A two-dimensional approach for sound attenuation of multi-chamber perforated resonator and its optimal design

Applied Acoustics 127 (2017) 105–117

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A two-dimensional approach for sound attenuation of multi-chamber perforated resonator and its optimal design Rong Guo ⇑, Li-ting Wang, Wen-bo Tang, Shuai Han School of Automotive Studies, Tongji University, Shanghai 201804, PR China Clean Energy Automotive Engineering Center, Tongji University, Shanghai 201804, PR China

a r t i c l e

i n f o

Article history: Received 4 December 2016 Received in revised form 17 May 2017 Accepted 25 May 2017

Keywords: Two-dimensional Transfer matrix method Perforated resonator Transmission loss Optimization

a b s t r a c t A two-dimensional (2D) method is investigated to predict the acoustic performances of single-chamber perforated reactive resonators. The effect of non-planar wave propagation on the acoustic performance of acoustically short and long resonator is studied. A desirable resonance behavior appears in acoustically short chamber substantially below the cut-off frequency due to non-planar wave propagation. Adding inlet/outlet extensions has similar effects to that of reducing perforation rate on acoustic performances for short-length perforated resonators. Based on the 2D approach, a 2D transfer matrix method (TMM) is developed through solving the acoustical continuity functions under two outlet boundary conditions to predict the acoustic performances of multi-chamber perforated resonators (MCPRs) which can attenuate broadband noise. Comparisons between the calculations and tests show that the 2D TMM is much more accurate than one-dimensional approach within entire frequency range. In order to evaluate its engineering applicability, a new optimization procedure including a targeted transmission loss curve and a reasonable objective function is introduced to optimize the structure parameters of a MCPR with three chambers using a genetic algorithm. The result can meet the target well at desired frequencies under space constraint. The theoretical method developed in this work can be used for the calculation and optimization of MCPRs in various applications. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Air intake noise is one of the main vehicle noises, especially for turbocharged engine in transient conditions, which seriously affects the driving comfort [1]. The overall noise level of turbocharged engine is normally 2 dB(A)–3 dB(A) higher than naturally aspirated engine, resulting from the additional air borne noise which commonly distributes in 1.5–3 kHz [2]. However, traditional resonators (Helmholtz resonator, expansion muffler, quarter-wavelength tube etc.) cannot efficiently attenuate such noise within a broadband frequency range. A multi-chamber perforated resonator (MCPR) is a typical kind of silencer, which can attenuate broadband noise [3]. In addition, the advantages of a compact structure and a low pressure drop enable the resonator to meet the strict installation requirements. As shown in Fig. 1, an ideal acoustic performance can be achieved through adjusting the width, diameter and number of chambers or the diameter, number, thickness of perforated holes. The transfer ⇑ Corresponding author at: School of Automotive Studies, Tongji University, Shanghai 201804, PR China. E-mail address: [email protected] (R. Guo). http://dx.doi.org/10.1016/j.apacoust.2017.05.030 0003-682X/Ó 2017 Elsevier Ltd. All rights reserved.

matrix method (TMM) based on plane wave theory is to obtain the four-pole parameters of a resonator, which can be used to predict the transmission loss (TL). Guo [4] extended the TMM from a single chamber perforated resonator (SCPR) to a MCPR through multiplying the transfer matrixes of connected sections. The onedimensional (1D) TMM is widely used for the design and optimization of perforated and micro-perforated resonators. Shi [5] investigated the wave propagation in a periodic array of micro-perforated tube mufflers through 1D TMM. Chiu [6] used 1D TMM to evaluate the acoustic performance of MCPRs and obtained the best shape of a MCPR under space constraint through optimization with a genetic algorithm (GA). But the 1D method is limited to the cutoff frequencies of the resonator chambers. Whereas the finite element method (FEM) is able to predict the acoustic performances of resonators more precisely [7]. Yet as the number of chambers increases, the geometry becomes more complicated, hence the structure modeling and finite element calculation become inefficient. In addition, the structure optimization will be inconvenient if the acoustic performance is unsatisfying. Therefore, it is necessary to develop a theoretical method, which is simultaneously accurate and efficient to calculate and optimize the acoustic performance of MCPRs.

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Nomenclature Sn c hd, dh ht d1 d2 Do f J0, J1 k0 kx,S,n kr,S,n li lm lo

modal amplitudes in region S (A, B, C, D, E) sound speed in air diameter of the perforations thickness of the inner tube diameter of the inlet tube diameter of the outlet tube diameter of the chambers frequency Bessel functions of the first kind of order 0 and 1 sound wave number in air axial wave number in region S (A, B, C, D, E) radial wave number in region S (A, B, C, D, E) length of inlet extension length of perforation area length of outlet extension

L OBJ P r1 R T U v Y0, Y1

length of a resonator chamber objective function acoustic pressure radius of the inner tube radius of the chambers transfer matrix particle velocity air viscosity Bessel functions of the second kind of order 0 and 1 end correction coefficient perforation impedance porosity air density eigen functions in region S (A, B, C, D, E)

a

f

r q

£nS (r)

2. Two-dimensional approach for sound attenuation of a SCPR 2.1. Acoustic modeling of a SCPR with extended inlet/outlet As shown in Fig. 2, a SCPR with extended inlet/outlet can be divided into five sections: inlet A, extended inlet chamber B, perforated tube C, extended outlet chamber D and outlet E. The total length L is divided into an extended inlet of length li, a perforated tube of length lm, and an extended outlet of length lo. The radius of the inner tube and outer chamber are r1 and R. The Helmholtz equation of sound wave in axisymmetric tube is expressed as [13]

(

Fig. 1. Configuration of a MCPR.

