AbstractThe micro-perforated panel (MPP) structure has been widely used in various noise control applications, and thus its acousticperformance prediction has been receiving increasing attention. The acoustic performance of simpleMPP structures, such as aMPP sound absorber, has been predicted using an analytical calculationmethod.However, this is not a suitable approach towardpredicting the acoustic performance of complicated MPP structures, owing to the structural complexity of these structures.Moreover, the many perforations of submillimeter scale diameter render the MPP structures very difficult to analyze usingnumerical simulation. Thus, this study focused on two different simplified MPP simulation methods: the transfer admittancemethod and the equivalent fluid method, and their application on double-layer MPP structures. Based on the two simplifiedMPP simulation methods, the transmission loss value of the double-layer MPP mufflers with two sets of different structuralparameters was calculated, respectively. The predicted results were compared with the impedance tube measurements. Theresults revealed that the two simplifiedMPP simulation methods could effectively predict the acoustic performance of double-layer MPP structures. Moreover, the prediction based on the transfer admittance method can outperform the two simplifiedsimulation methods.
Keywords Double-layer MPP structure · Acoustic simulation · Transfer admittance · Porous material · Experimental test
1 Hubei Key Laboratory of Advanced Technology forAutomotive Components, Wuhan University of Technology,Wuhan 430070, China
2 Hubei Collaborative Innovation Center for AutomotiveComponents Technology, Wuhan University ofTechnology, Wuhan 430070, China
β Transfer admittanceK Diameter ratioZ Specific acoustic impedanceR Specific acoustic resistanceX Specific acoustic reactanceη Dynamic viscosityρ Air densityω Angular frequencyf Frequencyϕ Porosityσ Flow resistivityα∞ Tortuosityεe Correction lengthAv Viscous characteristic lengthAt Thermal characteristic length
1 Introduction
The micro-perforated panel (MPP) structure is very promis-ing as the basis for the next generation of sound absorbingconstructions. This structure has enough acoustic resistance
Simplified Method of Simulating Double-Layer Micro-Perforated Panel Structure 375
and sufficiently low acoustic reactance to provide goodabsorption properties by reducing the perforation diameterto submillimeter scale. The MPP combined with a uniformair-back cavity and a rigid back wall can be used to con-struct conventional single-layerMPP sound absorbers,whichwere first proposed by Maa [1]. Based on Maa’s pioneer-ing work, many theoretical methods have been proposed topredict the acoustic performance of MPP sound absorbers.The most common method is the acoustic-electric analogymethod [2–4], which has been used to calculate the normalsound absorption coefficient of many different types of MPPsound absorbers. The transfer matrix method [5, 6] has alsobeen used to predict the sound absorption performance of asingle-layer MPP sound absorber or a multilayer MPP soundabsorber. Additionally, the impedance transfer method [7, 8]has been used to calculate the surface acoustic impedance ofMPP sound absorbers. In summary, the acoustic performanceof simpleMPP sound absorbers can be easily predicted basedon the abovementioned analytical calculationmethods.How-ever, in practical applications, MPPs are more often appliedto a complicated muffler, with a complex interior structuralenvironment. Moreover, the incidence sound wave is alsomore frequently subject to an oblique incidence condition.Therefore, in these cases, it is not suitable to use any of thecommonly used analytical calculation methods to predict theoverall acoustic performance of the MPP structures.
To ensure the prediction accuracy, many scholars haveused the acoustic finite element method (FEM) to simulatethe acoustic performance of MPP structures. For example,Gerdes et al. [9] established a geometrical perforation modeland adopted a computational fluid dynamics (CFD) approachto calculate the acoustic transfer impedance of a MPP. Sub-sequently, they used the acoustic transfer impedance resultsto describe theMPP layer in the finite element (FE) model ofa MPP sound absorber. However, the large number of perfo-rations in theMPP increased the difficulty of the geometricalmodeling process, which in turn increased considerably thecomplexity of the entire simulation calculation.
