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Abstract

The one-dimensional analytical approach is developed to predict and analyze the acoustic attenuation performance of three-pass perforated tube muffler with perforated bulkhead(s) and validated by comparing the transmission loss results from one-dimensional analytical approach and finite element method. The one-dimensional analytical approach is then used to investigate the effect of perforated bulkhead(s) on the acoustic attenuation performance of three-pass perforated tube reactive and hybrid mufflers. The results indicated that perforates on the bulkhead may change the resonant frequency of the three-pass perforated tube reactive muffler, and the resonance frequency is shifted to higher frequency with the increase in the porosity of perforation. The sound-absorbing material filling in the middle chamber may improve the acoustic attenuation performance of the mufflers from middle to high frequencies, while the acoustic attenuation performance around the resonance frequencies is lowered somewhat; therefore, a flatter acoustic response may be formed for the hybrid muffler than the reactive muffler.

Keywords

Three-pass perforated tube muffler perforated bulkheads one-dimensional analytical approach acoustic attenuation performance hybrid mufflers

Introduction

Three-pass perforated tube mufflers are widely used in automotive exhaust systems to reduce the engine exhaust noise. Munjal^1,2 and Selamet et al.³ presented the one-dimensional (1D) frequency-domain approaches, and Dickey et al.⁴ developed the 1D time-domain approach to predict the acoustic attenuation performance of the typical three-pass perforated tube mufflers. For a prototype muffler, their predictions of transmission loss agree reasonably well with the experiments within the plane wave range. Ji and Selamet⁵ employed the three-dimensional (3D) boundary element method (BEM) to predict the acoustic attenuation performance of the typical three-pass perforated tube muffler, and the BEM predictions agree well with experiments for a prototype muffler in the frequency range of interest. Sánchez-Orgaz et al.⁶ proposed for the acoustic analysis of automotive silencers including a perforated duct with uniform axial mean flow and an outer chamber with heterogeneous absorbent material in terms of 3D finite element method (FEM). Recently, Ji and Fang⁷ developed the 1D analytical approach to predict the acoustic attenuation performance of the three-pass perforated tube muffler with end-resonator, and the predictions of transmission loss are validated by comparing the FEM and BEM results. The main advantage of 3D numerical methods is that the effect of 3D sound field in the muffler is considered, and the accurate prediction of acoustic attenuation performance of muffler may be obtained at higher frequencies. However, the 3D numerical methods will take long computational time for the acoustic analysis of muffler, while the 1D analytical approach may capture the acoustic response of muffler quickly and, therefore, became an effective tool for the acoustic analysis of muffler, especially in the stage of initial design of muffler.

In order to adjust the resonance frequency and acoustic response of the three-pass perforated tube muffler, the bulkhead(s) may be perforated to form a new kind of muffler called as the three-pass perforated tube muffler with perforated bulkheads. The objective of this study is to develop a 1D analytic approach to predict the acoustic attenuation performance of this kind muffler and then to investigate the effect of perforated bulkhead(s) on acoustic behavior of the reactive and hybrid mufflers.

Theory

Following the works of Selamet et al.,³ the relationship of sound pressures and particle velocities at the inlet ( $x = 0$ ) and outlet ( $x = L_{p}$ ) of the common perforated section in the middle chamber of mufflers shown in Figures 1 –3 may be expressed as

{\begin{matrix} \begin{matrix} p_{1} (0) \\ ρ_{0} c_{0} u_{1} (0) \\ p_{2} (0) \\ ρ_{0} c_{0} u_{2} (0) \end{matrix} \\ \begin{matrix} p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \\ p_{4} (0) \\ ρ_{c} c_{c} u_{4} (0) \end{matrix} \end{matrix}} = [R] {\begin{matrix} \begin{matrix} p_{1} (L_{p}) \\ ρ_{0} c_{0} u_{1} (L_{p}) \\ p_{2} (L_{p}) \\ ρ_{0} c_{0} u_{2} (L_{p}) \end{matrix} \\ \begin{matrix} p_{3} (L_{p}) \\ ρ_{0} c_{0} u_{3} (L_{p}) \\ p_{4} (L_{p}) \\ ρ_{c} c_{c} u_{4} (L_{p}) \end{matrix} \end{matrix}}

(1)

where $[R]$ can be derived from the matrix $[T]$ of equation (62) in Selamet et al.;³p is the sound pressure; u is the particle velocity; $ρ_{0}$ is the density of air; $c_{0}$ is the speed of sound; $ρ_{c}$ is the complex density of sound-absorbing material; $c_{c}$ is the complex speed of sound in the sound-absorbing material; and subscripts 1, 2, 3, and 4 represent the perforated inlet tube, center tube, outlet tube, and expansion chamber, respectively.

Figure 1.

Three-pass perforated tube muffler with left-perforated bulkhead (the case of $L_{c 2} \geq L_{c 3}$ and $L_{d 1} \geq L_{d 2}$ ).

Figure 2.

Three-pass perforated tube muffler with right-perforated bulkhead (the case of $L_{c 2} \geq L_{c 3}$ and $L_{d 1} \geq L_{d 2}$ ).

Figure 3.

Three-pass perforated tube muffler with both perforated bulkheads (the case of $L_{c 2} \geq L_{c 3}$ and $L_{d 1} \geq L_{d 2}$ ).

Muffler with left-perforated bulkhead

For uniform perforations in the perforated bulkhead, the velocity and sound pressure on the left and right sides of the perforate may be expressed as

\frac{(p_{L} - p_{R})}{u_{L}} = ρ_{0} c_{0} ζ_{l}, u_{L} = u_{R}

(2)

where $ζ_{l}$ is specific acoustic impedance of the left-perforated bulkhead and $p_{L}$ , $u_{L}$ , $p_{R}$ , and $u_{R}$ are the sound pressures and velocities at the left and right sides of the perforated bulkhead, respectively.

Equation (2) can be written as

{\begin{matrix} p_{L} \\ ρ_{0} c_{0} u_{L} \end{matrix}} = [\begin{matrix} 1 & \frac{ζ_{l}}{z_{n}} \\ 0 & \frac{1}{z_{n}} \end{matrix}] {\begin{matrix} p_{R} \\ ρ_{c} c_{c} u_{R} \end{matrix}} = [Z] {\begin{matrix} p_{R} \\ ρ_{c} c_{c} u_{R} \end{matrix}}

(3)

where $z_{n} = ρ_{c} c_{c} / ρ_{0} c_{0}$ is the impedance ratio of sound-absorbing material to air.

