On the variations of acoustic absorption peak with flow velocity in Micro-Perforated Panels at high level of excitation
OOn the variations of acoustic absorption peak with fl ow velocity in Micro ‐ Perforated
Panels at high level of excitation Rostand
Tayong,
Thomas
Dupont, and
Philippe
Leclaire
Laboratoire de Recherche en Mécanique et Acoustique (LRMA),
ISAT
Université de Bourgogne,
Nevers
France
The acoustic behavior of micro ‐ perforated panels (MPP) is studied theoretically andexperimentally at high level of pressure excitation. A model based on Forchheimer’s regime of fl owvelocity in the perforations is proposed. This model is valid at relatively high Reynolds numbers and low
Mach numbers.
The experimental method consists in measuring the acoustical pressure at three di ff erentpositions in an impedance tube, the two measurement positions usually considered in an impedance tube and one measurement in the vicinity of the rear surface of the MPP.
The impedance tube is equipped with a pressure driver instead of the usual loudspeaker and capable of delivering a high soundpressure level up to dB. Several
MPP specimens made out of steel and polypropylene were tested. Measurements using random noise or sinusoidal excitation in a frequency range between and were carried out on MPPs backed by air cavities. It was observed that the maximum of absorption can be a positive or a negative function of the fl ow velocity in the perforations. This suggests the existence of a maximum of absorption as a function of fl ow velocity. This behavior was predicted by the model and con fi rmed experimentally. Introduction
Micro ‐ perforated panels (referred to as MPPs), when associated with air cavities, are of great interestin noise reduction applications such as Helmholtz resonators.
They are robust and easy to manufacture andthey can be used in hostile temperature and pressure environments. Several models were developed in thelinear regime to describe their surface impedance, their absorption coe ffi cient and their transmission lossfactor . A simplified schematic of the main linear regime phenomena is depicted in Fig.1. In the case of high sound pressure excitation (Fig. it is thought that the jet formation (vorticity) atthe opening modi fi es signi fi cantly the absorption mechanisms. By measuring the velocity in the aperturewith the help of a hot wire, Ingard and
Ising showed that at high sound pressure levels the fl ow separates atthe outlet ori fi ce, forming a high velocity jet. During the in fl ow half ‐ cycle, the incident fl ow at the inlet of theori fi ce is essentially irrotational but highly rotational (in form of jetting) after exiting from the outlet ori fi ce.The acoustic particle velocity is increased sharply as the wave is squeezed into the minute perforations. Thenonlinear regime implies that the acoustic properties (mainly the impedance) are dependent upon theacoustic particle velocity either in front of the panel or into the aperture. During the other half of the cycle,the fl ow pattern is reversed. Recently,
Kraft et al. proposed a model for MPP for high sound pressure.
Theirmodel was derived from the measurements of single degree ‐ of ‐ freedom liners over a wide range of soundpressure levels. It gives a prediction for the combined linear and nonlinear resistance and reactance for aface sheet. The nonlinear parameter which is considered is the discharge coe ffi cient de fi ned as the ratio Absorption variations of Micro-perforated panels f the actual discharge (discharge that occurs and which is a ff ected by friction as the jet passes through the ori fi ce) divided by the ideal discharge (without friction). The discharge coe ffi cient value varies between and Kraft et al.5,6 used a fi xed discharge coe ffi cient of in their model assuming constant nonlinear behavior regardless of the fl ow conditions. Moreover, their resistive and reactive part of the plateimpedance are independent of frequency essentially assuming constant ori fi ce mass. Melling derived an expression for the resistive impedance in the nonlinear regime. The discharge coe ffi cient used by Melling is valid only for the case where the panel thickness is smaller than the ori fi ce diameter. However, this discharge coe ffi cient is highly frequency dependent. Later on,
Maa showed that the acoustic non ‐ linearity of apertures is an external phenomenon i.e. the internal impedance is independent of the sound intensity. He suggested a nonlinear impedance term expression for MPP with small open area ratio.
