,,3,4, where u is the velocity vector, is the density, is the control surface, n is the unit vector normal to the surface. Alying Eq. to each of the four control volumes yields where j t u j j x u j x f j0, j1,,3,4, 3 f j 4v j,4 d j, j1,,3; f 4 4d 1v 1,4 d v,4 d 3 v 3,4 d 4 d 1 d d ; 3 v j,4 is the radial velocity of duct j; d 1, d, d 3 are the diameters of the erforated tubes; d 4 is the diameter of the chamber or the external housing J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 1549

3 Similarly, integrating the momentum equation u j t u j u j 1 j over the finite control volume gives u j d 1 1 u t j u j n d j d0, j1,,3,4, 6 where j is the ressure in duct j. The x comonent of Eq. 6 is then u j t u u j j x 1 j 0, x j1,,3,4. 7 Equations 3 7 are linearized by substituting 0, u j U j ũ j, j 0 j retaining only the first-order terms, where U j is the mean velocity in duct j, the symbol denotes a fluctuating comonent, the subscrit 0 is the steady-state comonent. This yields ũ j t U j x 0 x 0f j 0, j1,,3,4, 8 for the continuity equation ũ j 0 t ũ j 0U j x j 0, j1,,3,4, 9 x for the momentum equation. Similarly, from isentroic relationshi c For uniform erforations in a duct of constant cross section, at any osition along the erforate, 1 v j,4 j 4, j1,,3, 11 0 c 0 j where c 0 is the seed of sound j is the secific acoustic imedance of the erforate of duct j, given in terms of the 5 mean flow, duct geometry, the erforate geometry. 1 3,8,16 Substituting the exressions for f j given, resectively, by Eqs. 4 10, ineq.8 emloying Eq. 11 yields 1 j c 0 t U j j c 0 x ũ j 0 x 4 d j for j4) 1 4 ũ 4 c 0 t 0 x 4 d 4 d 1 d d 3 m1 3 j 4 c 0 j 0, j1,,3, 1 d m c 0 m m Uon oerating Eqs with /t subtracting from it the corresonding Eq. 9 after oerating it with /x) yields 1 j c 0 t U j j c 0 tx 1M j j x 4 c 0 j d j j t 4 t 4M j j d j for the erforated ducts ( j1 to3, j x 4 x c 0 t 4 x 4 d 4 d 1 d d 3 d t d t d t 14 4 d 4 d 1 d d 3 d 1 d d t for the chamber ( j4). Substituting for the time harmonic motion x,tx ex it in Eqs yields 16 D1D D10 0 D D D D 3 D 6 9 D D where j M j 1M j ik j d 0, j1,,3, 18 j j3 1 1M j k 0 4ik 0 j1,,3, 19 j d j, 4M j j6 1M, j j d j j1,,3, 0 4ik 0 j9 1M, j j d j j1,,3, 1 4ik 0 d j / j j1 d 4 d 1 d d, 3 j1,,3, 1550 J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 1550