The three-pass perforated tube muffler with left-perforated bulkhead is shown in Figure 1. For the case of $L_{c 2} \geq L_{c 3}$ in the left cavity, the following expressions may be obtained by the transfer matrix method

\begin{matrix} {\begin{matrix} p_{2} (0) \\ - ρ_{0} c_{0} u_{2} (0) \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{1}) & j \sin (k_{0} l_{1}) \\ j \sin (k_{0} l_{1}) & \cos (k_{0} l_{1}) \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ \frac{[j \tan (k_{0} l_{2}) (S_{4} - S_{1})]}{S_{2}} & \frac{(S_{4} - S_{1} - S_{2})}{S_{2}} \end{matrix}] \\ \times [\begin{matrix} \cos (k_{0} l_{3}) & j \sin (k_{0} l_{3}) \\ j \sin (k_{0} l_{3}) & \cos (k_{0} l_{3}) \end{matrix}] {\begin{matrix} p_{5} \\ ρ_{0} c_{0} u_{5} \end{matrix}} \end{matrix}

(4)

{\begin{matrix} p_{6} \\ ρ_{0} c_{0} u_{6} \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{4}) & j \sin (k_{0} l_{4}) \\ j \sin (k_{0} l_{4}) & \cos (k_{0} l_{4}) \end{matrix}] {\begin{matrix} p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \end{matrix}}

(5)

\begin{matrix} {\begin{matrix} p_{7} \\ ρ_{0} c_{0} u_{7} \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{5}) & j \sin (k_{0} l_{5}) \\ j \sin (k_{0} l_{5}) & \cos (k_{0} l_{5}) \end{matrix}] [Z] \\ [\begin{matrix} \cos (k_{c} L_{a}) & j \sin (k_{c} L_{a}) \\ j \sin (k_{c} L_{a}) & \cos (k_{c} L_{a}) \end{matrix}] {\begin{matrix} p_{4} (0) \\ ρ_{c} c_{c} u_{4} (0) \end{matrix}} \end{matrix}

(6)

where $j = \sqrt{- 1}$ is the imaginary unit; $l_{1} = L_{a} + t_{bL} + L_{c 2} + δ_{c 2}$ ; $l_{2} = L_{c} - L_{c 2} - δ_{c 2}$ ; $l_{3} = L_{c 2} + δ_{c 2} - L_{c 3} - δ_{c 3}$ ; $l_{4} = L_{a} + t_{bL} + L_{c 3} + δ_{c 3}$ ; $l_{5} = L_{c 3} + δ_{c 3}$ ; $t_{bL}$ is the thickness of the left bulkhead; $δ_{c 2}$ and $δ_{c 3}$ are the end corrections of extensions of center and outlet tubes into the left cavity which can be calculated using expression (17) in Kang and Ji;⁸ $k_{0}$ is the wave number in the air; $k_{c}$ is the complex wave number in the sound-absorbing material; and $S_{1}$ , $S_{2}$ , $S_{3}$ , and $S_{4}$ are the cross-sectional areas of inlet, center, and outlet tubes and expansion chamber, respectively.

Similarly, the following expressions are obtained by the transfer matrix method for the case of $L_{c 2} < L_{c 3}$

{\begin{matrix} p_{2} (0) \\ - ρ_{0} c_{0} u_{2} (0) \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{6}) & j \sin (k_{0} l_{6}) \\ j \sin (k_{0} l_{6}) & \cos (k_{0} l_{6}) \end{matrix}] {\begin{matrix} p_{5} \\ ρ_{0} c_{0} u_{5} \end{matrix}}

(7)

\begin{matrix} {\begin{matrix} p_{6} \\ ρ_{0} c_{0} u_{6} \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{7}) & j \sin (k_{0} l_{7}) \\ j \sin (k_{0} l_{7}) & \cos (k_{0} l_{7}) \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ \frac{j \tan (k_{0} l_{8}) (S_{4} - S_{1})}{S_{4} - S_{1} - S_{3}} & \frac{S_{3}}{S_{4} - S_{1} - S_{3}} \end{matrix}] \\ \times [\begin{matrix} \cos (k_{0} l_{9}) & j \sin (k_{0} l_{9}) \\ j \sin (k_{0} l_{9}) & \cos (k_{0} l_{9}) \end{matrix}] {\begin{matrix} p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \end{matrix}} \end{matrix}

(8)

\begin{matrix} {\begin{matrix} p_{7} \\ ρ_{0} c_{0} u_{7} \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{10}) & j \sin (k_{0} l_{10}) \\ j \sin (k_{0} l_{10}) & \cos (k_{0} l_{10}) \end{matrix}] [Z] \\ [\begin{matrix} \cos (k_{c} L_{a}) & j \sin (k_{c} L_{a}) \\ j \sin (k_{c} L_{a}) & \cos (k_{c} L_{a}) \end{matrix}] {\begin{matrix} p_{4} (0) \\ ρ_{c} c_{c} u_{4} (0) \end{matrix}} \end{matrix}

(9)

where $l_{6} = L_{a} + t_{bL} + L_{c 2} + δ_{c 2}$ , $l_{7} = L_{c 3} + δ_{c 3} - L_{c 2} - δ_{c 2}$ , $l_{8} = L_{c} - L_{c 3} - δ_{c 3}$ , $l_{9} = L_{a} + t_{bL} + L_{c 3} + δ_{c 3}$ , and $l_{10} = L_{c 2} + δ_{c 2}$ .

Expressions (4)–(6) and (7)–(9) may be written as

{\begin{matrix} p_{5} \\ ρ_{0} c_{0} u_{5} \end{matrix}} = [A] {\begin{matrix} p_{2} (0) \\ ρ_{0} c_{0} u_{2} (0) \end{matrix}}

(10)

{\begin{matrix} p_{6} \\ ρ_{0} c_{0} u_{6} \end{matrix}} = [B] {\begin{matrix} p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \end{matrix}}

(11)

{\begin{matrix} p_{7} \\ ρ_{0} c_{0} u_{7} \end{matrix}} = [C] {\begin{matrix} p_{4} (0) \\ ρ_{c} c_{c} u_{4} (0) \end{matrix}}

(12)

On the interface, the continuity conditions of sound pressures and particle velocities are

p_{5} = p_{6} = p_{7}

(13)

ρ_{0} c_{0} u_{5} S_{5} = ρ_{0} c_{0} u_{6} S_{6} + ρ_{0} c_{0} u_{7} S_{7}

(14)

where $S_{5} = S_{4} - S_{1} - S_{2}$ , $S_{6} = S_{3}$ , and $S_{7} = S_{4} - S_{1} - S_{2} - S_{3}$ for the case of $L_{c 2} \geq L_{c 3}$ and $S_{5} = S_{2}$ , $S_{6} = S_{4} - S_{1} - S_{3}$ , and $S_{7} = S_{4} - S_{1} - S_{2} - S_{3}$ for the case of $L_{c 2} < L_{c 3}$ .

For the right cavity, the following expression may be obtained using the transfer matrix method

\begin{matrix} {\begin{matrix} p_{1} (L_{p}) \\ ρ_{0} c_{0} u_{1} (L_{p}) \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{11}) & j \sin (k_{0} l_{11}) \\ j \sin (k_{0} l_{11}) & \cos (k_{0} l_{11}) \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ \frac{j \tan (k_{0} l_{12}) (S_{4} - S_{3})}{S_{1}} & \frac{(S_{4} - S_{1} - S_{3})}{S_{1}} \end{matrix}] \\ \times [\begin{matrix} \cos (k_{0} l_{13}) & j \sin (k_{0} l_{13}) \\ j \sin (k_{0} l_{13}) & \cos (k_{0} l_{13}) \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ j \tan (k_{0} l_{14}) S_{a 1} & \frac{S_{2}}{(S_{4} - S_{1} - S_{3})} \end{matrix}] \\ \times [\begin{matrix} \cos (k_{0} l_{15}) & j \sin (k_{0} l_{15}) \\ j \sin (k_{0} l_{15}) & \cos (k_{0} l_{15}) \end{matrix}] {\begin{matrix} p_{2} (L_{p}) \\ - ρ_{0} c_{0} u_{2} (L_{p}) \end{matrix}} \end{matrix}

