Research ArticleAnalysis of Sound Absorption Performance of UnderwaterAcoustic Coating considering Different Poissonrsquos Ratio Values

Zhuhui Luo 1 Tao Li1 Yuanwei Yan2 and Zhou Zhou2

1School of Mechanical Engineering Hunan University of Technology Zhuzhou China2Zhuzhou Times New Materials Technology COLTD Zhuzhou China

Correspondence should be addressed to Zhuhui Luo 10925022zjueducn

Received 2 August 2021 Revised 2 September 2021 Accepted 4 September 2021 Published 24 September 2021

Academic Editor Azwan I Azmi

Copyright copy 2021 Zhuhui Luo et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Viscoelastic material acoustic coating plays an important role in noise and vibration control of underwater equipment edynamic mechanical properties of the viscoelastic material have a direct effect on the sound absorption performance of theacoustic coating e influence of Poissonrsquos ratio on sound absorption performance is studied A finite element model wasestablished to calculate the sound absorption performance of three typical acoustic coatings homogeneous acoustic coatingsAlberich acoustic coatings and trumpet cavity acoustic coatings and the influence of Poissonrsquos ratio on the sound absorptionperformance of the three kinds of acoustic coatings was analyzed e results show that when Poissonrsquos ratio varies from 049 to04999 the larger Poissonrsquos ratio is the larger the frequency of the first absorption peak is the smaller the absorption coefficientbelow the frequency of the first absorption peak is and the smaller the average absorption coefficient in the whole analysisfrequency range ise dynamic Poissonrsquos ratio with the change of frequency is obtained by interpolating the test results and staticPoissonrsquos ratio finite element calculation results e calculation results show that the dynamic Poissonrsquos ratio can get moreaccurate calculation resultsis work can provide a reference for researchers to set Poissonrsquos ratio in theoretical analysis and finiteelement analysis of acoustic coating

1 Introduction

Acoustic coatings of viscoelastic materials play an importantrole in noise and vibration control of underwater equipmente typical acoustic coating structure used in underwaterequipment is homogeneous acoustic coating [1 2] Alberichacoustic coating [3 4] and trumpet cavity-type coating [5]Homogeneous acoustic coating has the characteristics ofsimple structure convenient installation and maintenanceand low cost but its low-frequency sound absorption per-formance is poor Alberich coating originated in GermanyduringWorldWar II and has excellent low-frequency soundabsorption performance e structure of trumpet cavity-type coating is more complex and the low-frequency soundabsorption performance is also excellent Alberich acousticcoating and trumpet cavity-type coating belong to resonantcavity acoustic coating According to these two acousticcoating various forms of resonant cavity acoustic coating

are derived Sound absorption performance is one of themost important properties of acoustic coatings [6] In ad-dition to structure dynamic mechanical property parame-ters of materials used in acoustic coatings have a directimpact on sound absorption performance [7 8] ere aresix parameters to describe the dynamic mechanical prop-erties of viscoelastic materials Youngrsquos modulus shearmodel Poissonrsquos ratio Lamy constant bulk compressionmodulus and bulk longitudinal wave modulus Among thesix parameters only two parameters are independentamong which Youngrsquos modulus and Poissonrsquos ratio are thetwo commonly used parameters [9 10] Either throughfrequency expansion or through inversely solving acousticmeasurement such as sound velocity the Young modulus ofviscoelastic materials can be obtained Defined as the neg-ative value of the ratio of the transverse positive strain to theaxial positive strain when a material is under unidirectionaltension or compression Poissonrsquos ratio can be obtained

HindawiShock and VibrationVolume 2021 Article ID 4861877 11 pageshttpsdoiorg10115520214861877

either through measurement or calculation with the Youngmodulus shear modulus or bulk modulus Regardless theorder of magnitude for the accuracy of Poissonrsquos ratio ob-tained can reach only 001 [11 12]

e influence of Poissonrsquos ratio on the acoustic per-formance of coatings is seldom studied Since viscoelasticmaterials are approximately incompressible materialsPoissonrsquos ratio is difficult to be measured with high precision(the measurement accuracy is generally 001) According tothis actual situation researchers usually assume Poissonrsquosratio to be a value close to but less than 05 such as 049[9 13ndash20] 0493 [21] 0495 [15 19] 04997 [19] 049976[15] and 0499981 [22] It can be seen that Poissonrsquos ratio ofviscoelastic materials is set differently in the above literatures(although the difference in value is very small) is paper isto study whether and what effects such slight difference hason sound absorption performance In addition in theseliteratures Poissonrsquos ratio is set as a static value that does notchange with the frequency In literature [13] when studyingthe influence of Poissonrsquos loss factor on acoustic coveringlayer the real part of Poissonrsquos ratio is still regarded as a fixedvalue In fact Poissonrsquos ratio of viscoelastic materials varieswith frequency In this paper the finite element calculationof sound absorption performance of acoustic coating iscarried out and Poissonrsquos ratio varying with frequency isused in order to obtain more accurate calculation results

2 Poissonrsquos Ratio and the AttenuationConstant of Acoustic Waves

e Young modulus and Poissonrsquos ratio two parametersrequired to describe the properties of viscoelastic materialsare denoted as E and ] en the shear modulus is indicatedby G the Lamy constant is denoted as λ and the bulklongitudinal wave modulus is represented by S

G E

2(1 + ]) (1)

λ E]

(1 + ])(1 minus 2]) (2)

S λ + 2G (3)

For the viscoelastic material both Youngrsquos modulus andPoissonrsquos ratio are complex numbers as a result of whichthey are the related parameters In an isotropic infinitemedium the wave velocity of longitudinal waves cl and thewave velocity of shear waves ct are written as follows

cl

S

ρ

1113971

(4)

ct

G

ρ

1113971

(5)

where ρ is the density of the sound wave propagationmaterial According to ρ and sound velocity c the charac-teristic impedance of the material is expressed as follows

Z ρc (6)

e attenuation constant of longitudinal and shear wavesin the viscoelastic material αl and αt is expressed as follows

αl ω2cl

η (7)

αt ω2ct

η (8)

where ω represents the angular frequency of the sound waveand η represents the loss factor of the material For suchviscoelastic materials as rubber Poissonrsquos ratio approaches05 and 1 minus 2] is close to 0 According to Equations (2ndash8)even a slight change in Poissonrsquos ratio will have a significantimpact on the sound velocity of the longitudinal wave thusaffecting the characteristic impedance and attenuationconstant of the material and then on the characteristics ofthe acoustic coating in respect of sound absorption To someextent the attenuation constant can characterize only thecharacteristics of the material in terms of sound absorptionIn response to the acoustic coating with a limited size butcomplex structure it is necessary to conduct quantitativeanalysis of the impact caused by Poissonrsquos ratio on thecharacteristics of sound absorption

3 Acoustic Coating

e acoustic coatings analyzed in this paper are three typesof commonly used underwater equipment including ho-mogeneous coating Alberich acoustic coating and trumpetcavity-type coating Homogeneous coating is an equalthickness coating that does not contain any cavity structureWhen in use it is made into a rectangular block structureand pasted on the outer surface of underwater equipmentFigure 1 shows the underwater structure of the Alberichcoating applied In aquatic environment acoustic waves areincident perpendicularly along the ndashz axis to the coating laidon a steel plate with air behind Made of viscoelastic ma-terial the acoustic coating has cylindrical cavities uniformlyand periodically distributed inside Statistically the periodicspacing L1 in the X and Y directions is 30mm the diameterof the cavities denoted as d is 10mm the height of thecylindrical cavities represented by t2 is 46mm the thicknessof the hole sealing layers on both sides of the cavities in-dicated by t1 and t3 respectively is 2mm and the totalthickness of the coating is 50mm

Figure 2 shows the underwater structure of the trumpetcavity-type coating In aquatic environment acoustic wavesare incident perpendicularly along the ndashz axis to the coatinglaid on a steel plate with air behind Made of a viscoelasticmaterial the acoustic coating with a shape of trumpet cavityis periodically distributed in the X and Y directions ecavities are distributed in the form of an equilateral trianglee side length L2 of the equilateral triangle is 1474mmDenoted as d2 the diameter of the trumpet with a shape ofcavity tube is 13mm Referred to as d3 the diameter of thetrumpet mouth is 11mm Indicated by t5 the height of thecavities is 46mm Denoted as t4 and t6 respectively the

2 Shock and Vibration

thickness of the hole sealing layers on both sides of thecavities is 2mme total thickness of the coating is 50mm

4 Finite Element Model

e finite element method is a very effective method toanalyze the sound absorption performance of acousticcoating Since the 1990s Hennion and Easwaran [23] beganto study acoustic coatings by the finite element method(FEM) Nowadays commercial numerical simulation soft-ware (for example ANSYS [24] Virtual Lab [25] andCOMSOL [26 27]) can analyze the acoustic performance ofacoustic coatings on complex structures e finite elementmodules of the acoustic simulation software LMS VirtualLab 11 were applied to construct the finite element model forthe sound absorption performance of acoustic coatings asshown in Figure 3 is problem belongs to acoustic fluidstructure coupling problem e following assumptions aremade that is water is a homogeneous fluid without flow andviscosity and different adjacent solids are completely

connected Under the Blouch periodic boundary conditions[26] the periodic unit shown in Figures 1 and 2 was selectedfor calculation e finite element model is comprised ofwater acoustic coating and steel plate backing ere is noneed to model the air behind the steel plate e defaultsetting of the software is a total reflection boundary egrids of the three materials were divided into tetrahedralmesh with the size of themesh varying from one structure ofthe coatings to another e maximal side lengths of tet-rahedral meshes of water acoustic coating and steel plateare 23mm 1mm and 1mm respectively erefore thenumber of tetrahedral elements of Alberich acoustic coatingand trumpet cavity-type coating is 351464 and 74965 re-spectively Conode processing was conducted on the surfaceof contact between the coating and steel plate Denoted as S1S2 S3 and S4 respectively the four sides of the coating andthe steel plate were constrained by two degrees of freedom inthe X and Y directions Plane acoustic wave excitation wasapplied to one end of the water grid Field point 1 and fieldpoint 2 were added in the water meshers the distance

-Z

A

A-A

A

Y

X

Water Rubber Steel Air

d

Incidentsoundwave

t1 t2 t3 tS

L 1

L 1

L1

Period cell

Figure 1 Alberich acoustic coating structure

A-A

Y

X

A

A

Period cell

L2

Water Rubber Steel Air

Incidentsoundwave

-Z

t4 t5 t6 tS

d2

d 3

Figure 2 Structure of trumpet cavity-type coating

Shock and Vibration 3

between the two field points is denoted as s and the distancebetween field point 1 and interface between water andacoustic coating is denoted as zl

e sound pressure of incident and reflected waves inwater can be expressed as follows

pi PIejkz

pr PReminus jkz

(9)

where k represents the wave number PI and PR refer to theamplitudes of the incident wave pi and reflected wave pr onthe XOY plane of the datum plane respectively PR rPIand r stands for the reflection coefficient

erefore the sound pressure at the two field points isthe superposition of the incident and the reflected wavesen

p1 PIejkzl + PRe

minus jkzl

p2 PIejk zlminus s( ) + PRe

minus jk zlminus s( )(10)

e transfer function from field point 1 to field point 2 ofthe total sound field is written as follows

H12 p2

p1

PIejk zlminus s( ) + rPIe

minus jk zlminus s( )

PIejkzl + rPIe

minusjkzl (11)

e reflection coefficient r can be obtained using

r p2p1 minus e

minus jks

ejks

minus p2p1e

j2kxl (12)

During the process of simulation calculation the soundpressure p1 and p2 of field point 1 and 2 could be obtainedWith s and zl used to obtain the reflection coefficient r thesound absorption coefficient is expressed as

α 1 minus r2 (13)

In order to verify the accuracy of the calculation resultsof the finite element model a 50mm thick homogeneouscoating was selected from Ref [13] for calculation egeometric parameters and material parameters of thecoating are consistent with the literature at is to say theYoung modulus of rubber is E 140MPa the loss factor is023 the density is 1100 kgm3 Poissonrsquos ratio is 049 and 0

is taken as Poissonrsquos loss factor e Young modulus of thesteel plate is 216times105MPa Poissonrsquos ratio is 03 and thedensity is 7800 kgm3 e density of water is 1000 kgm3and the velocity of sound is 1489ms e above values areused for the material properties of water and steel plate later

We performed a mesh sensitivity analysis to representthe mesh dependency of the current problem e sidelength of tetrahedral meshers of water acoustic coating andsteel plate is set to 1mm 2mm 4mm and 6mm and thecalculation results of sound absorption coefficient are ob-served e results show that the water mesh size has noeffect on the results (the graphical results are not given in thepaper) Figure 4 shows the comparison drawn between thesound absorption coefficient calculated using the finite el-ement model under different mesh size settings and thecalculation results shown in Ref [13] e percentage errorsbetween the current results and Ref [13] are 23 2995 and 116 According to the results when the meshsize is set to 1mm and 2mm the sound absorption coef-ficient calculation results are basically consistent with thosein Ref [13] indicating the feasibility of applying the finiteelement model to conduct research on the setting of Pois-sonrsquos ratio

5 The Impact of Poissonrsquos Ratios on thePerformance of Acoustic Coatings inSound Absorption

In Section 3 the abovementioned finite element model isadopted to calculate the performance of the three acousticcoatings in sound absorption In the model Poissonrsquos ratio isset to 0490 0493 0496 0499 and 04999 according to theapproximate incompressible characteristics of viscoelasticmaterials and the values commonly used for researchSimilar to the modulus Poissonrsquos ratio is a complex numberIn the case of lower analytical frequency than 5000Hzhowever the imaginary part of Poissonrsquos ratio would besignificantly smaller than the real part indicating thatPoissonrsquos loss factor is as insignificant as its impact on thecharacteristics of sound absorption exhibited by varioustypes of acoustic coatings that can be ignored Moreover formost acoustic coatings the sound absorption coefficients areclose to 1 if the analysis frequency exceeds 5000Hzus the

Plane sound wave

Water

Anechoic coatingsSteel

Field point 1 Field point 2 X

Y

S1

S2

S3

S4

s

zl ZX

YO

O

Figure 3 Finite element model

4 Shock and Vibration

analytical frequency used in this paper ranges from 100 to5000Hz and Poissonrsquos ratio could be set as a real number

For simplicity purpose the modulus of materials was setin some research studies to a constant regardless of fre-quency for the analysis and calculation of the performance ofacoustic coatings in sound absorption In fact however itvaries with frequency

As shown in Figure 5 the static modulus is assigned withthe value shown in Ref [13] so that E 14times108 Pa and theloss factor is 023 Figure 5 shows the results of finite elementcalculation for the sound absorption coefficients of the threecoatings

Figure 6 shows the sound absorption coefficients of thethree coatings as obtained when the modulus of the materialis assigned with the modulus parameter of the carbon blackfilled nitrile rubber shown in Figure 7

According to Figures 5 and 6 with the increase inPoissonrsquos ratio the first peak of the sound absorption co-efficient curve shifts to high frequency As for the homo-geneous acoustic coating the same conclusion can be drawndespite no appearance of the sound absorption peak in thefigure e first peak of sound absorption shifts to lowfrequency which causes the sound absorption coefficient todecline with the rise of Poissonrsquos ratio when the frequencyfalls below the peak is is because the velocity of longi-tudinal sound increases and the attenuation constant de-creases with the rise of Poissonrsquos ratio as a result of whichthe sound absorption coefficient of the acoustic coatingdrops as shown in Equations (1)ndash(8) in Section 2

In figures 5(b) 5(c) 6(b) and 6(c) when Poissonrsquos ratiois 0499 and 04999 the characteristics of sound absorptioncoefficient curve are quite different from those whenPoissonrsquos ratio is 0490 0493 and 0496 which may be dueto the high sensitivity of sound absorption coefficient toPoissonrsquos ratio As shown in Equations (3) Poissonrsquos ratio isclose to 05 and the denominator in the calculation formulaof the Lamy constant tends to 0 A small change in Poissonrsquos

ratio can cause a great change in the Lamy constantresulting in a large change in sound absorption coefficientTable 1 shows the percentage of change rate of Poissonrsquosratio and the percentage of change rate of the Lamy constantwhen Youngrsquos modulus is 140MPa It can be seen that whenPoissonrsquos ratio is 0493 and 0496 the change rate of theLamy constant is relatively small while when Poissonrsquos ratiois 0499 and 04999 the change rate of the Lamy constant isvery large erefore when Poissonrsquos ratio is 0499 and04999 the sound absorption coefficient curve is quitedifferent

In order to further analyze the influence of Poissonrsquosratios on the sound absorption coefficient the modulus wasassigned with the static value and dynamic value Figure 8shows the average sound absorption coefficient of the threecoatings ranging from 100 to 5000Hz As shown in thefigure the average value of sound absorption decreases withthe increase in Poissonrsquos ratio regardless of whether themodulus value is static or dynamic and what type of acousticcoating it is Besides it can be seen from Figure 8 that thehomogeneous one has different characteristics to the othertwo acoustic coatings as manifested mainly in its smalleraverage sound absorption coefficient and the pattern ofcoefficient variations is different from that of the other two asreflected in the frequency curves of the coefficient in Fig-ures 5 and 6 is is because the two coatings follow morecomplicated mechanisms of acoustic energy loss which issimilar to the conversion from longitudinal waves to shearwaves Although the Alberich acoustic coating and thetrumpet cavity-type coating have good sound absorptioneffect when Poissonrsquos ratio is less than 0499 the soundabsorption performance decreases suddenly when Poissonrsquosratio increases from 0499 to 04999 e reason is shown inTable 2When Poissonrsquos ratio increases from 0499 to 04999Poissonrsquos ratio only increases by 00001 but the Lamyconstant does increase by an order of magnitude en thesound absorption coefficient decreases suddenly

According to Figures 5 6 and 8 the sound absorptioncoefficient calculated by FEM using the static modulus issignificantly lower than using the dynamic one which isbecause the dynamic modulus is lower than the static oneand the velocity of sound is lower accordingly Additionallycoupled with the higher loss factor the attenuation constantis reduced which can be calculated using (7) and (8)Consequently the average sound absorption coefficientincreases

6 Poissonrsquos Ratio Varying with Frequency

As shown in Figure 2 a standing wave tube method wasapplied to measure the vertical incidence sound absorptioncoefficient of the trumpet cavity-type acoustic coating Inaddition the modulus of the nitrile rubber used for acousticcoating was measured (Figure 7) and inputted into the finiteelement model Poissonrsquos ratio was set to 0496 0497 04980499 04992 04994 04995 04996 04997 and 04998respectively In the frequency range of 1000ndash5000Hz thesound absorption coefficients of one-third octave range ofthe acoustic coating were obtained by means of test and

