On an Integrated Transfer Matrix method for multiply connected mufflers

Journal of Sound and Vibration 331 (2012) 1926–1938

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On an Integrated Transfer Matrix method for multiply connected mufflers N.K. Vijayasree 1, M.L. Munjal n Facility for Research in Technical Acoustics (FRITA), Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India

a r t i c l e in f o

abstract

Article history: Received 7 April 2011 Received in revised form 1 December 2011 Accepted 3 December 2011 Handling Editor: L.G. Tham Available online 4 January 2012

The commercial automotive mufflers are generally of a complicated shape with multiply connected parts and complex acoustic elements. The analysis of such complex mufflers has always been a great challenge. In this paper, an Integrated Transfer Matrix method has been developed to analyze complex mufflers. Integrated transfer matrix relates the state variables across the entire cross-section of the muffler shell, as one moves along the axis of the muffler, and can be partitioned appropriately in order to relate the state variables of different tubes constituting the cross-section. The paper presents a generalized one-dimensional (1-D) approach, using the transfer matrices of simple acoustic elements, which are available from the literature. The present approach is robust and flexible owing to its capability to construct an overall matrix of the muffler with the transfer matrices of individual acoustic elements and boundary conditions, which can then be used to evaluate the transmission loss, insertion loss, etc. Results from the present approach have been validated through comparisons with the available experimental and three-dimensional finite element method (FEM) based results. The results show good agreement with both measurements and FEM analysis up to the cut-off frequency. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction Silencers are extensively used for the attenuation of exhaust noise in an automobile. An increasing demand for their efficient performance has led to the development of various acoustic elements like area discontinuities, perforated surfaces, baffles, etc. The area discontinuities, flow reversals and extended tubes introduced in the muffler attenuate noise essentially through the mechanism of impedance mismatch [1]. However, these elements also introduce substantial pressure drop in the flow. To overcome this drawback, perforated tubes are used. Sullivan and Crocker developed an analytical approach to predict the transmission loss of concentric tube resonators [2–4]. Further, Munjal et al. developed a generalized decoupling method to analyze perforated elements [5]. Peat gave a generalized eigenvalue approach to derive the transfer matrix for perforated surfaces [6]. Once perforated tubes could be analyzed, their use in mufflers gained momentum. These perforated tubes have been used ingeniously in combination with area expansions and flow reversals in order to obtain better sound attenuation with less back pressure [7]. Concurrently, mufflers with dissipative liners have been analyzed for better performance over a wide frequency range [8].

Abbreviations: ITM, Integrated Transfer Matrix; FEM, Finite Element Method n Corresponding author. Tel.: þ 91 80 2293 2303; fax: þ91 80 23600648. E-mail addresses: [email protected] (N.K. Vijayasree), [email protected] (M.L. Munjal). 1 Now with Bharat Heavy Electricals Limited (BHEL), Corporate Research & Development Division, Hyderabad 500093, India. 0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.12.003

N.K. Vijayasree, M.L. Munjal / Journal of Sound and Vibration 331 (2012) 1926–1938

