Enhancing the low frequency sound absorption of a perforated panel by parallel-arranged extended tubes

Applied Acoustics 102 (2016) 126–132

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Enhancing the low frequency sound absorption of a perforated panel by parallel-arranged extended tubes Dengke Li, Daoqing Chang, Bilong Liu ⇑ Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, 100190 Beijing, PR China

a r t i c l e

i n f o

Article history: Received 21 August 2015 Received in revised form 25 September 2015 Accepted 1 October 2015 Available online 17 October 2015 Keywords: Micro-perforated panel Extended tubes Sound absorption

a b s t r a c t This paper is concerned with the use of a perforated panel with extended tubes (PPET) to improve the sound absorption confined to low frequencies. In comparison with a micro-perforated panel (MPP), the sound absorption can be significantly improved by using the PPET at the expense of the bandwidth of the sound absorption. A particular configuration combining four parallel-arranged PPETs with different cavities is introduced to achieve a wider bandwidth of the sound absorption at low frequencies. The analysis is extended to the combination of three parallel-arranged PPETs and a MPP to further increase the bandwidth of the sound absorption. A theoretical model is described to predict the sound absorption coefficient and the simulated annealing method is introduced to the proposed absorbers, allowing optimization of the overall performance. The theory with experimental validations demonstrates that the proposed configurations offer a potential improvement of more than one octave in the bandwidth of the sound absorption at low frequencies. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Combination of perforated panel and porous material is widely used in noise control engineering [1]. The sound absorption of a perforated panel can be significantly improved by the acoustic resistance of porous material. Maa [2] proposed the MPP with sub-millimeter perforation, which could provide sufficient acoustic resistance and replace porous material in circumstances requiring fireproof and environmental quality. A formula for computing the acoustical properties of MPP was also derived in Maa’s paper. Although the MPP has a large bandwidth of the sound absorption, it has not quite lived up to its promises in the low frequency range in some engineering applications. Maa further proposed a double-deck MPP structure to achieve large sound absorption at lower frequencies [3]. Since this study, substantial research has been paid to the improvement of sound absorption at low frequencies through the MPP. Lee et al. [4] utilized the flexible vibration of a thin MPP to broaden the bandwidth of the sound absorption. Chang et al. [5] attached a piezoelectric material with a shunt circuit to the MPP to improve the low frequency sound absorbing performance. Wang et al. [6,7] investigated the coupling effect of the parallel-arranged MPP with different air cavities, and concluded that multi-resonant ⇑ Corresponding author. E-mail address: [email protected] (B. Liu). http://dx.doi.org/10.1016/j.apacoust.2015.10.001 0003-682X/Ó 2015 Elsevier Ltd. All rights reserved.

systems have potential to improve the bandwidth of the sound absorption. Alternatively, the absorption peak frequency can be tuned to lower frequency by increasing the thickness of the MPP, whilst the penalty of the weight and cost of the sound absorbing structure is more obvious. Lu et al. [8] designed a low-weight and low-cost device by attaching flexible tube bundles (100 mm) to the perforated/micro-perforated panel to achieve low frequency sound absorption. Yahya and Harjana [9] presented an idea using arrays of constrained short tubes to enhance the sound absorption of the MPP. They carried out experimental investigation into the influence of tube number, tube length and cavity depth on the sound absorption coefficient of the perforated panels. Though the idea of the perforated panel with extended tubes (PPET) has been reported previously, a theoretical model to the prediction and optimization of the sound absorption has not been found in the literature. This motivated us to carry out a theoretical investigation into the performance of the PPET. Based on this, optimized parameters are available to be obtained to achieve optimal performance for practical application. In this paper, a PPET that is designed for low-frequency (120–250 Hz) sound absorption in a constrained space (100 mm in depth) is investigated. The paper starts with a theoretical analysis of two particular designs of the PPET absorbers (four parallel-arranged PPETs, and three parallel-arranged PPETs combined with a MPP) in Section 2, and