0 5000 10000 15000 20000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Ref [13]1 mmerr=232 mmerr=29

4 mmerr=956 mmerr=116

Figure 4 Comparison of calculation results in this paper and Ref[13]

Shock and Vibration 5

calculation as shown in Table 2 and Figure 9 According tothe figure when Poissonrsquos ratio was set to the above valuesthe finite element calculation results of the sound absorptioncoefficient were starkly different from the measured results

As indicated by the results of Ref [12 28] Poissonrsquos ratiodrops with the rise of frequency According to this patternand Table 1 a linear interpolation method was adopted toobtain Poissonrsquos ratio which varies with frequency Taking1000Hz as an example the sound absorption coefficientmeasured is 0420 which is between the last two values of0442 and 0336 in the first column of the coefficients shownin the table e corresponding Poissonrsquos ratio is 04997 and

04998 respectively e linear interpolation was conductedto obtain Poissonrsquos ratio of 049972 e shaded numberslisted in the table represent the interval values of the in-terpolation Varying within the frequency of 1000ndash5000Hzthe interpolated dynamic Poissonrsquos ratios are presented inFigure 10 After linear fitting they were found consistentwith the characteristic as described in Ref [12 29] Morespecifically Poissonrsquos ratio at low frequencies and the log-arithmic frequency bears linear relationship

With the interpolated dynamic Poissonrsquos ratio inputtedinto the finite element model the calculation results wereobtained to indicate that the sound absorption coefficients

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06A

bsor

ptio

n co

effic

ient

s

Frequency (Hz)

049004930496

049904999

(a)

Abs

orpt

ion

coef

ficie

nts

049004930496

049904999

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

Frequency (Hz)

(b)

Abs

orpt

ion

coef

ficie

nts

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

Frequency (Hz)

049004930496

049904999

00

02

04

06

(c)

Figure 5When the modulus shows no change with frequency the sound absorption coefficient of the acoustic coating varies with Poissonrsquosratio (a) homogeneous coating (b) Alberich coating (c) trumpet cavity-type coating

6 Shock and Vibration

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10A

bsor

ptio

n co

effic

ient

s

Frequency (Hz)

049004930496

049904999

(a)

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

049004930496

049904999

(b)

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

049004930496

049904999

(c)

Figure 6 When the modulus varies with frequency the sound absorption coefficient of the acoustic coating varies with Poissonrsquos ratio(a) homogeneous coating (b) Alberich coating (c) trumpet cavity-type coating

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

0102030405060708090

100110120130

Mod

ulus

(MPa

)

Frequency (Hz)

Storage modulusLoss modulus

Figure 7 Dynamic modulus of carbon black filled nitrile rubber

Shock and Vibration 7

were well consistent with the test results as shown inFigure 9 e aforementioned dynamic Poissonrsquos ratios wereused to calculate the sound absorption characteristics of theAlberich coating made of the same nitrile rubber as thetrumpet cavity-type coating Figure 11 shows the calculationand measured results suggesting that the sound absorption

coefficients calculated using the static Poissonrsquos ratios aresignificantly deviant from the test results and that the onescalculated using the dynamic ratios are more consistent withthe test results In addition it can be seen from the figure thatdynamic Poissonrsquos ratio should be used to obtain moreaccurate results for the finite element calculation of the

Table 1 Poissonrsquos ratio Lamy constant and their rate of change

Poissonrsquos ratio Lamy constant Change rate of Poissonrsquos ratio Change rate of Lamy constant0490 23E+ 09 0 00493 33E+ 09 061 430496 58E+ 09 122 1520499 233E+ 10 184 91204999 233E+ 11 202 10034

049 0493 0496 0499 04999

00

01

02

03

04

05

06

07

08

09

Ave

rage

abso

rptio

n co

effic

ient

Poissons Ratio

Static modulus HomogenousStatic modulus AlberichStatic modulus Horn

Dynamic modulus HomogenousDynamic modulus AlberichDynamic modulus Horn

Figure 8 Variation of the average sound absorption coefficient with Poissonrsquos ratio

Table 2 Calculated and measured sound absorption coefficient

FrequencyPoissonrsquos ratio 1000 1250 1600 2000 2500 3150 4000 5000

Calculated

0496 0922 0935 0942 0935 0911 0866 0796 07180497 0917 0939 0957 0960 0946 0909 0841 07560498 0889 0922 0956 0978 0982 0963 0908 08260499 0772 0821 0879 0930 0969 0996 0994 095304992 0720 0769 0831 0888 0937 0978 0997 097904994 0642 0692 0756 0817 0874 0929 0973 098504995 0589 0638 0701 0764 0823 0884 0940 097004996 0524 0570 0631 0692 0752 0816 0883 093004997 0442 0483 0538 0595 0651 0715 0787 084704998 0336 0369 0414 0461 0509 0564 0630 0692

Measured 0420 0615 0815 0900 0985 0955 0830 0755

8 Shock and Vibration

1000 1250 1600 2000 2500 3150 4000 5000

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with constant Poissons ratioTestedCalculated with dynamic Poissons ratio

Figure 9 Calculation and measurement results of the sound absorption coefficient of the trumpet cavity-type acoustic coating

1000 1250 1600 2000 2500 3150 4000 500004960

04965

04970

04975

04980

04985

04990

04995

05000

05005

Poiss

ons

Ratio

Frequency (Hz)

InterpolatedFited

Figure 10 Interpolated Poissonrsquos ratio varying with frequency

1000 1250 1600 2000 2500 3150 4000 5000

02

03

04

05

06

07

08

09

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with Poissons Ratio = 0490Calculated with Poissons Ratio = 0493Calculated with Poissons Ratio = 0496Calculated with Poissons Ratio = 0499Calculated with Poissons Ratio = 04999Calculated with dynamic Poissons RatioMeasured

Figure 11 Calculation and measurement results of the sound absorption coefficients of the Alberich coating

Shock and Vibration 9

sound absorption characteristics of acoustic coatings Un-fortunately there is still no accurate and reliable methodavailable to get the dynamic Poissonrsquos ratios of viscoelasticmaterials [11]

7 Conclusion

(1) When Poissonrsquos ratio of homogeneous acousticcoating Alberich acoustic coating and trumpetcavity-type coating varies from 049 to 04999 thelarger Poissonrsquos ratio is the larger the frequency ofthe first absorption peak is the smaller the ab-sorption coefficient is in the frequency range belowthe first absorption peak and the smaller the averageabsorption coefficient is in the analysis frequencyrange

(2) More accurate calculation results of sound absorp-tion coefficient can be obtained by using Poissonrsquosratio varying with frequency In the analysis of thesound absorption performance of acoustic coatingsthe accurate dynamic Poissonrsquos ratio with the changeof frequency should be used

(3) Poissonrsquos ratio of viscoelastic materials varies in avery small numerical range such a small change hasa great influence on the sound absorption perfor-mance of acoustic coatings erefore it is urgent tostudy the accurate method to obtain the dynamicPoissonrsquos ratio

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the Scientific Re-search Project of Education Department of Hunan ProvinceChina (Project no 19C0589)

References

[1] L Alimonti and N Atalla ldquoEfficient modeling of flat andhomogeneous acoustic treatments for vibroacoustic finiteelement analysis Finite size correction by image sourcesrdquoJournal of Sound and Vibration vol 388 pp 201ndash215 2017

[2] X L Yao X Z Wang F Z Pang and L Q Sun ldquoStudy onAcoustic Characteristics of Underwater Coating StructurerdquoNoise and Vibration Control vol 30 no 4 pp 1ndash5 2010

[3] S M Ivansson ldquoErratumrdquo Journal of the Acoustical Society ofAmerica vol 131 no 4 p 3180 2012

[4] S M Ivansson ldquoNumerical design of Alberich anechoiccoatings with superellipsoidal cavities of mixed sizesrdquo Journalof the Acoustical Society of America vol 123 no 5 p 34662008

[5] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wave

normal incidence conditionrdquo IOP Conference Series Mate-rials Science and Engineering vol 522 no 1 Article ID012001 2019

[6] H Bai Z Zhan J Liu and Z Ren ldquoFrom local structure tooverall performance An overview on the design of an acousticcoatingrdquo Materials vol 12 no 16 2019

[7] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic wave by a doubly periodicstructure using the finite-element methodrdquo Journal of theAcoustical Society of America vol 88 2005

[8] T Meng ldquoSimplified model for predicting acoustic perfor-mance of an underwater sound absorption coatingrdquo Journal ofVibration and Control vol 20 no 3 pp 339ndash354 2014

[9] L M Lu J H Wen H G Zhao and X S Wen ldquoDynamicalmechanical property of viscoelastic materials and its effect onacoustic absorption of coatingrdquo Acta Physica Sinica vol 63no 15 2014

[10] Z Wei H Hou N Gao Y Huang and J Yang ldquoComplexYoungrsquos modulus measurement by incident wave extractingin a thin resonant barrdquo Journal of the Acoustical Society ofAmerica vol 142 no 6 pp 3436ndash3442 2017

[11] N W Tschoegl W G Knauss and I Emri ldquoPoissonrsquos ratio inlinear viscoelasticity - A critical reviewrdquo Mechanics of Time-Dependent Materials vol 6 no 1 pp 3ndash51 2002

[12] T Pritz ldquoFrequency dependences of complex moduli andcomplex Poissonrsquos ratio of real solid materialsrdquo Journal ofSound and Vibration vol 214 no 1 pp 83ndash104 1998

[13] J Zhong H Zhao H Yang J Yin and J Wen ldquoEffect ofPoissonrsquos loss factor of rubbery material on underwater soundabsorption of anechoic coatingsrdquo Journal of Sound and Vi-bration vol 424 pp 293ndash301 2018

[14] H Meng J Wen H Zhao L Lv and X Wen ldquoAnalysis ofabsorption performances of anechoic layers with steel platebackingrdquo Journal of the Acoustical Society of America vol 132no 1 pp 69ndash75 2012

[15] V Easwaran and M L Munjal ldquoAnalysis of reflectioncharacteristics of a normal incidence plane wave on resonantsound absorbers A finite element approachrdquo Journal of theAcoustical Society of America vol 93 no 3 pp 1308ndash13181993

[16] Z Jin Y Yin and B Liu ldquoEquivalent modulus method forfinite element simulation of the sound absorption of anechoiccoating backed with orthogonally rib-stiffened platerdquo Journalof Sound and Vibration vol 366 pp 357ndash371 2016

[17] L M Lu J H Wen H Zhao H Meng and X S Wen ldquoLow-frequency acoustic absorption of viscoelastic coating withvarious shapes of scatterersrdquo Wuli XuebaoActa PhysicaSinica vol 61 2012

[18] J Li and S Li ldquoTopology optimization of anechoic coating formaximizing sound absorptionrdquo Journal of Vibration andControl vol 24 no 11 pp 2369ndash2385 2018

[19] C Cai K C Hung and M S Khan ldquoSimulation-basedanalysis of acoustic absorbent lining subject to normal planewave incidencerdquo Journal of Sound and Vibration vol 291no 3-5 pp 656ndash680 2006

[20] S N Panigrahi C S Jog and M L Munjal ldquoMulti-focusdesign of underwater noise control linings based on finiteelement analysisrdquo Applied Acoustics vol 69 no 12pp 1141ndash1153 2008

[21] H Meng J Wen H Zhao and X Wen ldquoOptimization oflocally resonant acoustic metamaterials on underwater soundabsorption characteristicsrdquo Journal of Sound and Vibrationvol 331 no 20 pp 4406ndash4416 2012

10 Shock and Vibration

[22] M A Lewniska V G Kouznetsova J A W van DommelenA O Krushynska and M G D Geers ldquoe attenuationperformance of locally resonant acoustic metamaterials basedon generalised viscoelastic modellingrdquo International Journalof Solids and Structures vol 126 pp 163ndash174 2017

[23] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic-wave by a doubly periodicstructure using the finite-element method- application toalberich anechoic coatingrdquo Journal of the Acoustical Society ofAmerica vol 90 no 6 pp 3356ndash3367 1991

[24] J Villalobos-Luna P Lopez-Cruz R Ramirez-Valencia andF J Elizondo-Garza ldquoSimulation of acoustic wave behavior inducts and a plenum using finite elements package ANSYSrdquoJournal of the Acoustical Society of America vol 112 no 5p 2427 2002

[25] Y Xu G Zhang B Zhou H Wang and Q Tang ldquoAnalysis ofacoustic radiation problems using the cell-based smoothedradial point interpolation method with Dirichlet-to-Neu-mann boundary conditionrdquo Engineering Analysis withBoundary Elements vol 108 no Nov pp 447ndash458 2019

[26] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wavenormal incidence conditionrdquo in Proceedings of the 4th In-ternational Conference on Smart Material Research ICSMR2018 Sydney Australia November 2018

[27] Y E Hanfeng M Tao and L I Junjie ldquoSound absorptionperformance analysis of anechoic coatings under obliqueincidence condition based on COMSOLrdquo Journal of Vibrationand Shock vol 038 no 12 pp 213ndash218 2019

[28] T Pritz ldquoe Poissonrsquos loss factor of solid viscoelastic ma-terialsrdquo Journal of Sound and Vibration vol 306 no 3-5pp 790ndash802 2007

[29] S H Sohrabi and M J Ketabdari ldquoNumerical simulation of aviscoelastic sound absorbent coating with a doubly periodicarray of cavitiesrdquo Cogent Engineering vol 5 no 1 2018

Shock and Vibration 11

thickness of the hole sealing layers on both sides of thecavities is 2mme total thickness of the coating is 50mm

4 Finite Element Model

e finite element method is a very effective method toanalyze the sound absorption performance of acousticcoating Since the 1990s Hennion and Easwaran [23] beganto study acoustic coatings by the finite element method(FEM) Nowadays commercial numerical simulation soft-ware (for example ANSYS [24] Virtual Lab [25] andCOMSOL [26 27]) can analyze the acoustic performance ofacoustic coatings on complex structures e finite elementmodules of the acoustic simulation software LMS VirtualLab 11 were applied to construct the finite element model forthe sound absorption performance of acoustic coatings asshown in Figure 3 is problem belongs to acoustic fluidstructure coupling problem e following assumptions aremade that is water is a homogeneous fluid without flow andviscosity and different adjacent solids are completely

connected Under the Blouch periodic boundary conditions[26] the periodic unit shown in Figures 1 and 2 was selectedfor calculation e finite element model is comprised ofwater acoustic coating and steel plate backing ere is noneed to model the air behind the steel plate e defaultsetting of the software is a total reflection boundary egrids of the three materials were divided into tetrahedralmesh with the size of themesh varying from one structure ofthe coatings to another e maximal side lengths of tet-rahedral meshes of water acoustic coating and steel plateare 23mm 1mm and 1mm respectively erefore thenumber of tetrahedral elements of Alberich acoustic coatingand trumpet cavity-type coating is 351464 and 74965 re-spectively Conode processing was conducted on the surfaceof contact between the coating and steel plate Denoted as S1S2 S3 and S4 respectively the four sides of the coating andthe steel plate were constrained by two degrees of freedom inthe X and Y directions Plane acoustic wave excitation wasapplied to one end of the water grid Field point 1 and fieldpoint 2 were added in the water meshers the distance

-Z

A

A-A

A

Y

X

Water Rubber Steel Air

d

Incidentsoundwave

t1 t2 t3 tS

L 1

L 1

L1

Period cell

Figure 1 Alberich acoustic coating structure

A-A

Y

X

A

A

Period cell

L2

Water Rubber Steel Air

Incidentsoundwave

-Z

t4 t5 t6 tS

d2

d 3

Figure 2 Structure of trumpet cavity-type coating

Shock and Vibration 3

between the two field points is denoted as s and the distancebetween field point 1 and interface between water andacoustic coating is denoted as zl

e sound pressure of incident and reflected waves inwater can be expressed as follows

pi PIejkz

pr PReminus jkz

(9)

where k represents the wave number PI and PR refer to theamplitudes of the incident wave pi and reflected wave pr onthe XOY plane of the datum plane respectively PR rPIand r stands for the reflection coefficient

erefore the sound pressure at the two field points isthe superposition of the incident and the reflected wavesen

p1 PIejkzl + PRe

minus jkzl

p2 PIejk zlminus s( ) + PRe

minus jk zlminus s( )(10)

e transfer function from field point 1 to field point 2 ofthe total sound field is written as follows

H12 p2

p1

PIejk zlminus s( ) + rPIe

minus jk zlminus s( )

PIejkzl + rPIe

minusjkzl (11)

e reflection coefficient r can be obtained using

r p2p1 minus e

minus jks

ejks

minus p2p1e

j2kxl (12)

During the process of simulation calculation the soundpressure p1 and p2 of field point 1 and 2 could be obtainedWith s and zl used to obtain the reflection coefficient r thesound absorption coefficient is expressed as

α 1 minus r2 (13)

In order to verify the accuracy of the calculation resultsof the finite element model a 50mm thick homogeneouscoating was selected from Ref [13] for calculation egeometric parameters and material parameters of thecoating are consistent with the literature at is to say theYoung modulus of rubber is E 140MPa the loss factor is023 the density is 1100 kgm3 Poissonrsquos ratio is 049 and 0

is taken as Poissonrsquos loss factor e Young modulus of thesteel plate is 216times105MPa Poissonrsquos ratio is 03 and thedensity is 7800 kgm3 e density of water is 1000 kgm3and the velocity of sound is 1489ms e above values areused for the material properties of water and steel plate later

We performed a mesh sensitivity analysis to representthe mesh dependency of the current problem e sidelength of tetrahedral meshers of water acoustic coating andsteel plate is set to 1mm 2mm 4mm and 6mm and thecalculation results of sound absorption coefficient are ob-served e results show that the water mesh size has noeffect on the results (the graphical results are not given in thepaper) Figure 4 shows the comparison drawn between thesound absorption coefficient calculated using the finite el-ement model under different mesh size settings and thecalculation results shown in Ref [13] e percentage errorsbetween the current results and Ref [13] are 23 2995 and 116 According to the results when the meshsize is set to 1mm and 2mm the sound absorption coef-ficient calculation results are basically consistent with thosein Ref [13] indicating the feasibility of applying the finiteelement model to conduct research on the setting of Pois-sonrsquos ratio