The aim of the present work is to provide a two-dimensional (2D) TMM for the acoustic modeling and optimization of MCPRs. Theoretically, the 2D approach can be used to calculate the TLs of axisymmetric resonators. Selamet [8,9] predicted the TLs of both a single-chamber and a dual-chamber circular expansion muffler with extended inlet/outlet using a 2D weighted integral method. However, the equations to be solved in this approach will be complicated when many chambers are connected, since the transfer matrixes of the silencer elements are not considered and all continuity equations must be solved at one time. For resonators with perforations, the 2D approach was mostly applied to mufflers with a single chamber [10–12]. An effective and efficient 2D approach has still not been developed for the TL prediction of perforated reactive resonators especially MCPRs. In this study, a 2D analytical method using direct integral is applied to calculate the TL of a single-chamber perforated reactive resonator with extended inlet/outlet. The effect of non-planar wave propagation on the acoustic performance of acoustically short and long SCPRs is studied through the 2D approach. The transfer matrix of a perforated resonator is derived through solving the continuity equations under two different outlet conditions. Compared to the tests, the accuracy of 2D TMM is validated at both low and high frequencies. In addition, considering the engineering application, a new optimization procedure is introduced, including defining a targeted TL curve in a wide frequency and a reasonable objective function for the MCPRs’ design. At last, based on 2D TMM, the geometry of a MCPR with three chambers are optimized applying a genetic algorithm (GA) under space constraint.

@2 P @r 2

þ 1r

@P @r

2

2

þ @@xP2 þ k0 P ¼ 0

k0 ¼ x=c ¼ 2pf =c

ð1Þ

where P is the sound pressure; k0 is the wave number; f is the sound frequency; c is the sound velocity; r is the distance to the axis. Upon making use of the separation method of variables, the sound pressure is assumed as

Pðr; xÞ ¼

X Rn ðrÞX n ðxÞ

ð2Þ

n

Then, Eq. (1) can be divided into two independent wave equations [13]

8 2 2 < d XðxÞ ¼ kx XðxÞ 2 dx

: d2 RðrÞ þ 1 dr 2

(

r

dRðrÞ dr

2

þ kr RðrÞ ¼ 0

ð3Þ

Solutions of Eq. (3) is expressed as

X n ðxÞ ¼ Sþn ejkx;S;n x þ Sn ejkx;S;n x Rn ðrÞ ¼ /nS ðrÞ

ð4Þ

Substituting Eq. (4) to Eq. (2) yields

PS ¼

1 X

ðSþn ejkx;S;n xS þ Sn ejkx;S;n xS Þ/nS ðrÞ

ð5Þ

n¼0

Combing the linear momentum equation, the axial particle velocity is derived as

U x;S ¼

1 j @PS 1 X k ðSþ ejkx;S;n xS  Sn ejkx;S;n xS Þ/nS ðrÞ ¼ qx @x q0 x n¼0 x;S;n n

ð6Þ

where S stands for the five sections A, B, C, D, E; Sn+, Sn are the nth modal amplitudes corresponding to positive and negative x

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Fig. 2. Configuration of a perforated resonator.

directions; £nS (r) is the eigenfunction; kr,S,n is the nth radial wave number; kx,S,n is the nth axial wave number. For section A and E with straight tubes, the eigenfunctions are



/nA ðrÞ ¼ J 0 ðkr;A;n rÞ /nE ðrÞ ¼ J 0 ðkr;E;n rÞ

ð7Þ

For section B and D with concentric annular tubes, the eigenfunctions are



/nB ðrÞ ¼ J 0 ðkr;B;n rÞ  ½J 1 ðkr;B;n RÞ=Y 1 ðkr;B;n RÞY 0 ðkr;B;n rÞ

ð8Þ

/nD ðrÞ ¼ J 0 ðkr;D;n rÞ  ½J 1 ðkr;D;n RÞ=Y 1 ðkr;D;n RÞY 0 ðkr;D;n rÞ

For section C with concentric perforated tube, the eigenfunction is



/nC ðrÞ ¼

ð17aÞ

PB jx1 ¼0¼ P C jx2 ¼0 ; ðr 1 6 r 6 RÞ

ð17bÞ

PE jx3 ¼0¼ P C jx2 ¼lm ; ð0 6 r 6 r 1 Þ

ð17cÞ

PD jx3 ¼0¼ PC jx2 ¼lm ; ðr 1 6 r 6 RÞ

ð17dÞ

( U x;C jx2 ¼0 ¼

J 0 ðkr;C;n r 1 Þ þ jfðkr;C;n =k0 ÞJ 1 ðkr;C;n r 1 Þ J 0 ðkr;C;n r1 Þ  ½J 1 ðkr;C;n RÞ=Y 1 ðkr;C;n RÞY 0 ðkr;C;n r 1 Þ