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then followed by an optimal design in Section 3 and test validation in Section 4. Finally, conclusions are drawn in Section 5.

to Crandall [10], the equation of the aerial motion inside the tube can be written as:

jxquz 

2. Theoretical analysis 2.1. Acoustic impedance of the PPET The parallel-arranged PPETs consist of a perforated panel with a lattice of extended tubes in the back cavity, as shown in Fig. 1(a). The back cavity was partitioned into several sub-cavities with clapboards, and the cylindrical shell and clapboards are regarded as acoustic rigid. The cavity depth can be altered by changing the position of the rigid backing in the cavity. It is noted that a single PPET is not able to cover the sound absorption over a wider frequency range. Thus the paper is focused on the performance of parallel-arranged PPETs. Two particular configurations as illustrated in Fig. 1(b) and (c) are of interests in this analysis. One configuration is the four parallel-arranged PPETs with different tube parameters (tube lengths, diameters and perforations), and another one is the three parallel-arranged PPETs combined with a MPP. In Fig. 1, P i and Pr denote the incident and reflected sound waves respectively, D is the cavity depth, t p is the thickness of the clapboard, / is the ratio of each PPET or MPP surface area over that of the total area. In the following discussion it is assumed as /1 ¼ /2 ¼ /3 ¼ /4 ¼ 1=4. The effective diameter of the absorber is assumed as dm ¼ 100 mm, the effective cross-section area is 2

therefore given by Sm ¼ pðdm =2Þ . Fig. 2 illustrates one unit of the four parallel-arranged PPETs in Fig. 1(b). Sm =4 is the effective cross-section area of this unit, t and d0 are the length and inner diameter of the tube, respectively. t 0 and t 1 are the thickness of the perforated panel and the extended tubes. Consider acoustic waves propagating in the tubes, according


 @uz @P ¼ r @z r @r @r

g @

ð1Þ

where P is the sound pressure, z is the axis of the cylinder, x is the angular velocity, r is the radius vector, uz is the particle velocity, q and g are the density and viscous coefficient of the air. Assuming that the tube length is much smaller than the sound wavelength, Eq. (1) becomes

jxquz 


 @uz DP r ¼ t r @r @r

g @

ð2Þ

where DP is the difference of sound pressure at two ends of the tube. From Eq. (2), the particle velocity can be derived as:

uz ¼ 

DP 2

gtk



1

J 0 ðkrÞ J 0 ðkr 0 Þ



ð3Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k ¼ jxq=g, r 0 is the inner radius of the extended tube, J0
is the Bessel function of the zero order. The average velocity u across the tube cross-section is


¼ u

Z

Z

r0

r0

uz rdr 0

rdr

ð4Þ

0

Submitting Eq. (3) into Eq. (4), we can get the specific acoustic impedance of the tube.

" pffiffiffiffiffiffi #1 DP 2 J 1 ðx jÞ pffiffiffiffiffiffi ¼ jxq0 t 1  pffiffiffiffiffiffi Z¼
u x j J 0 ðx jÞ

ð5Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where x ¼ d0 =2 2pf q=g is the ratio between the inner radius and the viscous boundary layer thickness inside the tube, f is the

(a)

(b)

(c)

Fig. 1. (a) A three-dimensional configuration of the four parallel-arranged PPETs. (b) Schematic of the four parallel-arranged PPETs. (c) Schematic of the three parallelarranged PPETs combined with a MPP.