5 The Impact of Poissonrsquos Ratios on thePerformance of Acoustic Coatings inSound Absorption

In Section 3 the abovementioned finite element model isadopted to calculate the performance of the three acousticcoatings in sound absorption In the model Poissonrsquos ratio isset to 0490 0493 0496 0499 and 04999 according to theapproximate incompressible characteristics of viscoelasticmaterials and the values commonly used for researchSimilar to the modulus Poissonrsquos ratio is a complex numberIn the case of lower analytical frequency than 5000Hzhowever the imaginary part of Poissonrsquos ratio would besignificantly smaller than the real part indicating thatPoissonrsquos loss factor is as insignificant as its impact on thecharacteristics of sound absorption exhibited by varioustypes of acoustic coatings that can be ignored Moreover formost acoustic coatings the sound absorption coefficients areclose to 1 if the analysis frequency exceeds 5000Hzus the

Plane sound wave

Water

Anechoic coatingsSteel

Field point 1 Field point 2 X

Y

S1

S2

S3

S4

s

zl ZX

YO

O

Figure 3 Finite element model

4 Shock and Vibration

analytical frequency used in this paper ranges from 100 to5000Hz and Poissonrsquos ratio could be set as a real number

For simplicity purpose the modulus of materials was setin some research studies to a constant regardless of fre-quency for the analysis and calculation of the performance ofacoustic coatings in sound absorption In fact however itvaries with frequency

As shown in Figure 5 the static modulus is assigned withthe value shown in Ref [13] so that E 14times108 Pa and theloss factor is 023 Figure 5 shows the results of finite elementcalculation for the sound absorption coefficients of the threecoatings

Figure 6 shows the sound absorption coefficients of thethree coatings as obtained when the modulus of the materialis assigned with the modulus parameter of the carbon blackfilled nitrile rubber shown in Figure 7

According to Figures 5 and 6 with the increase inPoissonrsquos ratio the first peak of the sound absorption co-efficient curve shifts to high frequency As for the homo-geneous acoustic coating the same conclusion can be drawndespite no appearance of the sound absorption peak in thefigure e first peak of sound absorption shifts to lowfrequency which causes the sound absorption coefficient todecline with the rise of Poissonrsquos ratio when the frequencyfalls below the peak is is because the velocity of longi-tudinal sound increases and the attenuation constant de-creases with the rise of Poissonrsquos ratio as a result of whichthe sound absorption coefficient of the acoustic coatingdrops as shown in Equations (1)ndash(8) in Section 2

In figures 5(b) 5(c) 6(b) and 6(c) when Poissonrsquos ratiois 0499 and 04999 the characteristics of sound absorptioncoefficient curve are quite different from those whenPoissonrsquos ratio is 0490 0493 and 0496 which may be dueto the high sensitivity of sound absorption coefficient toPoissonrsquos ratio As shown in Equations (3) Poissonrsquos ratio isclose to 05 and the denominator in the calculation formulaof the Lamy constant tends to 0 A small change in Poissonrsquos

ratio can cause a great change in the Lamy constantresulting in a large change in sound absorption coefficientTable 1 shows the percentage of change rate of Poissonrsquosratio and the percentage of change rate of the Lamy constantwhen Youngrsquos modulus is 140MPa It can be seen that whenPoissonrsquos ratio is 0493 and 0496 the change rate of theLamy constant is relatively small while when Poissonrsquos ratiois 0499 and 04999 the change rate of the Lamy constant isvery large erefore when Poissonrsquos ratio is 0499 and04999 the sound absorption coefficient curve is quitedifferent

In order to further analyze the influence of Poissonrsquosratios on the sound absorption coefficient the modulus wasassigned with the static value and dynamic value Figure 8shows the average sound absorption coefficient of the threecoatings ranging from 100 to 5000Hz As shown in thefigure the average value of sound absorption decreases withthe increase in Poissonrsquos ratio regardless of whether themodulus value is static or dynamic and what type of acousticcoating it is Besides it can be seen from Figure 8 that thehomogeneous one has different characteristics to the othertwo acoustic coatings as manifested mainly in its smalleraverage sound absorption coefficient and the pattern ofcoefficient variations is different from that of the other two asreflected in the frequency curves of the coefficient in Fig-ures 5 and 6 is is because the two coatings follow morecomplicated mechanisms of acoustic energy loss which issimilar to the conversion from longitudinal waves to shearwaves Although the Alberich acoustic coating and thetrumpet cavity-type coating have good sound absorptioneffect when Poissonrsquos ratio is less than 0499 the soundabsorption performance decreases suddenly when Poissonrsquosratio increases from 0499 to 04999 e reason is shown inTable 2When Poissonrsquos ratio increases from 0499 to 04999Poissonrsquos ratio only increases by 00001 but the Lamyconstant does increase by an order of magnitude en thesound absorption coefficient decreases suddenly

According to Figures 5 6 and 8 the sound absorptioncoefficient calculated by FEM using the static modulus issignificantly lower than using the dynamic one which isbecause the dynamic modulus is lower than the static oneand the velocity of sound is lower accordingly Additionallycoupled with the higher loss factor the attenuation constantis reduced which can be calculated using (7) and (8)Consequently the average sound absorption coefficientincreases

6 Poissonrsquos Ratio Varying with Frequency

As shown in Figure 2 a standing wave tube method wasapplied to measure the vertical incidence sound absorptioncoefficient of the trumpet cavity-type acoustic coating Inaddition the modulus of the nitrile rubber used for acousticcoating was measured (Figure 7) and inputted into the finiteelement model Poissonrsquos ratio was set to 0496 0497 04980499 04992 04994 04995 04996 04997 and 04998respectively In the frequency range of 1000ndash5000Hz thesound absorption coefficients of one-third octave range ofthe acoustic coating were obtained by means of test and

0 5000 10000 15000 20000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Ref [13]1 mmerr=232 mmerr=29

4 mmerr=956 mmerr=116

Figure 4 Comparison of calculation results in this paper and Ref[13]

Shock and Vibration 5

calculation as shown in Table 2 and Figure 9 According tothe figure when Poissonrsquos ratio was set to the above valuesthe finite element calculation results of the sound absorptioncoefficient were starkly different from the measured results

As indicated by the results of Ref [12 28] Poissonrsquos ratiodrops with the rise of frequency According to this patternand Table 1 a linear interpolation method was adopted toobtain Poissonrsquos ratio which varies with frequency Taking1000Hz as an example the sound absorption coefficientmeasured is 0420 which is between the last two values of0442 and 0336 in the first column of the coefficients shownin the table e corresponding Poissonrsquos ratio is 04997 and

04998 respectively e linear interpolation was conductedto obtain Poissonrsquos ratio of 049972 e shaded numberslisted in the table represent the interval values of the in-terpolation Varying within the frequency of 1000ndash5000Hzthe interpolated dynamic Poissonrsquos ratios are presented inFigure 10 After linear fitting they were found consistentwith the characteristic as described in Ref [12 29] Morespecifically Poissonrsquos ratio at low frequencies and the log-arithmic frequency bears linear relationship

With the interpolated dynamic Poissonrsquos ratio inputtedinto the finite element model the calculation results wereobtained to indicate that the sound absorption coefficients

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06A

bsor

ptio

n co

effic

ient

s

Frequency (Hz)

049004930496

049904999

(a)

Abs

orpt

ion

coef

ficie

nts

049004930496

049904999

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

Frequency (Hz)

(b)

Abs

orpt

ion

coef

ficie

nts

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

Frequency (Hz)

049004930496

049904999

00

02

04

06

(c)

Figure 5When the modulus shows no change with frequency the sound absorption coefficient of the acoustic coating varies with Poissonrsquosratio (a) homogeneous coating (b) Alberich coating (c) trumpet cavity-type coating

6 Shock and Vibration

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10A

bsor

ptio

n co

effic

ient

s

Frequency (Hz)

049004930496

049904999

(a)

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

049004930496

049904999

(b)

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

049004930496

049904999

(c)

Figure 6 When the modulus varies with frequency the sound absorption coefficient of the acoustic coating varies with Poissonrsquos ratio(a) homogeneous coating (b) Alberich coating (c) trumpet cavity-type coating

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

0102030405060708090

100110120130

Mod

ulus

(MPa

)

Frequency (Hz)

Storage modulusLoss modulus

Figure 7 Dynamic modulus of carbon black filled nitrile rubber

Shock and Vibration 7

were well consistent with the test results as shown inFigure 9 e aforementioned dynamic Poissonrsquos ratios wereused to calculate the sound absorption characteristics of theAlberich coating made of the same nitrile rubber as thetrumpet cavity-type coating Figure 11 shows the calculationand measured results suggesting that the sound absorption

coefficients calculated using the static Poissonrsquos ratios aresignificantly deviant from the test results and that the onescalculated using the dynamic ratios are more consistent withthe test results In addition it can be seen from the figure thatdynamic Poissonrsquos ratio should be used to obtain moreaccurate results for the finite element calculation of the

Table 1 Poissonrsquos ratio Lamy constant and their rate of change

Poissonrsquos ratio Lamy constant Change rate of Poissonrsquos ratio Change rate of Lamy constant0490 23E+ 09 0 00493 33E+ 09 061 430496 58E+ 09 122 1520499 233E+ 10 184 91204999 233E+ 11 202 10034

049 0493 0496 0499 04999

00

01

02

03

04

05

06

07

08

09

Ave

rage

abso

rptio

n co

effic

ient

Poissons Ratio

Static modulus HomogenousStatic modulus AlberichStatic modulus Horn

Dynamic modulus HomogenousDynamic modulus AlberichDynamic modulus Horn

Figure 8 Variation of the average sound absorption coefficient with Poissonrsquos ratio

Table 2 Calculated and measured sound absorption coefficient

FrequencyPoissonrsquos ratio 1000 1250 1600 2000 2500 3150 4000 5000

Calculated

0496 0922 0935 0942 0935 0911 0866 0796 07180497 0917 0939 0957 0960 0946 0909 0841 07560498 0889 0922 0956 0978 0982 0963 0908 08260499 0772 0821 0879 0930 0969 0996 0994 095304992 0720 0769 0831 0888 0937 0978 0997 097904994 0642 0692 0756 0817 0874 0929 0973 098504995 0589 0638 0701 0764 0823 0884 0940 097004996 0524 0570 0631 0692 0752 0816 0883 093004997 0442 0483 0538 0595 0651 0715 0787 084704998 0336 0369 0414 0461 0509 0564 0630 0692

Measured 0420 0615 0815 0900 0985 0955 0830 0755

8 Shock and Vibration

1000 1250 1600 2000 2500 3150 4000 5000

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with constant Poissons ratioTestedCalculated with dynamic Poissons ratio

Figure 9 Calculation and measurement results of the sound absorption coefficient of the trumpet cavity-type acoustic coating

1000 1250 1600 2000 2500 3150 4000 500004960

04965

04970

04975

04980

04985

04990

04995

05000

05005

Poiss

ons

Ratio

Frequency (Hz)

InterpolatedFited

Figure 10 Interpolated Poissonrsquos ratio varying with frequency

1000 1250 1600 2000 2500 3150 4000 5000

02

03

04

05

06

07

08

09

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with Poissons Ratio = 0490Calculated with Poissons Ratio = 0493Calculated with Poissons Ratio = 0496Calculated with Poissons Ratio = 0499Calculated with Poissons Ratio = 04999Calculated with dynamic Poissons RatioMeasured

Figure 11 Calculation and measurement results of the sound absorption coefficients of the Alberich coating

Shock and Vibration 9

sound absorption characteristics of acoustic coatings Un-fortunately there is still no accurate and reliable methodavailable to get the dynamic Poissonrsquos ratios of viscoelasticmaterials [11]

7 Conclusion

(1) When Poissonrsquos ratio of homogeneous acousticcoating Alberich acoustic coating and trumpetcavity-type coating varies from 049 to 04999 thelarger Poissonrsquos ratio is the larger the frequency ofthe first absorption peak is the smaller the ab-sorption coefficient is in the frequency range belowthe first absorption peak and the smaller the averageabsorption coefficient is in the analysis frequencyrange

(2) More accurate calculation results of sound absorp-tion coefficient can be obtained by using Poissonrsquosratio varying with frequency In the analysis of thesound absorption performance of acoustic coatingsthe accurate dynamic Poissonrsquos ratio with the changeof frequency should be used

(3) Poissonrsquos ratio of viscoelastic materials varies in avery small numerical range such a small change hasa great influence on the sound absorption perfor-mance of acoustic coatings erefore it is urgent tostudy the accurate method to obtain the dynamicPoissonrsquos ratio

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the Scientific Re-search Project of Education Department of Hunan ProvinceChina (Project no 19C0589)

References

[1] L Alimonti and N Atalla ldquoEfficient modeling of flat andhomogeneous acoustic treatments for vibroacoustic finiteelement analysis Finite size correction by image sourcesrdquoJournal of Sound and Vibration vol 388 pp 201ndash215 2017

[2] X L Yao X Z Wang F Z Pang and L Q Sun ldquoStudy onAcoustic Characteristics of Underwater Coating StructurerdquoNoise and Vibration Control vol 30 no 4 pp 1ndash5 2010

[3] S M Ivansson ldquoErratumrdquo Journal of the Acoustical Society ofAmerica vol 131 no 4 p 3180 2012

[4] S M Ivansson ldquoNumerical design of Alberich anechoiccoatings with superellipsoidal cavities of mixed sizesrdquo Journalof the Acoustical Society of America vol 123 no 5 p 34662008

[5] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wave

normal incidence conditionrdquo IOP Conference Series Mate-rials Science and Engineering vol 522 no 1 Article ID012001 2019

[6] H Bai Z Zhan J Liu and Z Ren ldquoFrom local structure tooverall performance An overview on the design of an acousticcoatingrdquo Materials vol 12 no 16 2019

[7] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic wave by a doubly periodicstructure using the finite-element methodrdquo Journal of theAcoustical Society of America vol 88 2005

[8] T Meng ldquoSimplified model for predicting acoustic perfor-mance of an underwater sound absorption coatingrdquo Journal ofVibration and Control vol 20 no 3 pp 339ndash354 2014

[9] L M Lu J H Wen H G Zhao and X S Wen ldquoDynamicalmechanical property of viscoelastic materials and its effect onacoustic absorption of coatingrdquo Acta Physica Sinica vol 63no 15 2014

[10] Z Wei H Hou N Gao Y Huang and J Yang ldquoComplexYoungrsquos modulus measurement by incident wave extractingin a thin resonant barrdquo Journal of the Acoustical Society ofAmerica vol 142 no 6 pp 3436ndash3442 2017

[11] N W Tschoegl W G Knauss and I Emri ldquoPoissonrsquos ratio inlinear viscoelasticity - A critical reviewrdquo Mechanics of Time-Dependent Materials vol 6 no 1 pp 3ndash51 2002

[12] T Pritz ldquoFrequency dependences of complex moduli andcomplex Poissonrsquos ratio of real solid materialsrdquo Journal ofSound and Vibration vol 214 no 1 pp 83ndash104 1998

[13] J Zhong H Zhao H Yang J Yin and J Wen ldquoEffect ofPoissonrsquos loss factor of rubbery material on underwater soundabsorption of anechoic coatingsrdquo Journal of Sound and Vi-bration vol 424 pp 293ndash301 2018

[14] H Meng J Wen H Zhao L Lv and X Wen ldquoAnalysis ofabsorption performances of anechoic layers with steel platebackingrdquo Journal of the Acoustical Society of America vol 132no 1 pp 69ndash75 2012

[15] V Easwaran and M L Munjal ldquoAnalysis of reflectioncharacteristics of a normal incidence plane wave on resonantsound absorbers A finite element approachrdquo Journal of theAcoustical Society of America vol 93 no 3 pp 1308ndash13181993

[16] Z Jin Y Yin and B Liu ldquoEquivalent modulus method forfinite element simulation of the sound absorption of anechoiccoating backed with orthogonally rib-stiffened platerdquo Journalof Sound and Vibration vol 366 pp 357ndash371 2016

[17] L M Lu J H Wen H Zhao H Meng and X S Wen ldquoLow-frequency acoustic absorption of viscoelastic coating withvarious shapes of scatterersrdquo Wuli XuebaoActa PhysicaSinica vol 61 2012

[18] J Li and S Li ldquoTopology optimization of anechoic coating formaximizing sound absorptionrdquo Journal of Vibration andControl vol 24 no 11 pp 2369ndash2385 2018

[19] C Cai K C Hung and M S Khan ldquoSimulation-basedanalysis of acoustic absorbent lining subject to normal planewave incidencerdquo Journal of Sound and Vibration vol 291no 3-5 pp 656ndash680 2006

[20] S N Panigrahi C S Jog and M L Munjal ldquoMulti-focusdesign of underwater noise control linings based on finiteelement analysisrdquo Applied Acoustics vol 69 no 12pp 1141ndash1153 2008

[21] H Meng J Wen H Zhao and X Wen ldquoOptimization oflocally resonant acoustic metamaterials on underwater soundabsorption characteristicsrdquo Journal of Sound and Vibrationvol 331 no 20 pp 4406ndash4416 2012

10 Shock and Vibration

[22] M A Lewniska V G Kouznetsova J A W van DommelenA O Krushynska and M G D Geers ldquoe attenuationperformance of locally resonant acoustic metamaterials basedon generalised viscoelastic modellingrdquo International Journalof Solids and Structures vol 126 pp 163ndash174 2017

[23] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic-wave by a doubly periodicstructure using the finite-element method- application toalberich anechoic coatingrdquo Journal of the Acoustical Society ofAmerica vol 90 no 6 pp 3356ndash3367 1991

[24] J Villalobos-Luna P Lopez-Cruz R Ramirez-Valencia andF J Elizondo-Garza ldquoSimulation of acoustic wave behavior inducts and a plenum using finite elements package ANSYSrdquoJournal of the Acoustical Society of America vol 112 no 5p 2427 2002

[25] Y Xu G Zhang B Zhou H Wang and Q Tang ldquoAnalysis ofacoustic radiation problems using the cell-based smoothedradial point interpolation method with Dirichlet-to-Neu-mann boundary conditionrdquo Engineering Analysis withBoundary Elements vol 108 no Nov pp 447ndash458 2019

[26] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wavenormal incidence conditionrdquo in Proceedings of the 4th In-ternational Conference on Smart Material Research ICSMR2018 Sydney Australia November 2018

[27] Y E Hanfeng M Tao and L I Junjie ldquoSound absorptionperformance analysis of anechoic coatings under obliqueincidence condition based on COMSOLrdquo Journal of Vibrationand Shock vol 038 no 12 pp 213ndash218 2019

[28] T Pritz ldquoe Poissonrsquos loss factor of solid viscoelastic ma-terialsrdquo Journal of Sound and Vibration vol 306 no 3-5pp 790ndash802 2007

[29] S H Sohrabi and M J Ketabdari ldquoNumerical simulation of aviscoelastic sound absorbent coating with a doubly periodicarray of cavitiesrdquo Cogent Engineering vol 5 no 1 2018