To avoid the geometry modeling of massive perforations,Zuo et al. [10] first obtained the acoustic impedance of aMPPaccording to Maa’s formula and then simulated the MPP bydefining the transfer admittance relationships between bothof its sides such that the transmission loss of a single-layerMPP muffler could be predicted. From a different perspec-tive, Atalla [11] proposed that a rigid perforated plate couldbe modeled as an equivalent fluid by following the John-son–Allard approach, given that the equivalent tortuosityis appropriately defined. Thus, it becomes possible to usethe existing finite element models of rigid porous materi-als to model rigid MPPs. On the basis of Atalla’s idea, Hou[12, 13] further expanded the modeling method and demon-strated that a flexible MPP can be modeled in a similarmanner by using poro-elastic finite elements, which allow
the prediction of the panel’s structural resonances. The twoabovementioned simplified MPP simulation methods havebeen applied to the acoustic prediction of single-layer MPPstructures. However, this application has not yet been investi-gated for complicatedMPP structures, such as a double-layerMPP muffler. Moreover, many studies have focused only onone of these two simplified MPP simulation methods, andlittle effort has been put into investigating them simultane-ously and comparing their accuracy. To fill this research gap,the two abovementioned simplifiedMPP simulationmethodswere used, respectively, to predict the acoustic performanceof double-layer MPP mufflers with a wide silencing band-width.
In this study, first, the double-layer MPP structures wereintroduced. Then, the transmission loss of one double-layered MPP muffler was predicted using the transfer admit-tance method and the equivalent fluid method, respectively.The muffler was tested with an impedance tube to validatethe predicted results. Additionally, the other muffler with dif-ferent structural parameters was treated in a similar mannerto further validate the results.
2 Double-Layer MPP Structures
2.1 Double-Layer MPP Sound Absorber
To better understand the double-layer MPP muffler underinvestigation, a brief introduction to the double-layer MPPsound absorber will be given first. A conventional single-layer MPP sound absorber consists of a MPP, rigid backingwall, and air cavity between them. This type of MPPsound absorber offers an outstanding alternative to tradi-tional porous materials, but has an obvious disadvantage:it is effective only within a narrow band around its resonancefrequency, owing to its resonator nature, which typically ren-ders its sound absorption capacity inadequate for a generalpurpose absorber. In an effort to widen the sound absorp-tion frequency range, many studies have reported that theintroduction of extra absorption peaksmay be themost effec-tive and promising approach. Thus, compound MPP soundabsorbers have been repeatedly proposed. A double-layerMPP sound absorber is the most common compound MPPsound absorber andwas first proposed byMaa [1, 14]. There-after, several studies have followed with similar proposals[15, 16].Adouble-layeredMPP sound absorber has twoMPPlayers, two air cavities, and a rigid backingwall. The distancebetween the two MPP layers is D1 and the distance betweenthe inner layer MPP and the rigid wall is D2, as illustratedin Fig. 1. Note that, in this study, the two MPP layers hadthe same properties. Therefore, the acoustic performance ofthe double-layer MPP sound absorber depended on the panelthickness t, perforation diameter d, perforation ratio p (ratio
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376 W. Chen et al.
Fig. 1 Schematic diagram of double-layer MPP sound absorber
Fig. 2 Schematic diagram of double-layer MPP muffler
Table 1 Muffler’s MPP structural parameters and cavity depths
t (mm) d (mm) p (%) D1 (mm) D2 (mm)
0.5 0.58 8.26 9 1.8
of the perforation surface area to the total surface area of thepanel), and cavity depth D1 and D2.
2.2 Double-Layer MPPMuffler
Two double-layer MPP mufflers with different structuralparameters were selected as the research object. To clarifythe two simplified MPP simulation methods, one of the twomufflers was considered for a detailed illustration. The muf-fler was made out of stainless steel. The two MPP layers ofthe muffler were both straight cylindrical tubes and had thesame structural parameters. Similarly, the spacing betweenthe inner MPP and the outer MPP was D1 and the spac-ing between the outer MPP and the rigid wall was D2, asshown in Fig. 2. The detailed values of the MPP’s struc-tural parameters and the muffler’s cavity depths are listed inTable 1. Moreover, the MPP perforations were circular holesarranged in a square manner. Thus, theMPP perforation ratiowas p=πd2/(4b2), where b is the spacing between the centersof two adjacent perforations.
The transmission loss is determined by the structure ofthe muffler and is often used to evaluate its acoustic perfor-
mance [17–19]. Thus, we focused on the transmission loss.The transmission loss value of the double-layer MPPmufflerwas calculated and measured, as will be discussed below.