This expression is the ratio of the acoustic velocity inside the aperture divided by the product of sound speed and open area ratio. Morerecently,
Hersh et al. derived a model extended to multiple ori fi ces via the open area ratio assuming continuity of volume velocity and no interactions between the holes. The discharge coe ffi cient and some other parameters of the model were determined empirically. The aim of this work is to propose a model for micro ‐ perforated panels backed by an air cavity involving parameters that are easier to estimate than the discharge coe ffi cient. This model is based on Maa’s previous work on low sound pressure excitation, the use of dimensional analysis and Forchheimer’s law.
The fi rst section deals with the theoretical analysis of Maa’s
MPP model in the linear regime. The next section presents the derivation of the proposed model. In this section, a dimensionless parameter involved in the MPP behavior is introduced. An experimental and theoretical analysis shows that the maximum of absorption can be a positive or a negative function of the fl ow velocity in the perforations, suggesting the existence of a maximum of absorption as a function of fl ow velocity. This behavior was predicted by the proposed model and con fi rmed experimentally. The fi nal section describes the experimental setup, and o ff ers comments on the results. I. Linear regime model
The main mechanism of absorption in the linear regime is the conversion of the acoustical energy intoheat. In this regime (low sound pressure and velocity amplitudes), if the dimensions of the MPP (diameter ofholes, holes separation, thickness) are small with respect to the impinging acoustic wavelength, and if theaperture dimension (diameter of holes) is of the order of the viscous and thermal boundary layersthicknesses (Fig. the major part of the acoustical energy is dissipated through viscous and thermal e ff ects. Maa’s linear model
Based on the theory and equations of acoustical propagation in short and narrow circular tubes, Maa derived an equation of aerial motion given for one perforation by (cid:1862)(cid:2033)(cid:2025)(cid:1873) (cid:3398) (cid:2015)(cid:1870) (cid:2034)(cid:2034)(cid:1870) (cid:3428)(cid:1870) (cid:2034)(cid:2034)(cid:1870) (cid:1873)(cid:3432) (cid:3404) ∆(cid:1868)(cid:1860) , (1) where p is the pressure drop across the tube, h the length of the tube (which corresponds to the thickness of the MPP), η the dynamic viscosity, ρ the density of air, ω the angular frequency, r the radial coordinate and u the particle velocity in the perforation. Absorption variations of Micro-perforated panels IG. Simpli fi ed schematic of the linear regime (low sound pressure levels). By solving the equation with respect to the velocity, the speci fi c acoustic impedance of the short tubede fi ned as the ratio of p to the average velocity < u > over a cross ‐ sectional area of the tube is given by : (cid:1852) (cid:3043)(cid:3032)(cid:3045)(cid:3033) (cid:3404) ∆(cid:3043)(cid:2996)(cid:3048)(cid:2997) (cid:3404) (cid:1862)(cid:2033)(cid:2025) (cid:2868) (cid:1860) (cid:3428)1 (cid:3398) (cid:2870)(cid:3051)(cid:3493)(cid:2879)(cid:3037) (cid:3011) (cid:3117) (cid:4666)(cid:3051)(cid:3493)(cid:2879)(cid:3037)(cid:4667)(cid:3011) (cid:3116) (cid:4666)(cid:3051)(cid:3493)(cid:2879)(cid:3037)(cid:4667) (cid:3432) (cid:2879)(cid:2869) , (2) where (cid:1876) (cid:3404) (cid:1856). (cid:3495) (cid:3104)(cid:3096) (cid:3116) (cid:2872)(cid:3086) is a constant de fi ned as the ratio of ori fi ce diameter d to the viscous boundarylayer thickness of the air in the ori fi ce, J and J the Bessel functions of the fi rst kind of orders and An approximation of the above equation on the Bessel functions and valid for narrow tubes wasgiven by Maa1 as : (cid:1852) (cid:3043)(cid:3032)(cid:3045)(cid:3033) (cid:3404) 32(cid:2015)(cid:1860)(cid:1856) (cid:2870) (cid:3496)1 (cid:3397) (cid:1876) (cid:2870)
32 (cid:3397) (cid:1862)(cid:2033)(cid:2025) (cid:2868) (cid:1860) (cid:4680)1 (cid:3397) 1(cid:3493)3 (cid:2870) (cid:3397) (cid:1876) (cid:2870) /2(cid:4681). (3)
Due to the end radiating e ff ects at the aperture, an end correction factor proposed by Rayleigh should betaken into account twice (once for each end).
