Fractal Model for Acoustic Absorbing of Porous Fibrous...

Research ArticleFractal Model for Acoustic Absorbing of Porous FibrousMetal Materials

Weihua Chen12 Tianning Chen1 XiaopengWang1 Jiuhui Wu1 and Suobin Li1

1School of Mechanical Engineering and State Key Laboratory for Strength and Vibration of Mechanical StructuresXirsquoan Jiaotong University Xirsquoan Shaanxi 710049 China2College of Mechano-Electronic Engineering Lanzhou University of Technology Lanzhou 730050 China

Correspondence should be addressed to Xiaopeng Wang xpwangmailxjtueducn

Received 1 May 2016 Accepted 10 August 2016

Academic Editor Yuri S Karinski

Copyright copy 2016 Weihua Chen et al This 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

To investigate the changing rules between sound absorbing performance and geometrical parameters of porous fibrous metalmaterials (PFMMs) this paper presents a fractal acoustic model by incorporating the static flow resistivity based on Biot-Allardmodel Static flow resistivity is essential for an accurate assessment of the acoustic performance of the PFMM However it is quitedifficult to evaluate the static flow resistivity from the microstructure of the PFMM because of a large number of disordered poresIn order to overcome this difficulty we firstly established a static flow resistivity formula for the PFMM based on fractal theorySecondly a fractal acoustic model was derived on the basis of the static flow resistivity formula The sound absorption coefficientscalculated by the presented acoustic model were validated by the values of Biot-Allard model and experimental data Finally thevariation of the surface acoustic impedance the complex wave number and the sound absorption coefficient with the fractaldimensions were discussedThe research results can reveal the relationship between sound absorption and geometrical parametersand provide a basis for improving the sound absorption capability of the PFMMs

1 Introduction

Over the last two decades the propagation characteris-tics of acoustic wave in porous fibrous metal materials(PFMMs) have attracted considerable attention because oftheir potential engineering applications [1ndash3] The PFMM isa new type of multifunctional structural material and hasdemonstrated various novel physical properties particularlythe sound absorption under extreme conditions such as highsound pressure level and high temperature Due to theirexcellent sound absorption performance and distinctive heatresistance lightness and stiffness PFMMs are widely used inaircraft engine liners and combustion chambers for the rocketengines [4 5]

In 1956 Biot [6 7] constructed the constitutive equationsand fluctuation control equations of porous materials foranalyzing their acoustic properties Cox and DrsquoAntonio [8]studied the static flow resistivity of porous absorbers andsummarized several empirical and semiempirical formulasfor the parallel and perpendicular fibers random fibrous

arrangement polyester fibers and so forth These empiricalformulas can be used to evaluate the static flow resistivityKawasima [9] and Tarnow [10 11] established themicrostruc-ture acoustic model and studied the sound propagation andthe static flow resistivity of fibrous materials It was shownthat the static flow resistivity is important to describe soundpropagation and attenuation in fibrous materials Althoughthe viscous dissipation of sound wave in a porous materialdepends on the thermal exchanges due to the compressionand rarefaction instances between the air and the porouswalls the static flow resistivity dominates in viscous lossesAllard and Champoux [12] also found that the flow resistanceof porous fibrous materials is closely related to the porousstructure but the accuracy of calculation for the static flowresistivity depends on the extent of approximation of theactual microstructures In order to study the static flowresistivity of fibrous materials based on a great deal ofexperimental data Delany and Bazley [13] built an empiricalrelationship between the characteristic impedance and air-flow resistivity Sebaa et al [14] used the reflected acoustic

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 2890857 17 pageshttpdxdoiorg10115520162890857

2 Shock and Vibration

(a) (b)