3 Numerical Simulation Using TransferAdmittanceMethod
3.1 Transfer Admittance of MPP
The particle velocity and sound pressure relationshipsbetween both sides of the MPP are expressed by its trans-fer admittance, as follows:[
vn1
vn2
]�
[α1 α2
α4 α5
][p1p2
]+
[α3
α6
](1)
where vn1 and vn2 are the normal particle velocity on the innerand outer surfaces of theMPP, respectively; p1 and p2 are thesound pressure on the inner and outer surfaces of the MPP,respectively; α1, α2, α4, and α5 are the transfer admittancecoefficients; α1 �β, α2 �−β, α4 �−Kβ, α5 �Kβ, β is thetransfer admittance of theMPP, andK is the ratio of the innerdiameter to the outer diameter of the MPP tube; α3 and α6
are the sound source coefficients; in the acoustic calculationof the muffler α3 �α6 �0 [20].
Thus, to define the particle velocity and the sound pres-sure relationships between both sides of theMPP, the transferadmittance of theMPPmust first be calculated. According toMaa [1], whose basic idea is that the MPP is a parallel con-nection of the perforations, the acoustic transfer impedanceof the MPP is expressed as follows:
Z�R + j X (2)
with
R � 32ηt
pd2
⎡⎣
√1 +
k2
32+
√2k
8
d
t
⎤⎦ (3)
X � ωρt
p
⎡⎣1 +
1√9 + k2
2
+ 0.85d
t
⎤⎦ (4)
and
k � d
2
√ωρ
η(5)
where R is the specific acoustic resistance of the MPP;X is the specific acoustic reactance of the MPP; η is thedynamic viscosity, η=1.82×10−5 kg/(ms); ρ is the air den-sity, ρ =1.225 kg/m3; and ω=2π f represents the angularfrequency (f is the frequency of the incident acoustic wave).
Then, the transfer admittance of the MPP is expressed asfollows:
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Simplified Method of Simulating Double-Layer Micro-Perforated Panel Structure 377
Table 2 Acoustic impedance of MPP
Frequency (Hz) Acoustic resistance Acoustic reactance
10 11.8196 1.0791
20 12.4082 2.1553
30 12.8730 3.2286
… … …
6990 64.9178 674.9958
7000 64.9616 675.9426
Fig. 3 Schematic diagram of muffler’s air cavity cross section
β � 1
Z(6)
3.2 Transmission Loss Calculation
TheMPPs of the double-layer MPPmuffler can be simulatedby creating the corresponding transfer admittance properties.Because the acoustic transfer impedance of a MPP is differ-ent at different frequencies, an acoustic transfer impedancetable must be prepared before the simulation calculation.The acoustic transfer impedance table is obtained based oncomplicated calculations with Eqs. 3–5 and is presented inTable 2. The frequency range of the impedance table was setto approximately 10–7000 Hz, and the frequency step wasset to 10 Hz. As mentioned in Sect. 2.2, the structural param-eters of the inner layer MPP and outer layer MPP had thesame values; thus, the MPPs could be simulated using thesame impedance table.
Considering that theMPPs of this muffler weremetal pan-els with large rigidity, the panel vibration effect under anyacoustic loading could be ignored. The geometry model ofthe MPPs was not established, but the air cavity geometrymodel of the muffler was established, and the schematic dia-gram of its cross section is shown in Fig. 3. Surfaces 1 and2 are the inner and outer surfaces, respectively, of the innerlayer gap representing the original inner layer MPP. Surfaces3 and 4 are the inner and outer surfaces, respectively, of theouter layer gap representing the original outer layer MPP.