This correction is important when the perforation diametersare greater or of the order of the plate thickness . The radiating impedance for the end correction is givenby : (cid:1852) (cid:3045)(cid:3028)(cid:3052) (cid:3404) (cid:4666)(cid:1863) (cid:2868) (cid:1856)(cid:4667) (cid:2870) (cid:2868) (cid:1856)6(cid:2024) , (4) where k o is the wave number. The e ff ect of the vibration of the air particles on the ba ffl e in the vicinity of theaperture increases the thermo ‐ viscous frictions. To take this e ff ect into account, Ingard and
Labate proposed an additional factor on the resistive part of the tube impedance. This resistance is given by : (cid:1844) (cid:3020) (cid:3404) 12 (cid:3493)2(cid:2033)(cid:2025) (cid:2868) (cid:2015). (5) Under a certain number of assumptions, a process of homogenization can be applied, providing anexpression of the impedance for multiple perforations. The minimal distance between perforations must begreater than their diameters and smaller than the wavelength and so it is possible to consider that there is Absorption variations of Micro-perforated panels o interactions between the apertures and the absorption mechanism is dominant. The
MPP must be thinner than the wavelength to insure the continuity of the velocities on both sides of the plate. Within these assumptions, the visco ‐ thermal interactions between the fl uid the solid are taken into account through a viscosity correction function The total impedance of the MPP is then given by the impedance of one perforation divided by the open area ratio φ : Z MPP = Z perf + 2Z ray + 4R S Φ (6) II.
Model for the impedance of MPP at high sound pressure levels A. Variation of the impedance with MPP geometrical parameters
From experimental measurement using hot wires,
Ingard and
Ising have observed that over a halfperiod of the propagation of the high amplitude sound wave, the incident fl ow is irrotational while theout fl ow is a highly rotational jetting (Fig. During the other half period the fl ow pattern is reversed.According to Ingard , air current losses energy at the inlet due to friction on the panel surface along whichpart of the air current has to move when it is squeezed into the small area of the tube. They have alsoobserved that the ori fi ce resistance varies linearly with particle velocity. FIG. Simpli fi ed schematic of the nonlinear regime (High sound pressure levels). From the semi ‐ empirical approach of Ingard , we investigated a model based on dimensionalanalysis in order to study the in fl uence of the MPP geometrical parameters.
This model is found to be consistent with including the Forchheimer nonlinear fl ow regime in the linear model. This approach was used by Auregan and
Pachebat for the study of nonlinear acoustical behavior of rigid frame porous materials. Strictly speaking, the characteristic impedance is not de fi ned for nonlinear wave propagation.However, if the harmonic distortion is not too high ( fi rst harmonic at much higher amplitude than the following harmonics), it is possible to de fi ne an impedance. Ingard and
Ising proposed an expression where the real part of the impedance is given by Absorption variations of Micro-perforated panels (cid:1857)(cid:4668)(cid:1852) (cid:3014)(cid:3017)(cid:3017) (cid:4669) (cid:3015)(cid:3013) (cid:3404) (cid:1827)(cid:1873) (cid:3397) (cid:1828) , (7) where Re {Z MPP } is the MPP resistance, u the velocity in the perforation, A and B are constants. It is found thatthe variation of the reactance with u is fairly moderate. Melling explained that this remark about thenonlinear reactance (imaginary part of the impedance) is considered valid for thin plates. He stated that thereactance tends to an asymptotic limiting value (this would be con fi rmed by our experimental results) ofapproximately one ‐ half the linear regime value. Dividing u by the speed of sound c , the Mach number M = u/c is introduced and this expression canbe written in a normalized form as: (cid:1844)(cid:1857)(cid:4668)(cid:1878) (cid:3014)(cid:3017)(cid:3017) (cid:4669) (cid:3015)(cid:3013) (cid:3404) (cid:1853)(cid:1839) (cid:3397) (cid:1854) , (8) where z MPP is the normalized impedance, a and b are now dimensionless parameters that will depend on theMPP geometrical features and on the fl uid constants. This was already noticed by Maa in a recent work inwhich he observed that a is inversely proportional to the open area ratio φ while Hersh observed that theresistance is dependent on the ratio h/d. From the dimensional analysis , it is found that the variation seems to be a more complicatedcombination of these two behaviors : (cid:1853) (cid:3404) (cid:1837) (cid:3436)(cid:1856)(cid:1860)(cid:3440) (cid:3039) (cid:3436)(cid:1860)(cid:2025) (cid:2868) (cid:1855) (cid:2868) (cid:2020) (cid:3440) (cid:3040) (cid:2038) (cid:3041) , (9) where K, l, m and n are constants. However, we tend to believe that the constant K is quite related to theshape of the aperture edge and to the material itself. The in fl uence of the edges shapes on the resistive partof the ori fi ce impedance of a cylindrical tube has been studied by Atig et al. B. Constant determination from the
Forchheimer nonlinear fl ow model in the perforations The study of rigid frame porous materials at high sound pressure levels by Auregan and
Pachebat makes use of Forchheimer’s law , which states that for high Reynolds number (Re larger than unity), the fl ow resistivity increases linearly with the Reynolds number.
