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 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 59, 2017 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Zhuo Yang, Junjie Ba, Jing Pan 
Copyright © 2017, AIDIC Servizi S.r.l. 
ISBN 978-88-95608- 49-5; ISSN 2283-9216 

Research on the Sound Absorption Performance of Metal 

Rubber Material Based on BP Neural Network 

Yugang Chen 

Jiangxi Institute of Fashion Technology, Nanchang 330201, China 

28941204@qq.com 

In this paper, in order to study on the sound absorption performance of metal rubber material, the BP neural 

network method is used to fulfilling the requirements of data processing. The sound absorption performance of 

Metal Rubber material was studied theoretically and experimentally. It is feasible to model the sound 

absorption performance of metal rubber with uniform isotropic porosity. It provides a simple way to study the 

structural performance of metal rubber accurately. The structure constant is stable for metal rubber material 

with the same mean cavity diameters. It decreases with increase of the frequency and finally tends to a 

constant. The frequency factor in the structure constant is varied from 1 to 4/3, while the structure factor 

decreases in an exponentially decayed way with increase of mean cavity diameter. In the general frequency 

range, the compressive modules of the sound absorber samples is approximately constant when metal rubber 

material has the same structure parameters. The ratio of the real part to the imaginary part of the modules 

increases linearly with increase of mean cavity diameters. 

1. Introduction 

The control of noise in large powerful fan, internal-combustion engines, gas turbines and jet devices has 

become more serious with the development of modern technology (Izzheurov, 2009; Dong et al., 2011). It is 

essential to eliminate or control the noise known as Pipeline Muffler to improve working and living conditions. 

The study has been focused on the selection of appropriate porous sound- absorption materials, its properties 

and its mechanism for the practical application. As to sates the requirements of aerospace and national 

defence weaponry, it is urgent to develop a systematic research on porous sound-absorption material with 

properties of a wide temperature range, erosion resistance, high intensity, and a long life-span. However, most 

of currently used sound-absorption materials are not suitable to be used in the severe environment. Porous 

fibre materials, such as fibre glass and rock, are widely used as sound-absorption materials. Some new 

porous fibre materials, for instance, metal fibre material, also have been created to satisfy the certain 

requirement (Duplancic et al., 2017). 

In the work of Jiang et al. (2007), the sound-absorption properties of metal fibre materials were studied by 

using the theoretical formulas to calculate acoustic parameters, but experimental results were not in good 

agreement with the theoretical values at low frequency. Xi et al. (Xi et al., 2008) developed a physical model to 

describe the transmission of sound wave in fibber material with assumption of a number of long straight 

cylindrical fibres with irregular distribution in material. However, the theoretical model is very complex, 

although it can be used to obtain the effective density and the effective compression modulus. Metal rubber 

material is a kind of homogeneous porous elastic material. It is manufactured by stacking a specified quantity 

of tensile spiral wires into a mould and forming through a cold stamping process. Through this technology, 

metal rubber material has perfect permeability because of connective micro holes and crannies both inside 

and on the surface of the material. Thus, it can be used as a good sound-absorption material in the severe 

environment, because it integrates many favourable characteristics, such as high effective porosity (i.e., from 

0.13 to 0.95), few dead points, large working surface area, good loading capacity and stability, wide working 

temperature range, simple manufacturing process, etc. Therefore, it is meaningful to make a systematic 

research on its acoustic characteristics theoretically and experimentally. Although some sound-absorption 

parameters and the theoretical calculation of acoustic parameters of metal rubber have been studied (Guo et 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1759113

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Yugang Chen, 2017, Research on the sound absorption performance of metal rubber material based on bp neural 
network, Chemical Engineering Transactions, 59, 673-678  DOI:10.3303/CET1759113   

673



al., 2007), it is still essential to make a systemic study on its sound- absorption performance. In the present 

study, the formulas of sound absorption parameters of metal rubber material are developed based on an 

isotropic model. The structure constant for the same mean porosity diameter and the compressive modulus in 

the general frequency range are studied. The relationship of the sound absorption performance parameter, the 

frequency and the structure parameters is obtained for the further application of metal rubber in the sound-

absorption field. 