Shock and Vibration 11

between the two field points is denoted as s and the distancebetween field point 1 and interface between water andacoustic coating is denoted as zl

e sound pressure of incident and reflected waves inwater can be expressed as follows

pi PIejkz

pr PReminus jkz

(9)

where k represents the wave number PI and PR refer to theamplitudes of the incident wave pi and reflected wave pr onthe XOY plane of the datum plane respectively PR rPIand r stands for the reflection coefficient

erefore the sound pressure at the two field points isthe superposition of the incident and the reflected wavesen

p1 PIejkzl + PRe

minus jkzl

p2 PIejk zlminus s( ) + PRe

minus jk zlminus s( )(10)

e transfer function from field point 1 to field point 2 ofthe total sound field is written as follows

H12 p2

p1

PIejk zlminus s( ) + rPIe

minus jk zlminus s( )

PIejkzl + rPIe

minusjkzl (11)

e reflection coefficient r can be obtained using

r p2p1 minus e

minus jks

ejks

minus p2p1e

j2kxl (12)

During the process of simulation calculation the soundpressure p1 and p2 of field point 1 and 2 could be obtainedWith s and zl used to obtain the reflection coefficient r thesound absorption coefficient is expressed as

α 1 minus r2 (13)

In order to verify the accuracy of the calculation resultsof the finite element model a 50mm thick homogeneouscoating was selected from Ref [13] for calculation egeometric parameters and material parameters of thecoating are consistent with the literature at is to say theYoung modulus of rubber is E 140MPa the loss factor is023 the density is 1100 kgm3 Poissonrsquos ratio is 049 and 0

is taken as Poissonrsquos loss factor e Young modulus of thesteel plate is 216times105MPa Poissonrsquos ratio is 03 and thedensity is 7800 kgm3 e density of water is 1000 kgm3and the velocity of sound is 1489ms e above values areused for the material properties of water and steel plate later

We performed a mesh sensitivity analysis to representthe mesh dependency of the current problem e sidelength of tetrahedral meshers of water acoustic coating andsteel plate is set to 1mm 2mm 4mm and 6mm and thecalculation results of sound absorption coefficient are ob-served e results show that the water mesh size has noeffect on the results (the graphical results are not given in thepaper) Figure 4 shows the comparison drawn between thesound absorption coefficient calculated using the finite el-ement model under different mesh size settings and thecalculation results shown in Ref [13] e percentage errorsbetween the current results and Ref [13] are 23 2995 and 116 According to the results when the meshsize is set to 1mm and 2mm the sound absorption coef-ficient calculation results are basically consistent with thosein Ref [13] indicating the feasibility of applying the finiteelement model to conduct research on the setting of Pois-sonrsquos ratio

5 The Impact of Poissonrsquos Ratios on thePerformance of Acoustic Coatings inSound Absorption

In Section 3 the abovementioned finite element model isadopted to calculate the performance of the three acousticcoatings in sound absorption In the model Poissonrsquos ratio isset to 0490 0493 0496 0499 and 04999 according to theapproximate incompressible characteristics of viscoelasticmaterials and the values commonly used for researchSimilar to the modulus Poissonrsquos ratio is a complex numberIn the case of lower analytical frequency than 5000Hzhowever the imaginary part of Poissonrsquos ratio would besignificantly smaller than the real part indicating thatPoissonrsquos loss factor is as insignificant as its impact on thecharacteristics of sound absorption exhibited by varioustypes of acoustic coatings that can be ignored Moreover formost acoustic coatings the sound absorption coefficients areclose to 1 if the analysis frequency exceeds 5000Hzus the

Plane sound wave

Water

Anechoic coatingsSteel

Field point 1 Field point 2 X

Y

S1

S2

S3

S4

s

zl ZX

YO

O

Figure 3 Finite element model

4 Shock and Vibration

analytical frequency used in this paper ranges from 100 to5000Hz and Poissonrsquos ratio could be set as a real number

For simplicity purpose the modulus of materials was setin some research studies to a constant regardless of fre-quency for the analysis and calculation of the performance ofacoustic coatings in sound absorption In fact however itvaries with frequency

As shown in Figure 5 the static modulus is assigned withthe value shown in Ref [13] so that E 14times108 Pa and theloss factor is 023 Figure 5 shows the results of finite elementcalculation for the sound absorption coefficients of the threecoatings

Figure 6 shows the sound absorption coefficients of thethree coatings as obtained when the modulus of the materialis assigned with the modulus parameter of the carbon blackfilled nitrile rubber shown in Figure 7

According to Figures 5 and 6 with the increase inPoissonrsquos ratio the first peak of the sound absorption co-efficient curve shifts to high frequency As for the homo-geneous acoustic coating the same conclusion can be drawndespite no appearance of the sound absorption peak in thefigure e first peak of sound absorption shifts to lowfrequency which causes the sound absorption coefficient todecline with the rise of Poissonrsquos ratio when the frequencyfalls below the peak is is because the velocity of longi-tudinal sound increases and the attenuation constant de-creases with the rise of Poissonrsquos ratio as a result of whichthe sound absorption coefficient of the acoustic coatingdrops as shown in Equations (1)ndash(8) in Section 2

In figures 5(b) 5(c) 6(b) and 6(c) when Poissonrsquos ratiois 0499 and 04999 the characteristics of sound absorptioncoefficient curve are quite different from those whenPoissonrsquos ratio is 0490 0493 and 0496 which may be dueto the high sensitivity of sound absorption coefficient toPoissonrsquos ratio As shown in Equations (3) Poissonrsquos ratio isclose to 05 and the denominator in the calculation formulaof the Lamy constant tends to 0 A small change in Poissonrsquos

ratio can cause a great change in the Lamy constantresulting in a large change in sound absorption coefficientTable 1 shows the percentage of change rate of Poissonrsquosratio and the percentage of change rate of the Lamy constantwhen Youngrsquos modulus is 140MPa It can be seen that whenPoissonrsquos ratio is 0493 and 0496 the change rate of theLamy constant is relatively small while when Poissonrsquos ratiois 0499 and 04999 the change rate of the Lamy constant isvery large erefore when Poissonrsquos ratio is 0499 and04999 the sound absorption coefficient curve is quitedifferent

In order to further analyze the influence of Poissonrsquosratios on the sound absorption coefficient the modulus wasassigned with the static value and dynamic value Figure 8shows the average sound absorption coefficient of the threecoatings ranging from 100 to 5000Hz As shown in thefigure the average value of sound absorption decreases withthe increase in Poissonrsquos ratio regardless of whether themodulus value is static or dynamic and what type of acousticcoating it is Besides it can be seen from Figure 8 that thehomogeneous one has different characteristics to the othertwo acoustic coatings as manifested mainly in its smalleraverage sound absorption coefficient and the pattern ofcoefficient variations is different from that of the other two asreflected in the frequency curves of the coefficient in Fig-ures 5 and 6 is is because the two coatings follow morecomplicated mechanisms of acoustic energy loss which issimilar to the conversion from longitudinal waves to shearwaves Although the Alberich acoustic coating and thetrumpet cavity-type coating have good sound absorptioneffect when Poissonrsquos ratio is less than 0499 the soundabsorption performance decreases suddenly when Poissonrsquosratio increases from 0499 to 04999 e reason is shown inTable 2When Poissonrsquos ratio increases from 0499 to 04999Poissonrsquos ratio only increases by 00001 but the Lamyconstant does increase by an order of magnitude en thesound absorption coefficient decreases suddenly

According to Figures 5 6 and 8 the sound absorptioncoefficient calculated by FEM using the static modulus issignificantly lower than using the dynamic one which isbecause the dynamic modulus is lower than the static oneand the velocity of sound is lower accordingly Additionallycoupled with the higher loss factor the attenuation constantis reduced which can be calculated using (7) and (8)Consequently the average sound absorption coefficientincreases

6 Poissonrsquos Ratio Varying with Frequency

As shown in Figure 2 a standing wave tube method wasapplied to measure the vertical incidence sound absorptioncoefficient of the trumpet cavity-type acoustic coating Inaddition the modulus of the nitrile rubber used for acousticcoating was measured (Figure 7) and inputted into the finiteelement model Poissonrsquos ratio was set to 0496 0497 04980499 04992 04994 04995 04996 04997 and 04998respectively In the frequency range of 1000ndash5000Hz thesound absorption coefficients of one-third octave range ofthe acoustic coating were obtained by means of test and

0 5000 10000 15000 20000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Ref [13]1 mmerr=232 mmerr=29

4 mmerr=956 mmerr=116

Figure 4 Comparison of calculation results in this paper and Ref[13]

Shock and Vibration 5

calculation as shown in Table 2 and Figure 9 According tothe figure when Poissonrsquos ratio was set to the above valuesthe finite element calculation results of the sound absorptioncoefficient were starkly different from the measured results

As indicated by the results of Ref [12 28] Poissonrsquos ratiodrops with the rise of frequency According to this patternand Table 1 a linear interpolation method was adopted toobtain Poissonrsquos ratio which varies with frequency Taking1000Hz as an example the sound absorption coefficientmeasured is 0420 which is between the last two values of0442 and 0336 in the first column of the coefficients shownin the table e corresponding Poissonrsquos ratio is 04997 and

04998 respectively e linear interpolation was conductedto obtain Poissonrsquos ratio of 049972 e shaded numberslisted in the table represent the interval values of the in-terpolation Varying within the frequency of 1000ndash5000Hzthe interpolated dynamic Poissonrsquos ratios are presented inFigure 10 After linear fitting they were found consistentwith the characteristic as described in Ref [12 29] Morespecifically Poissonrsquos ratio at low frequencies and the log-arithmic frequency bears linear relationship

With the interpolated dynamic Poissonrsquos ratio inputtedinto the finite element model the calculation results wereobtained to indicate that the sound absorption coefficients

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06A

bsor

ptio

n co

effic

ient

s

Frequency (Hz)

049004930496

049904999

(a)

Abs

orpt

ion

coef

ficie

nts

049004930496

049904999

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

Frequency (Hz)

(b)

Abs

orpt

ion

coef

ficie

nts

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

Frequency (Hz)

049004930496

049904999

00

02

04

06

(c)

Figure 5When the modulus shows no change with frequency the sound absorption coefficient of the acoustic coating varies with Poissonrsquosratio (a) homogeneous coating (b) Alberich coating (c) trumpet cavity-type coating

6 Shock and Vibration

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10A

bsor

ptio

n co

effic

ient

s

Frequency (Hz)

049004930496

049904999

(a)

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

049004930496

049904999

(b)

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

049004930496

049904999

(c)

Figure 6 When the modulus varies with frequency the sound absorption coefficient of the acoustic coating varies with Poissonrsquos ratio(a) homogeneous coating (b) Alberich coating (c) trumpet cavity-type coating

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

0102030405060708090

100110120130

Mod

ulus

(MPa

)

Frequency (Hz)

Storage modulusLoss modulus

Figure 7 Dynamic modulus of carbon black filled nitrile rubber

Shock and Vibration 7

were well consistent with the test results as shown inFigure 9 e aforementioned dynamic Poissonrsquos ratios wereused to calculate the sound absorption characteristics of theAlberich coating made of the same nitrile rubber as thetrumpet cavity-type coating Figure 11 shows the calculationand measured results suggesting that the sound absorption

coefficients calculated using the static Poissonrsquos ratios aresignificantly deviant from the test results and that the onescalculated using the dynamic ratios are more consistent withthe test results In addition it can be seen from the figure thatdynamic Poissonrsquos ratio should be used to obtain moreaccurate results for the finite element calculation of the

Table 1 Poissonrsquos ratio Lamy constant and their rate of change

Poissonrsquos ratio Lamy constant Change rate of Poissonrsquos ratio Change rate of Lamy constant0490 23E+ 09 0 00493 33E+ 09 061 430496 58E+ 09 122 1520499 233E+ 10 184 91204999 233E+ 11 202 10034

049 0493 0496 0499 04999

00

01

02

03

04

05

06

07

08

09

Ave

rage

abso

rptio

n co

effic

ient

Poissons Ratio

Static modulus HomogenousStatic modulus AlberichStatic modulus Horn

Dynamic modulus HomogenousDynamic modulus AlberichDynamic modulus Horn

Figure 8 Variation of the average sound absorption coefficient with Poissonrsquos ratio

Table 2 Calculated and measured sound absorption coefficient

FrequencyPoissonrsquos ratio 1000 1250 1600 2000 2500 3150 4000 5000

Calculated

0496 0922 0935 0942 0935 0911 0866 0796 07180497 0917 0939 0957 0960 0946 0909 0841 07560498 0889 0922 0956 0978 0982 0963 0908 08260499 0772 0821 0879 0930 0969 0996 0994 095304992 0720 0769 0831 0888 0937 0978 0997 097904994 0642 0692 0756 0817 0874 0929 0973 098504995 0589 0638 0701 0764 0823 0884 0940 097004996 0524 0570 0631 0692 0752 0816 0883 093004997 0442 0483 0538 0595 0651 0715 0787 084704998 0336 0369 0414 0461 0509 0564 0630 0692

Measured 0420 0615 0815 0900 0985 0955 0830 0755

8 Shock and Vibration

1000 1250 1600 2000 2500 3150 4000 5000

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with constant Poissons ratioTestedCalculated with dynamic Poissons ratio

Figure 9 Calculation and measurement results of the sound absorption coefficient of the trumpet cavity-type acoustic coating

1000 1250 1600 2000 2500 3150 4000 500004960

04965

04970

04975

04980

04985

04990

04995

05000

05005

Poiss

ons

Ratio

Frequency (Hz)

InterpolatedFited

Figure 10 Interpolated Poissonrsquos ratio varying with frequency

1000 1250 1600 2000 2500 3150 4000 5000

02

03

04

05

06

07

08

09

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with Poissons Ratio = 0490Calculated with Poissons Ratio = 0493Calculated with Poissons Ratio = 0496Calculated with Poissons Ratio = 0499Calculated with Poissons Ratio = 04999Calculated with dynamic Poissons RatioMeasured

Figure 11 Calculation and measurement results of the sound absorption coefficients of the Alberich coating

Shock and Vibration 9

sound absorption characteristics of acoustic coatings Un-fortunately there is still no accurate and reliable methodavailable to get the dynamic Poissonrsquos ratios of viscoelasticmaterials [11]

7 Conclusion

(1) When Poissonrsquos ratio of homogeneous acousticcoating Alberich acoustic coating and trumpetcavity-type coating varies from 049 to 04999 thelarger Poissonrsquos ratio is the larger the frequency ofthe first absorption peak is the smaller the ab-sorption coefficient is in the frequency range belowthe first absorption peak and the smaller the averageabsorption coefficient is in the analysis frequencyrange

(2) More accurate calculation results of sound absorp-tion coefficient can be obtained by using Poissonrsquosratio varying with frequency In the analysis of thesound absorption performance of acoustic coatingsthe accurate dynamic Poissonrsquos ratio with the changeof frequency should be used

(3) Poissonrsquos ratio of viscoelastic materials varies in avery small numerical range such a small change hasa great influence on the sound absorption perfor-mance of acoustic coatings erefore it is urgent tostudy the accurate method to obtain the dynamicPoissonrsquos ratio

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the Scientific Re-search Project of Education Department of Hunan ProvinceChina (Project no 19C0589)

References

[1] L Alimonti and N Atalla ldquoEfficient modeling of flat andhomogeneous acoustic treatments for vibroacoustic finiteelement analysis Finite size correction by image sourcesrdquoJournal of Sound and Vibration vol 388 pp 201ndash215 2017

[2] X L Yao X Z Wang F Z Pang and L Q Sun ldquoStudy onAcoustic Characteristics of Underwater Coating StructurerdquoNoise and Vibration Control vol 30 no 4 pp 1ndash5 2010

[3] S M Ivansson ldquoErratumrdquo Journal of the Acoustical Society ofAmerica vol 131 no 4 p 3180 2012

[4] S M Ivansson ldquoNumerical design of Alberich anechoiccoatings with superellipsoidal cavities of mixed sizesrdquo Journalof the Acoustical Society of America vol 123 no 5 p 34662008

[5] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wave

normal incidence conditionrdquo IOP Conference Series Mate-rials Science and Engineering vol 522 no 1 Article ID012001 2019

[6] H Bai Z Zhan J Liu and Z Ren ldquoFrom local structure tooverall performance An overview on the design of an acousticcoatingrdquo Materials vol 12 no 16 2019

[7] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic wave by a doubly periodicstructure using the finite-element methodrdquo Journal of theAcoustical Society of America vol 88 2005

[8] T Meng ldquoSimplified model for predicting acoustic perfor-mance of an underwater sound absorption coatingrdquo Journal ofVibration and Control vol 20 no 3 pp 339ndash354 2014

[9] L M Lu J H Wen H G Zhao and X S Wen ldquoDynamicalmechanical property of viscoelastic materials and its effect onacoustic absorption of coatingrdquo Acta Physica Sinica vol 63no 15 2014

[10] Z Wei H Hou N Gao Y Huang and J Yang ldquoComplexYoungrsquos modulus measurement by incident wave extractingin a thin resonant barrdquo Journal of the Acoustical Society ofAmerica vol 142 no 6 pp 3436ndash3442 2017

[11] N W Tschoegl W G Knauss and I Emri ldquoPoissonrsquos ratio inlinear viscoelasticity - A critical reviewrdquo Mechanics of Time-Dependent Materials vol 6 no 1 pp 3ndash51 2002

[12] T Pritz ldquoFrequency dependences of complex moduli andcomplex Poissonrsquos ratio of real solid materialsrdquo Journal ofSound and Vibration vol 214 no 1 pp 83ndash104 1998

[13] J Zhong H Zhao H Yang J Yin and J Wen ldquoEffect ofPoissonrsquos loss factor of rubbery material on underwater soundabsorption of anechoic coatingsrdquo Journal of Sound and Vi-bration vol 424 pp 293ndash301 2018

[14] H Meng J Wen H Zhao L Lv and X Wen ldquoAnalysis ofabsorption performances of anechoic layers with steel platebackingrdquo Journal of the Acoustical Society of America vol 132no 1 pp 69ndash75 2012

[15] V Easwaran and M L Munjal ldquoAnalysis of reflectioncharacteristics of a normal incidence plane wave on resonantsound absorbers A finite element approachrdquo Journal of theAcoustical Society of America vol 93 no 3 pp 1308ndash13181993

[16] Z Jin Y Yin and B Liu ldquoEquivalent modulus method forfinite element simulation of the sound absorption of anechoiccoating backed with orthogonally rib-stiffened platerdquo Journalof Sound and Vibration vol 366 pp 357ndash371 2016

[17] L M Lu J H Wen H Zhao H Meng and X S Wen ldquoLow-frequency acoustic absorption of viscoelastic coating withvarious shapes of scatterersrdquo Wuli XuebaoActa PhysicaSinica vol 61 2012