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D

P

r1

Sa

t1

d0

r0

r1 t0

U

U2

U1

uz

dm 2

r0

U t

r

z

r1 tp 2 tp 2

Perforated panel

o

Sm 4

tp 2

t r0

tp 2

o

Clapboard

dm 2

(b)

(a)

Fig. 2. Schematics of one unit of the four parallel-arranged PPETs. (a) Side elevation; and (b) front elevation.

pffiffiffiffiffiffiffi frequency, j ¼ 1 is the imaginary unit. For Eq. (5), Crandall gave the limiting cases of the impedance when k < 1 and k > 10. Later Maa [1] found an approximate formula which could satisfy the case when 1 < k < 10 and it yields as:


qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ x2 =32 þ jxqt 1 þ 1= 9 þ x2 =2 Z ¼ 32gt=d0

ð6Þ

The end corrections of the resistance and mass reactance should be added to Eq. (6) according to Ingard [11], the additional resistance caused by the air flow friction on the surface panel is pffiffiffiffiffiffiffiffiffiffiffiffiffi Rs ¼ 2xqg=2. As far as the PPET is concerned, the additional resistance is produced by one side surface panel, so the corrected resistance is Rs =2. While the mass reactance due to the sound radiation at both ends is 0:85d0 . To derive the specific acoustic impedance of the PPET, the specific acoustic impedance of a single tube (plus end corrections) should be divided by the perforation ratio of the PPET. Then the relative (to the characteristic impedance qc in air) acoustic impedance of PPET is

Zp ¼

rp ¼

Z

rqc

¼ rp þ jxmp

32gt 2 cd0

rq

ð7Þ


1=2 pffiffiffi ! x2 2xd0 1þ þ 32 64t


xt x2 xmp ¼ 1þ 9þ rc 2

1=2

d0 þ 0:85 t

P P Sm ¼ uqc U qc 4

ð12Þ

By substituting Eqs. (10) and (11) into Eq. (12), the normal acoustic impedance of the back cavity can be written as:

ZD ¼

Sm =4 jSa tanðxðD  t 0 Þ=cÞ þ jðSa  NS1 Þ tanðxt 0 =cÞ

¼ j=ðd tanðxðD  t 0 Þ=cÞ þ ðd  r0 Þ tanðxt 0 =cÞÞ

ð13Þ

where d ¼ Sa =ðSm =4Þ is the effective cross-section area expansion ratio from the back cavity to the PPET, and r0 ¼ NS1 =ðSm =4Þ is the ratio of the outer cross-section area of extended tubes to the PPET. When t0 ¼ 0, the extended tubes have no extensions into the cavity, the cavity reactance is zD ¼ j cotðxD=cÞ=d. When t 0 ¼ D, the extended tubes reach to the rigid backing, then the cavity reactance is zD ¼ j cotðxD=cÞ=ðd  r0 Þ. Then the surface acoustic impedance of the PPET is

j ðd tanðxðD  t0 Þ=cÞ þ ðd  r0 Þ tanðxt 0 =cÞÞ

Z p ¼ r p þ jxmp 

ð9Þ

2.2. Acoustic impedance of the MPP

ð14Þ

!

ð10Þ

The admittance is defined as Y ¼ U=P, then the following relationship exists between the admittances

U1 þ U2 Y¼ ¼ Y1 þ Y2 P

ZD ¼

ð8Þ

where r ¼ NS0 =ðSm =4Þ is perforation ratio of the PPET, N denotes the number of the extended tubes, S0 ¼ pr20 is the inner crosssection area of the extended tube, c is the sound speed in air, qc is the characteristic impedance in air. Assuming that the acoustic waves have two possible propagations in the back cavity (Fig. 2(a)), the following relationship can be found in the volume velocities:

U ¼ U1 þ U2

S1 ¼ pr 21 is the outer cross-section area of the extended tube, t0 ¼ t  t0 is the length of the tube extension into the back cavity. The relative acoustic reactance of the cavity is defined as:

ð11Þ

where Y 1 ¼ jSa tanðxðDt0 Þ=cÞ=qc and Y 2 ¼ jðSa NS1 Þtanðxt 0 =cÞ=qc, and Sa is the effective cross-section area of the back cavity,

According to Maa [2], the surface acoustic impedance of the MPP is

Z mpp ¼




Z

rmpp qc

¼ r mpp þ jxmmpp  j cot 0

rmpp

x2 @ 1 þ mpp ¼ 2 32 rmpp qcdmpp 32gtmpp

0

!1=2

x2 xt xmmpp ¼ mpp @1 þ 9 þ mpp rmpp c 2 where xmpp ¼

dmpp 2



c

1 pffiffiffi 2xmpp dmpp A þ 32t mpp

!1=2

qffiffiffiffiffiffiffiffi 2pf q g . t mpp , dmpp ,

xDmpp

1 dmpp A þ 0:85 t mpp

ð15Þ

ð16Þ

ð17Þ

rmpp and Dmpp are the thickness,

diameter, perforation ratio and cavity depth of the MPP.