4 16 k , 3 k 0 f /c 0 is the wave number, f is the frequency. Equation 17 is decouled 8 solved as shown next. II. NUMERICAL DECOUPLING Let y 1 1, y, y 3 3, y 4 4, y 5 1, y 6, y 7 3, y 8 4, 4 where the symbol denotes d/dx. Substituting Eq. 4 in Eq. 17 then yields where yby, 5 y y 1, y, y 3, y 4, y 5, y 6, y 7, y 8 T, B suerscrit T denotes the transose. Let y, 8 where is a matrix whose columns are the eigenvectors of matrix B, is a transformation vector or a vector of generalized coordinates. Substituting Eq. 8 in Eq. 5 then gives 1 B, 9 where is a diagonal matrix comosed of the eigenvalues of the matrix B. Substituting the solution of Eq. 9, c 1 e 1 x,c e x,c 3 e 3 x,c 4 e 4 x, in Eq. 8 yields c 5 e 5 x,c 6 e 6 x,c 7 e 7 x,c 8 e 8 x, T, y c 1 e 1 x,c e x,c 3 e 3 x,c 4 e 4 x, c 5 e 5 x,c 6 e 6 x,c 7 e 7 x,c 8 e 8 x T Substituting Eq. 31 in Eq. 4, relacing j by 0 c 0 u j using Eq. 9, then rearranging the equation gives xc, where 1, 0 c 0 u 1,, 0 c 0 u, 3 3, 0 c 0 u 3, 4, 0 c 0 u 4 T, 33 c c 1, c, c 3, c 4, c 5, c 6, c 7, c 8 T, e 1 x 5 e x 58 e 8 x 11 e xl 1 x 1 e x 18e 8x ik 0 M 1 1 ik 0 M 1 ik 0 M e 1 x 6 e x 68 e 8 x 1 e 1 x e x 8e 8x ik 0 M 1 ik 0 M ik 0 M 8 71 e 1 x 7 e x 78 e 8 x m. 31 e 1 x 3 e x 38e 8x ik 0 M 3 1 ik 0 M 3 ik 0 M e 1 x 8 e x 88 e 8 x 41 e 1 x ik 0 M 4 1 III. BOUNDARY CONDITIONS 4 e x 48e 8x ik 0 M 4 ik 0 M The schematic of two tyical three-ass erforated element mufflers i where the erforated ducts do not extend into the end chambers ii where the erforated ducts extend into the end chambers are shown in Figs. 3, resectively. Two different segments can be identified: i exansion chamber ii end chambers. Boundary conditions for these segments are discussed in this section J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 1551

5 FIG.. A tyical three-ass erforated muffler. A. Exansion chamber At sections A B, because of the rigid walls the velocity u 4 0. Therefore, 0 c 0 u 4 A i tan k 0 l a 4, 0 c 0 u 4 B i tan k 0 l b 4. B. End cavities No extensions 36 At the junction C in the right end chamber REC see Fig. 1 REC, A 1 u 1 A u A REC u REC, 37 i tan k 0 l c REC 0 c 0 u REC, where A 1, A, A REC are the cross-sectional areas of ducts 1,, the end chamber at C, resectively. Also, 1 0 c 0 u 1 B cos k 0l b1 i sin k 0 l b1 i sin k 0 l b1 cos k 0 l b1 1 0 c 0 u 1 C 38 0 c 0 u cos k 0l b i sin k 0 l b B i sin k 0 l b cos k 0 l b 0 c 0 u, C 39 where l b1 l b t b 1, l b l b t b C,.45d d 1 d 4 d 3, 40 C 0.45d d d 4 d 3 are the end corrections for the exansion of ducts 1, resectively, into the end chamber at C. The end correction FIG. 3. A tyical three-ass erforated muffler with ducts extending into the end cavities. 155 J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 155

6 lengths are added to account for the lumed inertance at the ends of ducts 1 at location C corresond to the lumed inertance of a duct terminating into an infinite flange. 17 Combining Eqs gives where 1 0 c 0 u 1 B Q cos k 0l b1 Q 0 c 0 u, 41 B i sin k 0 l b1 i sin k 0 l b1 cos k 0 l b1 1 i A REC tan k A 0 l c A 1 A cos k 1 0l b i sin k 0 l b i sin k 0 l b cos k 0 l b. 4 Similarly, alying boundary conditions at junction D in the left end chamber LEC leads to 0 c 0 u A R 3 0 c 0 u 3, 43 A where R can be obtained from Q by relacing i l b by l a, ii t b by t a, iii l c by l d, iv A 1 by A A by A 3, v 1 by 3 by 4, vi the subscrit REC by LEC, as cos k R 0 l a1 i sin k 0 l a1 i sin k 0 l a1 cos k 0 l a1 i A LEC tan k A 0 l d ) A 3 A cos k 0 l a i sin k 0 l a i sin k 0 l a cos k 0 l a 1, 44 where l a1 l a t a D, l a l a t a 3, D 0.45d d d 4 d 1, d 3 d d 4 d 1, are the end corrections for the exansion of ducts 3, resectively, into the end cavity at D. C. End cavities With extensions A tyical three-ass muffler where the inlet, center, the outlet ducts extend into the end chamber is illustrated in Fig. 3. An enlarged view of its right end chamber is shown in Fig Case I (l 1 l ) At location E see Fig. 4 1 E REC E REC E A 1 0 c 0 u 1 E A REC A 1 0 c 0 u REC E A REC 0 c 0 u REC E Because of the rigid ends at locations C C, u REC 0. Hence, 0 c 0 u REC i tan k 0 l c l 1 REC E 48 0 c 0 u REC REC E i tan k 0 l. 49 Combining Eqs gives 1 i A REC tan k A 0 l c l 1 A RECA 1 1 A 1 0 c 0 u 1 E At location E, FIG. 4. An enlarged view of right end chamber. REC 0 c 0 u REC. 50 E REC E REC E E J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 1553