(15)

for the case of $L_{d 1} \geq L_{d 2}$ , and

\begin{matrix} {\begin{matrix} p_{1} (L_{p}) \\ ρ_{0} c_{0} u_{1} (L_{p}) \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{16}) & j \sin (k_{0} l_{16}) \\ j \sin (k_{0} l_{16}) & \cos (k_{0} l_{16}) \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ j \tan (k_{0} l_{17}) S_{a 2} & \frac{(S_{4} - S_{2} - S_{3})}{S_{1}} \end{matrix}] \\ \times [\begin{matrix} \cos (k_{0} l_{18}) & j \sin (k_{0} l_{18}) \\ j \sin (k_{0} l_{18}) & \cos (k_{0} l_{18}) \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ \frac{j \tan (k_{0} l_{19}) (S_{4} - S_{3})}{S_{4} - S_{2} - S_{3}} & \frac{S_{2}}{S_{4} - S_{2} - S_{3}} \end{matrix}] \\ \times [\begin{matrix} \cos (k_{0} l_{20}) & j \sin (k_{0} l_{20}) \\ j \sin (k_{0} l_{20}) & \cos (k_{0} l_{20}) \end{matrix}] {\begin{matrix} p_{2} (L_{p}) \\ - ρ_{0} c_{0} u_{2} (L_{p}) \end{matrix}} \end{matrix}

(16)

for the case of $L_{d 1} < L_{d 2}$ , where $l_{11} = L_{b} + t_{bR} + L_{d 1} + δ_{d 1}$ , $l_{12} = L_{d} - L_{d 1} - δ_{d 1}$ , $l_{13} = L_{d 1} + δ_{d 1} - L_{d 2} - δ_{d 2}$ , $l_{14} = L_{d 2} + δ_{d 2}$ , $l_{15} = L_{b} + t_{bR} + L_{d 2} + δ_{d 2}$ , $l_{16} = L_{b} + t_{bR} + L_{d 1} + δ_{d 1}$ , $l_{17} = L_{d 1} + δ_{d 1}$ , $l_{18} = L_{d 2} + δ_{d 2} - L_{d 1} - δ_{d 1}$ , $l_{19} = L_{d} - L_{d 2} - δ_{d 2}$ , $l_{20} = L_{b} + t_{bR} + L_{d 2} + δ_{d 2}$ , $S_{a 1} = (S_{4} - S_{1} - S_{2} - S_{3}) / (S_{4} - S_{1} - S_{3})$ , $S_{a 2} = (S_{4} - S_{1} - S_{2} - S_{3}) / S_{1}$ , $t_{bR}$ is the thickness of right bulkhead, and $δ_{d 1}$ and $δ_{d 2}$ are the end corrections of extensions of inlet and center tubes into the right cavity.

Expressions (15) and (16) are rewritten as

{\begin{matrix} p_{1} (L_{p}) \\ ρ_{0} c_{0} u_{1} (L_{p}) \end{matrix}} = [Tr] {\begin{matrix} p_{2} (L_{p}) \\ ρ_{0} c_{0} u_{2} (L_{p}) \end{matrix}}

(17)

For the expansion chamber, the following relation may be obtained

\frac{ρ_{c} c_{c} u_{4} (L_{p})}{p_{4} (L_{p})} = j \tan (k_{c} L_{b})

(18)

Substituting equations (13) and (18) into equation (1) yields

{\begin{matrix} p_{1} (0) \\ ρ_{0} c_{0} u_{1} (0) \\ p_{2} (0) \\ \begin{matrix} ρ_{0} c_{0} u_{2} (0) \\ p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \\ \begin{matrix} p_{4} (0) \\ ρ_{c} c_{c} u_{4} (0) \end{matrix} \end{matrix} \end{matrix}} = [P] {\begin{matrix} p_{1} (L_{p}) \\ ρ_{0} c_{0} u_{1} (L_{p}) \\ p_{2} (L_{p}) \\ \begin{matrix} ρ_{0} c_{0} u_{2} (L_{p}) \\ p_{3} (L_{p}) \\ ρ_{0} c_{0} u_{3} (L_{p}) \end{matrix} \end{matrix}}

(19)

where

\begin{matrix} P_{mn} = R_{mn} - \\ \frac{(B_{11} R_{5 n} + B_{12} R_{6 n} - C_{11} R_{7 n} - C_{12} R_{8 n}) [R_{m 7} + j R_{m 8} \tan (k_{c} L_{b})]}{Z}, \\ (m = 1, \dots, 8, n = 1, \dots, 6) \\ Z = B_{11} [R_{57} + j R_{58} \tan (k_{c} L_{b})] + B_{12} [R_{67} + j R_{68} \tan (k_{c} L_{b})] \\ - C_{11} [R_{77} + j R_{78} \tan (k_{c} L_{b})] - C_{12} [R_{87} + j R_{88} \tan (k_{c} L_{b})] \end{matrix}

Writing matrix $[P]$ as

[P] = [\begin{matrix} [P_{11}] & [P_{12}] & [P_{13}] \\ [P_{21}] & [P_{22}] & [P_{23}] \\ [P_{31}] & [P_{32}] & [P_{33}] \\ [P_{41}] & [P_{42}] & [P_{43}] \end{matrix}]

(20)

where $[P_{ij}]$ are the 2 × 2 matrices (i = 1,…, 4, j = 1,…, 3).

Substituting equation (17) into equation (19) then gives

\begin{matrix} {\begin{matrix} p_{1} (0) \\ ρ_{0} c_{0} u_{1} (0) \\ p_{2} (0) \\ \begin{matrix} ρ_{0} c_{0} u_{2} (0) \\ p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \\ \begin{matrix} p_{4} (0) \\ ρ_{c} c_{c} u_{4} (0) \end{matrix} \end{matrix} \end{matrix}} = [\begin{matrix} [P_{11} Tr + P_{12}] & [P_{13}] \\ [P_{21} Tr + P_{22}] & [P_{23}] \\ [P_{31} Tr + P_{32}] & [P_{33}] \\ [P_{41} Tr + P_{42}] & [P_{43}] \end{matrix}] \\ {\begin{matrix} p_{2} (L_{p}) \\ ρ_{0} c_{0} u_{2} (L_{p}) \\ p_{3} (L_{p}) \\ ρ_{0} c_{0} u_{3} (L_{p}) \end{matrix}} = [Q] {\begin{matrix} p_{2} (L_{p}) \\ ρ_{0} c_{0} u_{2} (L_{p}) \\ p_{3} (L_{p}) \\ ρ_{0} c_{0} u_{3} (L_{p}) \end{matrix}} \end{matrix}

(21)

Substituting equations (13) and (14) into equation (21) yields the following relationship between the inlet and outlet of muffler

{\begin{matrix} p_{1} (0) \\ ρ_{0} c_{0} u_{1} (0) \end{matrix}} = [T] {\begin{matrix} p_{3} (L_{p}) \\ ρ_{0} c_{0} u_{3} (L_{p}) \end{matrix}}

(22)

where

\begin{matrix} T_{mn} = Q_{m 1} [a_{n + 2} - \frac{(a_{2} b_{n + 2} + a_{2} b_{1} a_{n + 2})}{(b_{2} + a_{2} b_{1})}] \\ - Q_{m 2} \frac{(b_{n + 2} + b_{1} a_{n + 2})}{(b_{2} + a_{2} b_{1})} + Q_{m (n + 2)}, \begin{matrix} (m, n = 1, 2) \end{matrix} \\ a_{1} = B_{11} Q_{51} + B_{12} Q_{61} - A_{11} Q_{31} - A_{12} Q_{41} \\ a_{n} = \frac{[A_{11} Q_{3 n} + A_{12} Q_{4 n} - B_{11} Q_{5 n} - B_{12} Q_{6 n}]}{a_{1}}, \\ \begin{matrix} (n = 2, \dots, 4) \end{matrix} \\ b_{n} = S_{6} (B_{21} Q_{5 n} + B_{22} Q_{6 n}) + S_{7} (C_{21} Q_{7 n} + C_{22} Q_{8 n}) \\ - S_{5} (A_{21} Q_{3 n} + A_{22} Q_{4 n}), (\begin{matrix} n = 1, \dots, 4 \end{matrix}) \end{matrix}