Since the
Reynolds number is proportional tothe aperture diameter, the resistivity (also seen as the real part of the impedance per unit thickness) isdirectly proportional to the diameter. In fact, from expression (9), by considering l = m = and n = − , oneeasily fi nds the result given by Auregan and
Pachebat if the viscous characteristic length is taken equal tothe perforation radius (case of cylindrical pores). Finally, the model for the coe ffi cient of the real part of the MPP impedance is (cid:1844)(cid:1857)(cid:4668)(cid:1878) (cid:3014)(cid:3017)(cid:3017) (cid:4669) (cid:3015)(cid:3013) (cid:3404) (cid:1853)(cid:1839) (cid:3397) (cid:1854) (10) (cid:1853) (cid:3404) (cid:1837) (cid:3436)(cid:1856)(cid:1860)(cid:3440) (cid:2869) (cid:3436)(cid:1860)(cid:2025) (cid:2868) (cid:1855) (cid:2868) (cid:2020) (cid:3440) (cid:2869) (cid:2038) (cid:2879)(cid:2869) , a can be further simpli fi ed as (11)Absorption variations of Micro-perforated panels (cid:3404) (cid:1837) (cid:1856)(cid:1855) (cid:2868) (cid:2029)(cid:2038) , (12) where ν is the kinematic viscosity. Now, still following
Auregan and
Pachebat work, expression of b is related to the low sound excitation resistance value of the MPP.
They observed regimes : the linear, the transition and a weakly nonlinear regime. An appropriate expression is given by : (cid:1854) (cid:3404) (cid:4666)1 (cid:3397) (cid:2012)(cid:4667)(cid:1844)(cid:1857)(cid:4668)(cid:1878) (cid:3014)(cid:3017)(cid:3017) (cid:4669) (cid:3039)(cid:3036)(cid:3041)(cid:3032)(cid:3028)(cid:3045) . (13) In fact, b is the intercept with vertical axis (real part of impedance) and δ is a dimensionless parameteradapted to describe the high sound pressure regime. The parameters K and δ are to be determinedexperimentally. III.