2

2

U x;B jx1 ¼0 ; ðr 1 6 r 6 RÞ

U x;C jx2 ¼lm ¼

U x;E jx3 ¼0;ð06r6r1 Þ U x;D jx3 ¼0 ; ðr1 6 r 6 RÞ

ð17eÞ

ð17fÞ

Truncating the number of modes to a suitable value N, and then substituting the PS and US by Eqs. (5) and (6), Eq. (17) yields

ð10Þ

The perforation area of section C is expressed as a perforation impedance f (see Appendix A); J0 and J1 are the zeroth and first order Bessel function of the first kind; Y0 and Y1 are the zeroth and first order Bessel function of the second kind. The relation between radial and axial wave numbers can be expressed as

kr;S;n þ kx;S;n ¼ k0

U x;A jx1 ¼0;ð06r6r1 Þ

(

C 5 fJ 0 ðkr;C;n rÞ  ½J 1 ðkr;C;n RÞ=Y 1 ðkr;C;n RÞY 0 ðkr;C;n rÞg; r1 < r < R

With

2

PA jx1 ¼0 ¼ PC jx2 ¼0 ; ð0 6 r 6 r 1 Þ

J 0 ðkr;C;n rÞ; 0 < r < r 1 ð9Þ

C5 ¼

The sound pressure and particle velocity satisfy the continuity conditions at interfaces AC, BC, CD and CE, as follows

ð11Þ

With kr,A,n, kr,B,n, kr,C,n, kr,D,n, kr,E,n being the solutions of the boundary conditions

J 1 ðkr;A;n r 1 Þ ¼ 0

ð12Þ

J 1 ðkr;B;n r 1 Þ  ½J 1 ðkr;B;n RÞ=Y 1 ðkr;B;n RÞY 1 ðkr;B;n r 1 Þ ¼ 0

ð13Þ

N N X X ðAþn þ An Þ/A;n ðrÞ ¼ ðC þn þ C n Þ/C;n ðrÞ; ð0 6 r 6 r 1 Þ n¼0

N N X X ðBþn þ Bn Þ/B;n ðrÞ ¼ ðC þn þ C n Þ/C;n ðrÞ; ðr 1 6 r 6 RÞ n¼0

J 1 ðkr;D;n r 1 Þ  ½J 1 ðkr;D;n RÞ=Y 1 ðkr;D;n RÞY 1 ðkr;D;n r1 Þ ¼ 0

ð15Þ

J 1 ðkr;E;n r 1 Þ ¼ 0

ð16Þ

ð18bÞ

n¼0

N N X X ðEþn þ En Þ/E;n ðrÞ ¼ ðC þn ejkx;C;n lm þ C n ejkx;C;n lm Þ/C;n ðrÞ; ð0 6 r 6 r 1 Þ n¼0

n¼0

ð18cÞ N N X X ðDþn þ Dn Þ/D;n ðrÞ ¼ ðC þn ejkx;C;n lm þ C n ejkx;C;n lm Þ/C;n ðrÞ; ðr 1 6 r 6 RÞ n¼0

n¼0

ð18dÞ

J 0 ðkr;C;n r 1 Þ kr;C;n J 0 ðkr;C;n r 1 ÞY 1 ðkr;C;n RÞ  Y 0 ðkr;C;n r 1 ÞJ 1 ðkr;C;n RÞ ¼ þ jf J 1 ðkr;C;n r 1 Þ J 1 ðkr;C;n r 1 ÞY 1 ðkr;C;n RÞ  Y 1 ðkr;C;n r 1 ÞJ 1 ðkr;C;n RÞ k0 ð14Þ

ð18aÞ

n¼0

N X kx;C;n ðC þn  C n Þ/C;n ðrÞ ¼ n¼0

8 N X > > > kx;A;n ðAþn  An Þ/A;n ðrÞ; ð0 6 r 6 r 1 Þ > < n¼0

N > X > > > kx;B;n ðBþn  Bn Þ/B;n ðrÞ; ðr1 6 r 6 RÞ : n¼0

ð18eÞ

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R. Guo et al. / Applied Acoustics 127 (2017) 105–117

N X

kx;C;n ðC þn ejkx;C;n lm

n¼0

¼



8 m > < r m;P1 ¼ r m;P3 ¼ Nþ1 r 1 ; m ¼ 1; 2; . . . ; N þ 1 m r m;P2 ¼ r m;P4 ¼ r1 þ Nþ1 ðR  r 1 Þ; m ¼ 1; 2; . . . ; N þ 1 > :r m ¼ R; m ¼ 1; 2; . . . ; N þ 1

C n ejkx;C;n lm Þ/C;n ðrÞ

8 N X > > > k ðEþ  En Þ/E;n ðrÞ; ð0 6 r 6 r 1 Þ > > < n¼0 x;E;n n

m;U

ð18fÞ

N > X > > > kx;D;n ðDþn  Dn Þ/D;n ðrÞ; ðr1 6 r 6 RÞ > : n¼0

The direct integral procedure is to multiply both sides of Eq. (18) by rdr

Z N X ðAþn þ An Þ

r m;P1

Z N X ðC þn þ C n Þ

/A;n ðrÞrdr ¼

0

n¼0

r m;P1

/B;n ðrÞrdr ¼

0

n¼0

Z N X ðC þn þ C n Þ

ð19aÞ r m;P2

/C;n ðrÞrdr;