1927

The cascading method uses transfer matrices of simple acoustic elements connected in series to analyze relatively complex mufflers [9]. It is very convenient and fast in computation, but it lacks the flexibility of analyzing mufflers with ˚ multiply connected internal elements. Glav and Abom [10] gave a general formalism for analyzing acoustic two-port networks, but did not propose any method for automotive mufflers comprising of absorptive linings and perforated elements. Elnady et al. [11] suggested a new two-port segmentation approach, applying the scattering matrix formalism of ˚ Glav and Abom. This method is very effective in analyzing complex mufflers with perforated acoustic elements, both with and without mean flow. But the referred work does not include the analysis of mufflers with absorptive liners. Panigrahi and Munjal [12] have developed a volume synthesis (VS) algorithm which gives a generalized scheme for analysis of multifarious, commercially used mufflers. Their work includes both perforated elements and absorptive linings. But this approach is only suitable for stationary medium. Further, the algorithm does not hold good for mufflers consisting of acoustic elements like perforated cross baffles. The analysis of the performance of complex mufflers with onedimensional plane wave approach is very effective, although it is suitable at low frequencies below the cut-off frequency. The 3-D analysis methods like FEM and BEM, though suitable over a wide frequency domain, involve high precomputational effort and a high CPU time. Hence, the 1-D analytical and numerical methods have been continually improved upon. The present paper is a step in that direction. The Integrated Transfer Matrix (ITM) method presented here suggests a transfer matrix based approach to analyze a general commercial muffler. Here, the given muffler is divided into a number of sections based on area discontinuities, perforated surfaces, baffles, or absorptive linings. An integrated transfer matrix relates the state variables across the entire cross-section of the muffler shell, as one moves along the axis, and can be partitioned appropriately in order to relate the state variables of different tubes constituting the cross-section. An ‘n  n’ transfer matrix is obtained relating the upstream and downstream state variables. The size of the n  n matrix depends on the number of the individual acoustic elements which can be identified in each section, for which the transfer matrix relations are already available in the literature. The constituent muffler sections are then combined appropriately into a single final matrix, with the aid of the boundary conditions, to get the relation between the inlet and outlet state variables, and transmission loss can then be calculated. The present method, thus, gives a more comprehensive way of analyzing complex multiply connected muffler configurations with much less pre-computational effort and faster execution. It also deals effectively with perforated elements and absorptive linings, and incorporates the convective as well as dissipative effects of mean flow. 2. Outline of the Integrated Transfer Matrix (ITM) approach In the proposed approach, the given muffler is divided into a number of sections. The division of muffler into sections is carried out on the basis of the relative changes in the muffler geometry such as sudden area changes, perforated pipes, cross baffles or boundaries. Once the muffler is dissected into sections, each section is assigned a transfer matrix relating the upstream and downstream state variables of the section. The size of the state vector and hence the transfer matrix depends on the number of smallest individual elements constituting the cross-section. The present method is illustrated hereunder for three different muffler configurations, starting with a complex muffler with interacting perforated tubes, followed by the application of the method to mufflers with area discontinuities without and with interacting perforated tubes. 2.1. Illustration 1: A muffler with partially overlapping perforated ducts and a baffle The muffler shown in Fig. 1 consists of both perforated pipes and baffles without any area discontinuities in between. This muffler can be divided into 11 sections with three acoustic elements in each section depending on the presence of perforated pipes and baffles. Once the muffler is divided into sections, transfer matrices are written for each section relating the upstream state variables of each element of the section to the downstream state variables. Let the state variables of an element, acoustic pressure pi and volume velocity vi be clubbed into a state vector ½Si . As is clear from Fig. 1, there are three interacting ducts in each segment, the integrated state vector for which can be written as 2 3 p1 ð0Þ 6 7 6 v1 ð0Þ 7 2 3 6 7 S1,0 6 p2 ð0Þ 7 6 7 6S 7 (1) 6 7  4 2,0 5 6 v2 ð0Þ 7 6 7 S 3,0 6 p ð0Þ 7 4 3 5 v3 ð0Þ Let Pi,j_k be the 2  2 transfer matrix of uniform pipe of the ith acoustic element between the section (j_k), relating the state variables Si,j and Si,k. Thus, " # cosðkLj_k Þ jY sinðkLj_k Þ ½Pi,j_k  ¼ j (2) cosðkLj_k Þ Y sinðkLj_k Þ

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Fig. 1. Schematic of a muffler with partially overlapping perforated ducts and a baffle, divided into sections to illustrate the Integrated Transfer Matrix approach (points u and d represent the upstream and downstream points, respectively).

where k( ¼ o/c) and Y ( ¼ r0c0/S) denote wavenumber and characteristic impedance, respectively. The section (0_1) in Fig. 1 can be viewed as a combination of three rigid-wall pipes, for which the individual transfer matrices are known. The transfer matrix for the whole section can be written by arranging the three known rigid-pipe transfer matrices. Then, the transfer matrix between the section (0_1) can be written as 2

3 p1,0 6 v 7 7 2 2 3 6 1,0 6 7 S1,0 ½P1,0_1  6 p2,0 7 7 6 ½O 6S 7 6 7¼4 4 2,0 5 ¼ 6 22 6 v2,0 ð ¼ 0Þ 7 6 7 S3,0 ½O 22 6 p3,0 7 4 5 v3,0 ð ¼ 0Þ

½O22 ½P2,0_1  ½O22

32

3 S1,1 7 6 ½O22 54 S2,1 7 5 ½P3,0_1  S3,1 ½O22

(3)

where ½O represents a null matrix. Similarly, the section (1_2) consists of elements 1 and 2 interacting through perforated surface with rigid-pipe annulus as the third element. Let Ei,m,j_k be a 4  4 matrix representing the interaction of ducts i and m, between the sections j and k. The transfer matrix may be evaluated by means of the eigenanalysis of the coupled equations. Coefficients of the matrix may be evaluated from the coupled equations by means of Kar and Munjal’s generalized algorithm [13]. Thus, the overall transfer matrix for the section (1_2) consists of a 4  4 interacting matrix, [E], and a 2  2 straight uniform pipe matrix, [P], as given below: 2

3 " S1,1 ½E1,2,1_2 44 6S 7 ¼ 2,1 4 5 ½O24 S3,1

2 3 # S1,2 ½O42 6S 7 4 2,2 5 ½P3,1_2 22 S3,2

(4)

Similar transfer matrices can be obtained for the other sections with various acoustic elements. Let Fi,m,p,j_k be a 6  6 matrix representing the interaction of ducts i and m across the annulus p, between the sections j and k. The matrix can be evaluated by using the generalized algorithm of Ref. [13], as indicated above. And for the cross baffle, the transfer matrix for the flow-acoustic resistance offered by it can be written as Bj_k ¼