The finite element (FE) model of the muffler should beestablished after obtaining the geometrical model of the air
Fig. 4 Finite element (FE) muffler model
Fig. 5 Transmission loss calculated by numerical simulation based ontransfer admittance method
cavity. Generally, the length of the element is less than 1/6of the wavelength corresponding to the highest calculationfrequency. The investigated double-layer MPP muffler wasmainly used for high-frequency noise control. Thus, to ensurethe calculation accuracy, the maximum element size of theFEmodel was set to 2mm. Figure 4 shows the entire FEmuf-fler model. As can be seen, both ends of the muffler’s meshmodel represent the mesh model of the end connectors usedto fix the muffler for experimental measurement. Then, theFE model of the muffler was imported into the LMS acous-tics module of the Virtual Lab software. The inner layerMPPwas simulated by defining the transfer admittance propertiesbetween surfaces 1 and 2; that is, by inputting the aboveimpedance table and converting it to the transfer admittancecoefficients α1, α2, α4, and α5 at each frequency. Similarly,the outer layer MPP could also be simulated by definingthe transfer admittance properties between surfaces 3 and4. The fluid material properties of the air domains were themass density of 1.225 kg/m3 and the corresponding soundvelocity of 340 m/s. A unit velocity boundary condition wasdefined at the inlet of the FE model, while a full absorptionboundary condition was defined at its outlet. After perform-ing a harmonic response, the transmission loss value of thedouble-layer MPPmuffler was derived. The plot of the trans-mission loss curve is shown in Fig. 5.
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4 Numerical Simulation by Equivalent FluidMethod
4.1 Porous Material Model
A porous material consists of a frame and a fluid medium.Johnson-Champoux-Allard porous material models are cate-gorized into three types according to frame stiffness; namelythe rigid model, limp model, and elastic model [21].
The rigid porous material model ignores the frame defor-mation, and its acoustic performance depends on the fluidparameters (sound velocity, mass density, specific heat ratio,Prandtl number, and dynamic viscosity) and the pore param-eters (porosity, flow resistivity, tortuosity, and viscous andthermal characteristic lengths). In this case, sound absorp-tion was produced by the air’s viscous resistance, which issimilar to the sound absorption principle of the MPP. Basedon the rigid porous material model, the limp porous mate-rial model considers the influence of the inertial load onthe acoustic wave. Thus, it has an additional characteristicparameter (frame density), which refers to that of a rigidporous material model. Among the three models, the elasticporous material model is the most complicated. Consideringthe frame vibration effect under acoustic loading, the elasticmodel has three additional characteristic parameters (framedensity, Young’s modulus, and Poisson’s ratio), which referto those of the rigid porous material model.
4.2 MPP Equivalent Porous Material Model
A thin porous material structure may become equivalent toa MPP by transforming the MPP structural parameters tothe relevant parameters of the corresponding porous materialmodel. Typically, the fluid medium of the MPP structures isair. Thus, only the pore parameters need to be calculatedwhen establishing the equivalent porous material model fora rigid MPP. According to Atalla [11], the parameters aretransformed as follows:
ϕ � p (7)
σ � 32η
ϕd2(8)
α∞ � 1 +2εet
(9)
when the perforations are circular holes and arranged in asquare manner, the following relationships hold:
εe � 0.24√
πd2(1 − 1.14
√ϕ),√
ϕ < 0.4 (10)
Av � At � d
2(11)
Table 3 Parameters of MPP equivalent porous material model
ϕ (%) σ (Pa·s·m−2)
α∞ Av (mm) At (mm)
8.26 20,959.71 1.66 0.29 0.29
Fig. 6 FE model of muffler’s outlet
where ϕ is the porosity, σ is the flow resistivity, α∞ is thetortuosity, εe is the correction length, Av is the viscous char-acteristic length, and At is the thermal characteristic length.
The three porous material models correspond to threeMPP equivalent porous material models, respectively [12,13]. An appropriate MPP equivalent porous material modelwas selected based on the material properties of the MPP.During the parameter conversion, the panel density was con-sidered in the limp MPP equivalent porous material model,and the panel density, Young’s modulus, and Poisson’sratio were considered in the elastic MPP equivalent porousmaterial model.
4.3 Transmission Loss Calculation
TheMPPs of the double-layer MPP muffler have large rigid-ity. Thus, a rigid MPP equivalent porous material model waschosen for the muffler’s simulation calculation. Based onEqs. 7–11, the relevant parameters of the MPP equivalentporous material model were calculated and are presented inTable 3. Figure 6 shows the FE model of the muffler’s out-let. The equivalent porous material structure was modeledon the basis of the MPP geometry model. The FE modelwas imported into the acoustics module of the LMS VirtualLab software. The two layers of theMPPs were simulated bydefining the corresponding porous material properties. Thefluid material properties of the air domains and the boundaryconditionswere the same as thosementioned in Sect. 3.2. Themuffler’s transmission loss results obtained by the simulationcalculation are shown in Fig. 7.