2. Calculation of acoustic performance parameters of metal rubber  

The most important acoustic performance parameters of metal rubber are the sound absorption coefficient, a0, 

and the acoustic impedance coefficient, ZS. Compared with other porous sound absorption materials, metal 

rubber has the same performance on sound absorption. Its sound absorption coefficient increases with the 

increase of the frequency before it reaches the primo resonance sound absorption frequency, fT. Then, it 

varies in a certain range. The structure parameters determining the sound absorption performance of metal 

rubber involve the porosity, o, the metal wire diameter, d, and the thickness of the specimen, h. The micro 

porosities and crannies inside metal rubber material interconnect to each other in an irregular and isotropic 

way, which makes the fluid infiltrate equally in any direction. Though metal rubber material is porous, the wires 

inside are still rigid and non-compressed. In the study, only the movement of fluid inside material is considered. 

Therefore, the acoustic performance parameters, i.e., the characteristic impedance, ZT, and the propagation 

constant, it can be calculated as follows (Jiang et al., 2008): 

ˆ ˆ ( )
T

F W    (1) 

It’s provided by the adaptive weight law. So estimation error of the weight is 

ˆW W W    (2) 

The positive values Wmax as follows: 

maxF
W W

 (3) 

The adaptive weights law is defined as 

2 2
ˆ ˆ ( )

T
W kG z W z G    

  (4) 

 
2

2
( ) exp , for 1, 2, ,

j ji

ji j

ji

C
i H

b


 

  
  
 
    (5) 

In this space, the mth multidimensional receptive-field function is defined as 

1

( ) ( ), for 1, 2, ,
L

m ji j
j

m N  


   
  (6) 

The function can be written in a vector notation as 

1
( , , ) [ , ..., ]

T

m N
C b    

 (7) 

The weight memory space with N components can be expressed in a vector as 

1
[ , ,..., ]

T

m N
W W W W

  (8) 

where x is the structural constant of the material, po is the density of the air, k is the compressibility modulus of 

the material, W is the viscosity of the air, and w is the angular frequency. Eqs. (1) and (2) describe the 

relationship between the acoustic and the structure performance parameters of metal rubber material, 

respectively. It is noted that x and k are dependent on the frequency and structural parameters of metal rubber 

material, both of which can be obtained experimentally. When metal rubber is fixed on the rigid wall, its 

surface acoustic impedance coefficient, Zs, can be deduced as  

( )
T

y W  
  (9) 

From Eq. (3), the sound absorption coefficient, it can be calculated as 

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1 1 d
z x y 

  (10) 

2 2 1
z x  

  (11) 

The following tracking error dynamics is shown as:  

1 1 2 2 1d d d
z x y x y z y      

 (12) 

From (2) and (6), it can be obtained: 

1 1 1

2 2 1 2 1
( )

g
z x M Cx M G d M  

  
       

 (13) 

τ is selected as 

2 2 1
z z F    

  (14) 

Then we can get: 

2 1 2 2

1

2

T
V V z Mz 

  (15) 

where porn is characteristic impedance of the air. parameters into Eqs. (1), (2), (3) and (4) gives the sound 

absorption coefficient, ao. Substituting the structure performance acoustic impedance coefficient, Zs, and 3 

Determination of structure constant and compressibility modulus As discussed above, the structure constant 

and compressibility modulus of the metal rubber material are independent on the sound absorption 

performance parameters. They showed dependency on the frequency and structure performance parameters, 

i.e., the porosity and the diameter of metal wires, d. In order to study their dependency, a parameter named 

mean cavity diameter, defined as the da and ad (1)-(5) is introduced into the study. This parameter indicates 

the average size of the interior cavities. Eq. (5) shows the relationship among the porosity, the metal wire 

diameter and the mean cavity size inside the material as well. 