1

Z j_k

0

1



(5)

where Zj_k is the lumped impedance offered by the baffle to the axially moving plane wave in a moving medium [9]. From Eqs. (2), (4) and (5), the transfer matrix can be defined between any section by a proper combination of the matrices of individual elements of the section. The transfer matrices of different sections of the muffler of Fig. 1 are given below: 2

3 2 3 S1,2 S1,3 6S 7 6S 7 4 2,2 5 ¼ ½F1,3,2,2_3 66 4 2,3 5 S3,2 S3,3

(6)

N.K. Vijayasree, M.L. Munjal / Journal of Sound and Vibration 331 (2012) 1926–1938

2

3 " S1,3 ½P1,3_4 22 6S 7 ¼ 2,3 4 5 ½O42 S3,3 2

S1,4

2

3

½P1,4_5 

½O22

6 S 7 6 ½O 4 2,4 5 ¼ 4 22 S3,4 ½O22 2

2

3

S1,5

S1,6

2

3

½P1,5_6 

½P1,6_7 

S1,7

3

6S 7 4 2,7 5 ¼ S3,7

"

½P3,4_5 

½O22

½O22

½P3,5_6 

½O22

½O22

32

S1,6

32

S1,7

(7)

3 (8)

3 (9)

3

6 7 ½O22 7 54 S2,7 5 ½P3,6_7  S3,7

½P2,6_7  ½O22

½O24

S1,5

6 7 ½O22 7 54 S2,6 5 S3,6

½O22

½E1,2,7_8 44

32

6 7 ½O22 7 54 S2,5 5 S3,5

½O22

½B5_6 

6 S 7 6 ½O 4 2,6 5 ¼ 4 22 S3,6 ½O22 2

½O22

½P2,4_5 

6 S 7 6 ½O 4 2,5 5 ¼ 4 22 S3,5 ½O22 2

2 3 # S1,4 6S 7 4 2,4 5 ½E2,3,3_4 44 S3,4 ½O24

1929

2 3 # S1,8 6S 7 4 2,8 5 ½P3,7_8 22 S3,8 ½O42

3 2 3 S1,9 S1,8 6S 7 6S 7 4 2,8 5 ¼ ½F1,3,2,8_9 66 4 2,9 5 S3,8 S3,9

(10)

(11)

2

2

3 " S1,9 ½E1,2,9_10 44 6S 7 4 2,9 5 ¼ ½O24 S3,9 3 2 ½P1,10_11  S1,10 6S 7 6 ½O ¼ 2,10 4 5 4 22 S3,10 ½O22 2

(12)

2 3 # S1,10 6S 7 4 2,10 5 ½P3,9_10 22 S3,10 ½O42

½O22 ½P2,10_11  ½O22

32

3 S1,11 7 6 ½O22 54 S2,11 7 5 ½P3,10_11  S3,11 ½O22

(13)

(14)

Successively multiplying all the above eleven 6  6 matrices, the overall 6-port transfer matrix connecting the 0-section to the 11th section can be calculated. Thus, 2 3 2 32 3 p1,0 p1,11 T 11 T 12 T 13 T 14 T 15 T 16 6v 7 6 76 7 6 1,0 7 6 T 21 T 22 T 23 T 24 T 25 T 26 76 v1,11 7 6 7 6 76 7 6 p2,0 7 6 T 31 T 32 T 33 T 34 T 35 T 36 76 p2,11 7 6 7 6 76 7 (15) 6 7¼6 76 7 6 v2,0 7 6 T 41 T 42 T 43 T 44 T 45 T 46 76 v2,11 7 6 7 6 76 7 6 p3,0 7 6 T 51 T 52 T 53 T 54 T 55 T 56 76 p3,11 7 4 5 4 54 5 v3,0 v3,11 T 61 T 62 T 63 T 64 T 65 T 66 Inverting the above matrix yields 2

3 2 p1,11 R11 6v 7 6 R 6 1,11 7 6 21 6 7 6 6 p2,11 7 6 R31 6 7 6 6 7¼6 6 v2,11 7 6 R41 6 7 6 6 p3,11 7 6 R51 4 5 4 v3,11 R61

R12

R13

R14

R15

R22

R23

R24

R25

R32

R33

R34

R35

R42

R43

R44

R45

R52 R62

R53 R63

R54 R64

R55 R65

3 p1,0 76 v 7 R26 76 1,0 7 76 7 6 7 R36 7 76 p2,0 7 76 7 R46 76 v2,0 7 76 7 6 7 R56 7 54 p3,0 5 v3,0 R66 R16

32

(16)

Assuming the end-plates to be rigid, there are four boundary conditions at sections 0 and 11, v2,0 ¼ 0,

v3,0 ¼ 0,

v1,11 ¼ 0 and v2,11 ¼ 0

(17220)

Combining the above 6  6 matrix with the boundary conditions yields one single final matrix relating the inlet state variables, p1_0 , v1_0 , and outlet state variables, p3_11 , v3_11 .