5 Experimental Verification
To verify and compare the accuracy of the two simplifiedMPP simulation methods with regard to the acoustic sim-ulation of the double-layer MPP muffler, the transmissionloss value of the muffler was measured with an impedance
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Simplified Method of Simulating Double-Layer Micro-Perforated Panel Structure 379
Fig. 7 Transmission loss calculated by numerical simulation based onequivalent fluid method
Fig. 8 Experimental setup
tube using the two-load method [22, 23]. Figure 8 shows theexperimental setup in the laboratory. The impedance tubeused in this study was a B&K Type 4206 impedance tubewith an inner diameter of 29 mm and an effective measure-ment frequency range of 500–6400Hz. Because themuffler’sinlet and outlet diameters were not the same as the diame-ter of the impedance tube, two end connectors were made toconnect them seamlessly for the experiment (the end connec-tors were also considered in the numerical calculation). Fourflush-mounted 1/2 B&K free-field microphones were used tomeasure the sound pressure inside the duct. A loudspeakerdriven by a power amplifier was connected at one end of theimpedance tube as the excitation source, and a removablecap was placed at the other end of the tube to provide twodifferent termination conditions. The muffler sample to betested was located between the two tubes. A four-channelB&K type 3560-C signal analyzer platform and a personalcomputer equipped with the PULSE 8.0 software were usedfor data acquisition and signal processing. Finally, the trans-mission loss value of the muffler was obtained by testing.
The comparison between the numerical simulation resultsand the experimental results of transmission loss for thedouble-layer MPP muffler is presented in Fig. 9. As can
Fig. 9 Comparison of numerical simulations and experimental resultsof transmission loss for the double-layer MPP muffler: + indicatesexperimental measurement; – indicates prediction by transfer admit-tance method; · indicates prediction by equivalent fluid method
Table 4 Second muffler’s MPP structural parameters and cavity depths
t′ (mm) d′ (mm) p′ (%) D1′ (mm) D2
′ (mm)
0.5 0.5 4.91 9 1.8
be seen, the numerical results based on the two simplifiedMPP simulation methods are in relatively good agreementwith the experimental results, which indicates that the pro-posed transfer admittance method and the equivalent fluidmethod can both accurately predict the acoustic performanceof a double-layer MPP structure. It can also be seen that thetransmission loss curves based on the two simplified MPPsimulation methods are almost identical below 4000 Hz.Moreover, in frequencies above 4000 Hz, the transmissionloss results based on the transfer admittance method arecloser to the experimental results at the peaks and dips incomparison with the transmission loss results based on theequivalent fluid method.
To further validate the above results, another double-layerMPPmuffler with different structural parameters was treatedin a similarmanner. The detailed values of thismuffler’sMPPstructural parameters and cavity depths are listed in Table 4.Additionally, the comparison between the numerical simula-tion results and the experimental results of transmission lossis presented in Fig. 10. The comparison results are consis-tent with those mentioned above, which demonstrates thatthe above discussion is reasonable.
6 Conclusions
This study adopted two different simplified MPP simulationmethods to calculate the transmission loss of double-layerMPP mufflers. One method combined classical MPP soundabsorber theory withMPP transfer admittance and simulated
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Fig. 10 Comparison of numerical simulations and experimental resultsof transmission loss for second double-layer MPP muffler: + indicatesexperimental measurement;− indicates prediction by transfer admit-tance method; · indicates prediction by equivalent fluid method
a MPP by defining a set of transfer admittance coefficientsbetween its inner and outer surfaces. The other method con-sidered the effect of theMPPmaterial properties and replacedthe MPP with an equivalent porous material model. To ver-ify and compare their accuracy, the transmission loss value ofthe double-layerMPPmufflerswasmeasured experimentallywith an impedance tube. According to the comparison resultsbetween the two numerical simulations and the experimentaldata, conclusions may be illustrated as follows:
(1) The transfer admittance method and the equivalent fluidmethod both provide an easy and efficient approachtoward predicting the acoustic performance of double-layer MPP structures.
(2) Acoustic prediction based on the transfer admittancemethod may be more accurate than that based on theequivalent fluid method at the peaks and dips of TLcurve.
Acknowledgements This study was supported by the National KeyLaboratory Open Foundation of Tractor Power System (Grant No.SKT2017012) and the National Natural Science Foundation of China(Grant No. 51575410).
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