[18] J Li and S Li ldquoTopology optimization of anechoic coating formaximizing sound absorptionrdquo Journal of Vibration andControl vol 24 no 11 pp 2369ndash2385 2018

[19] C Cai K C Hung and M S Khan ldquoSimulation-basedanalysis of acoustic absorbent lining subject to normal planewave incidencerdquo Journal of Sound and Vibration vol 291no 3-5 pp 656ndash680 2006

[20] S N Panigrahi C S Jog and M L Munjal ldquoMulti-focusdesign of underwater noise control linings based on finiteelement analysisrdquo Applied Acoustics vol 69 no 12pp 1141ndash1153 2008

[21] H Meng J Wen H Zhao and X Wen ldquoOptimization oflocally resonant acoustic metamaterials on underwater soundabsorption characteristicsrdquo Journal of Sound and Vibrationvol 331 no 20 pp 4406ndash4416 2012

10 Shock and Vibration

[22] M A Lewniska V G Kouznetsova J A W van DommelenA O Krushynska and M G D Geers ldquoe attenuationperformance of locally resonant acoustic metamaterials basedon generalised viscoelastic modellingrdquo International Journalof Solids and Structures vol 126 pp 163ndash174 2017

[23] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic-wave by a doubly periodicstructure using the finite-element method- application toalberich anechoic coatingrdquo Journal of the Acoustical Society ofAmerica vol 90 no 6 pp 3356ndash3367 1991

[24] J Villalobos-Luna P Lopez-Cruz R Ramirez-Valencia andF J Elizondo-Garza ldquoSimulation of acoustic wave behavior inducts and a plenum using finite elements package ANSYSrdquoJournal of the Acoustical Society of America vol 112 no 5p 2427 2002

[25] Y Xu G Zhang B Zhou H Wang and Q Tang ldquoAnalysis ofacoustic radiation problems using the cell-based smoothedradial point interpolation method with Dirichlet-to-Neu-mann boundary conditionrdquo Engineering Analysis withBoundary Elements vol 108 no Nov pp 447ndash458 2019

[26] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wavenormal incidence conditionrdquo in Proceedings of the 4th In-ternational Conference on Smart Material Research ICSMR2018 Sydney Australia November 2018

[27] Y E Hanfeng M Tao and L I Junjie ldquoSound absorptionperformance analysis of anechoic coatings under obliqueincidence condition based on COMSOLrdquo Journal of Vibrationand Shock vol 038 no 12 pp 213ndash218 2019

[28] T Pritz ldquoe Poissonrsquos loss factor of solid viscoelastic ma-terialsrdquo Journal of Sound and Vibration vol 306 no 3-5pp 790ndash802 2007

[29] S H Sohrabi and M J Ketabdari ldquoNumerical simulation of aviscoelastic sound absorbent coating with a doubly periodicarray of cavitiesrdquo Cogent Engineering vol 5 no 1 2018

Shock and Vibration 11

analytical frequency used in this paper ranges from 100 to5000Hz and Poissonrsquos ratio could be set as a real number

For simplicity purpose the modulus of materials was setin some research studies to a constant regardless of fre-quency for the analysis and calculation of the performance ofacoustic coatings in sound absorption In fact however itvaries with frequency

As shown in Figure 5 the static modulus is assigned withthe value shown in Ref [13] so that E 14times108 Pa and theloss factor is 023 Figure 5 shows the results of finite elementcalculation for the sound absorption coefficients of the threecoatings

Figure 6 shows the sound absorption coefficients of thethree coatings as obtained when the modulus of the materialis assigned with the modulus parameter of the carbon blackfilled nitrile rubber shown in Figure 7

According to Figures 5 and 6 with the increase inPoissonrsquos ratio the first peak of the sound absorption co-efficient curve shifts to high frequency As for the homo-geneous acoustic coating the same conclusion can be drawndespite no appearance of the sound absorption peak in thefigure e first peak of sound absorption shifts to lowfrequency which causes the sound absorption coefficient todecline with the rise of Poissonrsquos ratio when the frequencyfalls below the peak is is because the velocity of longi-tudinal sound increases and the attenuation constant de-creases with the rise of Poissonrsquos ratio as a result of whichthe sound absorption coefficient of the acoustic coatingdrops as shown in Equations (1)ndash(8) in Section 2

In figures 5(b) 5(c) 6(b) and 6(c) when Poissonrsquos ratiois 0499 and 04999 the characteristics of sound absorptioncoefficient curve are quite different from those whenPoissonrsquos ratio is 0490 0493 and 0496 which may be dueto the high sensitivity of sound absorption coefficient toPoissonrsquos ratio As shown in Equations (3) Poissonrsquos ratio isclose to 05 and the denominator in the calculation formulaof the Lamy constant tends to 0 A small change in Poissonrsquos

ratio can cause a great change in the Lamy constantresulting in a large change in sound absorption coefficientTable 1 shows the percentage of change rate of Poissonrsquosratio and the percentage of change rate of the Lamy constantwhen Youngrsquos modulus is 140MPa It can be seen that whenPoissonrsquos ratio is 0493 and 0496 the change rate of theLamy constant is relatively small while when Poissonrsquos ratiois 0499 and 04999 the change rate of the Lamy constant isvery large erefore when Poissonrsquos ratio is 0499 and04999 the sound absorption coefficient curve is quitedifferent

In order to further analyze the influence of Poissonrsquosratios on the sound absorption coefficient the modulus wasassigned with the static value and dynamic value Figure 8shows the average sound absorption coefficient of the threecoatings ranging from 100 to 5000Hz As shown in thefigure the average value of sound absorption decreases withthe increase in Poissonrsquos ratio regardless of whether themodulus value is static or dynamic and what type of acousticcoating it is Besides it can be seen from Figure 8 that thehomogeneous one has different characteristics to the othertwo acoustic coatings as manifested mainly in its smalleraverage sound absorption coefficient and the pattern ofcoefficient variations is different from that of the other two asreflected in the frequency curves of the coefficient in Fig-ures 5 and 6 is is because the two coatings follow morecomplicated mechanisms of acoustic energy loss which issimilar to the conversion from longitudinal waves to shearwaves Although the Alberich acoustic coating and thetrumpet cavity-type coating have good sound absorptioneffect when Poissonrsquos ratio is less than 0499 the soundabsorption performance decreases suddenly when Poissonrsquosratio increases from 0499 to 04999 e reason is shown inTable 2When Poissonrsquos ratio increases from 0499 to 04999Poissonrsquos ratio only increases by 00001 but the Lamyconstant does increase by an order of magnitude en thesound absorption coefficient decreases suddenly

According to Figures 5 6 and 8 the sound absorptioncoefficient calculated by FEM using the static modulus issignificantly lower than using the dynamic one which isbecause the dynamic modulus is lower than the static oneand the velocity of sound is lower accordingly Additionallycoupled with the higher loss factor the attenuation constantis reduced which can be calculated using (7) and (8)Consequently the average sound absorption coefficientincreases

6 Poissonrsquos Ratio Varying with Frequency

As shown in Figure 2 a standing wave tube method wasapplied to measure the vertical incidence sound absorptioncoefficient of the trumpet cavity-type acoustic coating Inaddition the modulus of the nitrile rubber used for acousticcoating was measured (Figure 7) and inputted into the finiteelement model Poissonrsquos ratio was set to 0496 0497 04980499 04992 04994 04995 04996 04997 and 04998respectively In the frequency range of 1000ndash5000Hz thesound absorption coefficients of one-third octave range ofthe acoustic coating were obtained by means of test and

0 5000 10000 15000 20000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Ref [13]1 mmerr=232 mmerr=29

4 mmerr=956 mmerr=116

Figure 4 Comparison of calculation results in this paper and Ref[13]

Shock and Vibration 5

calculation as shown in Table 2 and Figure 9 According tothe figure when Poissonrsquos ratio was set to the above valuesthe finite element calculation results of the sound absorptioncoefficient were starkly different from the measured results

As indicated by the results of Ref [12 28] Poissonrsquos ratiodrops with the rise of frequency According to this patternand Table 1 a linear interpolation method was adopted toobtain Poissonrsquos ratio which varies with frequency Taking1000Hz as an example the sound absorption coefficientmeasured is 0420 which is between the last two values of0442 and 0336 in the first column of the coefficients shownin the table e corresponding Poissonrsquos ratio is 04997 and

04998 respectively e linear interpolation was conductedto obtain Poissonrsquos ratio of 049972 e shaded numberslisted in the table represent the interval values of the in-terpolation Varying within the frequency of 1000ndash5000Hzthe interpolated dynamic Poissonrsquos ratios are presented inFigure 10 After linear fitting they were found consistentwith the characteristic as described in Ref [12 29] Morespecifically Poissonrsquos ratio at low frequencies and the log-arithmic frequency bears linear relationship

With the interpolated dynamic Poissonrsquos ratio inputtedinto the finite element model the calculation results wereobtained to indicate that the sound absorption coefficients

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06A

bsor

ptio

n co

effic

ient

s

Frequency (Hz)

049004930496

049904999

(a)

Abs

orpt

ion

coef

ficie

nts

049004930496

049904999

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

Frequency (Hz)

(b)

Abs

orpt

ion

coef

ficie

nts

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

Frequency (Hz)

049004930496

049904999

00

02

04

06

(c)

Figure 5When the modulus shows no change with frequency the sound absorption coefficient of the acoustic coating varies with Poissonrsquosratio (a) homogeneous coating (b) Alberich coating (c) trumpet cavity-type coating

6 Shock and Vibration

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10A

bsor

ptio

n co

effic

ient

s

Frequency (Hz)

049004930496

049904999

(a)

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

049004930496

049904999

(b)

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

049004930496

049904999

(c)

Figure 6 When the modulus varies with frequency the sound absorption coefficient of the acoustic coating varies with Poissonrsquos ratio(a) homogeneous coating (b) Alberich coating (c) trumpet cavity-type coating

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

0102030405060708090

100110120130

Mod

ulus

(MPa

)

Frequency (Hz)

Storage modulusLoss modulus

Figure 7 Dynamic modulus of carbon black filled nitrile rubber

Shock and Vibration 7

were well consistent with the test results as shown inFigure 9 e aforementioned dynamic Poissonrsquos ratios wereused to calculate the sound absorption characteristics of theAlberich coating made of the same nitrile rubber as thetrumpet cavity-type coating Figure 11 shows the calculationand measured results suggesting that the sound absorption

coefficients calculated using the static Poissonrsquos ratios aresignificantly deviant from the test results and that the onescalculated using the dynamic ratios are more consistent withthe test results In addition it can be seen from the figure thatdynamic Poissonrsquos ratio should be used to obtain moreaccurate results for the finite element calculation of the

Table 1 Poissonrsquos ratio Lamy constant and their rate of change

Poissonrsquos ratio Lamy constant Change rate of Poissonrsquos ratio Change rate of Lamy constant0490 23E+ 09 0 00493 33E+ 09 061 430496 58E+ 09 122 1520499 233E+ 10 184 91204999 233E+ 11 202 10034

049 0493 0496 0499 04999

00

01

02

03

04

05

06

07

08

09

Ave

rage

abso

rptio

n co

effic

ient

Poissons Ratio

Static modulus HomogenousStatic modulus AlberichStatic modulus Horn

Dynamic modulus HomogenousDynamic modulus AlberichDynamic modulus Horn

Figure 8 Variation of the average sound absorption coefficient with Poissonrsquos ratio

Table 2 Calculated and measured sound absorption coefficient

FrequencyPoissonrsquos ratio 1000 1250 1600 2000 2500 3150 4000 5000

Calculated

0496 0922 0935 0942 0935 0911 0866 0796 07180497 0917 0939 0957 0960 0946 0909 0841 07560498 0889 0922 0956 0978 0982 0963 0908 08260499 0772 0821 0879 0930 0969 0996 0994 095304992 0720 0769 0831 0888 0937 0978 0997 097904994 0642 0692 0756 0817 0874 0929 0973 098504995 0589 0638 0701 0764 0823 0884 0940 097004996 0524 0570 0631 0692 0752 0816 0883 093004997 0442 0483 0538 0595 0651 0715 0787 084704998 0336 0369 0414 0461 0509 0564 0630 0692

Measured 0420 0615 0815 0900 0985 0955 0830 0755

8 Shock and Vibration

1000 1250 1600 2000 2500 3150 4000 5000

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with constant Poissons ratioTestedCalculated with dynamic Poissons ratio

Figure 9 Calculation and measurement results of the sound absorption coefficient of the trumpet cavity-type acoustic coating

1000 1250 1600 2000 2500 3150 4000 500004960

04965

04970

04975

04980

04985

04990

04995

05000

05005

Poiss

ons

Ratio

Frequency (Hz)

InterpolatedFited

Figure 10 Interpolated Poissonrsquos ratio varying with frequency

1000 1250 1600 2000 2500 3150 4000 5000

02

03

04

05

06

07

08

09

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with Poissons Ratio = 0490Calculated with Poissons Ratio = 0493Calculated with Poissons Ratio = 0496Calculated with Poissons Ratio = 0499Calculated with Poissons Ratio = 04999Calculated with dynamic Poissons RatioMeasured

Figure 11 Calculation and measurement results of the sound absorption coefficients of the Alberich coating

Shock and Vibration 9

sound absorption characteristics of acoustic coatings Un-fortunately there is still no accurate and reliable methodavailable to get the dynamic Poissonrsquos ratios of viscoelasticmaterials [11]

7 Conclusion

(1) When Poissonrsquos ratio of homogeneous acousticcoating Alberich acoustic coating and trumpetcavity-type coating varies from 049 to 04999 thelarger Poissonrsquos ratio is the larger the frequency ofthe first absorption peak is the smaller the ab-sorption coefficient is in the frequency range belowthe first absorption peak and the smaller the averageabsorption coefficient is in the analysis frequencyrange

(2) More accurate calculation results of sound absorp-tion coefficient can be obtained by using Poissonrsquosratio varying with frequency In the analysis of thesound absorption performance of acoustic coatingsthe accurate dynamic Poissonrsquos ratio with the changeof frequency should be used

(3) Poissonrsquos ratio of viscoelastic materials varies in avery small numerical range such a small change hasa great influence on the sound absorption perfor-mance of acoustic coatings erefore it is urgent tostudy the accurate method to obtain the dynamicPoissonrsquos ratio

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the Scientific Re-search Project of Education Department of Hunan ProvinceChina (Project no 19C0589)

References

[1] L Alimonti and N Atalla ldquoEfficient modeling of flat andhomogeneous acoustic treatments for vibroacoustic finiteelement analysis Finite size correction by image sourcesrdquoJournal of Sound and Vibration vol 388 pp 201ndash215 2017

[2] X L Yao X Z Wang F Z Pang and L Q Sun ldquoStudy onAcoustic Characteristics of Underwater Coating StructurerdquoNoise and Vibration Control vol 30 no 4 pp 1ndash5 2010

[3] S M Ivansson ldquoErratumrdquo Journal of the Acoustical Society ofAmerica vol 131 no 4 p 3180 2012

[4] S M Ivansson ldquoNumerical design of Alberich anechoiccoatings with superellipsoidal cavities of mixed sizesrdquo Journalof the Acoustical Society of America vol 123 no 5 p 34662008

[5] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wave

normal incidence conditionrdquo IOP Conference Series Mate-rials Science and Engineering vol 522 no 1 Article ID012001 2019

[6] H Bai Z Zhan J Liu and Z Ren ldquoFrom local structure tooverall performance An overview on the design of an acousticcoatingrdquo Materials vol 12 no 16 2019

[7] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic wave by a doubly periodicstructure using the finite-element methodrdquo Journal of theAcoustical Society of America vol 88 2005

[8] T Meng ldquoSimplified model for predicting acoustic perfor-mance of an underwater sound absorption coatingrdquo Journal ofVibration and Control vol 20 no 3 pp 339ndash354 2014

[9] L M Lu J H Wen H G Zhao and X S Wen ldquoDynamicalmechanical property of viscoelastic materials and its effect onacoustic absorption of coatingrdquo Acta Physica Sinica vol 63no 15 2014

[10] Z Wei H Hou N Gao Y Huang and J Yang ldquoComplexYoungrsquos modulus measurement by incident wave extractingin a thin resonant barrdquo Journal of the Acoustical Society ofAmerica vol 142 no 6 pp 3436ndash3442 2017

[11] N W Tschoegl W G Knauss and I Emri ldquoPoissonrsquos ratio inlinear viscoelasticity - A critical reviewrdquo Mechanics of Time-Dependent Materials vol 6 no 1 pp 3ndash51 2002

[12] T Pritz ldquoFrequency dependences of complex moduli andcomplex Poissonrsquos ratio of real solid materialsrdquo Journal ofSound and Vibration vol 214 no 1 pp 83ndash104 1998

[13] J Zhong H Zhao H Yang J Yin and J Wen ldquoEffect ofPoissonrsquos loss factor of rubbery material on underwater soundabsorption of anechoic coatingsrdquo Journal of Sound and Vi-bration vol 424 pp 293ndash301 2018

[14] H Meng J Wen H Zhao L Lv and X Wen ldquoAnalysis ofabsorption performances of anechoic layers with steel platebackingrdquo Journal of the Acoustical Society of America vol 132no 1 pp 69ndash75 2012

[15] V Easwaran and M L Munjal ldquoAnalysis of reflectioncharacteristics of a normal incidence plane wave on resonantsound absorbers A finite element approachrdquo Journal of theAcoustical Society of America vol 93 no 3 pp 1308ndash13181993

[16] Z Jin Y Yin and B Liu ldquoEquivalent modulus method forfinite element simulation of the sound absorption of anechoiccoating backed with orthogonally rib-stiffened platerdquo Journalof Sound and Vibration vol 366 pp 357ndash371 2016

[17] L M Lu J H Wen H Zhao H Meng and X S Wen ldquoLow-frequency acoustic absorption of viscoelastic coating withvarious shapes of scatterersrdquo Wuli XuebaoActa PhysicaSinica vol 61 2012

[18] J Li and S Li ldquoTopology optimization of anechoic coating formaximizing sound absorptionrdquo Journal of Vibration andControl vol 24 no 11 pp 2369ndash2385 2018

[19] C Cai K C Hung and M S Khan ldquoSimulation-basedanalysis of acoustic absorbent lining subject to normal planewave incidencerdquo Journal of Sound and Vibration vol 291no 3-5 pp 656ndash680 2006

[20] S N Panigrahi C S Jog and M L Munjal ldquoMulti-focusdesign of underwater noise control linings based on finiteelement analysisrdquo Applied Acoustics vol 69 no 12pp 1141ndash1153 2008