Muffler with right-perforated bulkhead

Three-pass perforated tube muffler with right-perforated bulkhead is shown in Figure 2. For the case of $L_{d 1} \geq L_{d 2}$ in the right cavity, the following expressions may be obtained by the transfer matrix method

\begin{matrix} {\begin{matrix} p_{1} (L_{p}) \\ ρ_{0} c_{0} u_{1} (L_{p}) \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{1}) & j \sin (k_{0} l_{1}) \\ j \sin (k_{0} l_{1}) & \cos (k_{0} l_{1}) \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ \frac{j \tan (k_{0} l_{2}) (S_{4} - S_{3})}{S_{1}} & \frac{(S_{4} - S_{3} - S_{1})}{S_{1}} \end{matrix}] \\ \times [\begin{matrix} \cos (k_{0} l_{3}) & j \sin (k_{0} l_{3}) \\ j \sin (k_{0} l_{3}) & \cos (k_{0} l_{3}) \end{matrix}] {\begin{matrix} p_{8} \\ ρ_{0} c_{0} u_{8} \end{matrix}} \end{matrix}

(23)

{\begin{matrix} p_{9} \\ ρ_{0} c_{0} u_{9} \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{4}) & j \sin (k_{0} l_{4}) \\ j \sin (k_{0} l_{4}) & \cos (k_{0} l_{4}) \end{matrix}] {\begin{matrix} p_{2} (L_{p}) \\ - ρ_{0} c_{0} u_{2} (L_{p}) \end{matrix}}

(24)

\begin{matrix} {\begin{matrix} p_{10} \\ ρ_{0} c_{0} u_{10} \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{5}) & j \sin (k_{0} l_{5}) \\ j \sin (k_{0} l_{5}) & \cos (k_{0} l_{5}) \end{matrix}] [\begin{matrix} 1 & \frac{ζ_{r}}{z_{n}} \\ 0 & \frac{1}{z_{n}} \end{matrix}] \\ [\begin{matrix} \cos (k_{c} L_{b}) & j \sin (k_{c} L_{b}) \\ j \sin (k_{c} L_{b}) & \cos (k_{c} L_{b}) \end{matrix}] {\begin{matrix} p_{4} (L_{p}) \\ - ρ_{c} c_{c} u_{4} (L_{p}) \end{matrix}} \end{matrix}

(25)

where $l_{1} = L_{b} + t_{bR} + L_{d 1} + σ_{d 1}$ , $l_{2} = L_{d} - L_{d 1} - σ_{d 1}$ , $l_{3} = L_{d 1} + σ_{d 1} - L_{d 2} - σ_{d 2}$ , $l_{4} = L_{b} + t_{bR} + L_{d 2} + σ_{d 2}$ , $l_{5} = L_{d 2} + σ_{d 2}$ , and $ζ_{r}$ is specific acoustic impedance of the right-perforated bulkhead.

Similarly, the following expressions are obtained by the transfer matrix method for the case of $L_{d 1} < L_{d 2}$

{\begin{matrix} p_{1} (L_{p}) \\ ρ_{0} c_{0} u_{1} (L_{p}) \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{6}) & j \sin (k_{0} l_{6}) \\ j \sin (k_{0} l_{6}) & \cos (k_{0} l_{6}) \end{matrix}] {\begin{matrix} p_{8} \\ ρ_{0} c_{0} u_{8} \end{matrix}}

(26)

\begin{matrix} {\begin{matrix} p_{9} \\ ρ_{0} c_{0} u_{9} \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{7}) & j \sin (k_{0} l_{7}) \\ j \sin (k_{0} l_{7}) & \cos (k_{0} l_{7}) \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ \frac{j \tan (k_{0} l_{8}) (S_{4} - S_{3})}{S_{4} - S_{3} - S_{2}} & \frac{S_{2}}{S_{4} - S_{3} - S_{2}} \end{matrix}] \\ \times [\begin{matrix} \cos (k_{0} l_{9}) & j \sin (k_{0} l_{9}) \\ j \sin (k_{0} l_{9}) & \cos (k_{0} l_{9}) \end{matrix}] {\begin{matrix} p_{2} (L_{p}) \\ - ρ_{0} c_{0} u_{2} (L_{p}) \end{matrix}} \end{matrix}

(27)

\begin{matrix} {\begin{matrix} p_{10} \\ ρ_{0} c_{0} u_{10} \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{10}) & j \sin (k_{0} l_{10}) \\ j \sin (k_{0} l_{10}) & \cos (k_{0} l_{10}) \end{matrix}] [\begin{matrix} 1 & \frac{ζ_{r}}{z_{n}} \\ 0 & \frac{1}{z_{n}} \end{matrix}] \\ [\begin{matrix} \cos (k_{c} L_{b}) & j \sin (k_{c} L_{b}) \\ j \sin (k_{c} L_{b}) & \cos (k_{c} L_{b}) \end{matrix}] {\begin{matrix} p_{4} (L_{p}) \\ - ρ_{c} c_{c} u_{4} (L_{p}) \end{matrix}} \end{matrix}

(28)

where $l_{6} = L_{b} + t_{bR} + L_{d 1} + σ_{d 1}$ , $l_{7} = L_{d 2} + σ_{d 2} - L_{d 1} - σ_{d 1}$ , $l_{8} = L_{d} - L_{d 2} - σ_{d 2}$ , $l_{9} = L_{b} + t_{bR} + L_{d 2} + σ_{d 2}$ , and $l_{10} = L_{d 1} + σ_{d 1}$ .

Expressions (23)–(28) may be written as

{\begin{matrix} p_{8} \\ ρ_{0} c_{0} u_{8} \end{matrix}} = [D] {\begin{matrix} p_{1} (L_{p}) \\ ρ_{0} c_{0} u_{1} (L_{p}) \end{matrix}}

(29)

{\begin{matrix} p_{9} \\ ρ_{0} c_{0} u_{9} \end{matrix}} = [E] {\begin{matrix} p_{2} (L_{p}) \\ ρ_{0} c_{0} u_{2} (L_{p}) \end{matrix}}

(30)

{\begin{matrix} p_{10} \\ ρ_{0} c_{0} u_{10} \end{matrix}} = [F] {\begin{matrix} p_{4} (L_{p}) \\ ρ_{c} c_{c} u_{4} (L_{p}) \end{matrix}}

(31)

On the interface, the sound pressure and velocity continuity conditions are

p_{8} = p_{9} = p_{10}

(32)

ρ_{0} c_{0} u_{8} S_{8} = ρ_{0} c_{0} u_{9} S_{9} + ρ_{0} c_{0} u_{10} S_{10}

(33)

where $S_{8} = S_{4} - S_{1} - S_{3}$ , $S_{9} = S_{2}$ , and $S_{10} = S_{4} - S_{1} - S_{2} - S_{3}$ for the case of $L_{d 1} \geq L_{d 2}$ and $S_{8} = S_{1}$ , $S_{9} = S_{4} - S_{2} - S_{3}$ , and $S_{10} = S_{4} - S_{1} - S_{2} - S_{3}$ for the case of $L_{d 1} < L_{d 2}$ .