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2.3. Sound absorption of the parallel-arranged PPET/MPP In the case of the four parallel-arranged PPETs (Fig. 1(b)), the surface acoustic impedance can be obtained as:


Z¼

/1 / / / þ 2 þ 3 þ 4 Z p1 Z p2 Z p3 Z p4

1 ð18Þ

where Z p1 , Z p2 , Z p3 and Z p4 are the surface impedance of each PPET. In the case of the three parallel-arranged PPETs combined with a MPP (Fig. 1(c)), the surface acoustic impedance of the compounded absorber can be written as:


Z¼

/1 / / / þ 2 þ 3 þ 4 Z p1 Z p2 Z p3 Z mpp

1 ð19Þ

where Z p1 , Z p2 , Z p3 and Z mpp are the surface acoustic impedance of the three parallel-arranged PPETs and MPP, respectively. Then the normal incidence sound absorption coefficient is given as:

a¼

4RealðZÞ 2

ð1 þ RealðZÞÞ þ ðImagðZÞÞ2

ð20Þ

3. Optimization by simulated annealing 3.1. Simulated annealing Simulated annealing algorithm is an optimization method that mimics the metal annealing process [12,13]. It uses a temperature parameter to control the search of optimal value and this temperature parameter typically starts off high and is slowly ‘‘cooled” or ‘‘lowered” in the every iteration process. The simulated annealing algorithm is widely used in the global optimization problems, such as the optimization of multiple-layer sound absorber [14], multichamber muffler design [15] and parameter inverse problem [16]. The flow diagram of ‘‘simulated annealing” optimization is shown in Fig. 3. The algorithm starts with a random initial solution (X). In each iteration process, a new solution (X 0 ) is generated in the neighborhood area of the current solution (X). If the condition ‘‘Df ¼ f ðX 0 Þ  f ðXÞ 6 0” is satisfied, then the current solution is replaced with the new one. Otherwise, if the condition ‘‘Df ¼ f ðX 0 Þ  f ðXÞ > 0” is satisfied, the new solution should be

Fig. 3. Flow diagram of the simulated annealing algorithm.

accepted with the probability ‘‘PbðX 0 Þ ¼ expðDf =CTÞ > u”, where ‘‘C” and ‘‘T” are the Boltzmann constant and the current temperature, respectively, and ‘‘u” is a random number ‘‘rand(0,1)”. The temperature T is an essential factor that determines the acceptance of the configuration of worse solution. If ‘‘T ¼ 0” and ‘‘Df ¼ f ðX 0 Þ  f ðXÞ > 0”, then ‘‘PbðX 0 Þ ¼ expðDf =CTÞ ¼ 0”, which is always less than u (rand (0,1)), suggesting that the new solution would never be accepted. On the other hand, the larger is T, the larger probability is to accept a worse solution. Therefore, the new solution is accepted when Df 6 0, while the solution with Df > 0 prevents the objective function from being trapped in the local optimum. The inner loop terminates after Lp iterations, and the end of the inner loop will lead to the decay of the current temperature as:

T iþ1 ¼ eT i

ð21Þ

where e is a cooling factor and 0 < e < 1. While, the algorithm terminates when the lowest temperature T min is reached and the best solution does not improve during a successive iteration xmax . 3.2. Optimization of the four parallel-arranged PPETs For a single PPET, the optimization vector X contains four parameters, the tube length, tube diameter, perforation ratio and the cavity depth. For the four parallel-arranged PPETs, sixteen parameters can influence the sound absorption performance, so the optimization vector X can be written as:

X ¼ ðt1 ; t 2 ; t3 ; t 4 ; d1 ; d2 ; d3 ; d4 ; r1 ; r2 ; r3 ; r4 ; D1 ; D2 ; D3 ; D4 Þ

ð22Þ

Suppose that the objective of optimization is achieving the maximum average sound absorption in the low frequency range of 120–250 Hz. The ‘‘simulated annealing” optimization method is then adopted to obtain the maximum mean optimization result. We assume that the thickness of the perforated panel, the clapboard and the extended tube is 2 mm, 3 mm and 0.5 mm, respectively. According to the analyses in Section 2.3, the objective function and the assumed constraints for the parameters in the vector X are illustrated as follows:

Pn

Max : hai ¼

i¼1

aðf i ; XÞ n

;

i ¼ 1; 2; 3 . . . n

ð23Þ

Fig. 4. Comparison of the sound absorption of the optimized PPET. The optimized parameters for the single PPET are t = 20.0 mm, d = 1.40 mm, D = 100 mm and r = 3.31%, while that of the four parallel-arranged PPETs are listed in Table 1. T 0 ¼ 1, T min ¼ 0:0001, e ¼ 0:95, Lp ¼ 25, xmax ¼ 250.

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D. Li et al. / Applied Acoustics 102 (2016) 126–132

Table 1 The optimized parameters of the four parallel-arranged PPETs. Samples

PPET1

Parameters

d1 (mm)

t1 (mm)

r1 (%)

D1 (mm)

Optimized

5.10

51.0

3.26

100

Samples

PPET3

Parameters

d3 (mm)

t3 (mm)

r3 (%)

D3 (mm)

3.10

40.0

5.31

100

Optimized

PPET2 d2 (mm)

t2 (mm)

r2 (%)

D2 (mm)

3.50

40.0

3.74

100

d4 (mm)

t4 (mm)

r4 (%)

D4 (mm)

2.90

20.0

4.43

100

PPET4

Fig. 5. Influence of the inner diameter of the extended tube on the sound absorption of the four parallel-arranged PPETs.

Fig. 6. Influence of the thickness of the clapboard on the sound absorption of the four parallel-arranged PPETs.

8 d1 ; d2 ; d3 ; d4 2 ½0:01; 7:00 mm > > > < t ; t ; t ; ½40; 60 mm;t 2 ½20; 60 mm 1 2 3 4 Subject to : > D1 ; D2 ; D3 ; D4 2 ½0:01; 100 mm > > : r1 ; r2 ; r3 ; r4 2 ½0:01; 6%

ð24Þ

where aðf i ; XÞ and hai are the sound absorption at the frequency f i and the mean sound absorption in the range of 120–250 Hz. The step size is defined to be 1, so n equals 130. The sound absorption coefficients of the optimized single and four parallel-arranged PPETs are illustrated in Fig. 4. The optimized

Fig. 7. Comparison of the sound absorption of the three parallel-arranged PPETs combined with a MPP, T 0 ¼ 1, T min ¼ 0:0001, e ¼ 0:95, Lp ¼ 25, xmax ¼ 250. The optimized parameters for the single MPP are tmpp ¼ 2:00 mm, dmpp ¼ 0:80 mm, Dmpp ¼ 100 mm and rmpp ¼ 1:27%, while that of the three parallel-arranged PPETs with a MPP are listed in Table 2.