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This method can alternatively be implemented by writing all the available Eqs. (16)–(20) of the muffler in the Gaussian form: 3 2 2 3 32 p 0 0 R11 R12 R13 R14 R15 R16 1 0 0 1,0 6 7 6 6 R21 R22 R23 R24 R25 R26 7 v 0 1 0 0 76 1,0 7 6 0 7 6 7 7 6 6 76 p 7 6 0 7 6 R31 R32 R33 R34 R35 R36 7 0 0 1 0 76 2,0 7 6 6 6 7 76 6 R41 R42 R43 R44 R45 R46 v2,0 7 7 6 0 0 0 1 7 0 7 6 6 7 76 7 6 6 7 76 p3,0 7 6 p3,11 7 6 R51 R52 R53 R54 R55 R56 0 0 0 0 76 6 7 6 6 7 76 ¼ (21) 7 6R v3,0 7 6 0 0 0 0 7 v3,11 7 6 61 R62 R63 R64 R65 R66 7 76 7 6 6 6 7 76 6 p1,11 7 6 0 7 6 0 0 0 1 0 0 0 0 0 0 7 7 6 6 7 76 6 7 6 0 7 v1,11 7 6 0 7 0 0 0 0 1 0 0 0 0 76 6 7 7 6 6 6 7 76 7 4 0 0 0 0 0 0 0 1 0 0 56 4 p2,11 5 4 0 5 v2,11 0 0 0 0 0 0 0 0 0 0 1

Inverting the coefficient matrix by means of Gaussian Elimination with pivoting, using a standard subroutine/function yields 2 3 2 3 p1,0 0 6v 7 6 0 7 6 1,0 7 6 7 6 7 6 7 6 p2,0 7 6 0 7 6 7 6 7 6v 7 6 0 7 6 2,0 7 6 7 6 7 6 7 6 p3,0 7 6 p3,11 7 6 7 6 7 (22) 6 7 ¼ ½A6 7 6 v3,0 7 6 v3,11 7 6 7 6 7 6 p1,11 7 6 0 7 6 7 6 7 6 7 6 0 7 6 v1,11 7 6 7 6 7 6 7 6p 7 4 0 5 4 2,11 5 v2,11 0

whence the desired 2  2 transfer matrix [T] may be obtained as follows: " # " #" # " # p3,11 p3,11 p1,0 A15 A16 ¼ ¼ ½T v1,0 v3,11 v3,11 A25 A26

(23)

Having obtained the transfer matrix, the transmission loss can now be easily calculated as [9] " #   Y out 1=2 T 11 þ T 12 =Y out þY in T 21 þðY in =Y out ÞT 22  TL ¼ 20 log   Y in 2

(24)

2.2. Illustration 2: Muffler with area discontinuities The muffler configuration of Fig. 1 has continuous acoustic elements without any area discontinuities. The present ITM approach is now illustrated for muffler configurations with area discontinuities, as shown in Fig. 2. The muffler configuration can be divided into five sections with a total of nine acoustic elements as shown in Fig. 2. First, transfer matrices for each of the five sections may be constituted as follows: 2 3 2 3 2 3 32 S1,1 S1,1 S1,0 ½P1,0_1  ½O22 ½O22 6S 7 6 6 7 6 S2,1 7 ¼ ½A6 S2,1 7 6 2,0 7 ¼ 4 ½O22 ½P2,0_1  ½O22 7 7 (25) 56 4 5 4 5 4 5 S3,0 S3,1 S3,1 ½O22 ½O22 ½P3,0_1  "

S3,1

"

# ¼

S4,1

½P3,1_2 

½O22

½O22

½P4,1_2 

#"

S3,2

#

" ¼ ½B

S4,2

S3,2

# (26)

S4,2

(27)

½S5,2  ¼ ½P5,2_3 ½S5,3  ¼ ½C½S5,3  "

S6,3 S7,3

"

# ¼

½P6,3_4 

½O22

½O22

½P7,3_4 

#"

S6,4 S7,4

#

" ¼ ½D

S6,4 S7,4

# (28)

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Fig. 2. Schematic of a muffler with multiple area discontinuities.

2

3 2 S7,4 ½P7,4_5  6S 7 6 6 9,4 7 ¼ 4 ½O22 4 5 S8,4 ½O22

½O22

32 S7,5 6S 9,5 ½O22 7 56 4 S8,5 ½P8,4_5  ½O22

½P9,4_5  ½O22

3

2

3 S7,5 7 6 7 7 ¼ ½E6 S9,5 7 5 4 5 S8,5

(29)

The following boundary conditions can be applied at the area discontinuities at sections 1–4: p1,1 ¼ p2,1 ¼ p4,1 ;

v1,1 þv2,1 ¼ v4,1

(30-32)

p3,2 ¼ p4,2 ¼ p5,2 ;

v3,2 þv4,2 ¼ v5,2

(33-35)

p6,3 ¼ p7,3 ¼ p5,3 ;

v7,3 þv6,3 ¼ v5,3

(36-38)

p6,4 ¼ p9,4 ¼ p8,4 ;

v9,4 þv8,4 ¼ v6,4

(39-41)