0

n¼0

ðr1 6 r m;P2 6 RÞ Z N X ðEþn þ En Þ

r m;P3

/E;n ðrÞrdr ¼

0

n¼0

(

Bþn ¼ Bn e2jkx;B;n li

ð19bÞ

N X ðC þn ejkx;C;n lm þ C n ejkx;C;n lm Þ

Z

r m;P3

(1) The incoming wave is planar, and the magnitudes satisfy A+0 = 1; A+1,2,. . .,N = 0; (2) The outlet is an anechoic termination, which means E 0,1,. . .,N = 0; (3) The high-order acoustic wave propagation in the outlet decays seriously, only zero-order acoustic wave remains, which means E+1,2,. . .,N = 0. Assuming that

X ¼ ½A0 . . . AN ; B0 . . . BN ; C þ0 . . . C þN ; C 0 . . . C N ; D0 . . . DN ; Eþ0 . . . EþN 

n¼0



/C;n ðrÞrdr; ð0 6 r m;P3 6 r 1 Þ ð19cÞ

r m;P4

N X

/D;n ðrÞrdr ¼

r1

n¼0

ðC þn ejkx;C;n lm þ C n ejkx;C;n lm Þ

n¼0

Z



r m;P4

/C;n ðrÞrdr; ðr 1 6 r m;P4 6 RÞ

r1

ð19dÞ N X

kx;C;n ðC þn  C n Þ

n¼0

Z

r m;U

/C;n ðrÞrdr

0

8 N X > Rr > > kx;A;n ðAþn  An Þ 0 m;U /A;n ðrÞrdr; ð0 6 rm;U 6 r 1 Þ > > > > > n¼0 > > > > n¼0 > > > > > N X > > þ  R r m;U > > : þ kx;B;n ðBn  Bn Þ r1 /B;n ðrÞrdr; ðr 1 6 r m;U 6 RÞ

ð19eÞ

kx;C;n ðC þn ejkx;C;n lm  C n ejkx;C;n lm Þ

n¼0

Z

rm;U

An+

En

Substitute specific values for and in Eq. (19) according to assumptions (1) and (2), a series of constant values can be calculated. Place all the constant values on the right-hand side of Eq. (19) and combine them into a column vector named as b. Substitute Bn and Dn for Bn+ and Dn+ according to Eq. (21), and 6(N + 1) unknowns shown in the vector X are determined. Place all the segments about these unknowns on the left-hand side of Eq. (19). Then define a [6(N + 1)  6(N + 1)] matrix named as a containing the coefficient of each unknown in the column vector X. Eq. (19) can be written as a linear equation aX = b. Then the solution can be obtained as X = a1b. According to the assumption (2), E 0,1,. . .,N = 0. Then substitute specific values for An+ and En+ in Eq. (23) and then the TL is obtained

 P  þ  þ  N Aþ ejkx;A;n li  P A  A0    n¼0 n     ¼ 20log ¼ 20log TL ¼ 20log10  þ   10 PN 10  þ   PE E0 Eþ ejkx;E;n lo  n¼0 n

¼ 20log10 jEþ0 j

ð23Þ

To solve the above equations, the infinite series of amplitudes need to be truncated to a suitable number. For the frequency range and geometry discussed in this study, the result is sufficiently accurate when N > 5 so that N = 6 is used in the remainder of the study.

n¼0

N X

T

ð22Þ

0

Z N X ðDþn þ Dn Þ

ð21Þ

Dþn ¼ Dn e2jkx;D;n lo

0

n¼0

r m;P2

Nþ1

The simplification of Eq. (19) can refer to Appendix B. Using the boundary conditions in section B and D at x = -li and x = lm + lo, the relations between the positive and negative modal amplitudes yield

Three assumptions are made to determine the TL of a SCPR with extended inlet/outlet:

/C;n ðrÞrdr;

ð0 6 r m;P1 6 r 1 Þ Z N X ðBþn þ Bn Þ

ð20Þ

/C;n ðrÞrdr

0

8 N X > Rr > > kx;E;n ðEþn  En Þ 0 m;U /E;n ðrÞrdr; ð0 6 r m;U 6 r1 Þ > > > > > n¼0 > > > > n¼0 > > > > > N > R rm;U > X þ  > > : þ kx;D;n ðDn  Dn Þ r1 /D;n ðrÞrdr; ðr1 6 rm;U 6 RÞ

2.2. The effect of non-planar wave propagation on the acoustic performance of SCPR

ð19fÞ

n¼0

In order to have enough integral equations to calculate all the nth modal amplitudes, each region is divided into N + 1 parts in the radial direction. The outer radius of each part is used as the upper limit of integral in Eq. (19). So each equation in Eq. (19) will be calculated N + 1 times by changing the upper limit of integral, with