2.1 Determination of the material structure constant  

According to the theoretical calculation formulas of acoustic performance parameter of metal rubber material, 

the structural constant, x, is of much concern in the research. It also has a very close relationship with the 

material structure. Actually, it is a key factor for correction between the theoretical and experimental values. It 

is evaluated as where Zs and it can be measured by standing wave tube measuring method. Thus, the 

structural constant can be calculated by using of these two parameters.  

In the measurement of structural constant, the mean cavity diameters of metal rubber specimen, da, axe with 

0.42 mm and 0.21 mm, respectively. There are four types of specimen: o=0.81, d=0.1 mm; Q=0.68, d=0.2 mm; 

W=0.68, d=0.1 mm; a=0.59, d=0.15 mm. With the consideration of the resonance occurred at the high 

frequency, the feasible measuring range of frequency is taken from 200 Hz to 2000 Hz. Fig. 1 shows the test 

results. It is shown from Fig. 1 that the structural constant with the same mean cavity diameter has a good 

consistency. It means that when the specimen has the same mean cavity diameter, the structural constant of 

metal rubber is the same. When the mean cavity diameter varies, the structure constant will decrease with the 

increase of the frequency and finally to a fixed value. Thereby, the structural constant of metal rubber can be 

described as X=X1-X8, C8 where Xi, is the frequency factor which is related to the angular frequency; Xe is 

the structure factor dependent on the structure performance of the material. The frequency factor decreases 

with increase of the angular frequency initially, and then goes to a constant between 1 and 4/3. The structure 

constant has little effect on low frequency absorption. Furthermore, the frequency factor functions mainly at 

the range of low frequency, so let it equal to 1. When metal rubber works at high frequency, the structure 

factor, Xe, approximates to the measured value, which is fitted to obtain its relation with mean cavity diameter, 

shown in Fig. 2. It is noted that the ratio of the mean cavity diameter, da, and the reference diameter, da, 

represents the relative mean cavity diameter. In the current case, da=1.0 mm. Fig. 2 shows that the structure 

factor of metal rubber material decreases with increase of the mean cavity diameter, following an 

exponentially decayed trend as Eq. (10) is the empirical formula to calculate the structure factor of metal 

rubber. It indicates that the relationship among structure factor, frequency and structure parameters.  

675



       

Figure 1: Structural constants with mean cavity          Figure 2: Structure factor vs. relative mean 

diameter da=0.42 mm and 0.21 mm, respectively       cavity diameter 

2.2 Determination of the compressibility modulus  

Compressibility modulus, k, obtained experimentally, depicts the expansion and the compression performance 

of the air in porous materials. It is expressed with a complex format as Lz which shows the compressibility 

modulus of metal impedance and the propagation constant of metal risibility modulus for different structure 

factors Po is the balance pressure of the air. Rubber is determined by the characteristic rubber. The 

experimental results of comparing, in which kT=Po. The frequency has little effect on the compressibility 

modules at the testing frequency range (200-2000 Hz) because the modules are close to a constant if 

structure factors are the same. With consideration of practical application of metal rubber material, its acoustic 

Reynolds number, (da) is usually greater than 1. In the experiment, the real part of relative compressibility 

modules k/kT varies from 1.2 to 1.4. Thus, the compressibility modules can be expressed ashes k=kT where k' 

is the compressibility modules of tube (or slot) according to Kirchhoff Theory. The ratio of the real part to the 

imaginary part of compressibility modules of the material is the same as that of tube (or slot). Assuming that 

compressibility modules for different structure factors is a constant and neglecting the effect of the frequency, 

the ratio of the real part to imaginary part of compressibility modules can be described as a function of the 

mean cavity diameter, expresses as: 

2 1 1 1 2 2 2 2 2
( ) ( )

T T T T

g
V z z z z z f F z G d       

  (16) 

The ideal weight W from (10) and expressed as 

( )
T

F W  
  (17) 

Fig. 3 shows the experimental results of the ratio of the real part to the compressibility modules vs. the relative 

mean cavity diameters. It can make comparison of the linearly fitting results with the experimental results that 

point is less than 20%. 