Absorption coefficient of a MPP backed by a cavity Under the same assumption on harmonic amplitudes, it is possible to de fi ne a cavity impedance for high sound pressure levels and it is given by the usual expression (normalized form) : (cid:1878) (cid:3030) (cid:3404) (cid:3398)(cid:1862)(cid:1855)(cid:1867)(cid:1872)(cid:4666)(cid:1863) (cid:2868) (cid:1830) (cid:3030) (cid:4667) , (14) where Dc is the air gap thickness and the normalized surface impedance of the total system (MPP coupled toan air cavity) is given by the superposition of the two impedances : (cid:1878) (cid:3046) (cid:3404) (cid:1878) (cid:3014)(cid:3017)(cid:3017) (cid:3397) (cid:1878) (cid:3030) . The re fl ection coe ffi cient is given by the usual formula (cid:1844) (cid:3404) (cid:3053) (cid:3294) (cid:2879)(cid:2869)(cid:3053) (cid:3294) (cid:2878)(cid:2869) (16)(15) and the acoustic absorption coe ffi cient can be calculated from (cid:2009)(cid:4666)(cid:2033)(cid:4667) (cid:3404) 1 (cid:3398) |(cid:1844)(cid:4666)(cid:2033)(cid:4667)| (cid:2870) . (17) When the real and the imaginary part of the total impedance zs are separated, another expression of the absorption coe ffi cient can be given by : (cid:2009) (cid:3404) (cid:2872)(cid:3019)(cid:3032)(cid:4668)(cid:3053) (cid:3294) (cid:4669)(cid:4666)(cid:2869)(cid:2878)(cid:3019)(cid:3032)(cid:4668)(cid:3053) (cid:3294) (cid:4669)(cid:4667) (cid:3118) (cid:2878)(cid:4666)(cid:3010)(cid:3040)(cid:4668)(cid:3053) (cid:3294) (cid:4669)(cid:4667) (cid:3118) (18) IV.
Maximum absorption as a function of Mach number By inserting the expression of the real part of the MPP impedance in the equation for the absorption coe ffi cient we have (cid:2009) (cid:3404) (cid:2872)(cid:4666)(cid:3028)(cid:3014)(cid:2878)(cid:3029)(cid:4667)(cid:4666)(cid:2869)(cid:2878)(cid:3028)(cid:3014)(cid:2878)(cid:3029)(cid:4667) (cid:3118) (cid:2878)(cid:4666)(cid:3010)(cid:3040)(cid:4668)(cid:3027) (cid:3294) (cid:4669)(cid:4667) (cid:3118) (19) The maximum of absorption is obtained for Im{zs } = : (cid:2009) (cid:3014) (cid:3404) (cid:2872)(cid:4666)(cid:3028)(cid:3014)(cid:2878)(cid:3029)(cid:4667)(cid:4666)(cid:2869)(cid:2878)(cid:3028)(cid:3014)(cid:2878)(cid:3029)(cid:4667) (cid:3118) (20)Absorption variations of Micro-perforated panels This expression is now di ff erentiated with respect to M in order to study the variations with the Mach number (and implicitly with the incident sound pressure level) (cid:3105)(cid:3080) (cid:3262) (cid:3105)(cid:3014) (cid:3404) (cid:2872)(cid:3028)(cid:4666)(cid:2869)(cid:2879)(cid:3028)(cid:3014)(cid:2879)(cid:3029)(cid:4667)(cid:4666)(cid:2869)(cid:2878)(cid:3028)(cid:3014)(cid:2878)(cid:3029)(cid:4667) (cid:3119) (21)
This result shows that a critical value of the Mach number exists for which the absorption coe ffi cient isextremum (Fig. Since a = , this last expression provides a limit Mach number M c given by: From the study of the variations of the absorption coe ffi cient function, it is found that the peak of absorption (maximum of absorption with respect to frequency) increases with the Mach number, reaches a maximum as the Mach number approaches its critical value and then decreases for M increasing beyond the critical value M c . It is worth noticing that Maa predicted this behavior. In the present article, we propose a more re fi ned model, we de fi ne a critical value for the Mach number and include experimental data.
Evidently, this behavior can be observed only if the critical value is above the linear/nonlinear regime limit. Indeed, the experimental results will show that in some cases, a value for M c will not be identi fi ed if it is located in the linear range. In this case the MPP absorption peak will only decrease with the increasing soundpressure level. It is also worth noticing that α M (M c ) = for any MPP with Mc located in the nonlinear domain. This result shows potential applications in the design of MPPs. (cid:1839) (cid:3404) (cid:2869)(cid:2879)(cid:3029)(cid:3028) (22)
FIG. Absorption coe ffi cient at the resonance frequency versus the acoustical Mach
Number in front of the MPP. V. Experiments A. MPP samples
The measurements are performed on strong copolymer ‐ made MPP and steel ‐ made MPP (Fig. and Table
I).