For the SCPR with extended inlet/outlet, the structure parameters are chosen from Fang’s study [7]. Fig. 3 compares the TL calculated by 2D method using Eq. (22) with the test data in Fang’s study. Results of FEM and 1D TMM are also given for comparison. The acoustic FEM simulation is implemented through a commercial simulation software LMS.Virtual.lab. The calculation of 1D TMM is based on the formulas provided in the previous work [4]. Apparently, 1D TMM cannot accurately predict the TL when the frequency exceeds the cutoff value of planar wave, which is 1226 Hz, whereas results of the other two methods show good consistency with the test data in entire frequency range. Besides, the 2D method is much more timesaving than FEM. Thus, the 2D

R. Guo et al. / Applied Acoustics 127 (2017) 105–117

analytical method has the advantages of high accuracy and efficiency in calculating the TLs of SCPRs. The configuration of SCPR is acoustically similar to that of expansion muffler, which will have non-planar sound waves at the area discontinuities under the cut-off frequency. In fact, the length of the chamber determines how significant that the nonplanar sound waves which cannot be considered in 1D TMM will affect the acoustic performance of the expansion muffler. The 2D approach is able to predict the multi-dimensional sound propagation at the area discontinuities of SCPR. Therefore, in this section, the effect of length on the acoustic performance of SCPR will be studied through 2D approach. Defining the structure parameters of a SCPR as: R = 45 mm, li = lo = 0 mm, r1 = 22.5 mm. Comparisons of TLs between two structures with different lm/R are shown in Fig. 4. The onedimensional repeating dome behavior prevails in the SCPR with a long-length chamber, while only one resonance peak appears well

109

below the cut-off frequency when the length of chamber becomes short. The present study employs finite element method through a commercial software LMS.Virtual.lab to investigate this phenomenon caused by multi-dimensional waves. The pressure fields inside the two structures have been determined for several frequencies. Fig. 5 shows the internal pressure contours for the lm/ R = 6 chamber at three frequencies of TL troughs and three frequencies of TL peaks. These figures illustrate that the sound waves are planar throughout the chamber at very low frequencies. As frequency increases, some non-planar waves begin to appear at the area discontinuities. The non-planar mode decays exponentially with distance and only planar behavior is observed away from the area transitions. With the frequency becoming closer to the cut-off value, the multi-dimensional effects spread throughout the chamber. Contours for the lm/R = 1 chamber, shown in Fig. 6, illustrate the multi-dimensional propagation do not decay sufficiently can be dominant even at low frequencies, due to the short

Fig. 3. Transmission loss of the SCPR (The hole diameter hd = 4.98 mm; the perforation thickness ht = 0.9 mm; the perforation rate r = 8.4%; r1 = 24.5 mm; R = 82.2 mm; li = lo = 0 mm and lm = 257.2 mm).

Fig. 4. Transmission loss of two SCPR structures with different lm/R (r = 10%).

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(a) 625Hz

(d) 1000Hz

(b) 1950Hz

(e) 2300Hz

(c) 2550Hz

(f) 2700Hz

Fig. 5. The pressure (dB) contours for the lm/R = 6 chamber, (a–c) TL troughs, (d–f) TL peaks.

(a) 1000 Hz before resonance

(b) 1850 Hz resonance

(c) 2800 Hz after resonance

Fig. 6. The pressure (dB) contours for the lm/R = 1 chamber.

Fig. 7. TL predictions for SCPRs with short-length chambers (r = 10%).

length of the chamber. This radial mode causes the resonance even at almost half that of the cut-off frequency. Analytical TL curves for a number of short-length SCPRs with different chamber lengths are shown in Fig. 7. The dome is gradu-

ally replaced by a resonance peak as the chamber length decreases. Structures with an lm/R ratio of less than 1.6 will have no complete dome, while the resonators can be seen as acoustically short type. In fact, as for a SCPR, the length of chamber is not the only

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R. Guo et al. / Applied Acoustics 127 (2017) 105–117

parameter that affects the multi-dimensional wave propagation. Fig. 8 shows that structures with lower perforation rate are easier to be dominant by the non-planar behaviors. Thus, it’s barely possible to define an accurate boundary to distinguish between SCPRs with acoustically short and long chamber. It is well known that, due to the quarterwave tube effect, the inlet and outlet extensions can significantly improve the acoustic performance of a SCPR at mid frequencies as shown in Fig. 9. However, when it comes to an acoustically short chamber, the effects could be totally different. It can be found in Fig. 10, increasing the inlet/outlet extension length will only lower the resonance frequency and bring down the amplitude of TL for an acoustically short SCPR. Whereas the same effect can be achieved by reducing perforation rate to an appropriate value. Since that the length of each chamber in MCPRs applied to the intake system of turbocharged engine is normally around 30 mm (the chamber radius is normally around 45 mm and the perforation rate around 15%), which is acoustically short enough. There is no need to add

inlet/outlet extensions in the MCPR for consideration of acoustical improvement. 3. Two-dimensional transfer matrix method for sound attenuation of a MCPR 3.1. Acoustic modeling of a MCPR In 2D analytical method, the waves in the inlet/outlet are assumed as planar, hence an acoustic transfer matrix can be acquired



PA

qcU A



 ¼

T 11

T 12

T 21

T 22



PE

qcU E

 ð24Þ

In order to determine the four-pole parameters of the transfer matrix, Eq. (19) should be solved twice under two different outlet conditions, while the incoming wave of the inlet is planar with A+0 = 1; A+1,2,. . .,N = 0

Fig. 8. TL predictions for SCPRs with short-length chambers (lm/R = 1.6).