7 A REC A 1 0 c 0 u REC E A REC A 1 A 0 c 0 u REC E A 0 c 0 u E. Combining Eqs. 49, 51, 5 yields 5 REC 0 c 0 u REC E i A RECA 1 A A REC A 1 1 A tan k 0 l A REC A 0 c 0 u. 53 E Also, the state variables REC u REC at either ends of the duct segment E E are related by REC 0 c 0 u REC E cos k 0 l 1 l i sin k 0 l 1 l REC 0 c 0 u REC E. i sin k 0 l 1 l cos k 0 l 1 l 54 0 c 0 u B cos k 0l b4 i sin k 0 l b4 i sin k 0 l b4 cos k 0 l b4 0 c 0 u, E 57 resectively, where l b3 l b t b l 1 1 l b4 l b t b l, 1 are end corrections for the exansion of ducts 1 into the chamber at E E. Thus, in view of Eqs. 56, 57, 53, the matrix Q defined by Eq. 41 becomes Q cos k 0l b3 i sin k 0 l b3 i sin k 0 l b3 cos k 0 l b3 i A REC tan k A 0 l c l 1 A RECA 1 1 A 1 cos k 0 l 1 l i sin k 0 l 1 l i sin k 0 l 1 l cos k 0 l 1 l i A RECA 1 A A REC A 1 1 A tan k 0 l A REC A cos k 1 0l b4 i sin k 0 l b4 i sin k 0 l b4 cos k 0 l b4. 58 In view of Eqs. 50, 53, 54, 1 0 c 0 u 1 E i A REC tan k A 0 l c l 1 A RECA 1 1 A 1 cos k 0 l 1 l i sin k 0 l 1 l i sin k 0 l 1 l cos k 0 l 1 l i A RECA 1 A A REC A 1 1 A tan k 0 l A REC A 0 c 0 u. 55 E In the resence of ducts extending into the end cavities, Eqs become cos k 0l b3 i sin k 0 l b3 1 0 c 0 u 1 B i sin k 0 l b3 cos k 0 l b3 1 0 c 0 u 1 E 56. Case II (l 1 <l ) Following the rocedure illustrated for Case I, it can be shown that 0 c 0 u E i A REC tan k A 0 l c l A RECA A cos k 0 l l 1 i sin k 0 l l 1 i sin k 0 l l 1 cos k 0 l l 1 i A RECA 1 A A REC A A 1 tan k 0 l 1 A REC A 1 0 c 0 u E Equation 59 can also be obtained from Eq. 55 by switching subscrits 1, E E. The matrix Q is then given by 1554 J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 1554