The other four boundary conditions follow from the rigid end-plates where the normal velocities would be zero for acoustic elements 2, 3, 7 and 9 at sections 0 and 5: v2,0 ¼ 0,

v3,0 ¼ 0,

v9,5 ¼ 0 and v8,5 ¼ 0

(42-45)

The 38 linear algebraic equations represented by Eqs. (25)–(45) above may now be rearranged in the following single matrix which relates the input state variables p1,0, v1,0 to the output state variables p8,5, v8,5 as illustrated earlier in the matrix equation (21): 2 3 2 3 p1,0 0 6v 7 6 7 ^ 6 1,0 7 6 7 6 7 6 0 7 6 ^ 7 6 7 6 7 6 7 6 7 6 p8,5 7 6 p5,3 7 6 7 7 ¼ (46) ½T3838 6 6v 7 6 v5,3 7 6 8,5 7 6 7 6 7 6 7 6 0 7 6 ^ 7 6 7 6 7 6 7 6p 7 4 ^ 5 4 9,5 5 v9,5 0 381 381

By inverting the above matrix, transfer matrix can be obtained between the inlet (point u) and outlet (point d) state variables, which can then be used to calculate the transmission loss by means of Eq. (24). 2.3. Illustration 3: Muffler with non-overlapping perforated elements and area discontinuities The muffler configuration of Fig. 3 consists of both perforated pipes and cross baffles. The total muffler can be divided into 17 sections with three interacting acoustic elements in each section. The muffler can be visualized as one with

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Fig. 3. Schematic of a muffler with perforated elements, cross baffles and area discontinuities (adopted from Ref. [11]).

continuous acoustic elements up to section 13 and another with area discontinuities. The transfer matrices for the sections from 0 to 13 are written as done before for the muffler of Fig. 1. These have been listed in the Appendix as Eqs. (A1)–(A13). Successively multiplying these thirteen 6  6 matrices, we can calculate the 6-port transfer matrix connecting the 0-section to the 13th section: 2 3 2 32 3 p1,0 p1,13 T 11 T 12 T 13 T 14 T 15 T 16 6 v 7 6 76 v 7 6 1,0 7 6 T 21 T 22 T 23 T 24 T 25 T 26 76 1,13 7 6 7 6 76 7 6 p2,0 7 6 T 31 T 32 T 33 T 34 T 35 T 36 76 p2,13 7 6 7 6 76 7 (47) 6 7¼6 76 7 6 v2,0 7 6 T 41 T 42 T 43 T 44 T 45 T 46 76 v2,13 7 6 7 6 76 7 6 p3,0 7 6 T 51 T 52 T 53 T 54 T 55 T 56 76 p3,13 7 4 5 4 54 5 v3,0 v3,13 T 61 T 62 T 63 T 64 T 65 T 66 The two-port transfer matrices of the remaining elements may be used to close the model. For the uniform pipe section between sections 13 and 16 " # " # p1,16 p1,13 ¼ ½P1,13216  (48) v1,13 v1,16 Considering transverse plane waves in the end chambers as modeled by Elnady et al. [11] 2 3 # TM " # " " 0 # 1 0 6 p3,13 p1,16 7 ¼ 1 1 4 ðVariable 5½B15_16 ½P2,13_14  0 v3,13 v1,16 Z6 area ductÞ

(49)

For transverse plane wave propagation in an elliptical (variable-area) cross-section duct, the transfer matrix may be derived as per Refs. [14,15], and inserted in Eq. (49) above. Multiplying the five 2  2 transfer matrices of Eqs. (48) and (49) yields " # " 0 #" 0 # p3,13 p1,13 T 11 T 012 ¼ (50) v03,13 v1,13 T 021 T 022 Further, it may be noted that p2,13 ¼ p3,13 ¼ p03,13

(51-52)

v2,13 v3,13 ¼ v03,13

(53)