                               

Figure 3: The ratio of the real part to the imaginary          Figure 4: The Structure of Neurons 

part of modulus vs. relative mean cavity diameters 

3. BP neural network model and algorithm 

Neural network model originates in neurobiology. In neural networks, many different neurons axon terminals 

can enter a single neuron dendrite and the formation of synapses (Tang et al., 2008). All of the different 

676



sources of synaptic release of neurotransmitter can change the membrane potential of neurons to produce the 

same effect. It can be seen that the dendrite of neurons in the input information from different sources can be 

integrated. Figure symbol description is shown in Table 1 (Huang et al., 2014). 

Table 1: Mathematical models symbol description 

Symbol Description significance 

𝑥1,𝑥2,…, 𝑥𝑛 Enter part of neurons (send information on the first level) 

𝜃𝑖 Threshold of neurons 

𝑦𝑖 Output neuron 

f[𝑢1] Excitation function 

  (18) 

  (19) 

4. Comparison of the theoretical and experimental results of sound absorption performance 
parameters 

Structural constant and compressive modulus is analysed experimentally, and formula of acoustic 

characteristic parameters as well as sound absorption performance parameter are obtained based on the 

relationship between structure constant, compressive modulus and material characteristic parameters. The 

good agreement between the calculated and experimental results of sound absorption coefficient testifies the 

effectiveness of theoretical method and experimental operation. The mathematical model of sound absorption 

parameters of MR at resonance were proposed for the application of MR. The calculated results shows good 

agreement with the experimental ones which indicates that the method to investigate the acoustic parameters 

at resonance is effective and accurate. Sound absorption performances of single layer and bilayer structures 

are presented and the calculation formulas of parameters of these structures are deduced based on formula of 

acoustic characteristic parameters. Effects of material thickness, flow resistivity, and air layer thickness on 

sound absorption performances of single layer and bilayer structures are investigated by comparing the 

acoustic absorption performances of single layer and bilayer structures. In the study, four categories of metal 

rubber samples are tested: Sample 1: Q=0.86, d=0.1 mm, h=20 mm; Sample 2: Q=0.81, d=0.1 mm, h=20 mm; 

Sample 3:=0.72, d=0.1 mm, h=21 mm; Sample 4: Q=0.62, d=0.1 mm, h=21 mm. Using a standing wave tube 

measuring method, the sound absorption coefficients of metal rubber are tested. The calculated and 

experimental results are shown in Fig. 7 and Fig. 8, respectively. 

Compared with other porous sound absorption materials, metal rubber has the same performance on sound 

absorption. Its sound absorption coefficient increases with the increase of the frequency before it reaches the 

primo resonance sound absorption frequency, fT. Then, it varies in a certain range. The structure parameters 

determining the sound absorption performance of metal rubber involve the porosity, o, the metal wire diameter, 

d, and the thickness of the specimen, h. The micro porosities and crannies inside metal rubber material 

interconnect to each other in an irregular and isotropic way, which makes the fluid infiltrate equally in any 

direction. Though metal rubber material is porous, the wires inside are still rigid and non-compressed. 

   

Figure 5: Comparison of the calculated and experimental sound absorption coefficients of sample 3 

i

j

iiji
xwu  

 













 

j

iijii
wfufy 

677



 

Figure 6: Comparison experimental of the calculated and sound absorption coefficients of sample 4 

5. Conclusion 

In this paper, in order to study on the sound absorption performance of metal rubber material, the BP neural 

network method is used to fulfilling the requirements of data processing. The experiment result shows the 

simulation platform for rubber material sound absorption performance can be improved by using the big data 

fusion platform. The frequency factor in the structure constant is varied from 1 to 4/3, while the structure factor 

decreases in an exponentially decayed way with increase of mean cavity diameter. In the general frequency 

range, the compressive modules of the sound absorber samples are approximately constant when metal 

rubber material has the same structure parameters. The ratio of the real part to the imaginary part of the 

modules increases linearly with increase of mean cavity diameters. 

Acknowledgments  

The authors acknowledge the Humanities and Social Sciences Program of Jiangxi province in 2014 (YS1440) 

and 2015 Jiangxi social science "12th Five-Year" program (15YS21). 

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