8 Q cos k 0l b3 i sin k 0 l b3 i sin k 0 l b3 cos k 0 l b3 i A RECA 1 A A A REC A 1 tan k 0 l 1 A REC A 1 1 cos k 0 l l 1 i sin k 0 l l 1 i sin k 0 l l 1 cos k 0 l l 1 1 i A REC tan k A 0 l c l A RECA A cos k 1 0l b4 i sin k 0 l b4 i sin k 0 l b4 cos k 0 l b4. 60 The matrix R defined by Eq. 43 may now be obtained from Q following the rocedure similar to that illustrated in Sec. III B. In the absence of ducts extending into the end chambers, Eqs can readily be shown to reduce to Eq. 4. IV. TRANSFER MATRIX SOLUTION The transfer matrix T is defined as xa T xb, 61 which relates the state vector at A to that at B. By substituting xx A xx B in Eq. 3 gives Tx A x B 1. 6 Equation 61 subject to aroriate boundary conditions is the solution of the couled Eq. 17, which is illustrated next. Substituting Eq. 36 in Eq. 61 yields 1 1 TR 0 c 0 u 1 0 c 0 u 3 0 c 0 u 3 A 0 c 0 u 1, 63 0 c 0 u 3 0 c 0 u 3B where TR is the rank reduced transfer matrix given in terms of the elements of T by T 8n i tan k 0 l a T 7n T m7 i tan k 0 l b T m8 TR mn T mn, m,n1,...,6. 64 T 87 i tan k 0 l b T 88 i tan k 0 l a T 77 i tan k 0 l b T 78 Substituting Eqs in Eq. 63 yields where 1 0 c 0 u 1 T overall 3 A 0 c 0 u 3, 65 B T overall TR 13 TR 11 QTR 1 TR 1 Q TR RTR 31 QRTR 3 1 TR 3 RTR is the overall transfer matrix. Transmission loss TL is then evaluated by 18 TL0 log 10 1 A 1 A 11 T 1 T 1 T 3T 1M 1 1M Comuter rograms were develoed in C to redict the transmission loss of a three-ass muffler with without the erforated ducts extending into the end cavity. The results obtained from the aroach described in Secs. II IV are resented next. V. RESULTS AND DISCUSSION A. Prototye muffler A rototye three-ass muffler with a clear cast acrylic external housing of diameter d m with three erforated brass ies forming the three asses of inner diameter d 1 d d m thickness t m was fabricated to facilitate controlled exeriments. End cavities of lengths l c 0.15 m l d 0.10 m were searated from the central section by aluminum baffles of thickness t a t b m. An interior duct orosity of was obtained by drilling 448 holes of d h m diameter in each duct over the central region such that l m l a l b m. The dimensions of the fabricated rototye were chosen to dulicate a Chrysler roduction muffler. The three erforated brass tubes are held in osition by two identical baffles as shown in Fig. 5. The transmission loss characteristics of the fabricated rototye are measured in an imedance tube setu. The details of the setu are described elsewhere. 19 Figure 6 comares the theoretical redictions deicted by solid line exerimental results deicted by solid symbols for the fabricated rototye. The exerimental results agree reasonably well with the theoretical results u to about 900 Hz. Higher-order acoustic modes start roagating beyond this frequency, limiting the resent analysis. Note, this frequency is somewhat lower than the limit for the first diametral mode in a circular duct f first diametral mode 1555 J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 1555

9 FIG. 5. Schematic of the baffle late. FIG. 6. Transmission loss of the three-ass erforated muffler: theory versus exeriments. l 0.74 m, l a l b m, l c 0.15 m, l d 0.10 m, t a t b m, d 1 d d m, d m, d h m, t m, 0.045, 1.18 kg/m 3, c m/s, M J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 1556