Fig. 9. TLs for SCPRs with acoustically long chambers (r = 10%).

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R. Guo et al. / Applied Acoustics 127 (2017) 105–117

Fig. 10. TLs for SCPRs with acoustically short chambers.

Fig. 11. MCPRs with two chambers, (a) the configuration and (b) the two tested prototypes.

Table 1 Structure parameters of the MCPRs. Parameters (unit)

d (mm)

D0 (mm)

li1, li2 (mm)

lm1 (mm)

lm2 (mm)

lo1, lo2 (mm)

ht (mm)

hd (mm)

r1 (%)

r2 (%)

Prototype 1 Prototype 2

45 45

90 90

5 5

40 40

20 20

5 5

2 2

3 3

7 15

7 15

R. Guo et al. / Applied Acoustics 127 (2017) 105–117

113

+ (1) Total reflection end with E 0,1..N = E0,1,. . .,N;  (2) Anechoic end with E0,1,. . .,N = 0;

With the first outlet condition, it can be determined that

T 11 ¼

T 21 ¼

pA 1 þ A0 þ ¼ j pE 2Eþ0 E0;1;...N ¼E0;1;...N

qcU A pE

¼

ð25Þ

1  A0 þ j 2Eþ0 E0;1;...N ¼E0;1;...N

ð26Þ

With the second outlet condition, it can be determined that

T 12 ¼

pA 1 þ A0 ¼ jE ¼0  T 11 0;1;...N qcU E Eþ0

ð27Þ

T 22 ¼

qcU A 1  A0 ¼ jE ¼0  T 21 0;1;...N qcU E Eþ0

ð28Þ

Considering the extended inlet/outlet, the total transfer matrix can be written as

 ½T ¼

cosðk0 liÞ

j sinðk0 liÞ



j sinðk0 liÞ cosðk0 liÞ

T 11

T 12

T 21

T 22



cosðk0 loÞ

j sinðk0 loÞ



j sinðk0 loÞ cosðk0 loÞ ð29Þ

Then the integrated transfer matrix of a MCPR can be denoted as [4]

 ½T T  ¼ ½T N ½T i ½T 1  ¼

     T i;11 T i;12 T 1;11 T 1;12 T N;11 T N;12  ... T N;21 T N;22 T i;21 T i;22 T 1;21 T 1;22 ð30Þ

The TL is calculated as

TLM ¼ 20log10

   1  T T;11 þ T T;12 þ T T;21 þ T T;22  2

Fig. 12. Sketch of the TL test, (a) the test scheme and (b) the test set-up: 1. data acquisition system; 2. power amplifier; 3. loudspeaker; 4. impedance tube; 5. resonator; 6. microphone.

ð31Þ 4. Optimal design of the MCPR

3.2. Validation of the 2D TMM The acoustic modeling of a MCPR based on 1D approach that is inaccurate at high frequencies has already been studied. In this section, in order to evaluate the applicability and accuracy of the TMM derived from the 2D approach validated above. Two MCPRs that respectively consists of two chambers and has different frequencies of sound attenuation are discussed. Profiles of the resonators are shown in Fig. 11 and the corresponding structure parameters are listed in Table 1. The two-load technique [4] is applied in the TL measurement by means of an impedance tube, as shown in Fig. 12. Based on the requirements for the outlet, the first condition of the outlet is an open end, and the other condition is an end which is filled up with insulation cotton [4]. Fig. 13 compares the test results with TLs calculated by 2D and 1D TMM. The TL curve of 1D method has the same trend with that of the test in a frequency range bellows the cutoff value of 2424 Hz (Prototype 1). But the resonance peaks still have minor deviations. Furthermore, obvious mismatches will appear as the sound attenuation frequency moves to a higher range (Prototype 2). In comparison, TLs calculated by 2D TMM always agree well the test results. The minor discrepancies may result from the inaccuracy of perforation impedance and geometrical imperfections in the experimental set-up. Nevertheless, the 2D TMM is validated to be able to predict the acoustic performances of MCPRs at both low and high frequencies.

In the last section, a 2D TMM based on the 2D approach is developed to predict the acoustic performances of MCPRs which can attenuate broadband noise. Furthermore, optimal design to acquire better acoustic performances is an essential segment for the resonator design in engineering applications, since specific structural parameters are obtained with optimization based on 2D approach. For the optimization of resonators, Yedeg [15] developed the material distribution method for topology optimization in order to obtain high transmission loss for a broad range of frequencies. Guo [16] used least square method to obtain a desired TL curve. Chiu [17] used GA to optimize multi-chamber mufflers with plug-inlet tube. Besides GA, simulated annealing method [18] is also widely used in muffler optimizations. Chang [19,20] applied simulated annealing method to optimize multi-chamber mufflers conjugated with open-ended perforated intruding inlet tubes and multi-chamber mufflers with reverse-flow ducts. However, all the above optimizations are based on 1D approach, which is not accurate at high frequencies. Xiang [21] used a 2D analytical method in combination with GA to optimize an expansion muffler with three chambers and got satisfying results. Hence, it is theoretically feasible to optimize the shape of MCPRs based on 2D TMM introduced in this study. And in this section, the original TL of a MCPR is calculated by the 2D TMM, and a GA with its simplicity and universal applicability is chosen for the optimization of a MCPR.