For the left cavity, the following expression may be obtained using the transfer matrix method

\begin{matrix} {\begin{matrix} p_{2} (0) \\ - ρ_{0} c_{0} u_{2} (0) \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{11}) & j \sin (k_{0} l_{11}) \\ j \sin (k_{0} l_{11}) & \cos (k_{0} l_{11}) \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ \frac{j \tan (k_{0} l_{12}) (S_{4} - S_{1})}{S_{2}} & \frac{S_{4} - S_{1} - S_{2}}{S_{2}} \end{matrix}] \\ \times [\begin{matrix} \cos (k_{0} l_{13}) & j \sin (k_{0} l_{13}) \\ j \sin (k_{0} l_{13}) & \cos (k_{0} l_{13}) \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ j \tan (k_{0} l_{14}) S_{b 1} & \frac{S_{3}}{(S_{4} - S_{1} - S_{2})} \end{matrix}] \\ \times [\begin{matrix} \cos (k_{0} l_{15}) & j \sin (k_{0} l_{15}) \\ j \sin (k_{0} l_{15}) & \cos (k_{0} l_{15}) \end{matrix}] {\begin{matrix} p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \end{matrix}} \end{matrix}

(34)

for the case of $L_{c 2} \geq L_{c 3}$ , and

\begin{matrix} {\begin{matrix} p_{2} (0) \\ - ρ_{0} c_{0} u_{2} (0) \end{matrix}} = [\begin{matrix} \cos (k_{0} l_{16}) & j \sin (k_{0} l_{16}) \\ j \sin (k_{0} l_{16}) & \cos (k_{0} l_{16}) \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ j \tan (k_{0} l_{17}) S_{b 2} & \frac{(S_{4} - S_{1} - S_{3})}{S_{2}} \end{matrix}] \\ \times [\begin{matrix} \cos (k_{0} l_{18}) & j \sin (k_{0} l_{18}) \\ j \sin (k_{0} l_{18}) & \cos (k_{0} l_{18}) \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ \frac{j \tan (k_{0} l_{19}) (S_{4} - S_{1})}{S_{4} - S_{1} - S_{3}} & \frac{S_{3}}{S_{4} - S_{1} - S_{3}} \end{matrix}] \\ \times [\begin{matrix} \cos (k_{0} l_{20}) & j \sin (k_{0} l_{20}) \\ j \sin (k_{0} l_{20}) & \cos (k_{0} l_{20}) \end{matrix}] {\begin{matrix} p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \end{matrix}} \end{matrix}

(35)

for the case of $L_{c 2} < L_{c 3}$ , where $l_{11} = L_{a} + t_{bL} + L_{c 2} + δ_{c 2}$ , $l_{12} = L_{c} - L_{c 2} - δ_{c 2}$ , $l_{13} = L_{c 2} + δ_{c 2} - L_{c 3} - δ_{c 3}$ , $l_{14} = L_{c 3} + δ_{c 3}$ , $l_{15} = L_{a} + t_{bL} + L_{c 3} + δ_{c 3}$ , $l_{16} = L_{a} + t_{bL} + L_{c 2} + δ_{c 2}$ , $l_{17} = L_{c 2} + δ_{c 2}$ , $l_{18} = L_{c 3} + δ_{c 3} - L_{c 2} - δ_{c 2}$ , $l_{19} = L_{c} - L_{c 3} - δ_{c 3}$ , $l_{20} = L_{a} + t_{bL} + L_{c 3} + δ_{c 3}$ , $S_{b 1} = (S_{4} - S_{1} - S_{2} - S_{3}) / (S_{4} - S_{1} - S_{2})$ , and $S_{b 2} = (S_{4} - S_{1} - S_{2} - S_{3}) / S_{2}$ .

Expressions (34) and (35) may be rewritten as

{\begin{matrix} p_{2} (0) \\ ρ_{0} c_{0} u_{2} (0) \end{matrix}} = [Tl] {\begin{matrix} p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \end{matrix}}

(36)

For the expansion chamber, we may have

\frac{ρ_{c} c_{c} u_{4} (0)}{p_{4} (0)} = - j \tan (k_{c} L_{a})

(37)

Substituting equations (32) and (37) into equation (1) then gives

{\begin{matrix} p_{1} (0) \\ ρ_{0} c_{0} u_{1} (0) \\ p_{2} (0) \\ \begin{matrix} ρ_{0} c_{0} u_{2} (0) \\ p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \end{matrix} \end{matrix}} = [P] {\begin{matrix} p_{1} (L_{p}) \\ ρ_{0} c_{0} u_{1} (L_{p}) \\ p_{2} (L_{p}) \\ \begin{matrix} ρ_{0} c_{0} u_{2} (L_{p}) \\ p_{3} (L_{p}) \\ ρ_{0} c_{0} u_{3} (L_{p}) \end{matrix} \end{matrix}}

(38)

where

\begin{matrix} P_{mn} = R_{mn} + (R_{m 7} - \frac{R_{m 8} F_{11}}{F_{12}}) a_{n}, \\ \begin{matrix} (m = 1, \dots, 6, n = 1, 2, 5, 6) \end{matrix} \\ P_{m 3} = R_{m 3} + (R_{m 7} - \frac{R_{m 8} F_{11}}{F_{12}}) a_{3} + \frac{R_{m 8} E_{11}}{F_{12}}, \\ (m = 1, \dots, 6) \\ P_{m 4} = R_{m 4} + (R_{m 7} - \frac{R_{m 8} F_{11}}{F_{12}}) a_{4} + \frac{R_{m 8} E_{12}}{F_{12}}, \\ \begin{matrix} (m = 1, \dots, 6) \end{matrix} \\ a_{0} = \frac{[R_{88} + j R_{78} \tan (k_{c} L_{a})] F_{11}}{F_{12}} - [R_{87} + j R_{77} \tan (k_{c} L_{a})] \\ a_{n} = \frac{[R_{8 n} + j R_{7 n} \tan (k_{c} L_{a})]}{a_{0}}, \begin{matrix} (n = 1, 2, 5, 6) \end{matrix} \\ a_{3} = \frac{[R_{83} + R_{88} E_{11} / F_{12} + j \tan (k_{c} L_{a}) (R_{73} + R_{78} E_{11} / F_{12})]}{a_{0}} \\ a_{4} = \frac{[R_{84} + R_{88} E_{12} / F_{12} + j \tan (k_{c} L_{a}) (R_{74} + R_{78} E_{12} / F_{12})]}{a_{0}} \end{matrix}

Substituting equations (32) and (33) into equation (38) yields

{\begin{matrix} p_{1} (0) \\ ρ_{0} c_{0} u_{1} (0) \\ p_{2} (0) \\ \begin{matrix} ρ_{0} c_{0} u_{2} (0) \\ p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \end{matrix} \end{matrix}} = [Q] {\begin{matrix} p_{2} (L_{p}) \\ ρ_{0} c_{0} u_{2} (L_{p}) \\ p_{3} (L_{p}) \\ ρ_{0} c_{0} u_{3} (L_{p}) \end{matrix}}