Figure 1 The PFMM samples (a) photograph of the samples (b) SEM photo of micrographic surface

wave to measure the static flow resistivity of porous materialand showed that the reflected wave is much more sensitiveto the static flow resistivity than to porosity Fellah et al [15]introduced an acoustical transmission method to measurepermeability of porous materials at low-frequency rangeOn the basis of a temporal model of the direct and inversescattering problem the proposed method is simple rapidand efficient Experimental and numerical validation resultsof this method were presented Sadouki et al [16] establishedan acoustical method to measure static viscous permeabilityof porous absorbing materials This method was derivedwithout a priori estimation of the porosity but by simplyusing the transmitted waves Van der Kelen [17] investigatedthe static flow resistivity of two porousmaterials (fibrous glasswool and melamine foam) and the measurement methodand inverse estimation were employed to determine theanisotropic flow resistivity The research results showed thatthe homogeneities of the density and the static flow resistivityfor the two materials are different but seem to have nocorrelation with each other However the sound propagationpathways in the PFMM are tortuous because of the irregularporous structure and there have been no theoretical modelsproperly demonstrating the variation rules of the static flowresistivity with porous structures and pathways Thereforeit is essential to present a quantitative description of thedisordered pores in the PFMMs for evaluating the soundabsorption performance

Wang et al [18] studied the microstructure features ofFeCrAl fiber materials and obtained the fractal dimensionsby the scanning electron microscope (SEM) photographsThe research results showed that the porous structure of thePFMM demonstrated obvious fractal features Figure 1(a)displays the photographs of the PFMM samples and theirmicrostructures (fibrous clusters (at macrolevel) and fibers(at microlevel)) are shown in Figure 1(b)

Based on the foregoing discussion it can be found that aquantitative description of the disordered PFMM structureis crucial to the evaluation of the static flow resistivityAllard and Atalla [19] established the equivalent fluid modelconsidering the impacts of heat conduction air viscosityand the structure factor on sound propagation Howeverthese parameters are difficult to measure directly The static

flow resistivity is a key parameter which is a function ofthe complex porous structure and porosity Based on thesimilarity of aperiodic characters between the PFMM andsandstone or islands on the earth they are analogous forporous structure and the arrangement of pores [20 21]Fractal techniques have been used in diverse engineeringapplications involving physical phenomenon in disorderedstructures andmultiple scales [22 23]Therefore it is possibleto obtain the static flow resistivity formula of the PFMM by afractal analysis on the porous microstructures

On the basis of Biot-Allardmodel and the proposed staticflow resistivity formula a fractal acoustic model was pre-sented to investigate the acoustic performance of the PFMMWe focus our attention on the derivation of a fractal acousticmodel for the PFMM The sound propagation pathwaysand porous microstructure are characterized by two fractaldimensions one describing the tortuosity of the capillarypathways and the other relating the size of the capillary flowpathways to their quantity in the PFMM [24] The fractalflow resistivity formula yields an analytical expression forthe static flow resistivity which is a function of the twofractal dimensions and porosity of the PFMM Moreoverthe relationship between sound absorption performanceand geometrical parameters can be obtained by the fractalacoustic model The organization of the paper is as followsIn Section 2 the derivation of the fractal acoustic modeland the static flow resistivity formula are presented andthe calculation of the tortuosity fractal dimension the porearea fractal dimension the maximum pore diameter and thefibrous cluster radius is carried out in Section 3 In Section 4the sound absorption is validated by experimental data andthe influences of the fractal dimensions on the specific surfaceacoustic impedance the complex wave number and thesound absorption coefficient are discussed The conclusionsare drawn in Section 5

2 Acoustic Models

21 Introduction of Biot-Allard Model For the Biot-Allardmodel a layer of porous material is assumed to be a layer of

Shock and Vibration 3

equivalent fluidwith the effective density 120588 and bulkmodulus119870 The effective density 120588 can be written as [19]

120588 = 120572infin1205880 (1 + 1205901206011198951205961205880120572infin119866119888 (119904)) (1)

where119866119888(119904) = minus(119904radicminus1198954)(1198691(119904radicminus119895)1198690(119904radicminus119895))[1minus(2119904radicminus119895)(1198691(119904radicminus119895)1198690(119904radicminus119895))] and 119904 = 119888(81205961205880120572infin120590120601)12 120572infin is thetortuosity of the PFMM 1205880 is the static density of air 120590 isthe static flow resistivity 120601 is the porosity 120596 is the angularfrequency and 119888 is the shape factor In (1) the value ofstatic flow resistivity is usually measured but the relationshipbetween the static flow resistivity and geometrical parametersof the PFMM cannot be revealed

In addition the effective bulk modulus 119870 can beexpressed as [19]