All the sample panels have an external diameter of mm and all the holes are well separated from each other (no interactions between the apertures) and are evenly distributed over the panel area. The mounting conditions of the MPP inside the tube are closer to the clamped conditions than to the simply supported conditions. Absorption variations of Micro-perforated panels IG. Perforated panel
Sample.
TABLE I. Micro ‐ perforated panel characteristics. h (mm) d (mm) Φ (%) Density (Kg/m ) 900
21 × 10 MaterialStrong copolymer
Strong copolymer
Steel
MPP1
MPP2
MPP3 B. Impedance tube and data acquisition A schematic of the impedance tube used is shown in Fig. It is a rigid circular plane ‐ wave tube with a diameter of mm. Plane wave propagation is assumed below the cut ‐ o ff frequency (1.7 KHz). At the left hand side, a compression driver JBL model is mounted as the source of excitation. A transition piece provides a continuity transition between the circular section of the compression driver and the circular crosssection of the plane ‐ wave tube. At the right hand side of the tube, a soundproof plunger is used as the rigid backing wall. By moving the plunger along the longitudinal axis of the tube, one is able to create an air cavitybehind the MPP sample.
The
MPP sample is mounted between the speaker and the plunger. Three ¼ microphones are used to perform the signal detection. Two microphones are used to calculate the surface impedance of the sample by the standard impedance tube measurement technique . The third microphone (reference micro in Fig. acts as a reference microphone to get the level of pressure at the sample surface. When performing high sound pressure measurements inside an impedance tube, it is important to check the FIG. Schematic of the impedance tube used for the measurements. Absorption variations of Micro-perforated panels ABLE
II.
Dimensionless parameters of the micro ‐ perforated panels. K − − − δ MPP2
MPP3 resulting standing wave. In fact, up to a certain level, depending on the frequency excitation and the sample parameters, the standing wave seems to saturate and therefore the linear propagation hypothesis would nolonger be valid. E ff ects of bifurcation may occur. These phenomena of saturation and bifurcation wereobserved and shown by Maa and
Liu . A fi rst excitation is done with a periodic random noise signal in orderto have a general view of the absorption coe ffi cient curve and locate the viscous peak position. The result is used to determine the resonant frequency and to perform a second excitation (sine excitation) at frequencies concentrated around the viscous absorption peak(s). The amplitude of the source is adjusted such that the sound pressure level measured by the reference microphone is set at the desired level. The
SPL (Sound
Pressure
Level) is varied from ‐ dB at the face of the MPP monitored using the reference microphone. A phase and amplitude calibration method is used to correct the transfer function between the measurement microphones. The measurement are carried out at high sound pressure levels. However, assuming the plane wave hypothesis, one can measure the pressures and velocities on any section of the tube using the two microphone method. See for instance
Dalmont for more details. The acoustic velocity on the panel surface is given by: (cid:1873) (cid:3404) (cid:1862) (cid:3043) (cid:3117) (cid:3027) (cid:3116) (cid:3009)(cid:3030)(cid:3042)(cid:3046)(cid:4666)(cid:3038) (cid:3116) (cid:3039) (cid:3117) (cid:4667)(cid:2879)(cid:2913)(cid:2925)(cid:2929) (cid:4666)(cid:3038) (cid:3116) (cid:3039) (cid:3118) (cid:4667)(cid:2929)(cid:2919)(cid:2924) (cid:4666)(cid:3038) (cid:3116) (cid:3046)(cid:4667) (23) where l =s+l as in Fig. Z c is the characteristic impedance of air, k is the wave number, p is the pressureon microphone and l (resp. l ) is the distance from microphone (resp. to the panel sample. The valuesof the velocity shown in the experimental results are the viscous peak corresponding particle velocity. VI.
Results and comments In this section, the measurements are performed taking a single MPP with an air cavity behind and arigid wall. The dimensionless parameters K and δ used to fully determine expression (8) are given in Table
II)
Fig. shows the comparison between experimental results and the present model simulations for theresistance as a function of the Mach number for all the
MPP samples.
The air cavity depth is mm. Theexperiments and the present model are in good agreement for the high excitation levels. This result pointsout the fact that for high sound pressure levels, the dependency of the resistance and the Mach number islinear.