[21] H Meng J Wen H Zhao and X Wen ldquoOptimization oflocally resonant acoustic metamaterials on underwater soundabsorption characteristicsrdquo Journal of Sound and Vibrationvol 331 no 20 pp 4406ndash4416 2012

10 Shock and Vibration

[22] M A Lewniska V G Kouznetsova J A W van DommelenA O Krushynska and M G D Geers ldquoe attenuationperformance of locally resonant acoustic metamaterials basedon generalised viscoelastic modellingrdquo International Journalof Solids and Structures vol 126 pp 163ndash174 2017

[23] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic-wave by a doubly periodicstructure using the finite-element method- application toalberich anechoic coatingrdquo Journal of the Acoustical Society ofAmerica vol 90 no 6 pp 3356ndash3367 1991

[24] J Villalobos-Luna P Lopez-Cruz R Ramirez-Valencia andF J Elizondo-Garza ldquoSimulation of acoustic wave behavior inducts and a plenum using finite elements package ANSYSrdquoJournal of the Acoustical Society of America vol 112 no 5p 2427 2002

[25] Y Xu G Zhang B Zhou H Wang and Q Tang ldquoAnalysis ofacoustic radiation problems using the cell-based smoothedradial point interpolation method with Dirichlet-to-Neu-mann boundary conditionrdquo Engineering Analysis withBoundary Elements vol 108 no Nov pp 447ndash458 2019

[26] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wavenormal incidence conditionrdquo in Proceedings of the 4th In-ternational Conference on Smart Material Research ICSMR2018 Sydney Australia November 2018

[27] Y E Hanfeng M Tao and L I Junjie ldquoSound absorptionperformance analysis of anechoic coatings under obliqueincidence condition based on COMSOLrdquo Journal of Vibrationand Shock vol 038 no 12 pp 213ndash218 2019

[28] T Pritz ldquoe Poissonrsquos loss factor of solid viscoelastic ma-terialsrdquo Journal of Sound and Vibration vol 306 no 3-5pp 790ndash802 2007

[29] S H Sohrabi and M J Ketabdari ldquoNumerical simulation of aviscoelastic sound absorbent coating with a doubly periodicarray of cavitiesrdquo Cogent Engineering vol 5 no 1 2018

Shock and Vibration 11

calculation as shown in Table 2 and Figure 9 According tothe figure when Poissonrsquos ratio was set to the above valuesthe finite element calculation results of the sound absorptioncoefficient were starkly different from the measured results

As indicated by the results of Ref [12 28] Poissonrsquos ratiodrops with the rise of frequency According to this patternand Table 1 a linear interpolation method was adopted toobtain Poissonrsquos ratio which varies with frequency Taking1000Hz as an example the sound absorption coefficientmeasured is 0420 which is between the last two values of0442 and 0336 in the first column of the coefficients shownin the table e corresponding Poissonrsquos ratio is 04997 and

04998 respectively e linear interpolation was conductedto obtain Poissonrsquos ratio of 049972 e shaded numberslisted in the table represent the interval values of the in-terpolation Varying within the frequency of 1000ndash5000Hzthe interpolated dynamic Poissonrsquos ratios are presented inFigure 10 After linear fitting they were found consistentwith the characteristic as described in Ref [12 29] Morespecifically Poissonrsquos ratio at low frequencies and the log-arithmic frequency bears linear relationship

With the interpolated dynamic Poissonrsquos ratio inputtedinto the finite element model the calculation results wereobtained to indicate that the sound absorption coefficients

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06A

bsor

ptio

n co

effic

ient

s

Frequency (Hz)

049004930496

049904999

(a)

Abs

orpt

ion

coef

ficie

nts

049004930496

049904999

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

Frequency (Hz)

(b)

Abs

orpt

ion

coef

ficie

nts

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

Frequency (Hz)

049004930496

049904999

00

02

04

06

(c)

Figure 5When the modulus shows no change with frequency the sound absorption coefficient of the acoustic coating varies with Poissonrsquosratio (a) homogeneous coating (b) Alberich coating (c) trumpet cavity-type coating

6 Shock and Vibration

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10A

bsor

ptio

n co

effic

ient

s

Frequency (Hz)

049004930496

049904999

(a)

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

049004930496

049904999

(b)

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

049004930496

049904999

(c)

Figure 6 When the modulus varies with frequency the sound absorption coefficient of the acoustic coating varies with Poissonrsquos ratio(a) homogeneous coating (b) Alberich coating (c) trumpet cavity-type coating

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

0102030405060708090

100110120130

Mod

ulus

(MPa

)

Frequency (Hz)

Storage modulusLoss modulus

Figure 7 Dynamic modulus of carbon black filled nitrile rubber

Shock and Vibration 7

were well consistent with the test results as shown inFigure 9 e aforementioned dynamic Poissonrsquos ratios wereused to calculate the sound absorption characteristics of theAlberich coating made of the same nitrile rubber as thetrumpet cavity-type coating Figure 11 shows the calculationand measured results suggesting that the sound absorption

coefficients calculated using the static Poissonrsquos ratios aresignificantly deviant from the test results and that the onescalculated using the dynamic ratios are more consistent withthe test results In addition it can be seen from the figure thatdynamic Poissonrsquos ratio should be used to obtain moreaccurate results for the finite element calculation of the

Table 1 Poissonrsquos ratio Lamy constant and their rate of change

Poissonrsquos ratio Lamy constant Change rate of Poissonrsquos ratio Change rate of Lamy constant0490 23E+ 09 0 00493 33E+ 09 061 430496 58E+ 09 122 1520499 233E+ 10 184 91204999 233E+ 11 202 10034

049 0493 0496 0499 04999

00

01

02

03

04

05

06

07

08

09

Ave

rage

abso

rptio

n co

effic

ient

Poissons Ratio

Static modulus HomogenousStatic modulus AlberichStatic modulus Horn

Dynamic modulus HomogenousDynamic modulus AlberichDynamic modulus Horn

Figure 8 Variation of the average sound absorption coefficient with Poissonrsquos ratio

Table 2 Calculated and measured sound absorption coefficient

FrequencyPoissonrsquos ratio 1000 1250 1600 2000 2500 3150 4000 5000

Calculated

0496 0922 0935 0942 0935 0911 0866 0796 07180497 0917 0939 0957 0960 0946 0909 0841 07560498 0889 0922 0956 0978 0982 0963 0908 08260499 0772 0821 0879 0930 0969 0996 0994 095304992 0720 0769 0831 0888 0937 0978 0997 097904994 0642 0692 0756 0817 0874 0929 0973 098504995 0589 0638 0701 0764 0823 0884 0940 097004996 0524 0570 0631 0692 0752 0816 0883 093004997 0442 0483 0538 0595 0651 0715 0787 084704998 0336 0369 0414 0461 0509 0564 0630 0692

Measured 0420 0615 0815 0900 0985 0955 0830 0755

8 Shock and Vibration

1000 1250 1600 2000 2500 3150 4000 5000

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with constant Poissons ratioTestedCalculated with dynamic Poissons ratio

Figure 9 Calculation and measurement results of the sound absorption coefficient of the trumpet cavity-type acoustic coating

1000 1250 1600 2000 2500 3150 4000 500004960

04965

04970

04975

04980

04985

04990

04995

05000

05005

Poiss

ons

Ratio

Frequency (Hz)

InterpolatedFited

Figure 10 Interpolated Poissonrsquos ratio varying with frequency

1000 1250 1600 2000 2500 3150 4000 5000

02

03

04

05

06

07

08

09

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with Poissons Ratio = 0490Calculated with Poissons Ratio = 0493Calculated with Poissons Ratio = 0496Calculated with Poissons Ratio = 0499Calculated with Poissons Ratio = 04999Calculated with dynamic Poissons RatioMeasured

Figure 11 Calculation and measurement results of the sound absorption coefficients of the Alberich coating

Shock and Vibration 9

sound absorption characteristics of acoustic coatings Un-fortunately there is still no accurate and reliable methodavailable to get the dynamic Poissonrsquos ratios of viscoelasticmaterials [11]

7 Conclusion

(1) When Poissonrsquos ratio of homogeneous acousticcoating Alberich acoustic coating and trumpetcavity-type coating varies from 049 to 04999 thelarger Poissonrsquos ratio is the larger the frequency ofthe first absorption peak is the smaller the ab-sorption coefficient is in the frequency range belowthe first absorption peak and the smaller the averageabsorption coefficient is in the analysis frequencyrange

(2) More accurate calculation results of sound absorp-tion coefficient can be obtained by using Poissonrsquosratio varying with frequency In the analysis of thesound absorption performance of acoustic coatingsthe accurate dynamic Poissonrsquos ratio with the changeof frequency should be used

(3) Poissonrsquos ratio of viscoelastic materials varies in avery small numerical range such a small change hasa great influence on the sound absorption perfor-mance of acoustic coatings erefore it is urgent tostudy the accurate method to obtain the dynamicPoissonrsquos ratio

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the Scientific Re-search Project of Education Department of Hunan ProvinceChina (Project no 19C0589)

References

[1] L Alimonti and N Atalla ldquoEfficient modeling of flat andhomogeneous acoustic treatments for vibroacoustic finiteelement analysis Finite size correction by image sourcesrdquoJournal of Sound and Vibration vol 388 pp 201ndash215 2017

[2] X L Yao X Z Wang F Z Pang and L Q Sun ldquoStudy onAcoustic Characteristics of Underwater Coating StructurerdquoNoise and Vibration Control vol 30 no 4 pp 1ndash5 2010

[3] S M Ivansson ldquoErratumrdquo Journal of the Acoustical Society ofAmerica vol 131 no 4 p 3180 2012

[4] S M Ivansson ldquoNumerical design of Alberich anechoiccoatings with superellipsoidal cavities of mixed sizesrdquo Journalof the Acoustical Society of America vol 123 no 5 p 34662008

[5] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wave

normal incidence conditionrdquo IOP Conference Series Mate-rials Science and Engineering vol 522 no 1 Article ID012001 2019

[6] H Bai Z Zhan J Liu and Z Ren ldquoFrom local structure tooverall performance An overview on the design of an acousticcoatingrdquo Materials vol 12 no 16 2019

[7] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic wave by a doubly periodicstructure using the finite-element methodrdquo Journal of theAcoustical Society of America vol 88 2005

[8] T Meng ldquoSimplified model for predicting acoustic perfor-mance of an underwater sound absorption coatingrdquo Journal ofVibration and Control vol 20 no 3 pp 339ndash354 2014

[9] L M Lu J H Wen H G Zhao and X S Wen ldquoDynamicalmechanical property of viscoelastic materials and its effect onacoustic absorption of coatingrdquo Acta Physica Sinica vol 63no 15 2014

[10] Z Wei H Hou N Gao Y Huang and J Yang ldquoComplexYoungrsquos modulus measurement by incident wave extractingin a thin resonant barrdquo Journal of the Acoustical Society ofAmerica vol 142 no 6 pp 3436ndash3442 2017

[11] N W Tschoegl W G Knauss and I Emri ldquoPoissonrsquos ratio inlinear viscoelasticity - A critical reviewrdquo Mechanics of Time-Dependent Materials vol 6 no 1 pp 3ndash51 2002

[12] T Pritz ldquoFrequency dependences of complex moduli andcomplex Poissonrsquos ratio of real solid materialsrdquo Journal ofSound and Vibration vol 214 no 1 pp 83ndash104 1998

[13] J Zhong H Zhao H Yang J Yin and J Wen ldquoEffect ofPoissonrsquos loss factor of rubbery material on underwater soundabsorption of anechoic coatingsrdquo Journal of Sound and Vi-bration vol 424 pp 293ndash301 2018

[14] H Meng J Wen H Zhao L Lv and X Wen ldquoAnalysis ofabsorption performances of anechoic layers with steel platebackingrdquo Journal of the Acoustical Society of America vol 132no 1 pp 69ndash75 2012

[15] V Easwaran and M L Munjal ldquoAnalysis of reflectioncharacteristics of a normal incidence plane wave on resonantsound absorbers A finite element approachrdquo Journal of theAcoustical Society of America vol 93 no 3 pp 1308ndash13181993

[16] Z Jin Y Yin and B Liu ldquoEquivalent modulus method forfinite element simulation of the sound absorption of anechoiccoating backed with orthogonally rib-stiffened platerdquo Journalof Sound and Vibration vol 366 pp 357ndash371 2016

[17] L M Lu J H Wen H Zhao H Meng and X S Wen ldquoLow-frequency acoustic absorption of viscoelastic coating withvarious shapes of scatterersrdquo Wuli XuebaoActa PhysicaSinica vol 61 2012

[18] J Li and S Li ldquoTopology optimization of anechoic coating formaximizing sound absorptionrdquo Journal of Vibration andControl vol 24 no 11 pp 2369ndash2385 2018

[19] C Cai K C Hung and M S Khan ldquoSimulation-basedanalysis of acoustic absorbent lining subject to normal planewave incidencerdquo Journal of Sound and Vibration vol 291no 3-5 pp 656ndash680 2006

[20] S N Panigrahi C S Jog and M L Munjal ldquoMulti-focusdesign of underwater noise control linings based on finiteelement analysisrdquo Applied Acoustics vol 69 no 12pp 1141ndash1153 2008

[21] H Meng J Wen H Zhao and X Wen ldquoOptimization oflocally resonant acoustic metamaterials on underwater soundabsorption characteristicsrdquo Journal of Sound and Vibrationvol 331 no 20 pp 4406ndash4416 2012

10 Shock and Vibration

[22] M A Lewniska V G Kouznetsova J A W van DommelenA O Krushynska and M G D Geers ldquoe attenuationperformance of locally resonant acoustic metamaterials basedon generalised viscoelastic modellingrdquo International Journalof Solids and Structures vol 126 pp 163ndash174 2017

[23] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic-wave by a doubly periodicstructure using the finite-element method- application toalberich anechoic coatingrdquo Journal of the Acoustical Society ofAmerica vol 90 no 6 pp 3356ndash3367 1991

[24] J Villalobos-Luna P Lopez-Cruz R Ramirez-Valencia andF J Elizondo-Garza ldquoSimulation of acoustic wave behavior inducts and a plenum using finite elements package ANSYSrdquoJournal of the Acoustical Society of America vol 112 no 5p 2427 2002

[25] Y Xu G Zhang B Zhou H Wang and Q Tang ldquoAnalysis ofacoustic radiation problems using the cell-based smoothedradial point interpolation method with Dirichlet-to-Neu-mann boundary conditionrdquo Engineering Analysis withBoundary Elements vol 108 no Nov pp 447ndash458 2019

[26] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wavenormal incidence conditionrdquo in Proceedings of the 4th In-ternational Conference on Smart Material Research ICSMR2018 Sydney Australia November 2018

[27] Y E Hanfeng M Tao and L I Junjie ldquoSound absorptionperformance analysis of anechoic coatings under obliqueincidence condition based on COMSOLrdquo Journal of Vibrationand Shock vol 038 no 12 pp 213ndash218 2019

[28] T Pritz ldquoe Poissonrsquos loss factor of solid viscoelastic ma-terialsrdquo Journal of Sound and Vibration vol 306 no 3-5pp 790ndash802 2007

[29] S H Sohrabi and M J Ketabdari ldquoNumerical simulation of aviscoelastic sound absorbent coating with a doubly periodicarray of cavitiesrdquo Cogent Engineering vol 5 no 1 2018

Shock and Vibration 11

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10A

bsor

ptio

n co

effic

ient

s

Frequency (Hz)

049004930496

049904999

(a)

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

049004930496

049904999

(b)

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

00

02

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

049004930496

049904999

(c)

Figure 6 When the modulus varies with frequency the sound absorption coefficient of the acoustic coating varies with Poissonrsquos ratio(a) homogeneous coating (b) Alberich coating (c) trumpet cavity-type coating

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

0102030405060708090

100110120130

Mod

ulus

(MPa

)

Frequency (Hz)

Storage modulusLoss modulus

Figure 7 Dynamic modulus of carbon black filled nitrile rubber

Shock and Vibration 7

were well consistent with the test results as shown inFigure 9 e aforementioned dynamic Poissonrsquos ratios wereused to calculate the sound absorption characteristics of theAlberich coating made of the same nitrile rubber as thetrumpet cavity-type coating Figure 11 shows the calculationand measured results suggesting that the sound absorption

coefficients calculated using the static Poissonrsquos ratios aresignificantly deviant from the test results and that the onescalculated using the dynamic ratios are more consistent withthe test results In addition it can be seen from the figure thatdynamic Poissonrsquos ratio should be used to obtain moreaccurate results for the finite element calculation of the

Table 1 Poissonrsquos ratio Lamy constant and their rate of change

Poissonrsquos ratio Lamy constant Change rate of Poissonrsquos ratio Change rate of Lamy constant0490 23E+ 09 0 00493 33E+ 09 061 430496 58E+ 09 122 1520499 233E+ 10 184 91204999 233E+ 11 202 10034

049 0493 0496 0499 04999

00

01

02

03

04

05

06

07

08

09

Ave

rage

abso

rptio

n co

effic

ient

Poissons Ratio

Static modulus HomogenousStatic modulus AlberichStatic modulus Horn

Dynamic modulus HomogenousDynamic modulus AlberichDynamic modulus Horn

Figure 8 Variation of the average sound absorption coefficient with Poissonrsquos ratio

Table 2 Calculated and measured sound absorption coefficient

FrequencyPoissonrsquos ratio 1000 1250 1600 2000 2500 3150 4000 5000

Calculated

0496 0922 0935 0942 0935 0911 0866 0796 07180497 0917 0939 0957 0960 0946 0909 0841 07560498 0889 0922 0956 0978 0982 0963 0908 08260499 0772 0821 0879 0930 0969 0996 0994 095304992 0720 0769 0831 0888 0937 0978 0997 097904994 0642 0692 0756 0817 0874 0929 0973 098504995 0589 0638 0701 0764 0823 0884 0940 097004996 0524 0570 0631 0692 0752 0816 0883 093004997 0442 0483 0538 0595 0651 0715 0787 084704998 0336 0369 0414 0461 0509 0564 0630 0692

Measured 0420 0615 0815 0900 0985 0955 0830 0755

8 Shock and Vibration

1000 1250 1600 2000 2500 3150 4000 5000

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with constant Poissons ratioTestedCalculated with dynamic Poissons ratio

Figure 9 Calculation and measurement results of the sound absorption coefficient of the trumpet cavity-type acoustic coating

1000 1250 1600 2000 2500 3150 4000 500004960

04965

04970

04975

04980

04985

04990

04995

05000

05005

Poiss

ons

Ratio

Frequency (Hz)