119870 = 1205741198750120574 minus (120574 minus 1) 119865 (1198612120596) (2)

where 119865(1198612120596) = [1+ (12059012060111989512059611986121205880120572infin)119866119888(119861119904)]minus1 1198612 is Prandtlnumber 120574 is the air specific heat ratio and 1198750 is the staticpressure of air For air at 18∘C 1198750 = 10132 times 105 Pa 1205880 =1213 kg mminus3 120574 = 14 and 1198612 = 071

Moreover the surface acoustic impedance119885s the charac-teristic impedance 119885c and the complex wave number 119896 canbe calculated respectively as follows [19]

119885c = radic119870120588119896 = 119894120596radic 120588119870119885s = 119885c coth (119896ℎ)

(3)

where ℎ is the thickness of the PFMM sample Eventuallythe sound absorption coefficient 120572 at normal incidence isobtained as

120572 = 1 minus 10038161003816100381610038161003816100381610038161003816119885s minus 12058801198880119885s + 12058801198880

100381610038161003816100381610038161003816100381610038162 (4)

In Biot-Allard model the change rule between the effec-tive density 120588 and the static flow resistivity 120590 is revealedby (1) The static flow resistivity is used to predict thepropagation constant and characteristic impedance of porousmedia through the empirical formulas In this paper afractal acoustic model was presented based on the static flowresistivity with the tortuosity fractal dimension the pore areafractal dimension and the porosity of the PFMM

22 The Fractal Acoustic Model

221 Fractal Geometry Theory In general regular objectssuch as point curve surface and cube are described byinteger dimensions 0 1 2 and 3 in Euclidean geometry Eachdimension is a measurement of an object such as the lengthof a line the area of a surface and the volume of a cube How-ever a number of objects such as rough surface coastline

mountain river and island on the earth are disordered andirregular and they do not follow the Euclidean descriptiondue to the scale-dependent measurements of length areaand volume [21] These objects are called fractals and canbe described using a nonintegral dimension called fractaldimension The measurable quantity of a fractal object119872(120576)is related to the length scale 120576 by the form [21]

119872(120576) prop 120576119863 (5)

where119872(120576) means the length of a line the area of a surfacethe volume of a cube or the weight of an object 119863 isthe fractal dimension Equation (5) demonstrates the ruleof self-similarity which means the value of 119863 remains aconstant over a range of length scales 120576 Many geometricstructures such as Koch curve and Sierpinski gasket are theexamples of exact self-similar fractals However exact self-similarity objects in nature are seldom observed and thefractal description of many general objects is obtained basedon statistical self-similarity [23] The fractal dimension 119863used in this paper refers to the statistical fractal dimensionwhich is usually estimated briefly as the slope of linear fitof data on a log-log plot of average measurement 119872(120576)against the length scale 120576 The concept of local statisticalsimilarity introduced above is used in the following sectionto demonstrate a geometric description of porous structurefor the PFMM

222 Fractal Characteristics of the PFMM Theporous struc-ture of the PFMM is related to the fiber diameter and lengthand the arrangement of fibers In reality the fiber shape andagglomeration in the PFMM are irregular and disordered Asshown in Figure 2 1198710-sized cylindrical PFMM with variouspore sizes can be regarded as a bundle of tortuous capillarytubes having different cross-sectional areas The diameter ofa capillary and its length along the incident sound wave aredefined as 119889 and 119871 t(119889) respectively shown in Figure 2(a)Due to the tortuous nature of the capillary 119871 t(119889) is greaterthan or equal to 1198710 The tortuosity of capillary depends onits diameter 119889 the smaller the capillary diameter the greaterits tortuous length as illustrated schematically in Figures 2(a)and 2(b)

The scaling relationship of the length 119871 and the tortuosityfractal dimension 119863t by using the formula 119871 t(120576) = 1205761minus119863t119871119863t0was given by Wheatcraft and Tyler [25] where 120576 is thelength scale of measurement By analogy with this modelthe relationship between the diameter and the length ofcapillaries can be used to exhibit similar fractal scaling lawas