parameters obtained by the simulated annealing are summarized in Table 1. The target frequency range used in this optimization is 120–250 Hz. Four resonance peaks can be observed at frequencies 134 Hz, 160 Hz, 190 Hz and 226 Hz for the four parallel-arranged PPETs. The absorption coefficients at the resonance frequencies are well above 0.9 and the effective sound absorption bandwidth (the bandwidth of the sound absorption coefficient higher than 0.7) is about 125 Hz. While the effective sound absorption bandwidth for the single PPET is about 80 Hz. It is evident that the proposed absorber can significantly improve the absorption bandwidth when compared with that of the single PPET. Fig. 5 shows the influence of the inner diameter of the extended tube on the sound absorption of the four parallel-arranged PPETs. If the tube inner diameter is much less than that of the optimized, the resistance will be too large to achieve high sound absorption. On the other hand, if the tube diameter is too large, the resistance of the combined absorber is too small to maintain a satisfied sound absorption bandwidth. The influence of the thickness of the clapboard on the sound absorption of the four parallel-arranged PPETs is plotted in Fig. 6. When the thickness of the clapboard increases, the effective cavity depth is reduced, and the resonance peaks shift to the high frequency. 3.3. Optimization of the three parallel-arranged PPETs with a MPP In the case of the three parallel-arranged PPETs and a MPP, the frequency range used in the optimization is 120–450 Hz. The thickness of the perforated panel, the clapboard, the extended tubes and

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D. Li et al. / Applied Acoustics 102 (2016) 126–132 Table 2 The optimized parameters of the three parallel-arranged PPETs combined with a MPP. Samples

PPET1

Parameters

d1 (mm)

t1 (mm)

r1 (%)

D1 (mm)

Optimized

2.90

20.0

2.08

100

Samples

PPET3

Parameters

d3 (mm)

t3 (mm)

r3 (%)

D3 (mm)

1.70

10.0

3.74

100

Optimized

PPET2

Samples

PPET1

PPET2

Parameters

d1 (mm)

r1

d2 (mm)

r2

(%)

d4 (mm)

r4

(%)

d3 (mm)

r3

(%)

5.10 5.20

3.26 3.24

3.50 4.20

3.74 3.53

3.10 3.20

5.31 5.32

2.90 3.20

4.43 4.51

PPET3

PPET4 (%)

Table 4 The optimized and selected tube parameters (tube inner diameter and perforation ratio) of the three parallel-arranged PPETs combined with a MPP. Samples

PPET1

Parameters

d1 (mm)

r1

Optimized Selected

2.90 3.20

PPET2

PPET3

r2

(%)

d2 (mm)

2.08 2.05

2.50 3.20

MPP

r3

(%)

d3 (mm)

rmpp

(%)

dmpp (mm)

3.60 3.69

1.70 2.30

3.74 3.81

1.20 1.20

2.02 2.02

(%)

the micro-perforated panel are assumed as 2 mm, 3 mm, 0.5 mm and 2 mm, respectively. The optimization vector can be written as:

X ¼ ½t 1 ; t2 ; t 3 ; d1 ; d2 ; d3 ; dmpp ; r1 ; r2 ; r3 ; rmpp ; D1 ; D2 ; D3 ; Dmpp 

ð25Þ

For the three parallel-arranged PPETs combined with a MPP, the objective function and assumed constraints for the parameters in the vector X are illustrated as follows:

Pn

Max : hai ¼

i¼1

aðf i ; XÞ n

;

t2 (mm)

r2 (%)

D2 (mm)

2.50

20.0

3.60

100

dmpp (mm)

tmpp (mm)

rmpp (%)

Dmpp (mm)

1.20

2.00

2.03

100

MPP

Table 3 The optimized and selected tube parameters (tube inner diameter and perforation ratio) of the four parallel-arranged PPETs.

Optimized Selected

d2 (mm)

i ¼ 1; 2; 3 . . . n

8 d1 ; d2 ; d3 2 ½0:01; 7:00 mm; dmpp 2 ½0:01; 3:00 mm > > > < t ; t 2 ½20; 40 mm; t 2 ½10; 40mm 1 2 3 Subject to : > D1 ; D2 ; D3 2 ½0:01; 100 mm; Dmpp 2 ½0:01; 100 mm > > : r1 ; r2 ; r3 2 ½0:01; 6%; rmpp 2 ½0:01; 3%