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4.1. Objective of optimization In engineering application, a MCPR with at least two chambers is usually used to attenuate wideband noise. It is necessary to optimize the structure parameters to acquire better acoustic performances. The acoustic performance in a wide frequency range from 1500 Hz to 3500 Hz is considered here. Xiang [21] introduced the average TL as the optimization objective OBJ. However, this objective equalized the importance of TLs within entire frequency range, which usually cannot be applied to situations when varying TL amplitudes at different frequency ranges are required. Therefore, Guo [16] defined a TL target curve considering the varying degrees of noise problems at different frequency ranges. Then he used nonlinear least square method in combination with 1D TMM to optimize a MCPR. Although the optimized results can be exactly close to the target curve, the principle of the nonlinear least square method itself is to minimize the difference between the calculation and target rather than to make the former better. Therefore, apart from considering the nonuniform TL target, it is also significant to improve the TL amplitudes as much as possible to acquire the optimal results.

Based on the prominent noise problem distributed in 2000– 3000 Hz in previous study [4], a non-uniformly distributed TL target curve with a specific amplitude emphasis of 55 dB in 2000– 3000 Hz based on the characteristics of intake noise tested in semi-anechoic room is determined as shown in Fig. 14. Obviously, the acoustic performance is more desirable when the TL is higher than the target value. On the other hand, occasions that the TL bellows the curve should be punished. The optimization procedure can then be transformed into a minimization problem. Consider the balance between the two aspects to make the TL distribution relatively uniform, the fitness value at each frequency obj(f) is expressed as

( objðf Þ ¼

2

½tarðf Þ  TLðf Þ ; TLðf Þ < tarðf Þ ½TLðf Þ  tarðf Þ

1=2

; TLðf Þ > tarðf Þ

ð32Þ

where tar(f) and TL(f) are respectively the targeted (Fig. 14) and calculated TL values at frequency f. The TL(f) is calculated through the 2D TMM developed above, where the geometrical design parameters are used. The obj(f) which is a reward function is calculated based on the difference between tar(f) and TL(f) at each frequency.

Fig. 13. TL curves of the MCPRs, (a) the TL of Prototype 1 and (b) the TL of Prototype 2.

R. Guo et al. / Applied Acoustics 127 (2017) 105–117

Obj(f) gives the negative reward when the actual TL is less than targeted TL. The reason why the exponent is 2 is to avoid occasions that the TL values are too small in the entire frequency range. The less the actual TL is, the more the punishment is. Obj(f) gives the positive reward when the actual TL is greater than targeted TL. The reason why the exponent is 1/2 is to avoid occasions that the TL values are too small in other frequency range. The greater the actual TL is, the less the reward is. Then the total objective is the sum of the objective at each frequency

Z

3500

OBJ ¼

objðf Þdf

ð33Þ

1500

The ultimate objective of optimal design is to make function OBJ up to the maximum.

of function using genetic algorithm. Gaoptimget is used to obtain values of genetic algorithm options structure. Gaoptimset is used to create the genetic algorithm options structure. The GA optimization is carried out by calling the key functions and set relevant parameters. The GA optimization flow chart is shown in Fig. 15. In the optimization, considering that there are 6 design variables, the population size is 50 including 2 elite individuals. The maximum number of generations is defined as 150 to guarantee that the best fitness will converge to a constant value. In order to reduce the chance that the optimization trapping in local optima, the crossover and mutation ratio are set as 0.8 and 0.2. Given that, MCPR with more chambers is more likely to have higher amplitude and wider bandwidth of sound attenuation. Because the frequency bandwidth in Fig. 14 reaches 2000 Hz and there is a very high amplitude requirement especially in

4.2. Optimization procedure For a MCPR, the diameter of inner tube is often determined by the tube, to which the resonator is connected. The chamber diameter is usually limited to the radial space constraint. The thickness of holes is subjected to the thickness of inner tube. Hence, here the above three parameters can be set as constant values which are the same as that of the MCPRs in last Section: r1 = 22.5 mm, R = 45 mm and ht = 2 mm. Since the lengths of inlet and outlet extensions have similar effects to that of the perforation rate for acoustically short chambers, here these lengths are defined as li = 0 mm and lo = 0 mm. The diameters of perforation holes in different chambers are identical for manufacture convenience. Comparing to the non-linear square method, which is advantageous in function approximation, GA is a powerful stochastic search method for maximization and involves the use of optimal search strategies based on the notion of natural selection. Hence, GA is very appropriate in this study. The genetic algorithm repeatedly modifies a population of individual solutions. At each step, the genetic algorithm selects individuals at random from the current population to be parents and uses them to produces the children for the next generation. Over successive generations, the population ‘‘evolves” toward an optimal solution. The genetic algorithm is carried out by GA toolbox in the commercial software MATLAB. There are three key functions in GA toolbox, named as ga, gaoptimget and gaoptimset. Ga function is used to find the minimum

115

Fig. 15. The flow chart of GA optimization on MCPRs.

Fig. 14. Transmission loss target curve.