119871 t (119889) = 1198891minus119863t119871119863t0 (6)

where 119871 t(119889) is the total length of the tortuous capillary tubeand 119863t is the tortuosity fractal dimension of the capillarypath which lies in the range 1 lt 119863t lt 2 A higher valueof119863t corresponds to a highly tortuous capillary Note that thecapillary pathway is a straight pathwhen the tortuosity fractaldimension 119863t = 1 It corresponds to a highly tortuous linethat sprawls a plane if the tortuosity fractal dimension119863t = 2


Incident sound wave

d

L0

LＮ

(a)

d

LＮ

(b)

Figure 2 Schematic of sound propagation pathways of the PFMM (a)The d-sized capillaries (b) the illustration of the decrease in tortuositywith increasing capillary diameter

The tortuosity of the capillary pathways can be described by(6) which reflects the fractal properties of porous structure

On the other hand in order to describe the porousstructure of the PFMM it is also important to calculate thenumber of capillary pathways with the pore diameter 119889 byconsidering a cross section of the PFMM perpendicular tothe flow direction According to the fractal theory the totalnumber of the islands whose area 119860 is greater than 119886 canbe expressed by 119873(119860 gt 119886) sim 119886minus1198632 where 119863 is the fractaldimension [21] Assuming that the pores in the PFMM areanalogous to the islands on the earth the cumulative d-sizepores also follow the same fractal scaling law [24]

119873(119871 ge 119889) = (119889max119889 )119863f (7)

where 119863f is the pore area fractal dimension and 119889max is themaximum diameter of pores

223 The Static Flow Resistivity Formula As shown in Fig-ure 1(b) the PFMM is composed of disorderedmetallic fibersconnectedwith the sintered pointsThe fibrousmaterialsmaybe regarded as transverse isotropic Furthermore because thedensity and stiffness of the fibers are much larger than thoseof the air in the porous materials the propagation processfor the sound in the PFMM can be treated as motionlessWe assume that the sound wave remains in a plane whenit is incident normally on the PFMM In this paper a staticprocess for the sound propagation is applied

As shown in Figure 3 the porous material is placed ina pipe and a pressure difference across the porous materialinduces a steady flow of air The porous material is regardedas an infinite periodic structure and sound wave propagatesunidirectionally from left to right Based on the assumptionof the static process the airflow velocity in the pores is smallso the nonlinear effects are not taken into account The staticflow resistivity of a homogeneous isotropic porousmaterial isgiven by [19]

120590 = 1199012 minus 1199011119881ℎ (8)

where 1199012 is the incident pressure 1199011 is the exit pressure1199012 minus 1199011 is a pressure difference across the porous material

ℎV

p1p2

Figure 3 A differential pressure 1199012 minus1199011 induces steady airflow119881 ofair per unit area of material

119881 is the mean flow velocity of air per unit area The soundpropagation through the PFMM mainly depends upon thestatic flow resistivity the incident sound pressure differenceand the air viscosity

In this section we aim to obtain an analytical formularelating the static flow resistivity to the geometrical param-eters because the static flow resistivity can be quantified bythe tortuosity fractal dimension 119863t the pore area fractaldimension 119863f and the porosity 120601 For a unit cell which iscomposed of tubes with various cross-sectional areas thetotal volumetric airflow rate119876 can be obtained by integratingthe individual flow rate 119902(119889) over the whole pores The flowrate 119902(119889) through a single tortuous capillary is obtained bymodifying the well-known Hagen-Poiseuille equation [26]

119902 (119889) = 119892q Δ119875119871 t (119889)1198894120583 (9)

where 120583 is the viscosity of air and 119892q is a geometric factorbased on the shape of the capillary cross section Further thetotal flow rate 119876 through the cross section of the PFMM isobtained by [27]

119876 = 119892q Δ119875120583119871119863t0119863f3 + 119863t minus 119863f

1198893+119863tmax (10)


Table 1 The physical parameters of air used in this paper

Air properties 1198790 (K) 1205880 (kgmminus3) 1198750 (Pa) 120583 (Pasdots)Numerical values 293 1213 1013 times 105 18 times 10minus5

In addition the total flow rate 119876 passing through thePFMM can be calculated by Darcyrsquos law

119876 = 119860119881 (11)

where 119860 is the total area of the cross section of the PFMMCompared with the static flow resistivity formula 119881 is alsothe mean flow velocity of air per unit area of material Thethickness of the PFMM is set as 1198710 Substituting (10) and (11)into (8) the static flow resistivity formula can be obtained as