(39)

where

\begin{matrix} Q_{m 1} = P_{m 3} + \frac{P_{m 1} (E_{11} - D_{12} b_{1})}{D_{11}} + P_{m 2} b_{1}, (m = 1, \dots, 6) \\ Q_{m 2} = P_{m 4} + \frac{P_{m 1} (E_{12} - D_{12} b_{2})}{D_{11}} + P_{m 2} b_{2}, (m = 1, \dots, 6) \\ Q_{m 3} = P_{m 5} - \frac{P_{m 1} D_{12} b_{3}}{D_{11}} + P_{m 2} b_{3}, (m = 1, \dots, 6) \\ Q_{m 4} = P_{m 6} - \frac{P_{m 1} D_{12} b_{4}}{D_{11}} + P_{m 2} b_{4}, (m = 1, \dots, 6) \\ b_{0} = D_{22} S_{8} + (\frac{F_{22} F_{11}}{F_{12}} - F_{21}) (a_{2} - \frac{a_{1} D_{12}}{D_{11}}) S_{10} - \frac{D_{21} D_{12} S_{8}}{D_{11}} \\ b_{1} = \frac{[E_{21} S_{9} + F_{22} E_{11} S_{10} / F_{12} + (F_{21} - F_{22} F_{11} / F_{12}) (a_{3} + a_{1} E_{11} / D_{11}) S_{10} - D_{21} E_{11} S_{8} / D_{11}]}{b_{0}} \\ b_{2} = \frac{[E_{22} S_{9} + F_{22} E_{12} S_{10} / F_{12} + (F_{21} - F_{22} F_{11} / F_{12}) (a_{4} + a_{1} E_{12} / D_{11}) S_{10} - D_{21} E_{12} S_{8} / D_{11}]}{b_{0}} \\ b_{3} = \frac{(F_{21} - F_{22} F_{11} / F_{12}) a_{5} S_{10}}{b_{0}} \\ b_{4} = \frac{(F_{21} - F_{22} F_{11} / F_{12}) a_{6} S_{10}}{b_{0}} \end{matrix}

Matrix $[Q]$ may be rewritten as

[Q] = [\begin{matrix} [Q_{11}] & [Q_{12}] \\ [Q_{21}] & [Q_{22}] \\ [Q_{31}] & [Q_{32}] \end{matrix}]

(40)

where $[Q_{ij}]$ are the 2 × 2 matrices $(i = 1, \dots, 3, \begin{matrix} j = 1, \dots, 2 \end{matrix})$ .

Finally, substituting equation (36) into equation (39) obtains the following relationship between the inlet and outlet of muffler

{\begin{matrix} p_{1} (0) \\ ρ_{0} c_{0} u_{1} (0) \end{matrix}} = [T] {\begin{matrix} p_{3} (L_{p}) \\ ρ_{0} c_{0} u_{3} (L_{p}) \end{matrix}}

(41)

where $[T] = [Q_{11}] {([Q_{21}] - [Tl] [Q_{31}])}^{- 1} ([Tl] [Q_{32}] - [Q_{22}]) + [Q_{12}]$ .

Muffler with both perforated bulkheads

Three-pass perforated tube muffler with both perforated bulkheads is shown in Figure 3. Based on prior analyses of sections “Muffler with left-perforated bulkhead” and “Muffler with right-perforated bulkhead,” expressions (10)–(12), (29)–(31) as well as the boundary condition expressions (13), (14), (32), and (33) are still valid.

Substituting equations (14) and (33) into equation (1) yields

{\begin{matrix} p_{1} (0) \\ ρ_{0} c_{0} u_{1} (0) \\ p_{2} (0) \\ \begin{matrix} ρ_{0} c_{0} u_{2} (0) \\ p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \\ \begin{matrix} p_{4} (0) \\ ρ_{c} c_{c} u_{4} (0) \end{matrix} \end{matrix} \end{matrix}} = [P] {\begin{matrix} p_{1} (L_{p}) \\ ρ_{0} c_{0} u_{1} (L_{p}) \\ p_{2} (L_{p}) \\ \begin{matrix} ρ_{0} c_{0} u_{2} (L_{p}) \\ p_{3} (L_{p}) \\ ρ_{0} c_{0} u_{3} (L_{p}) \end{matrix} \end{matrix}}

(42)

where

\begin{matrix} P_{m 1} = R_{m 1} + \frac{R_{m 8} D_{21} S_{8}}{(F_{22} S_{10})} + (R_{m 7} - \frac{F_{21} R_{m 8}}{F_{22}}) b_{1}, \begin{matrix} (m = 1, \dots, 8) \end{matrix} \\ P_{m 2} = R_{m 2} + \frac{R_{m 8} D_{22} S_{8}}{(F_{22} S_{10})} + (R_{m 7} - \frac{F_{21} R_{m 8}}{F_{22}}) b_{2}, \begin{matrix} (m = 1, \dots, 8) \end{matrix} \\ P_{m 3} = R_{m 3} - \frac{R_{m 8} E_{21} S_{9}}{(F_{22} S_{10})} + (R_{m 7} - \frac{F_{21} R_{m 8}}{F_{22}}) b_{3}, \begin{matrix} (m = 1, \dots, 8) \end{matrix} \\ P_{m 4} = R_{m 4} - \frac{R_{m 8} E_{22} S_{9}}{(F_{22} S_{10})} + (R_{m 7} - \frac{F_{21} R_{m 8}}{F_{22}}) b_{4}, (m = 1, \dots, 8) \\ P_{m 5} = R_{m 5} + (R_{m 7} - \frac{F_{21} R_{m 8}}{F_{22}}) b_{5}, (m = 1, \dots, 8) \\ P_{m 6} = R_{m 6} + (R_{m 7} - \frac{F_{21} R_{m 8}}{F_{22}}) b_{6}, (m = 1, \dots, 8) \\ a_{n} = (B_{21} R_{5 n} + B_{22} R_{6 n}) S_{6} + (C_{21} R_{7 n} + C_{22} R_{8 n}) S_{7} \\ - (A_{21} R_{3 n} + A_{22} R_{4 n}) S_{5}, (n = 1, \dots, 8) \\ b_{0} = \frac{F_{21} a_{8}}{F_{22}} - a_{7} \\ b_{1} = \frac{[a_{1} + D_{21} S_{8} a_{8} / (F_{22} S_{10})]}{b_{0}} \\ b_{2} = \frac{[a_{2} + D_{22} S_{8} a_{8} / (F_{22} S_{10})]}{b_{0}} \\ b_{3} = \frac{[a_{3} - E_{21} S_{9} a_{8} / (F_{22} S_{10})]}{b_{0}} \\ b_{4} = \frac{[a_{4} - E_{22} S_{9} a_{8} / (F_{22} S_{10})]}{b_{0}} \\ b_{5} = \frac{a_{5}}{b_{0}} \\ b_{6} = \frac{a_{6}}{b_{0}} \end{matrix}

Substituting equations (13) and (32) into equation (42) yields

{\begin{matrix} p_{1} (0) \\ ρ_{0} c_{0} u_{1} (0) \\ p_{2} (0) \\ \begin{matrix} ρ_{0} c_{0} u_{2} (0) \\ p_{3} (0) \\ ρ_{0} c_{0} u_{3} (0) \end{matrix} \end{matrix}} = [Q] {\begin{matrix} p_{1} (L_{p}) \\ ρ_{0} c_{0} u_{1} (L_{p}) \\ p_{3} (L_{p}) \\ ρ_{0} c_{0} u_{3} (L_{p}) \end{matrix}}