The slope is di ff erent from one MPP to another, revealing therefore the fact that this slope, in thesame measurement conditions, depends on the MPP geometrical parameters.
This result also shows that thevalue of constant K is related to the type of material (1.43 × − for the copolymer samples and × − for the steel sample). As already mentioned, we tend to believe that the constant K is quite related to theshape of the aperture edge and to the thermal properties of the material considered. Fig. shows the comparison between experimental results and the present model simulations for themaximum absorption coe ffi cient (viscous peak) versus the Mach number for all the
MPP samples.
The air ca ‐ Absorption variations of Micro-perforated panels IG. Surface impedance resistive part as a function of the Mach number in front of the MPP sample.
Aircavity depth of mm. vity depth is mm. The experiments and the simulation are in fairly good agreement. This con fi rms the factthat depending on the value of the limit Mach number, with the increase of sound levels, the viscous peakwill in a fi rst phase rise up to a maximum value before decreasing. This limit
Mach number point isobservable on MPP1 and
MPP3 but not on MPP2. In fact, if this limit Mach number is low enough, theabsorption peak will solely decrease with the increase of sound pressure levels. FIG. Absorption coe ffi cient at resonance as a function of the Mach number in front of the MPP sample.
Air cavity depth of mm. Fig. shows the comparison between experimental and simulations results (Maa8 , Hersh and thepresent model) for the absorption coe ffi cients of MPP1 at dB in front of the panel (referencemicrophone) in the [200 ‐ Hz] frequency range for an air cavity depth of mm. The simulation of thepresent model and the measurement are in very good agreement. The fi rst peak (around Hz) is theviscous peak and the second peak (around Hz) is the result of the structural response of a panel coupled Absorption variations of Micro ‐ perforated panels an air cavity. This structural response is described in the appendix. The presented models and themeasurements are in fairly good agreement except for the Hersh high sound model which does not fi t wellwith the experiment results. FIG. Comparison between the absorption coe ffi cients of MPP1 at dB (U=0.325m/s) on the reference microphone. Air cavity depth of mm. Fig. and Fig. show the comparison between experimental and simulations results of the surface impedance (resistance (a) and reactance (b)) of MPP1 at dB in front of the panel (reference microphone) in the [200 ‐ Hz] frequency range for an air cavity depth of mm. The present model and the measurements are in very good agreement. On Fig.
Maa’s model underestimates the nonlinear resistance whereas
Hersh’s model agrees well with the experimental results except around the structural response frequency. On Fig. the models are in good agreement with the experimental results except for Hersh’s model.
FIG. Comparison between the surface impedances (resistance (a) and reactance (b)) of MPP1 at dB (U=0.325m/s) on the reference microphone. Air cavity depth of mm. Fig. shows the comparison between experimental and simulation results (Maa , Hersh and the present model) for the absorption coe ffi cients of MPP3 at dB in front of the panel (reference micropho ‐ Absorption variations of Micro-perforated panels e) in the [200 ‐ Hz] frequency range for an air cavity depth of mm. The present model and themeasurement results are in very good agreement. FIG.
Comparison between the absorption coe ffi cients of MPP3 at dB (U=0.337m/s) on the reference microphone. Air cavity depth of mm. Fig. and
Fig. show the comparison between experimental and simulations results for the surface impedance (resistance (a) and reactance (b)) of MPP3 at dB in front of the panel (reference microphone) in the [200 ‐ Hz] frequency range for an air cavity depth of mm. On Fig. for the resistance part, below the present model and the measurement results are in very good agreement. Yet beyond
Hz, the agreement is not good. This may imply considering a certain high frequency ‐ dependency of the nonlinear parameter for a more accurate prediction. However, as mentioned, for relatively high frequency, the absorption seems to be much more in fl uenced by the imaginary part of the impedance. Maa’s model underestimates the measurement results while
Hersh’s model tendency is goodcompared to the measurement results. On Fig. for the reactance part, the present model and
Maa’smodel both agree accurately with the measurement results.
Conclusion A model for the high sound pressure of Micro ‐ Perforated
Panels (MPP) when backed by an air cavityhas been proposed and tested experimentally. A dimensionless parameter and a limit Mach number werefound and used.