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These three equations along with v2,0 ¼ 0 provide the four equations or boundary conditions required to reduce the 6  6 matrix to the desired 2  2 matrix of the entire configuration: 2 3 2 32 3 p1,0 0 0 1 0 0 0 T 11 T 12 T 13 T 14 T 15 T 16 0 6 7 6 76 7 0 0 76 v1,0 7 6 0 7 6 0 1 0 0 T 21 T 22 T 23 T 24 T 25 T 26 6 7 6 76 7 6 6 0 0 1 0 T 31 T 32 T 33 T 34 T 35 T 36 6 7 7 0 0 76 p2,0 7 6 0 7 6 7 6 7 6 76 7 6 0 0 0 1 T 41 T 42 T 43 T 44 T 45 T 46 0 0 76 v2,0 7 6 0 7 6 7 6 76 7 6 6 0 0 0 0 T 51 T 52 T 53 T 54 T 55 T 56 6 7 7 0 0 76 p1,13 7 6 p3,0 7 6 7 6 7 6 76 7 6 0 0 0 0 T 61 T 62 T 63 T 64 T 65 T 66 0 0 76 v1,13 7 6 v3,0 7 6 7¼6 76 7 (54) 0 0 76 p 60 0 0 0 6 7 1 0 0 0 0 0 T 11 T 12 76 2,13 7 6 0 7 6 7 6 7 6 76 v 7 6 2,13 7 6 0 7 60 0 0 0 0 1 0 0 0 0 T 021 T 022 7 6 7 6 76 7 6 60 0 0 0 6 7 7 p 0 0 0 1 0 1 0 1 76 3,13 7 6 0 7 6 7 6 7 6 76 7 60 0 0 0 76 v3,13 7 6 0 7 0 0 1 0 1 0 0 0 6 7 6 76 0 7 6 60 0 0 0 7 6 7 0 0 0 0 1 0 1 0 7 4 54 p1,13 5 4 0 5 0 v 0 0 0 1 0 0 0 0 0 0 0 0 0 1,13 Inverting the coefficient matrix by means of Gaussian elimination with pivoting, using a standard subroutine/function yields the overall transfer matrix of the entire muffler configuration. Thus, Transmission Loss, Insertion Loss and Noise Reduction (or Level Difference) can be evaluated [9]. 3. Validation of the Integrated Transfer Matrix approach 3.1. Configuration 1 As a first example, a fairly simple reflective muffler has been taken (see Fig. 2). This configuration is chosen to validate the ITM method for mufflers with multiple area discontinuities. The results obtained with the present approach have been compared with the 1-D closed-form solution, obtained using the cascading method [9], in Fig. 4. It can be seen that the

Fig. 4. Comparison of TL for muffler of Fig. 2 evaluated from the closed-form solution and the present ITM method: the two curves overlap each other.

results are almost identical for both the approaches. This is of course expected since both the methods presume onedimensional wave propagation through the muffler, the only difference being the solution method. But, as the complexity of the muffler elements increases, the cascading method becomes increasingly difficult. The present approach, however, gives exact solutions for multiply connected mufflers with comparatively less computational effort. 3.2. Configuration 2 The muffler configuration of Fig. 5 is a relatively complex muffler with multiple connectivity. The muffler is analyzed using the present approach up to plane ‘A’. The impedance at plane ‘B’ has been transferred to plane ‘A’ and this has been used in the analytical formulation derived for the rest of the muffler. The outer elliptical shell has been assumed to be an equivalent circular shell.

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Fig. 5. Schematic of a complex multiply connected reflective muffler.

Fig. 6. Comparison of TL spectrum from 3-D FEM analysis with that from the present ITM method at 600 1C with proper end-corrections.

The transmission loss calculated by the present approach has been validated by means of three-dimensional finite element analysis using a 3-D FEM (SYSNOISETM) software [16]. The total muffler has been modeled and meshed in IDEAS and then the mesh is exported to SYSNOISE. The pre-computational effort and CPU time taken for 3-D analysis is quite high, which is one of the major disadvantages for these numerical methods. The analysis has been carried out at typical engine exhaust temperature (about 600 1C), where the sonic velocity would be 600 m/s and medium density would be 0.5 kg/m3. It may be observed from Fig. 6 that 1-D ITM analysis, with proper end corrections matches quite well with the 3-D FEM results up to about 1600 Hz. Incidentally, the first diametral mode calculated for an elliptical section with symmetry about major axis would get cut on at 1781 Hz. This is consistent with the peak in the 3-D FEM curve shown in Fig. 6. Obviously, the distances in Fig. 5 are too short for the evanescent waves to decay off sufficiently. 3.3. Configuration 3 The muffler in Fig. 7 is a combination muffler with both reflective and absorptive elements. As can be observed from the figure, the sound waves get dissipated in the perforated tubes and then interact with the absorptive lining through the annular perforated tube. The flow resistivity of the lining has been taken as 16,000 Pa s/m2. The same material properties have been applied to the model in the 3-D FEM software. The absorptive lining has been assigned complex sound speed and density, calculated from the material’s complex characteristic impedance and wavenumber, the formulae for which are taken from chapter 6 of Ref. [9]. Results from the present approach and 3-D FEM analysis have been compared in Fig. 8. It is observed that the results agree reasonably well in the low frequency range, but deviate at higher frequencies owing to the three-dimensional effects. 3.4. Configuration 4 Fig. 9 shows another complex multiply connected muffler. The predicted TL has been shown in Fig. 10. It can be seen that the predicted results match well with the FEM results. Of course, there are slight deviations at the

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1935

Fig. 7. Schematic of lined duct with annular air-gap between the central airway and the absorptive material.

Fig. 8. Comparison of TL spectrum from the 3-D FEM analysis with that of the ITM method.