10 1.84c 0 /d max 10 Hz, for d max chamber m a sound seed of 344 m/s. This limit, however, shifts to higher frequencies as the seed of sound increases or as the gas temerature rises since the seed of sound varies in direct roortion to the square root of the gas temerature. At very low frequencies 50 Hz, the imedance tube termination is not fully anechoic results in some discreancy between theoretical redictions exerimental results. The overall behavior resembles that of a simle exansion chamber of length l l l a l b because of the central section, with troughs occurring at frequency intervals of c 0 /l 50 Hz, eaks at odd multiles of the frequency c 0 /4l 60 Hz), with a suerimosed resonance near 40 Hz, contributed by the resence of the two end cavities. Note, the erforate imedance for quiescent medium, ik 0 t0.75d h /, 68 was used in Fig. 6. Here, t is the duct thickness d h is the erforate diameter. This same imedance is used in all the figures that follow. B. Production mufflers The transmission loss characteristics of two roduction three-ass mufflers used in Chrysler vehicles were measured in the imedance tube setu. The mufflers were later cut oen to obtain the geometric details. The exansion chamber the end cavities of both mufflers are ellitical in cross section. For theoretical treatment, ellitical cross sections were relaced by circular ducts of equal cross-sectional area. The theoretical exerimental results for the first muffler are shown in Fig. 7. As in Fig. 6, the muffler shows an exansion chamber behavior with a suerimosed resonance at about 50 Hz. The resonance may be attributed to the combined effect of the end cavities. Theoretical results comare reasonably well u to about 700 Hz, a frequency dictated rimarily by the roagation of higher-order modes. Discreancy between theory exeriments may also be attributed to the fact that the exact geometry the louver size its distribution are not known a riori; only a rough estimate of the hysical dimensions of the muffler has been used in the comutations. Also, the cross-sectional area of the inlet outlet ducts are not exactly uniform as assumed by the theoretical model. The inlet center erforated ducts of the second muffler extend into the right the left end chambers, resectively. The two baffle lates of this muffler have two holes each, aroximately m in diameter. Since the resent theoretical method cannot hle baffle holes which would coule the exansion chamber the end cavities directly, these holes were lugged before the muffler was mounted in the imedance tube setu. Theoretical exerimental results for this muffler are comared in Fig. 8. As in Figs. 6 7, two resonant eaks occur at low frequencies, due mainly to the resence of the end cavities. The agreement between the theory exeriments is reasonable at low frequencies. Similar to the muffler of Fig. 7, the discreancy between theory exeriments may artially be attributed to the fact that the exact geometry, the ore size its distribution are not known a riori; only a rough estimate of the hysical dimensions of the muffler has been used in the comutations. The theory assumes that there is a common length l over which the inlet, center, the outlet ducts are erforated. This articular muffler, however, has erforations which are nonuniformly distributed. Therefore, an average erforated length has been used in the comutations. FIG. 7. Transmission loss of a Chrysler muffler theory versus exeriments. l 0.8 m, l a 0.0 m, l b 0.03 m, l c 0.10 m, l d 0.15m, t a t b m, d 1 d d m, d m, d h m, t m, 0.09, 1.18 kg/m 3, c m/s, M J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 1557

11 FIG. 8. Transmission loss of a Chrysler muffler theory versus exeriments. l m, l a 0.05m, l b 0.05 m, l c 0.5 m, l d m, l.05 m, l 0m, l 0.11 m, l 3 0m, t a t b m, d.048m, d m, d m, d m, d h m, t m, 0.09, 1.18 kg/m 3, c m/s, M0. This may also contribute to the differences between the theoretical redictions the exerimental results. C. Parametric study Similar to Ref. 14, the variations of the arameters which affect the erformance of the three-ass muffler are studied in this section. The rototye muffler is used as the baseline case in all the comarisons that follow see solid lines in Figs The sensitivity of the three-ass muffler to variations in duct orosity is shown in Fig. 9. The effect of orosity is marginal until about 600 Hz beyond which exansion chamber behavior dominates. At frequencies greater than 600 Hz the transmission loss is greater for the muffler with lower duct orosities. At low frequencies 300 Hz, where the resonances are due to the end cavity, the effect is just the oosite. These frequency limits deend, to a large extent, on the geometry of the muffler. FIG. 9. Effect of orosity on the erformance of a three-ass muffler. l 0.74 m, l a l b m, l c 0.15 m, l d 0.10 m, t a t b m, d 1 d d m, d m, d h m, t m, 1.18 kg/m 3, c m/s, M J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 1558