These are suitable to predict the impedance of the system at high sound pressure levels.The results and the analysis showed that the model and the experiments are in good agreement. Thetheoretical work revealed the fact that with the increase of sound pressure level (or Mach number), theviscous absorption peak will in a fi rst phase rise to a maximum value and then decrease in a second phase.Experimentally, it was noticed that this latter result can clearly be observed if the value of the limit Machnumber is above the linear regime limit of the MPP. If the limit Mach number is below the linear regimelimit, the viscous absorption peak will only decrease with increasing sound pressure level. Finally, it wasshown that MPP are very sensitive to the incident sound pressure. Thermal e ff ects around the MPP systemtends to play an important role with the increase of sound intensities. Further work will consist ininvestigating these e ff ects and to properly understand their in fl uence on the MPP impedance for high sound
Absorption variations of Micro-perforated panels 12
IG.
Comparison between the surface impedances (resistance (a) and reactance (b)) of MPP3 at dB(U=0.337m/s) on the reference microphone. Air cavity depth of mm. intensities. It may also be interesting to investigate the apertures interactions when submitted to relativelyhigh sound pressure amplitudes. Acknowledgments
The support grant for this work was provided by the Conseil
Régional de Bourgogne.
APPENDIX A: model including the structural behavior of the panel In the absorption test of the copolymer micro ‐ perforated panel, an additional sound absorption peak was found around the frequency of due to the panel vibration fi rst mode. Theoretical and experimental studies were done by Ford and
McCormick and also Frommhold et al. on panel absorbers without perforations. From these studies, it was found that this additional absorption peak is due to the panel ‐ cavity resonance. Ford and
McCormick gave an expression of the structural impedance of such system (membrane coupled to an air cavity). This expression is considered and used in this paper when the structural resonant frequency of the panel is not beyond the frequency range of study (200 ‐ Only the fi rst structural mode is considered since the other structural frequency modes are always out of thefrequency range studied. Since the
MPP backed by an air cavity system can be considered analogous to anelectrical circuit, it is well assumed that the normalized acoustic impedance of the whole system Zs is given by: (cid:1852) (cid:3046) (cid:3404) (cid:3027) (cid:3262)(cid:3265)(cid:3265) (cid:3027) (cid:3297)(cid:3284)(cid:3277) (cid:3027) (cid:3262)(cid:3265)(cid:3265) (cid:2878)(cid:3027) (cid:3297)(cid:3284)(cid:3277) (cid:3397) (cid:1852) (cid:3030) , (A1) where Z MPP is the MPP surface impedance. Z c is the air cavity impedance. Z vib is the structural impedance of an air cavity backed membrane (without perforations) well described by Ford and
McCormick as: (cid:1852) (cid:3049)(cid:3036)(cid:3029) (cid:3404) (cid:3005) (cid:3293)(cid:3284)(cid:3282) (cid:3003) (cid:3288)(cid:3289) (cid:3093)(cid:3104)(cid:3035) (cid:3120) (cid:3397) (cid:1862)(cid:2033)(cid:1839) (cid:3043) (cid:1827) (cid:3040)(cid:3041) (cid:3397) (cid:2869)(cid:3037)(cid:3104) (cid:4666) (cid:3005) (cid:3293)(cid:3284)(cid:3282) (cid:3003) (cid:3288)(cid:3289) (cid:3035) (cid:3120) (cid:3397) (cid:3096) (cid:3116) (cid:3030) (cid:3116)(cid:3118) (cid:3005) (cid:3278) (cid:4667) , (A2)Absorption variations of Micro-perforated panels 13 where A mn and B mn are the modal constants, Mp the mass per unit area of panel, h is the lateral dimension(thickness of the panel), D c is the air cavity depth and D rig the bending sti ff ness of the panel. Where (cid:1830) (cid:3045)(cid:3036)(cid:3034) (cid:3404) (cid:3006)(cid:3035) (cid:3119) (cid:2869)(cid:2870)(cid:4666)(cid:2869)(cid:2879)(cid:3100) (cid:3291)(cid:3118) (cid:4667) (A3) where E is the Young modulus (Table
I), ξ =10% and ν p =0.3. Since our
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