Fig. 9. Schematic of a complex, multiply connected muffler (adopted from Ref. [12]).

higher frequencies due to the three dimensional effects, which are not considered in the present one dimensional ITM approach. 3.5. Configuration 5 The muffler of Fig. 3 has been analyzed using the present ITM approach. This muffler consists of both perforated pipes and cross baffles [11]. The muffler breakdown into sections has been shown in Fig. 3. The muffler has been analyzed for both with and without mean flow. The introduction of mean flow has been done by using the Flow Resistance model as

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Fig. 10. Comparison of TL spectrum for Fig. 9 from 3-D FEM analysis with that of the present ITM method.

Fig. 11. Validation of the ITM method with measurements (adopted from Ref. [11]) for the muffler of Fig. 3 for stationary medium.

Fig. 12. Validation of the ITM method with measurements (adopted from Ref. [11]) for the muffler of Fig. 3 at M ¼ 0.15.

discussed by Elnady et al. [11]. The grazing and bias mean flows have been calculated using the inbuilt fsolve function of MATLAB as per the flow network developed in Ref. [11]. The transmission loss curves obtained by the Integrated Transfer Matrix approach, both with and without mean flow have been compared with the experimental results in Figs. 11 and 12. It is clear that both give similar results for stationary

N.K. Vijayasree, M.L. Munjal / Journal of Sound and Vibration 331 (2012) 1926–1938

1937

medium, below the cut-off frequency which works out to be 564 Hz with respect to the major axis of 290 mm at 27 1C [11]. However, turbulent mean flow seems to affect the decay of evanescent modes of shorter wavelengths adversely. This is probably the reason the 1-D predictions start deviating from the measured values from 350 Hz onwards. Another reason could be the inadequacy of the perforate impedance expressions with mean flow at higher flow at higher frequencies. This proves the efficacy of the present approach in analyzing complex mufflers with any degree of accuracy, easily and effectively. 4. Conclusions An Integrated Transfer Matrix (ITM) approach for calculating the transmission loss of complex, multiply connected mufflers with/without absorptive lining has been presented, duly incorporating the convective as well as dissipative effects of mean flow. Results obtained by the present approach have been compared with the available experimental results and 3-D FEM results. The results are shown to agree quite well within the plane wave limit. The discrepancies observed at higher frequencies are due to the three dimensional effects, which are neglected in the present 1-D approach. The ITM approach extends the conventional transfer matrix method to the multiply connected muffler configurations which would not be amenable to cascading. Moreover, the ITM approach helps to analyze acoustically lined ducts, lined expansion chambers and lined plenums with equal ease, making use of the control volume approach developed originally for perforated elements.

Acknowledgment Financial support of the Department of Science and Technology of the Government of India for the Facility for Research in Technical Acoustics (FRITA) is gratefully acknowledged. Appendix A. Transfer matrices for sections 0 to 13 of the muffler configuration of Fig. 3 The transfer matrices for the sections 0 through to 13, for the muffler configuration of Fig. 3 are given below, making use of the Integrated Transfer Matrix method illustrated before for the muffler configuration of Fig. 1: 32 2 3 2 3 S1,1 ½P1,0_1  ½O22 ½O22 S1,0 7 6 S 7 6 ½O 6 ½P2,0_1  ½O22 54 S2,1 7 (A1) 4 2,0 5 ¼ 4 5 22 S3,0 ½O22 ½O22 ½P3,0_1  S3,1 2

S1,1

3

6S 7 4 2,1 5 ¼ S3,1 2

S1,2

3

"

½P1,1_2 22 ½O42

2

½P1,2_3 

6 S 7 6 ½O 4 2,2 5 ¼ 4 22 S3,2 ½O22 3 2 ½P1,3_4  S1,3 6 S 7 6 ½O 4 2,3 5 ¼ 4 22 S3,3 ½O22 2

2

S1,4

3

2

½P1,3_4 

6 S 7 6 ½O 4 2,4 5 ¼ 4 22 S3,4 ½O22 2

S1,5

3

"

½O22 ½P2,2_3  ½O22 ½O22 ½B3_4  ½O22 ½O22 ½P2,4_5  ½O22

½E12,5_6 44 6S 7 4 2,5 5 ¼ ½O24 S3,5 3 2 ½P1,6_7  S1,6 6 S 7 6 ½O ¼ 2,6 4 5 4 22 S3,6 ½O22 2

2 3 # S1,2 6S 7 4 2,2 5 ½E23,1_2 44 S3,2 ½O24

½O22

32

S1,3

3

6 7 ½O22 7 54 S2,3 5 S3,3

½O22

(A3)

½P3,2_3 

32

3 S1,4 7 6 ½O22 54 S2,4 7 5 ½P3,3_4  S3,4 ½O22

½O22

32

S1,5

(A4)

3

6 7 ½O22 7 54 S2,5 5 ½P3,4_5  S3,5

2 3 # S1,6 ½O42 6S 7 4 2,6 5 ½P 3,5_6 22 S3,6

½O22 ½P2,6_7 

(A2)