12 FIG. 10. Effect of the length of end cavity on the erformance of a three-ass muffler. l 0.74 m, l a l b m, t a t b m, d 1 d d m, d m, d h m, t m, 0.045, 1.18 kg/m 3, c m/s, M0. Figure 10 shows the effect of the length of the end cavities on the transmission loss characteristics. Increasing the length of the end cavities shifts the eaks in the transmission loss to lower frequencies. As noted in Figs. 5 9, the end cavities contribute to the resonant eaks, since they, together with the erforated ducts, constitute aroximately a Helmholtz resonator. It is well known that the resonance frequency of a Helmholtz resonator is inversely roortional to the square root of the length of the cylindrical cavity, which in this case is the length of end cavities for fixed crosssectional area. This exlains the shift observed in Fig. 10. Figure 11 comares the muffler of the baseline case with a muffler whose end chambers are identical only l c is varied made equal to l d ). Since the total length of the end cavity for the latter case is reduced, the resonance eaks shift to higher frequencies. The effect, however, is marginal beyond 400 Hz where the simle exansion chamber behavior dominates. The effect of the length of exansion chamber is FIG. 11. Effect of the length (l c ) of one of the end cavities on the erformance of a three-ass muffler. l 0.74 m, l a l b m, l d 0.10 m, t a t b 0.017m, d 1 d d m, d m, d h m, t m, 0.045, 1.18 kg/m 3, c m/s, M J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 1559

13 FIG. 1. Effect of the length (l c ) of the exansion chamber on the erformance of a three-ass muffler. l a l b m, l c 0.15 m, l d 0.10 m, t a t b 0.017m, d 1 d d m, d m, d h m, t m, 0.045, 1.18 kg/m 3, c m/s, M0. shown in Fig. 1. Increase in the exansion chamber length l l a l b (l a l b were retained the same shifts the eaks troughs to lower frequencies. The effect is minimal at low frequencies below 300 Hz where end cavities dominate. The effect of increase in the diameter of the exansion chamber is shown in Fig. 13. An increase in the diameter increases the transmission loss at higher frequencies 600 Hz. The resence of the middle chamber exansion chamber, as discussed earlier, results in a simle exansion chamber behavior. The troughs for a simle exansion chamber occur at frequency intervals of f c 0 /l 50 Hz. It is well known that the transmission loss for a simle exansion chamber increases with increase in the diameter ratio. This exlains the behavior observed at frequency 50 Hz. At lower frequencies, the Helmholtz behavior of the end chamber becomes imortant. Increase in the exansion chamber diameter increases the diameter of the end cavities, since FIG. 13. Effect of the diameter of the exansion chamber on the erformance of a three-ass muffler. l 0.74 m, l a l b m, l c 0.15 m, l d 0.10 m, t a t b m, d 1 d d m, d h m, t m, 0.045, 1.18 kg/m 3, c m/s, M J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 1560