(43)

where

\begin{matrix} Q_{m 1} = P_{m 1} + P_{m 4} d_{1} - (P_{m 3} + P_{m 4} d_{3}) \frac{(c_{1} + c_{4} d_{1})}{(c_{3} + c_{4} d_{3})}, (m = 1, \dots, 6) \\ Q_{m 2} = P_{m 2} + P_{m 4} d_{2} - (P_{m 3} + P_{m 4} d_{3}) \frac{(c_{2} + c_{4} d_{2})}{(c_{3} + c_{4} d_{3})}, (m = 1, \dots, 6) \\ Q_{m 3} = P_{m 5} + P_{m 4} d_{4} - (P_{m 3} + P_{m 4} d_{3}) \frac{(c_{5} + c_{4} d_{4})}{(c_{3} + c_{4} d_{3})}, (m = 1, \dots, 6) \\ Q_{m 4} = P_{m 6} + P_{m 4} d_{5} - (P_{m 3} + P_{m 4} d_{3}) \frac{(c_{6} + c_{4} d_{5})}{(c_{3} + c_{4} d_{3})}, (m = 1, \dots, 6) \\ c_{n} = B_{11} P_{5 n} + B_{12} P_{6 n} - C_{11} P_{7 n} - C_{12} P_{8 n}, (n = 1, \dots, 6) \\ d_{0} = F_{12} (\frac{E_{22} S_{9}}{F_{22} S_{10}} + F_{21} b_{4} / F_{22}) - F_{11} b_{4} + E_{12} \\ d_{1} = \frac{[F_{11} b_{1} + F_{12} (D_{21} S_{8} / F_{22} S_{10} - F_{21} b_{1} / F_{22})]}{d_{0}} \end{matrix}

\begin{matrix} d_{2} = \frac{[F_{11} b_{2} + F_{12} (D_{22} S_{8} / F_{22} S_{10} - F_{21} b_{2} / F_{22})]}{d_{0}} \\ d_{3} = \frac{[F_{11} b_{3} - F_{12} (E_{21} S_{9} / F_{22} S_{10} + F_{21} b_{3} / F_{22}) - E_{11}]}{d_{0}} \\ d_{4} = \frac{(F_{11} b_{5} - F_{12} F_{21} b_{5} / F_{22})}{d_{0}} \\ d_{5} = \frac{(F_{11} b_{6} - F_{12} F_{21} b_{6} / F_{22})}{d_{0}} \end{matrix}

Substituting equation (13) into equation (32) obtains the following relationship between the inlet and outlet of the muffler

{\begin{matrix} p_{1} (0) \\ ρ_{0} c_{0} u_{1} (0) \end{matrix}} = [T] {\begin{matrix} p_{3} (L_{p}) \\ ρ_{0} c_{0} u_{3} (L_{p}) \end{matrix}}

(44)

where

\begin{matrix} T_{mn} = Q_{m (n + 2)} + Q_{m 2} f_{n + 1} \\ - \frac{(Q_{m 1} + Q_{m 2} f_{1}) (e_{n + 2} + e_{2} f_{n + 1})}{(e_{1} + e_{2} f_{1})}, \begin{matrix} (m, n = 1, 2) \end{matrix} \\ e_{n} = A_{11} Q_{3 n} + A_{12} Q_{4 n} - B_{11} Q_{5 n} - B_{12} Q_{6 n}, \begin{matrix} (n = 1, \dots, 4) \end{matrix} \\ f_{0} = D_{12} - E_{12} d_{2} + (E_{11} + E_{12} d_{3}) \frac{(c_{2} + c_{4} d_{2})}{(c_{3} + c_{4} d_{3})} \\ f_{1} = \frac{[E_{12} d_{1} - D_{11} - (E_{11} + E_{12} d_{3}) (c_{1} + c_{4} d_{1}) / (c_{3} + c_{4} d_{3})]}{f_{0}} \\ f_{2} = \frac{[E_{12} d_{4} - (E_{11} + E_{12} d_{3}) (c_{5} + c_{4} d_{4}) / (c_{3} + c_{4} d_{3})]}{f_{0}} \\ f_{3} = \frac{[E_{12} d_{5} - (E_{11} + E_{12} d_{3}) (c_{6} + c_{4} d_{5}) / (c_{3} + c_{4} d_{3})]}{f_{0}} \end{matrix}

Acoustic properties of perforation and sound-absorbing material

For the perforated facing, one side of which is the air and another side is the sound-absorbing material, the following expressions for the specific acoustic impedance of perforation are adopted⁹

ζ_{p} = \frac{[R_{h} + j k_{0} {t_{w} + 0.5 α [1 + (k_{c} / k_{0}) z_{n}] d_{h}}]}{ϕ}

(45)

R_{h} = (1 + \frac{t_{w}}{d_{h}}) \sqrt{\frac{8 k_{0} μ}{z_{0}}}

(46)

where $R_{h}$ is the specific acoustic resistance of a hole, $t_{w}$ is the wall thickness of perforated tube, $α$ is the end correction coefficient of a single hole, $d_{h}$ is the hole diameter, $ϕ$ is the porosity of perforation, $μ$ is the dynamic viscosity of air, and $z_{0}$ is the acoustic impedance of air.

Ji and Fang¹⁰ reviewed the existing expressions for the end correction coefficient and acoustic impedance and indicated that following Ingard’s¹¹ expression for the end correction coefficient may be the most suitable for the prediction of acoustic attenuation behavior of perforated tube silencers; therefore, it is adopted in this work

α = \frac{4}{π^{2}} \frac{1}{{(ζ η)}^{0.5}} \sum_{m = 0}^{\infty} {\sum_{n = 0}^{\infty}}^{'} ε_{mn} \frac{J_{1}^{2} (π \sqrt{{(m ξ)}^{2} + {(n η)}^{2}})}{{[m^{2} (h / b) + n^{2} (b / h)]}^{3 / 2}}

(47)

where $ξ = d_{h} / b$ , $η = d_{h} / h$ , b, and h are the distances between two neighboring holes in two directions, and $J_{1}$ is the Bessel function of the first kind of order 1. And the prime on the summation sign implies that the term representing the (0, 0) fundamental mode is excluded, $ε_{mn} = 1$ if $m \neq 0$ and $n \neq 0$ or $ε_{mn} = 0.5$ if otherwise.

Lee and Selamet¹² presented the following expressions to calculate specific impedance and wave number of fibrous material with filling destiny $ρ_{f} = 100 g / L$

\frac{z_{c}}{z_{0}} = (1 + a_{1} f^{- b_{1}}) - j a_{2} f^{- b_{2}}

(48)

\frac{k_{c}}{k_{0}} = (1 + a_{3} f^{- b_{3}}) - j a_{4} f^{- b_{4}}

(49)

where $z_{c}$ is the specific characteristic impedance of sound-absorbing material, $z_{0}$ is the acoustic impedance of air, $k_{c}$ is the complex wave number of sound-absorbing material, $k_{0}$ is the wave number, f is the sound frequency; $a_{1} = 33.2$ , $b_{1} = 0.7523$ , $a_{2} = 28.32$ , $b_{2} = 0.6512$ , $a_{3} = 39.20$ , $b_{3} = 0.6841$ , $a_{4} = 38.39$ , and $b_{4} = 0.6285$ .