32 3 S1,7 ½O22 7 6 ½O22 54 S2,7 7 5 ½P3,6_7  S3,7

(A5)

(A6)

(A7)

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3 2 ½P1,7_8  S1,7 6 S 7 6 ½O ¼ 2,7 4 5 4 22 S3,7 ½O22 2

3 2 ½P1,8_9  S1,8 6 S 7 6 ½O 4 2,8 5 ¼ 4 22 S3,8 ½O22 2

2

S1,9

3

"

6S 7 4 2,9 5 ¼ S3,9 2

S1,10

3

2

½O42 ½P1,10_11 

6S 7 6 4 2,10 5 ¼ 4 ½O22 S3,10 ½O22 2

S1,11

3

2

½P1,11_12 

6S 7 6 4 2,11 5 ¼ 4 ½O22 S3,11 ½O22 2

S1,12

3

2

½P1,12_13 

6S 7 6 4 2,12 5 ¼ 4 ½O22 S3,12 ½O22

½O22

½B7_8  ½O22

(A8)

32

3 S1,9 7 6 ½O22 54 S2,9 7 5 ½P3,8_9  S3,9

(A9)

2 3 # S1,10 6S 7 4 2,10 5 ½E9:10,1_2 44 S3,10

(A10)

½O22

½O22

½P2,8_9  ½O22

½P1,9_10 22

32

3 S1,8 7 6 ½O22 54 S2,8 7 5 ½P3,7_8  S3,8

½O22

½O24

½O22 ½P2,10_11  ½O22 ½O22 ½B11_12 

½O22

S1,11

3

6 7 ½O22 7 54 S2,11 5 ½P3,10_11  S3,11 ½O22

32

S1,12

½P3,11_12 

½O22

½O22

32

(A11)

3

6 7 ½O22 7 54 S2,12 5 S3,12

½O22

½P2,12_13  ½O22

32

S1,13

(A12)

3

6 7 ½O22 7 54 S2,13 5 ½P3,12_13  S3,13

(A13)

These 13 matrices have been multiplied successively to obtain the 6-port transfer matrix connecting the 0-section to the 13th section. This has been incorporated in Eq. (47) in the text. References [1] M.L. Munjal, Velocity ratio cum transfer matrix method for the evaluation of a muffler with mean flow, Journal of Sound and Vibration 39 (1975) 105–119. [2] J.W. Sullivan, M.J. Crocker, Analysis of concentric tube resonators having unpartitioned cavities, Journal of Acoustical Society of America 64 (1978) 207–215. [3] J.W. Sullivan, A method of modeling perforated tube muffler components. I: theory, Journal of Acoustical Society of America 66 (1979) 721–778. [4] J.W. Sullivan, A method of modeling perforated tube muffler components. II: Applications, Journal of Acoustical Society of America 66 (1979) 779–788. [5] M.L. Munjal, K. Narayana Rao, A.D. Sahasrabudhe, Aeroacoustic analysis of perforated muffler components, Journal of Sound and Vibration 114 (2) (1987) 173–188. [6] K.S. Peat, A numerical decoupling analysis of perforated pipe silencer elements, Journal of Sound and Vibration 123 (2) (1988) 199–212. [7] M.L. Munjal, Analysis of a flush-tube three-pass perforated muffler by means of transfer matrices, International Journal of Acoustics and Vibration 2 (2) (1997) 63–68. [8] S.N. Panigrahi, M.L. Munjal, Comparison of various methods for analyzing lined circular ducts, Journal of Sound and Vibration 285 (2005) 905–923. [9] M.L. Munjal, Acoustics of Ducts and Mufflers, Wiley, New York, 1987. ˚ [10] R. Glav, M. Abom, A general formalism for analyzing acoustic 2-port networks, Journal of Sound and Vibration 202 (5) (1997) 739–747. ˚ [11] T. Elnady, M. Abom, S. Allam, Modeling perforates in mufflers using two ports, Journal of Vibration and Acoustics 132 (6) (2010) 061010. [12] S.N. Panigrahi, M.L. Munjal, A generalized scheme for analysis of multifarious commercially used mufflers, Applied Acoustics 68 (2007) 660–681. [13] T. Kar, M.L. Munjal, Generalized analysis of a muffler with any number of interacting ducts, Journal of Sound and Vibration 285 (2005) 585–596. [14] A. Mimani, M.L. Munjal, Transverse plane-wave analysis of short elliptical end chamber and expansion-chamber mufflers, International Journal of Acoustics and Vibration 15 (1) (2010) 24–38. [15] A. Mimani, M.L. Munjal, Transverse plane wave analysis of short elliptical chamber mufflers—an analytical approach, Journal of Sound and Vibration 330 (2011) 1472–1489. [16] SYSNOISE Users’ Manual, Rev. 5.6. LMS International, Belgium, 2003.