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Fig. 16. Configuration of the three-chamber perforated resonator.

Table 2 Design variables, ranges for optimization and optimal results.

n¼

Variable (unit)

Interval

Initial value

Optimal value

lm1, lm2 (mm) hd (mm) r1, r2, r3 (%)

[10, 40] [1, 4] [5, 30]

25, 35 3 5, 15, 25

35, 33 1.1 29, 9.5, 17

2000–3000 Hz, a MCPR with two chambers can barely meet the TL target after several times of trials. Therefore, a MCPR with three chambers is chosen as optimization object in this study as shown in Fig. 16. The design variables are framed in the figure. The parameters r1, R and ht are set as constant values based on previous discussions. The diameters of perforation holes in different chambers are identical for manufacture convenience. Hence, the design variables include the lengths of perforation area, the diameter of the perforations and porosity. Besides, only two of the three chamber lengths are design variables since the installation space is constrained in the engine compartment. The number of perforated holes is much more useful in engineering applications of resonators instead of porosity and it can be calculated by the following expression derived from Eq. (C.1) (see Appendix C)

4lmd1 r 2

hd

ð34Þ

where n is the number of perforated holes; hd is the diameter of perforated holes; lm is the length of perforation area; d1 is the diameter of perforation area, which equals to the diameter of inlet tube; r is porosity of perforation area. Constrained by the installation space of the resonator in the engine compartment, the total chamber length is determined as 3 X lmi ¼ 100 ðmmÞ

ð35Þ

i¼1

where i indicates the ith chamber of the resonator. The intervals and optimal values of the design variables to be optimized are listed in Table 2. Comparison of the target, original and optimal TL curves is shown in Fig. 17. It can be seen apparently that, under a certain space constraint, the optimized resonator’s TL can meet the target much better than the original TL in a wide frequency range. The objective defined above is proven to be reasonable and the optimization based on 2D TMM turns out to be effective and efficient.

Fig. 17. The target and optimized TL curves.

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R. Guo et al. / Applied Acoustics 127 (2017) 105–117

5. Conclusions

Appendix B

This paper presents a 2D TMM that can be used to accurately predict and optimize the acoustic performances of MCPRs within a broad frequency range. The main contributions are summarized as:

The integrals of Eqs. 19(a)–19(f) can be analytically determined in light of [12] For the zeroth-order Bessel function of the first kind,

1. A timesaving 2D approach is derived to calculate the TL of a single-chamber perforated reactive resonator, and the result is in good accordance with that of the test and FEM. 2. Through analyzing the effect of non-planar wave propagation on the acoustic performance of SCPR, it is concluded that, there is resonance behavior of acoustically short chambers substantially below the cut-off frequency, which may be desirable for practical designs. For an acoustically short chamber, increasing the lengths of inlet and outlet extensions has similar effects to that of reducing the perforation rate on the acoustic performance. 3. The transfer matrix of a perforated resonator is acquired by solving the acoustical continuity functions under two outlet boundary conditions. Multiplying the transfer matrixes of connected chambers, a 2D TMM is developed to calculate the TLs of MCPRs. Two MCPR prototypes that respectively attenuate low and high frequency noises are tested to validate the accuracy of the 2D approach. It turns out that TLs calculated by 2D TMM show better agreement with tests than that of 1D approach. 4. Considering the engineering application of the 2D approach, a new optimization procedure of MCPRs is presented. A targeted TL curve is determined based on the acoustic requirement of the resonator for turbocharged engines. Defining an appropriate objective function, a GA in combination with 2D TMM is applied to optimize the structure parameters of a MCPR with three chambers. The result shows that the resonator can meet the acoustic requirement under space constraint. It is concluded that the 2D TMM and optimization process are very effective and efficient for the analysis and design of MCPRs.

Acknowledgements This research is supported by the Fundamental Research Funds for the Central Universities (2016). Authors gratefully acknowledge the resources provided by the Clean Energy Automotive Engineering Center of Tongji University. Appendix A The perforation impedance model applied in this study follows Ji’s expression as [14]

f ¼ ð1=rÞðR0 þ jX 0 Þ

ðA:1Þ

With

(

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R0 ¼ ð1 þ ht=dh Þ 8k0 v =qc X 0 ¼ k0 ðht þ adh Þ

ðA:2Þ

where v is the kinetic viscosity of air; and a is the end correction coefficient of holes, expressed as

8 h 0:5 i 0:5 > < 0:85 1  2:34 rp 6 0:25 ; 0 < rp h a¼ r 0:5 i r 0:5 > : 0:668 1  1:9 p 6 0:5 ; 0:25 < p

ðA:3Þ

Z

r0

(

J 0 ðkrÞrdr ¼

0

Z

r 20 =2

k¼0

J 1 ðkr 0 Þr0 =k k – 0

ðB:1Þ

For the zeroth-order Bessel function of the second kind, r0

0

Y 0 ðkrÞrdr ¼ r0 Y 1 ðkr 0 Þ=k þ 2=ðpk2 Þ

ðB:2Þ

Appendix C Porosity

r¼

r can be calculated by the following expression

2 2 n  p  hd nðhdÞ 2 ¼ lm  p  d1 4lmd1

ðC:1Þ

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