The sound absorption coefficients of the optimized single MPP and three parallel-arranged PPETs with a MPP are plotted in Fig. 7. While the optimized parameters obtained by simulated annealing for the three parallel-arranged PPETs with the MPP are summarized in Table 2. Three resonance peaks observed at 160 Hz, 210 Hz, 285 Hz correspond to three PPETs, while the resonance peaks occur at 385 Hz corresponds to the MPP. The absorption coefficients at the resonance frequencies are well above 0.8 and the effective sound absorption bandwidth is about 290 Hz. In comparison with that of a single MPP, the improvement of the bandwidth is about 60 Hz and below 240 Hz. In this frequency range, it is not available to utilize the MPP to main a reasonable bandwidth in a constraint space. 4. Experimental validation Due to the manufacture constraint in reality, it is not available to select the tube inner diameters and perforation ratios exactly the same as the optimized parameters in Tables 1 and 2. Therefore approximate parameters as shown in Tables 3 and 4 are used for the experimental validation. The absorption difference resulted from this approximation is trivial and can be neglected. The measurements are carried out in an impedance tube (B&K 4206) with a diameter of 100 mm according to ISO 10534-2 [17], and the sample is installed at the end of the impedance tube, as shown in Fig. 8. The back cavity is made of 3D printing method, and the extended tubes are made of copper. The perforated panel used in experiments are plastic plate. The thickness of the perforated panel, extended tubes, clapboard and micro-perforated panel is 2 mm, 0.5 mm, 3 mm and 2 mm, respectively. The measured and predicted results of the four parallelarranged PPETs are illustrated in Fig. 9. It should be noted that each PPET occupies 1/4 of the total area. Reasonable agreement is obtained between the predicted and measured absorption curves. It is worthy to note that the perforated panel and the back cavity must be well-sealed to observe the four resonance peaks. The measured and predicted results of the three parallelarranged PPETs combined with a MPP are plotted in Fig. 10, where

PULSE B&K 3160

Computer

Power amplifier

Dimensions Unit mm Rigid backing Tube Microphone 1

100 3 Sound source 100 Clapboard

Test sample

Imepedance tube Fig. 8. Experiments set up.

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D. Li et al. / Applied Acoustics 102 (2016) 126–132

5. Conclusions The feasibility of using parallel-arranged PPETs for the sound absorption at the low frequencies is investigated. A theoretical model is described to predict the sound absorption and the simulated annealing method is employed to obtain the optimized parameters. Based on this, two types of parallel-arranged absorbers are proposed. One is the four parallel-arranged PPETs and the effective sound absorption covers the frequency range from 125 Hz to 250 Hz. The other one is the three parallel-arranged PPETs combined with a MPP and the effective sound absorption is from 150 Hz to 440 Hz. The results imply that the proposed absorbers show great superiority over conventional absorbers in the low frequency range. Consequently, the method described in the paper is useful for the design of sound absorber at low frequencies. Fig. 9. Sound absorption coefficient of the three parallel PPET with a MPP. Red circle line: predicted result. Blue triangle line: measured result. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Acknowledgements The authors greatly acknowledge the support provided by the National Basic Research Program of China (973 Program, 2012CB720204), and the National Natural Science Foundation of China (Grant No. 11374326). References

Fig. 10. Sound absorption coefficient of the three parallel-arranged PPETs combined with a MPP. Red circle line: predicted result. Blue triangle line: measured result. Black square line: experimental result in Ref. [8]. Pink diamond line: experimental result in Ref. [9]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the experimental results in Refs. [8,9] are shown for a comparison. Reasonable agreement is also obtained between the predicted and measured results. The test absorber in Ref. [8] is composed of a perforated panel with flexible tube bundles (100 mm). In comparison with that of the test result in Ref. [8], the proposed absorber can significantly improve the low frequency sound absorption below 400 Hz in the same space constraint (100 mm). Compared with that of the test absorber in Ref. [9] which contained the parallel arrangement of MPP with array of constrained short tubes, the proposed absorber can also achieve a good sound absorption in the low frequency range within the same space limit (100 mm).

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