Results and discussion

Validation

The three-pass perforated tube mufflers with perforated bulkheads are shown in Figures 1 –3, and the relative positions of inlet, center, and outlet tubes are shown in Figure 4. The dimensions of the muffler are selected as follows: $d_{1} = 46.2 mm$ , $d_{2} = 40.4 mm$ , and $d_{3} = 45.5 mm$ for the outside diameters of the inlet, center, and outlet tubes, respectively; $d_{4} = 157.0 mm$ for the inner diameter of expansion chamber; $L_{c} = 148.0 mm$ and $L_{d} = 250.0 mm$ for the lengths of left and right chambers; $L_{c 2} = 112.0 mm$ , $L_{c 3} = 0$ , $L_{d 1} = 205.0 mm$ , and $L_{d 2} = 0$ for the tube extensions into the left and right chambers; $L_{a} = 25.0 mm$ and $L_{b} = 50.0 mm$ for the lengths of solid portions of the perforated tubes in the center chamber; $L_{p} = 105.0 mm$ for the length of common perforated section with porosity $ϕ = 9 %$ ; $d_{h} = 2.49 mm$ for the hole diameter; and $t_{w} = 1.0 mm$ for the wall thickness of three-perforated tubes. The speed of sound $c_{0} = 344 m / s$ .

Figure 4.

The relative positions of inlet, center, and outlet tubes in the muffler.

Figures 5 –7 compare the transmission loss predictions of the reactive and hybrid (with fibrous material filling in the center chamber) mufflers with left-perforated bulkhead, right-perforated bulkhead, and both perforated bulkheads, respectively, from the 1D analytical approach and 3D FEM.

Figure 5.

Transmission loss of the reactive and hybrid mufflers with left-perforated bulkhead (porosity $ϕ_{L} = 4 %$ , hole diameter $d_{hL} = 3.5 mm$ , and thickness $t_{bL} = 1.5 mm$ ).

Figure 6.

Transmission loss of the reactive and hybrid mufflers with right-perforated bulkhead (porosity $ϕ_{R} =4 %$ , hole diameter $d_{hR} = 3.5 mm$ , and thickness $t_{bR} = 1.5 mm$ ).

Figure 7.

Transmission loss of the reactive and hybrid mufflers with both perforated bulkheads (porosity $ϕ_{L} = ϕ_{R} = 4 %$ , hole diameter $d_{hL} = d_{hR} = 3.5 mm$ , and thickness $t_{bL} = t_{bR} = 1.5 mm$ ).

For the reactive mufflers with perforated bulkhead(s), Figures 5 –7 indicate that the 1D analytical solution agrees reasonably well with the FEM predictions up to about 950 Hz and deviates at higher frequencies. The differences between transmission loss predictions from the 1D analytical approach and 3D FEM may be attributed to the local nonplanar wave effects, and the deviation at higher frequencies may be attributed to the effect of higher order modes. The cutoff frequency is determined as 964 Hz using FEM calculation for this muffler. Beyond the cutoff frequency, the multidimensional wave may propagate and affect the acoustic attenuation performance of the mufflers; therefore, the 1D analytical approach is not suitable anymore. For the hybrid mufflers with perforated bulkhead(s), reasonable good agreements are observed between both approaches in the frequency range of interest. Since the existence of sound-absorbing material, the effect of higher order modes is limited even beyond the cutoff frequency, and good agreements were observed in the whole frequency range for hybrid silencers.

Acoustic performance analysis of muffler

Figure 8 compares the transmission loss predictions of the reactive mufflers with solid bulkheads and one and two perforated bulkheads from the 1D analytical approach. Compared to the muffler with solid bulkheads, the perforated bulkhead(s) changed the resonance frequency and acoustic response throughout frequency of interest.

Figure 8.

Transmission loss predictions of the three-pass perforated tube muffler with solid, left-perforated, right-perforated, and both perforated bulkheads.

The effect of perforate porosity on the acoustic attenuation performance of the reactive muffler with left-perforated bulkhead (holes diameter $d_{hL} = 3.5 mm$ and thickness $t_{bL} = 1.5 mm$ ) is shown in Figure 9. It may be seen that the resonance frequency contributed by the left end-cavity is very sensitive to the porosity of perforation and shifts to higher frequency with the increase in the porosity of perforation. The influence of perforation on the left-perforated bulkhead on the resonance frequency of right end-cavity is negligible.

Figure 9.

Effect of porosity on acoustic attenuation performance of three-pass perforated tube muffler with left-perforated bulkhead.

Figure 10 shows the effect of different porosities of the right-perforated bulkhead (hole diameter $d_{hR} = 3.5 mm$ and thickness $t_{bR} = 1.5 mm$ ) on acoustic attenuation characteristics of the muffler. Figure 10 demonstrated that the resonance frequency contributed by right end-cavity moves to higher frequency with the increase in the porosity of perforation. For the cases of $ϕ_{R} = 4 %$ and $ϕ_{R} = 16 %$ , the right end-cavity resonance frequency closes to the left end-cavity resonance frequency and, therefore, leads to a single wider resonant peak.

Figure 10.

Effect of porosity on acoustic attenuation performance of three-pass perforated tube muffler with right-perforated bulkhead.

Figure 11 illustrates the effect of different porosities of the reactive mufflers with both perforated bulkheads (hole diameter $d_{hL} = d_{hR} = 3.5 mm$ and thickness $t_{bL} = t_{bR} = 1.5 mm$ ) on acoustic attenuation behavior of the muffler. The resonance frequencies shift to higher frequencies with the increase in the porosity of perforation.

Figure 11.

Effect of porosity on acoustic attenuation performance of three-pass perforated tube muffler with both perforated bulkheads.

Figures 12 –14 examine the effect of sound-absorbing material filling in the middle chamber on the acoustic attenuation performance of the mufflers. It may be seen that the sound-absorbing material decreases the transmission loss around the resonance frequencies and improve the acoustic attenuation at most frequencies. As a result, the transmission loss curve of the hybrid muffler is flatter than that of the corresponding reactive muffler.

Figure 12.

Effect of sound-absorbing on acoustic attenuation performance of three-pass perforated tube muffler with left-perforated bulkhead.

Figure 13.

Effect of sound-absorbing on acoustic attenuation performance of three-pass perforated tube muffler with right-perforated bulkhead.

Figure 14.

Effect of sound-absorbing on acoustic attenuation performance of three-pass perforated tube muffler with both perforated bulkheads.

Conclusion

The 1D analytical approach based on the plane wave propagation is presented to determine and analyze the acoustic behaviors of three-perforated tube reactive and hybrid mufflers with perforated bulkhead(s). The transmission loss predictions from the 1D analytical approach and FEM are compared to validate the 1D approach for the reactive and hybrid mufflers. The results indicated that 1D analytical approach agrees reasonably well with FEM until up to the plane wave cutoff frequency. Perforates on the bulkheads may change the resonant frequency of the three-pass perforated tube mufflers, and the resonance frequency is shifted to higher frequency with the increase in the porosity of perforation. Increasing the porosity of the bulkheads within a certain range may improve the acoustic attenuation behavior of the muffler in specific frequency range. The sound-absorbing material filling in the middle chamber decreases the acoustic attenuation around the resonance frequencies and improves the acoustic attenuation performance at most frequencies to lead a flatter acoustic response of the hybrid mufflers than the corresponding reactive mufflers. The effects of mean flow on the acoustic attenuation performance of mufflers with perforated components are important and may be considered using the proper expression of perforate impedance.

Footnotes

Academic Editor: Hongwei Wu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This article is funded by The National High Technology Research and Development Program of China (“863” Program) and the funding number is 2014AA041502.

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Three-pass perforated tube muffler with perforated bulkheads

Abstract

Keywords

Introduction

Theory

Muffler with left-perforated bulkhead

Muffler with right-perforated bulkhead

Muffler with both perforated bulkheads

Acoustic properties of perforation and sound-absorbing material

Results and discussion

Validation

Acoustic performance analysis of muffler

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References