InterpolatedFited

Figure 10 Interpolated Poissonrsquos ratio varying with frequency

1000 1250 1600 2000 2500 3150 4000 5000

02

03

04

05

06

07

08

09

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with Poissons Ratio = 0490Calculated with Poissons Ratio = 0493Calculated with Poissons Ratio = 0496Calculated with Poissons Ratio = 0499Calculated with Poissons Ratio = 04999Calculated with dynamic Poissons RatioMeasured

Figure 11 Calculation and measurement results of the sound absorption coefficients of the Alberich coating

Shock and Vibration 9

sound absorption characteristics of acoustic coatings Un-fortunately there is still no accurate and reliable methodavailable to get the dynamic Poissonrsquos ratios of viscoelasticmaterials [11]

7 Conclusion

(1) When Poissonrsquos ratio of homogeneous acousticcoating Alberich acoustic coating and trumpetcavity-type coating varies from 049 to 04999 thelarger Poissonrsquos ratio is the larger the frequency ofthe first absorption peak is the smaller the ab-sorption coefficient is in the frequency range belowthe first absorption peak and the smaller the averageabsorption coefficient is in the analysis frequencyrange

(2) More accurate calculation results of sound absorp-tion coefficient can be obtained by using Poissonrsquosratio varying with frequency In the analysis of thesound absorption performance of acoustic coatingsthe accurate dynamic Poissonrsquos ratio with the changeof frequency should be used

(3) Poissonrsquos ratio of viscoelastic materials varies in avery small numerical range such a small change hasa great influence on the sound absorption perfor-mance of acoustic coatings erefore it is urgent tostudy the accurate method to obtain the dynamicPoissonrsquos ratio

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the Scientific Re-search Project of Education Department of Hunan ProvinceChina (Project no 19C0589)

References

[1] L Alimonti and N Atalla ldquoEfficient modeling of flat andhomogeneous acoustic treatments for vibroacoustic finiteelement analysis Finite size correction by image sourcesrdquoJournal of Sound and Vibration vol 388 pp 201ndash215 2017

[2] X L Yao X Z Wang F Z Pang and L Q Sun ldquoStudy onAcoustic Characteristics of Underwater Coating StructurerdquoNoise and Vibration Control vol 30 no 4 pp 1ndash5 2010

[3] S M Ivansson ldquoErratumrdquo Journal of the Acoustical Society ofAmerica vol 131 no 4 p 3180 2012

[4] S M Ivansson ldquoNumerical design of Alberich anechoiccoatings with superellipsoidal cavities of mixed sizesrdquo Journalof the Acoustical Society of America vol 123 no 5 p 34662008

[5] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wave

normal incidence conditionrdquo IOP Conference Series Mate-rials Science and Engineering vol 522 no 1 Article ID012001 2019

[6] H Bai Z Zhan J Liu and Z Ren ldquoFrom local structure tooverall performance An overview on the design of an acousticcoatingrdquo Materials vol 12 no 16 2019

[7] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic wave by a doubly periodicstructure using the finite-element methodrdquo Journal of theAcoustical Society of America vol 88 2005

[8] T Meng ldquoSimplified model for predicting acoustic perfor-mance of an underwater sound absorption coatingrdquo Journal ofVibration and Control vol 20 no 3 pp 339ndash354 2014

[9] L M Lu J H Wen H G Zhao and X S Wen ldquoDynamicalmechanical property of viscoelastic materials and its effect onacoustic absorption of coatingrdquo Acta Physica Sinica vol 63no 15 2014

[10] Z Wei H Hou N Gao Y Huang and J Yang ldquoComplexYoungrsquos modulus measurement by incident wave extractingin a thin resonant barrdquo Journal of the Acoustical Society ofAmerica vol 142 no 6 pp 3436ndash3442 2017

[11] N W Tschoegl W G Knauss and I Emri ldquoPoissonrsquos ratio inlinear viscoelasticity - A critical reviewrdquo Mechanics of Time-Dependent Materials vol 6 no 1 pp 3ndash51 2002

[12] T Pritz ldquoFrequency dependences of complex moduli andcomplex Poissonrsquos ratio of real solid materialsrdquo Journal ofSound and Vibration vol 214 no 1 pp 83ndash104 1998

[13] J Zhong H Zhao H Yang J Yin and J Wen ldquoEffect ofPoissonrsquos loss factor of rubbery material on underwater soundabsorption of anechoic coatingsrdquo Journal of Sound and Vi-bration vol 424 pp 293ndash301 2018

[14] H Meng J Wen H Zhao L Lv and X Wen ldquoAnalysis ofabsorption performances of anechoic layers with steel platebackingrdquo Journal of the Acoustical Society of America vol 132no 1 pp 69ndash75 2012

[15] V Easwaran and M L Munjal ldquoAnalysis of reflectioncharacteristics of a normal incidence plane wave on resonantsound absorbers A finite element approachrdquo Journal of theAcoustical Society of America vol 93 no 3 pp 1308ndash13181993

[16] Z Jin Y Yin and B Liu ldquoEquivalent modulus method forfinite element simulation of the sound absorption of anechoiccoating backed with orthogonally rib-stiffened platerdquo Journalof Sound and Vibration vol 366 pp 357ndash371 2016

[17] L M Lu J H Wen H Zhao H Meng and X S Wen ldquoLow-frequency acoustic absorption of viscoelastic coating withvarious shapes of scatterersrdquo Wuli XuebaoActa PhysicaSinica vol 61 2012

[18] J Li and S Li ldquoTopology optimization of anechoic coating formaximizing sound absorptionrdquo Journal of Vibration andControl vol 24 no 11 pp 2369ndash2385 2018

[19] C Cai K C Hung and M S Khan ldquoSimulation-basedanalysis of acoustic absorbent lining subject to normal planewave incidencerdquo Journal of Sound and Vibration vol 291no 3-5 pp 656ndash680 2006

[20] S N Panigrahi C S Jog and M L Munjal ldquoMulti-focusdesign of underwater noise control linings based on finiteelement analysisrdquo Applied Acoustics vol 69 no 12pp 1141ndash1153 2008

[21] H Meng J Wen H Zhao and X Wen ldquoOptimization oflocally resonant acoustic metamaterials on underwater soundabsorption characteristicsrdquo Journal of Sound and Vibrationvol 331 no 20 pp 4406ndash4416 2012

10 Shock and Vibration

[22] M A Lewniska V G Kouznetsova J A W van DommelenA O Krushynska and M G D Geers ldquoe attenuationperformance of locally resonant acoustic metamaterials basedon generalised viscoelastic modellingrdquo International Journalof Solids and Structures vol 126 pp 163ndash174 2017

[23] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic-wave by a doubly periodicstructure using the finite-element method- application toalberich anechoic coatingrdquo Journal of the Acoustical Society ofAmerica vol 90 no 6 pp 3356ndash3367 1991

[24] J Villalobos-Luna P Lopez-Cruz R Ramirez-Valencia andF J Elizondo-Garza ldquoSimulation of acoustic wave behavior inducts and a plenum using finite elements package ANSYSrdquoJournal of the Acoustical Society of America vol 112 no 5p 2427 2002

[25] Y Xu G Zhang B Zhou H Wang and Q Tang ldquoAnalysis ofacoustic radiation problems using the cell-based smoothedradial point interpolation method with Dirichlet-to-Neu-mann boundary conditionrdquo Engineering Analysis withBoundary Elements vol 108 no Nov pp 447ndash458 2019

[26] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wavenormal incidence conditionrdquo in Proceedings of the 4th In-ternational Conference on Smart Material Research ICSMR2018 Sydney Australia November 2018

[27] Y E Hanfeng M Tao and L I Junjie ldquoSound absorptionperformance analysis of anechoic coatings under obliqueincidence condition based on COMSOLrdquo Journal of Vibrationand Shock vol 038 no 12 pp 213ndash218 2019

[28] T Pritz ldquoe Poissonrsquos loss factor of solid viscoelastic ma-terialsrdquo Journal of Sound and Vibration vol 306 no 3-5pp 790ndash802 2007

[29] S H Sohrabi and M J Ketabdari ldquoNumerical simulation of aviscoelastic sound absorbent coating with a doubly periodicarray of cavitiesrdquo Cogent Engineering vol 5 no 1 2018

Shock and Vibration 11

were well consistent with the test results as shown inFigure 9 e aforementioned dynamic Poissonrsquos ratios wereused to calculate the sound absorption characteristics of theAlberich coating made of the same nitrile rubber as thetrumpet cavity-type coating Figure 11 shows the calculationand measured results suggesting that the sound absorption

coefficients calculated using the static Poissonrsquos ratios aresignificantly deviant from the test results and that the onescalculated using the dynamic ratios are more consistent withthe test results In addition it can be seen from the figure thatdynamic Poissonrsquos ratio should be used to obtain moreaccurate results for the finite element calculation of the

Table 1 Poissonrsquos ratio Lamy constant and their rate of change

Poissonrsquos ratio Lamy constant Change rate of Poissonrsquos ratio Change rate of Lamy constant0490 23E+ 09 0 00493 33E+ 09 061 430496 58E+ 09 122 1520499 233E+ 10 184 91204999 233E+ 11 202 10034

049 0493 0496 0499 04999

00

01

02

03

04

05

06

07

08

09

Ave

rage

abso

rptio

n co

effic

ient

Poissons Ratio

Static modulus HomogenousStatic modulus AlberichStatic modulus Horn

Dynamic modulus HomogenousDynamic modulus AlberichDynamic modulus Horn

Figure 8 Variation of the average sound absorption coefficient with Poissonrsquos ratio

Table 2 Calculated and measured sound absorption coefficient

FrequencyPoissonrsquos ratio 1000 1250 1600 2000 2500 3150 4000 5000

Calculated

0496 0922 0935 0942 0935 0911 0866 0796 07180497 0917 0939 0957 0960 0946 0909 0841 07560498 0889 0922 0956 0978 0982 0963 0908 08260499 0772 0821 0879 0930 0969 0996 0994 095304992 0720 0769 0831 0888 0937 0978 0997 097904994 0642 0692 0756 0817 0874 0929 0973 098504995 0589 0638 0701 0764 0823 0884 0940 097004996 0524 0570 0631 0692 0752 0816 0883 093004997 0442 0483 0538 0595 0651 0715 0787 084704998 0336 0369 0414 0461 0509 0564 0630 0692

Measured 0420 0615 0815 0900 0985 0955 0830 0755

8 Shock and Vibration

1000 1250 1600 2000 2500 3150 4000 5000

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with constant Poissons ratioTestedCalculated with dynamic Poissons ratio

Figure 9 Calculation and measurement results of the sound absorption coefficient of the trumpet cavity-type acoustic coating

1000 1250 1600 2000 2500 3150 4000 500004960

04965

04970

04975

04980

04985

04990

04995

05000

05005

Poiss

ons

Ratio

Frequency (Hz)

InterpolatedFited

Figure 10 Interpolated Poissonrsquos ratio varying with frequency

1000 1250 1600 2000 2500 3150 4000 5000

02

03

04

05

06

07

08

09

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with Poissons Ratio = 0490Calculated with Poissons Ratio = 0493Calculated with Poissons Ratio = 0496Calculated with Poissons Ratio = 0499Calculated with Poissons Ratio = 04999Calculated with dynamic Poissons RatioMeasured

Figure 11 Calculation and measurement results of the sound absorption coefficients of the Alberich coating

Shock and Vibration 9

sound absorption characteristics of acoustic coatings Un-fortunately there is still no accurate and reliable methodavailable to get the dynamic Poissonrsquos ratios of viscoelasticmaterials [11]

7 Conclusion

(1) When Poissonrsquos ratio of homogeneous acousticcoating Alberich acoustic coating and trumpetcavity-type coating varies from 049 to 04999 thelarger Poissonrsquos ratio is the larger the frequency ofthe first absorption peak is the smaller the ab-sorption coefficient is in the frequency range belowthe first absorption peak and the smaller the averageabsorption coefficient is in the analysis frequencyrange

(2) More accurate calculation results of sound absorp-tion coefficient can be obtained by using Poissonrsquosratio varying with frequency In the analysis of thesound absorption performance of acoustic coatingsthe accurate dynamic Poissonrsquos ratio with the changeof frequency should be used

(3) Poissonrsquos ratio of viscoelastic materials varies in avery small numerical range such a small change hasa great influence on the sound absorption perfor-mance of acoustic coatings erefore it is urgent tostudy the accurate method to obtain the dynamicPoissonrsquos ratio

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the Scientific Re-search Project of Education Department of Hunan ProvinceChina (Project no 19C0589)

References

[1] L Alimonti and N Atalla ldquoEfficient modeling of flat andhomogeneous acoustic treatments for vibroacoustic finiteelement analysis Finite size correction by image sourcesrdquoJournal of Sound and Vibration vol 388 pp 201ndash215 2017

[2] X L Yao X Z Wang F Z Pang and L Q Sun ldquoStudy onAcoustic Characteristics of Underwater Coating StructurerdquoNoise and Vibration Control vol 30 no 4 pp 1ndash5 2010

[3] S M Ivansson ldquoErratumrdquo Journal of the Acoustical Society ofAmerica vol 131 no 4 p 3180 2012

[4] S M Ivansson ldquoNumerical design of Alberich anechoiccoatings with superellipsoidal cavities of mixed sizesrdquo Journalof the Acoustical Society of America vol 123 no 5 p 34662008

[5] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wave

normal incidence conditionrdquo IOP Conference Series Mate-rials Science and Engineering vol 522 no 1 Article ID012001 2019

[6] H Bai Z Zhan J Liu and Z Ren ldquoFrom local structure tooverall performance An overview on the design of an acousticcoatingrdquo Materials vol 12 no 16 2019

[7] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic wave by a doubly periodicstructure using the finite-element methodrdquo Journal of theAcoustical Society of America vol 88 2005

[8] T Meng ldquoSimplified model for predicting acoustic perfor-mance of an underwater sound absorption coatingrdquo Journal ofVibration and Control vol 20 no 3 pp 339ndash354 2014

[9] L M Lu J H Wen H G Zhao and X S Wen ldquoDynamicalmechanical property of viscoelastic materials and its effect onacoustic absorption of coatingrdquo Acta Physica Sinica vol 63no 15 2014

[10] Z Wei H Hou N Gao Y Huang and J Yang ldquoComplexYoungrsquos modulus measurement by incident wave extractingin a thin resonant barrdquo Journal of the Acoustical Society ofAmerica vol 142 no 6 pp 3436ndash3442 2017

[11] N W Tschoegl W G Knauss and I Emri ldquoPoissonrsquos ratio inlinear viscoelasticity - A critical reviewrdquo Mechanics of Time-Dependent Materials vol 6 no 1 pp 3ndash51 2002

[12] T Pritz ldquoFrequency dependences of complex moduli andcomplex Poissonrsquos ratio of real solid materialsrdquo Journal ofSound and Vibration vol 214 no 1 pp 83ndash104 1998

[13] J Zhong H Zhao H Yang J Yin and J Wen ldquoEffect ofPoissonrsquos loss factor of rubbery material on underwater soundabsorption of anechoic coatingsrdquo Journal of Sound and Vi-bration vol 424 pp 293ndash301 2018

[14] H Meng J Wen H Zhao L Lv and X Wen ldquoAnalysis ofabsorption performances of anechoic layers with steel platebackingrdquo Journal of the Acoustical Society of America vol 132no 1 pp 69ndash75 2012

[15] V Easwaran and M L Munjal ldquoAnalysis of reflectioncharacteristics of a normal incidence plane wave on resonantsound absorbers A finite element approachrdquo Journal of theAcoustical Society of America vol 93 no 3 pp 1308ndash13181993

[16] Z Jin Y Yin and B Liu ldquoEquivalent modulus method forfinite element simulation of the sound absorption of anechoiccoating backed with orthogonally rib-stiffened platerdquo Journalof Sound and Vibration vol 366 pp 357ndash371 2016

[17] L M Lu J H Wen H Zhao H Meng and X S Wen ldquoLow-frequency acoustic absorption of viscoelastic coating withvarious shapes of scatterersrdquo Wuli XuebaoActa PhysicaSinica vol 61 2012

[18] J Li and S Li ldquoTopology optimization of anechoic coating formaximizing sound absorptionrdquo Journal of Vibration andControl vol 24 no 11 pp 2369ndash2385 2018

[19] C Cai K C Hung and M S Khan ldquoSimulation-basedanalysis of acoustic absorbent lining subject to normal planewave incidencerdquo Journal of Sound and Vibration vol 291no 3-5 pp 656ndash680 2006

[20] S N Panigrahi C S Jog and M L Munjal ldquoMulti-focusdesign of underwater noise control linings based on finiteelement analysisrdquo Applied Acoustics vol 69 no 12pp 1141ndash1153 2008

[21] H Meng J Wen H Zhao and X Wen ldquoOptimization oflocally resonant acoustic metamaterials on underwater soundabsorption characteristicsrdquo Journal of Sound and Vibrationvol 331 no 20 pp 4406ndash4416 2012

10 Shock and Vibration

[22] M A Lewniska V G Kouznetsova J A W van DommelenA O Krushynska and M G D Geers ldquoe attenuationperformance of locally resonant acoustic metamaterials basedon generalised viscoelastic modellingrdquo International Journalof Solids and Structures vol 126 pp 163ndash174 2017

[23] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic-wave by a doubly periodicstructure using the finite-element method- application toalberich anechoic coatingrdquo Journal of the Acoustical Society ofAmerica vol 90 no 6 pp 3356ndash3367 1991

[24] J Villalobos-Luna P Lopez-Cruz R Ramirez-Valencia andF J Elizondo-Garza ldquoSimulation of acoustic wave behavior inducts and a plenum using finite elements package ANSYSrdquoJournal of the Acoustical Society of America vol 112 no 5p 2427 2002

[25] Y Xu G Zhang B Zhou H Wang and Q Tang ldquoAnalysis ofacoustic radiation problems using the cell-based smoothedradial point interpolation method with Dirichlet-to-Neu-mann boundary conditionrdquo Engineering Analysis withBoundary Elements vol 108 no Nov pp 447ndash458 2019

[26] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wavenormal incidence conditionrdquo in Proceedings of the 4th In-ternational Conference on Smart Material Research ICSMR2018 Sydney Australia November 2018

[27] Y E Hanfeng M Tao and L I Junjie ldquoSound absorptionperformance analysis of anechoic coatings under obliqueincidence condition based on COMSOLrdquo Journal of Vibrationand Shock vol 038 no 12 pp 213ndash218 2019

[28] T Pritz ldquoe Poissonrsquos loss factor of solid viscoelastic ma-terialsrdquo Journal of Sound and Vibration vol 306 no 3-5pp 790ndash802 2007