14 FIG. 14. Effect of the duct extensions into the end cavities on the erformance of a three-ass muffler. l 0.74 m, l a l b m, l c 0.15 m, l d 0.10 m, t a t b m, d 1 d d m, d m, d h m, t m, 0.045, 1.18 kg/m 3, c m/s, M0. For the muffler with extensions into the end cavity, l m, l 0m, l 0.16 m, l 3 0m. the Helmholtz resonance frequency is inversely roortional to the square root of the cavity cross-sectional area for a given length which in the resent case is the cross-sectional area of the end cavity, the resonant eaks shift to lower frequencies when the diameter of the end cavity is increased. Also, as the diameter of the exansion chamber increases, the exansion chamber behavior becomes more ronounced. Figure 14 shows the effect of ducts extending into the end chambers. The resence of duct extensions in the end cavity shifts the eaks troughs to lower frequencies; the effect, however, is marginal. Also, an additional eak is introduced at about 800 Hz. ACKNOWLEDGMENTS The first two authors areciate the suort by Chrysler Challenge Fund for this study. Thanks extend to numerous Chrysler eole, including P. Keller, D. Wilmot, K. Underwood, J. Hirschey, D. Aley, R. Buckley for their suort, articiation, time. We would also like to thank Professor M. L. Munjal of the Center of Excellence for Technical Acoustics of the Indian Institute of Science who rovided Ref. 15 er our request. VI. CONCLUDING REMARKS A quasi-one-dimensional aroach is resented to analyze three-ass mufflers with erforated ducts using the numerical decouling. The aroach is further develoed to include mufflers with ducts extended into the end cavities. Theoretical exerimental results are first comared for a fabricated rototye three-ass muffler. This is followed by a similar comarison for two roduction mufflers. Theoretical results comare reasonably well with exeriments. Duct orosity is shown to have only a marginal effect until about 600 Hz beyond which the transmission loss is greater for a muffler with lower duct orosity. Increasing the length of the end cavities is found to shift the transmission loss eaks to lower frequencies. The effect is similar when the diameter of the exansion chamber is increased. Increase in the exansion chamber diameter is also shown to increase the transmission loss at frequencies greater than 600 Hz at some lower frequencies. Finally, the resence of duct extensions in the end cavity is shown to shift the eaks troughs to lower frequencies; the effect, however, is marginal. 1 J. W. Sullivan M. J. Crocker, Analysis of concentric-tube resonators having unartitioned cavities, J. Acoust. Soc. Am. 64, J. W. Sullivan, A method of modeling erforated tube muffler comonents. I: theory, J. Acoust. Soc. Am. 66, J. W. Sullivan, A method of modeling erforated tube muffler comonents. II: alications, J. Acoust. Soc. Am. 66, K. Jayaraman K. Yam, Decouling aroach to modeling erforated tube muffler comonents, J. Acoust. Soc. Am. 69, P. T. Thawani K. Jayaraman, Modelling alications of straightthrough resonators, J. Acoust. Soc. Am. 73, K. N. Rao M. L. Munjal, A generalized decouling method for analyzing erforated element mufflers, in Proceedings of the 1984 Nelson Acoustic Conference, Wisconsin M. L. Munjal, K. N. Rao, A. D. Sahasrabudhe, Aeroacoustic analysis of erforated muffler comonents, J. Sound Vib. 114, K. S. Peat, A numerical decouling analysis of erforated ie silencers, J. Sound Vib. 13, G. R. Gogate M. L. Munjal, Analytical exerimental aeroacoustic studies of oen-ended three-duct erforated elements used in mufflers, J. Acoust. Soc. Am. 97, E. Dokumaci, Matrizant aroach to acoustic analysis of erforated multile ie mufflers carrying mean flow, J. Sound Vib. 191, E. Dokumaci, A subsystem aroach for acoustic analysis of mufflers having identical erforated ies, J. Sound Vib. 193, J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 1561

15 1 D. F. Ross, A finite element analysis of erforated comonent acoustic systems, J. Sound Vib. 79, N. S. Dickey, A. Selamet, J. M. Novak, Multi-ass erforated tube silencers: a comutational aroach, J. Sound Vib. 11, M. L. Munjal, Analysis of a flush-tube three-ass erforated element muffler by means of transfer matrices, Int. J. Acoust. Vib., M. L. Munjal, Analysis of extended-tube three-ass erforated element muffler by means of transfer matrices, in Proceedings of the Fifth International Congress on Sound Vibration, Adelaide, South Australia, December 1997, Vol. III, M. L. Munjal, Acoustics of Ducts Mufflers Wiley, New York, U. Ingard, On the theory design of acoustic resonators, J. Acoust. Soc. Am. 5, A. Selamet V. Easwaran, Wave roagation attenuation in Herschel Venturi tubes, J. Acoust. Soc. Am. 101, A. Selamet, N. S. Dickey, J. M. Novak, The Herschel Quincke tube: a theoretical, comutational, exerimental investigation, J. Acoust. Soc. Am. 96, J. Acoust. Soc. Am., Vol. 105, No. 3, March 1999 Selamet et al.: Three-ass mufflers 156