[29] S H Sohrabi and M J Ketabdari ldquoNumerical simulation of aviscoelastic sound absorbent coating with a doubly periodicarray of cavitiesrdquo Cogent Engineering vol 5 no 1 2018

Shock and Vibration 11

1000 1250 1600 2000 2500 3150 4000 5000

04

06

08

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with constant Poissons ratioTestedCalculated with dynamic Poissons ratio

Figure 9 Calculation and measurement results of the sound absorption coefficient of the trumpet cavity-type acoustic coating

1000 1250 1600 2000 2500 3150 4000 500004960

04965

04970

04975

04980

04985

04990

04995

05000

05005

Poiss

ons

Ratio

Frequency (Hz)

InterpolatedFited

Figure 10 Interpolated Poissonrsquos ratio varying with frequency

1000 1250 1600 2000 2500 3150 4000 5000

02

03

04

05

06

07

08

09

10

Abs

orpt

ion

coef

ficie

nts

Frequency (Hz)

Calculated with Poissons Ratio = 0490Calculated with Poissons Ratio = 0493Calculated with Poissons Ratio = 0496Calculated with Poissons Ratio = 0499Calculated with Poissons Ratio = 04999Calculated with dynamic Poissons RatioMeasured

Figure 11 Calculation and measurement results of the sound absorption coefficients of the Alberich coating

Shock and Vibration 9

sound absorption characteristics of acoustic coatings Un-fortunately there is still no accurate and reliable methodavailable to get the dynamic Poissonrsquos ratios of viscoelasticmaterials [11]

7 Conclusion

(1) When Poissonrsquos ratio of homogeneous acousticcoating Alberich acoustic coating and trumpetcavity-type coating varies from 049 to 04999 thelarger Poissonrsquos ratio is the larger the frequency ofthe first absorption peak is the smaller the ab-sorption coefficient is in the frequency range belowthe first absorption peak and the smaller the averageabsorption coefficient is in the analysis frequencyrange

(2) More accurate calculation results of sound absorp-tion coefficient can be obtained by using Poissonrsquosratio varying with frequency In the analysis of thesound absorption performance of acoustic coatingsthe accurate dynamic Poissonrsquos ratio with the changeof frequency should be used

(3) Poissonrsquos ratio of viscoelastic materials varies in avery small numerical range such a small change hasa great influence on the sound absorption perfor-mance of acoustic coatings erefore it is urgent tostudy the accurate method to obtain the dynamicPoissonrsquos ratio

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the Scientific Re-search Project of Education Department of Hunan ProvinceChina (Project no 19C0589)

References

[1] L Alimonti and N Atalla ldquoEfficient modeling of flat andhomogeneous acoustic treatments for vibroacoustic finiteelement analysis Finite size correction by image sourcesrdquoJournal of Sound and Vibration vol 388 pp 201ndash215 2017

[2] X L Yao X Z Wang F Z Pang and L Q Sun ldquoStudy onAcoustic Characteristics of Underwater Coating StructurerdquoNoise and Vibration Control vol 30 no 4 pp 1ndash5 2010

[3] S M Ivansson ldquoErratumrdquo Journal of the Acoustical Society ofAmerica vol 131 no 4 p 3180 2012

[4] S M Ivansson ldquoNumerical design of Alberich anechoiccoatings with superellipsoidal cavities of mixed sizesrdquo Journalof the Acoustical Society of America vol 123 no 5 p 34662008

[5] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wave

normal incidence conditionrdquo IOP Conference Series Mate-rials Science and Engineering vol 522 no 1 Article ID012001 2019

[6] H Bai Z Zhan J Liu and Z Ren ldquoFrom local structure tooverall performance An overview on the design of an acousticcoatingrdquo Materials vol 12 no 16 2019

[7] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic wave by a doubly periodicstructure using the finite-element methodrdquo Journal of theAcoustical Society of America vol 88 2005

[8] T Meng ldquoSimplified model for predicting acoustic perfor-mance of an underwater sound absorption coatingrdquo Journal ofVibration and Control vol 20 no 3 pp 339ndash354 2014

[9] L M Lu J H Wen H G Zhao and X S Wen ldquoDynamicalmechanical property of viscoelastic materials and its effect onacoustic absorption of coatingrdquo Acta Physica Sinica vol 63no 15 2014

[10] Z Wei H Hou N Gao Y Huang and J Yang ldquoComplexYoungrsquos modulus measurement by incident wave extractingin a thin resonant barrdquo Journal of the Acoustical Society ofAmerica vol 142 no 6 pp 3436ndash3442 2017

[11] N W Tschoegl W G Knauss and I Emri ldquoPoissonrsquos ratio inlinear viscoelasticity - A critical reviewrdquo Mechanics of Time-Dependent Materials vol 6 no 1 pp 3ndash51 2002

[12] T Pritz ldquoFrequency dependences of complex moduli andcomplex Poissonrsquos ratio of real solid materialsrdquo Journal ofSound and Vibration vol 214 no 1 pp 83ndash104 1998

[13] J Zhong H Zhao H Yang J Yin and J Wen ldquoEffect ofPoissonrsquos loss factor of rubbery material on underwater soundabsorption of anechoic coatingsrdquo Journal of Sound and Vi-bration vol 424 pp 293ndash301 2018

[14] H Meng J Wen H Zhao L Lv and X Wen ldquoAnalysis ofabsorption performances of anechoic layers with steel platebackingrdquo Journal of the Acoustical Society of America vol 132no 1 pp 69ndash75 2012

[15] V Easwaran and M L Munjal ldquoAnalysis of reflectioncharacteristics of a normal incidence plane wave on resonantsound absorbers A finite element approachrdquo Journal of theAcoustical Society of America vol 93 no 3 pp 1308ndash13181993

[16] Z Jin Y Yin and B Liu ldquoEquivalent modulus method forfinite element simulation of the sound absorption of anechoiccoating backed with orthogonally rib-stiffened platerdquo Journalof Sound and Vibration vol 366 pp 357ndash371 2016

[17] L M Lu J H Wen H Zhao H Meng and X S Wen ldquoLow-frequency acoustic absorption of viscoelastic coating withvarious shapes of scatterersrdquo Wuli XuebaoActa PhysicaSinica vol 61 2012

[18] J Li and S Li ldquoTopology optimization of anechoic coating formaximizing sound absorptionrdquo Journal of Vibration andControl vol 24 no 11 pp 2369ndash2385 2018

[19] C Cai K C Hung and M S Khan ldquoSimulation-basedanalysis of acoustic absorbent lining subject to normal planewave incidencerdquo Journal of Sound and Vibration vol 291no 3-5 pp 656ndash680 2006

[20] S N Panigrahi C S Jog and M L Munjal ldquoMulti-focusdesign of underwater noise control linings based on finiteelement analysisrdquo Applied Acoustics vol 69 no 12pp 1141ndash1153 2008

[21] H Meng J Wen H Zhao and X Wen ldquoOptimization oflocally resonant acoustic metamaterials on underwater soundabsorption characteristicsrdquo Journal of Sound and Vibrationvol 331 no 20 pp 4406ndash4416 2012

10 Shock and Vibration

[22] M A Lewniska V G Kouznetsova J A W van DommelenA O Krushynska and M G D Geers ldquoe attenuationperformance of locally resonant acoustic metamaterials basedon generalised viscoelastic modellingrdquo International Journalof Solids and Structures vol 126 pp 163ndash174 2017

[23] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic-wave by a doubly periodicstructure using the finite-element method- application toalberich anechoic coatingrdquo Journal of the Acoustical Society ofAmerica vol 90 no 6 pp 3356ndash3367 1991

[24] J Villalobos-Luna P Lopez-Cruz R Ramirez-Valencia andF J Elizondo-Garza ldquoSimulation of acoustic wave behavior inducts and a plenum using finite elements package ANSYSrdquoJournal of the Acoustical Society of America vol 112 no 5p 2427 2002

[25] Y Xu G Zhang B Zhou H Wang and Q Tang ldquoAnalysis ofacoustic radiation problems using the cell-based smoothedradial point interpolation method with Dirichlet-to-Neu-mann boundary conditionrdquo Engineering Analysis withBoundary Elements vol 108 no Nov pp 447ndash458 2019

[26] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wavenormal incidence conditionrdquo in Proceedings of the 4th In-ternational Conference on Smart Material Research ICSMR2018 Sydney Australia November 2018

[27] Y E Hanfeng M Tao and L I Junjie ldquoSound absorptionperformance analysis of anechoic coatings under obliqueincidence condition based on COMSOLrdquo Journal of Vibrationand Shock vol 038 no 12 pp 213ndash218 2019

[28] T Pritz ldquoe Poissonrsquos loss factor of solid viscoelastic ma-terialsrdquo Journal of Sound and Vibration vol 306 no 3-5pp 790ndash802 2007

[29] S H Sohrabi and M J Ketabdari ldquoNumerical simulation of aviscoelastic sound absorbent coating with a doubly periodicarray of cavitiesrdquo Cogent Engineering vol 5 no 1 2018

Shock and Vibration 11

sound absorption characteristics of acoustic coatings Un-fortunately there is still no accurate and reliable methodavailable to get the dynamic Poissonrsquos ratios of viscoelasticmaterials [11]

7 Conclusion

(1) When Poissonrsquos ratio of homogeneous acousticcoating Alberich acoustic coating and trumpetcavity-type coating varies from 049 to 04999 thelarger Poissonrsquos ratio is the larger the frequency ofthe first absorption peak is the smaller the ab-sorption coefficient is in the frequency range belowthe first absorption peak and the smaller the averageabsorption coefficient is in the analysis frequencyrange

(2) More accurate calculation results of sound absorp-tion coefficient can be obtained by using Poissonrsquosratio varying with frequency In the analysis of thesound absorption performance of acoustic coatingsthe accurate dynamic Poissonrsquos ratio with the changeof frequency should be used

(3) Poissonrsquos ratio of viscoelastic materials varies in avery small numerical range such a small change hasa great influence on the sound absorption perfor-mance of acoustic coatings erefore it is urgent tostudy the accurate method to obtain the dynamicPoissonrsquos ratio

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the Scientific Re-search Project of Education Department of Hunan ProvinceChina (Project no 19C0589)

References

[1] L Alimonti and N Atalla ldquoEfficient modeling of flat andhomogeneous acoustic treatments for vibroacoustic finiteelement analysis Finite size correction by image sourcesrdquoJournal of Sound and Vibration vol 388 pp 201ndash215 2017

[2] X L Yao X Z Wang F Z Pang and L Q Sun ldquoStudy onAcoustic Characteristics of Underwater Coating StructurerdquoNoise and Vibration Control vol 30 no 4 pp 1ndash5 2010

[3] S M Ivansson ldquoErratumrdquo Journal of the Acoustical Society ofAmerica vol 131 no 4 p 3180 2012

[4] S M Ivansson ldquoNumerical design of Alberich anechoiccoatings with superellipsoidal cavities of mixed sizesrdquo Journalof the Acoustical Society of America vol 123 no 5 p 34662008

[5] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wave

normal incidence conditionrdquo IOP Conference Series Mate-rials Science and Engineering vol 522 no 1 Article ID012001 2019

[6] H Bai Z Zhan J Liu and Z Ren ldquoFrom local structure tooverall performance An overview on the design of an acousticcoatingrdquo Materials vol 12 no 16 2019

[7] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic wave by a doubly periodicstructure using the finite-element methodrdquo Journal of theAcoustical Society of America vol 88 2005

[8] T Meng ldquoSimplified model for predicting acoustic perfor-mance of an underwater sound absorption coatingrdquo Journal ofVibration and Control vol 20 no 3 pp 339ndash354 2014

[9] L M Lu J H Wen H G Zhao and X S Wen ldquoDynamicalmechanical property of viscoelastic materials and its effect onacoustic absorption of coatingrdquo Acta Physica Sinica vol 63no 15 2014

[10] Z Wei H Hou N Gao Y Huang and J Yang ldquoComplexYoungrsquos modulus measurement by incident wave extractingin a thin resonant barrdquo Journal of the Acoustical Society ofAmerica vol 142 no 6 pp 3436ndash3442 2017

[11] N W Tschoegl W G Knauss and I Emri ldquoPoissonrsquos ratio inlinear viscoelasticity - A critical reviewrdquo Mechanics of Time-Dependent Materials vol 6 no 1 pp 3ndash51 2002

[12] T Pritz ldquoFrequency dependences of complex moduli andcomplex Poissonrsquos ratio of real solid materialsrdquo Journal ofSound and Vibration vol 214 no 1 pp 83ndash104 1998

[13] J Zhong H Zhao H Yang J Yin and J Wen ldquoEffect ofPoissonrsquos loss factor of rubbery material on underwater soundabsorption of anechoic coatingsrdquo Journal of Sound and Vi-bration vol 424 pp 293ndash301 2018

[14] H Meng J Wen H Zhao L Lv and X Wen ldquoAnalysis ofabsorption performances of anechoic layers with steel platebackingrdquo Journal of the Acoustical Society of America vol 132no 1 pp 69ndash75 2012

[15] V Easwaran and M L Munjal ldquoAnalysis of reflectioncharacteristics of a normal incidence plane wave on resonantsound absorbers A finite element approachrdquo Journal of theAcoustical Society of America vol 93 no 3 pp 1308ndash13181993

[16] Z Jin Y Yin and B Liu ldquoEquivalent modulus method forfinite element simulation of the sound absorption of anechoiccoating backed with orthogonally rib-stiffened platerdquo Journalof Sound and Vibration vol 366 pp 357ndash371 2016

[17] L M Lu J H Wen H Zhao H Meng and X S Wen ldquoLow-frequency acoustic absorption of viscoelastic coating withvarious shapes of scatterersrdquo Wuli XuebaoActa PhysicaSinica vol 61 2012

[18] J Li and S Li ldquoTopology optimization of anechoic coating formaximizing sound absorptionrdquo Journal of Vibration andControl vol 24 no 11 pp 2369ndash2385 2018

[19] C Cai K C Hung and M S Khan ldquoSimulation-basedanalysis of acoustic absorbent lining subject to normal planewave incidencerdquo Journal of Sound and Vibration vol 291no 3-5 pp 656ndash680 2006

[20] S N Panigrahi C S Jog and M L Munjal ldquoMulti-focusdesign of underwater noise control linings based on finiteelement analysisrdquo Applied Acoustics vol 69 no 12pp 1141ndash1153 2008

[21] H Meng J Wen H Zhao and X Wen ldquoOptimization oflocally resonant acoustic metamaterials on underwater soundabsorption characteristicsrdquo Journal of Sound and Vibrationvol 331 no 20 pp 4406ndash4416 2012

10 Shock and Vibration

[22] M A Lewniska V G Kouznetsova J A W van DommelenA O Krushynska and M G D Geers ldquoe attenuationperformance of locally resonant acoustic metamaterials basedon generalised viscoelastic modellingrdquo International Journalof Solids and Structures vol 126 pp 163ndash174 2017

[23] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic-wave by a doubly periodicstructure using the finite-element method- application toalberich anechoic coatingrdquo Journal of the Acoustical Society ofAmerica vol 90 no 6 pp 3356ndash3367 1991

[24] J Villalobos-Luna P Lopez-Cruz R Ramirez-Valencia andF J Elizondo-Garza ldquoSimulation of acoustic wave behavior inducts and a plenum using finite elements package ANSYSrdquoJournal of the Acoustical Society of America vol 112 no 5p 2427 2002

[25] Y Xu G Zhang B Zhou H Wang and Q Tang ldquoAnalysis ofacoustic radiation problems using the cell-based smoothedradial point interpolation method with Dirichlet-to-Neu-mann boundary conditionrdquo Engineering Analysis withBoundary Elements vol 108 no Nov pp 447ndash458 2019

[26] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wavenormal incidence conditionrdquo in Proceedings of the 4th In-ternational Conference on Smart Material Research ICSMR2018 Sydney Australia November 2018

[27] Y E Hanfeng M Tao and L I Junjie ldquoSound absorptionperformance analysis of anechoic coatings under obliqueincidence condition based on COMSOLrdquo Journal of Vibrationand Shock vol 038 no 12 pp 213ndash218 2019

[28] T Pritz ldquoe Poissonrsquos loss factor of solid viscoelastic ma-terialsrdquo Journal of Sound and Vibration vol 306 no 3-5pp 790ndash802 2007

[29] S H Sohrabi and M J Ketabdari ldquoNumerical simulation of aviscoelastic sound absorbent coating with a doubly periodicarray of cavitiesrdquo Cogent Engineering vol 5 no 1 2018

Shock and Vibration 11

[22] M A Lewniska V G Kouznetsova J A W van DommelenA O Krushynska and M G D Geers ldquoe attenuationperformance of locally resonant acoustic metamaterials basedon generalised viscoelastic modellingrdquo International Journalof Solids and Structures vol 126 pp 163ndash174 2017

[23] A C Hladkyhennion and J N Decarpigny ldquoAnalysis of thescattering of a plane acoustic-wave by a doubly periodicstructure using the finite-element method- application toalberich anechoic coatingrdquo Journal of the Acoustical Society ofAmerica vol 90 no 6 pp 3356ndash3367 1991

[24] J Villalobos-Luna P Lopez-Cruz R Ramirez-Valencia andF J Elizondo-Garza ldquoSimulation of acoustic wave behavior inducts and a plenum using finite elements package ANSYSrdquoJournal of the Acoustical Society of America vol 112 no 5p 2427 2002

[25] Y Xu G Zhang B Zhou H Wang and Q Tang ldquoAnalysis ofacoustic radiation problems using the cell-based smoothedradial point interpolation method with Dirichlet-to-Neu-mann boundary conditionrdquo Engineering Analysis withBoundary Elements vol 108 no Nov pp 447ndash458 2019

[26] Z Zhang L Li Y Huang and Q Huang ldquoSound absorptionperformance of underwater anechoic coating in plane wavenormal incidence conditionrdquo in Proceedings of the 4th In-ternational Conference on Smart Material Research ICSMR2018 Sydney Australia November 2018

[27] Y E Hanfeng M Tao and L I Junjie ldquoSound absorptionperformance analysis of anechoic coatings under obliqueincidence condition based on COMSOLrdquo Journal of Vibrationand Shock vol 038 no 12 pp 213ndash218 2019

[28] T Pritz ldquoe Poissonrsquos loss factor of solid viscoelastic ma-terialsrdquo Journal of Sound and Vibration vol 306 no 3-5pp 790ndash802 2007

[29] S H Sohrabi and M J Ketabdari ldquoNumerical simulation of aviscoelastic sound absorbent coating with a doubly periodicarray of cavitiesrdquo Cogent Engineering vol 5 no 1 2018

Shock and Vibration 11