120590 = 120583119860119871119863tminus10 (3 + 119863t minus 119863f)119892q119863f1198893+119863tmax (12)

This demonstrates that the static flow resistivity 120590 is afunction of the tortuous fractal dimension 119863t the pore areafractal dimension119863f and the geometrical parameters 119860 1198710and 119889max In addition aiming to describe the convolute extentof sound propagating pathway the concept of tortuosity ispresented As shown in Figure 4 the tortuosity of the sinteredfibrous metal material is defined as [19]

120572infin = 1cos2120593 = (119871 t (119889)1198710 )

2 (13)

where 120593 is the angle between the axis of the pores and thesurface And then (6) can be modified as

119871 t (119889)1198710 = (1198710119889 )119863tminus1 (14)

Substituting (14) into (13) the tortuosity formula relatedto the tortuosity fractal dimension can be deduced

120572infin = (1198710119889 )2(119863tminus1) (15)

Then substituting (15) into (1) and setting 1198710119889 = 119879 wecan get the effective density formula as

120588 = 12058801198792(119863tminus1) (1 + 12059012060111989512059612058801198792(119863tminus1)119866119888 (119904)) (16)

Therefore the sound absorption performance of thePFMM can be calculated by using (2) (3) (4) (12) and (16)The physical parameters of air for the acoustic calculation inthis paper are shown in Table 1

3 Determination of 119889max 119863t 119863f and 11987731 Evaluation of 119889max The architecture of the PFMM canbe thought to be built with cylindrical fibers and fibrousclusters The small pores are distributed between fibers and

Incident sound u

L0

LＮ(d)

Figure 4 Schematic illustration of the PFMM with pores ofconstant and equal radius with respect to the normal of surface

Figure 5 Schematic of the irregular porous shapes in the PFMMsample

the large ones are formed between fibrous clusters It can beseen from Figure 5 that the porous structures demonstratedifferent shapes that is triangular quadrilateral pentagonaland so on The calculation of the effective density 120588 was per-formed for cylindrical tubes having different cross sectionscorresponding to different shape factors [19]The shape factorof equilateral triangle is set as 111 For the presented PFMMsample the unit cell is assumed to be an equilateral triangleand its porosity is analyzed

Due to the structural similarity between the PFMM andbidispersed porous media the maximum pore diameter 119889maxcan be approximately given by [27]

119889max = 119877radic2(1 minus 1206011198941 minus 120601 minus 1) (17)

where 119877 is the radius of fibrous clusters 120601119894 is the porosity offibrous cluster and 120601 is the porosity of PFMM sample

32 Determination of 119863t and119863f

321 Evaluation of 119863t In this section we present twomethods for the calculation of119863t one is the determination byemploying the fractal relationship between the total distancethe soundwave travels and the straight-line distance along the119909-axis and the other one is the box-countingmethod Figure 6


0

L0

x

y

LＮ

Figure 6 The illustration of sound propagation pathway 119871 t versusthe thickness 1198710 of the sample

plots an x-y coordinate system on the fractal pathway and thesound wave propagates along the 119909-axis According toMonteCarlo simulations Wheatcraft and Tyler [25] presented anaverage fractal travel distance 119871 t(120582) versus the scale 120582 bythe fractal random walk model to simulate the dispersity inheterogeneous media They obtained the tortuosity fractaldimension119863t = 1081

For the present analysis it is assumed that the fractaldimension is a constant when the value of 119909 is assigned from0 to 1198710 The relationship between 119871 t and 1198710 can be obtainedby (6) By taking natural logarithms on both sides of (6) the119863t formula can be obtained as

119863t = ln 119871 t minus ln 119889ln (1198710119889) (18)

To evaluate the tortuosity fractal dimension 119863t 1198710 isset as the thickness of sample 119871 t and 119889 are measured bythe microstructure of the PFMM sample For instance theapproximate measured parameters 119871 t = 32mm 119889 = 05mmand 1198710 = 25mm can be used for our considered samplesThe calculated value of the tortuosity fractal dimension119863t is1063

Next the tortuosity fractal dimension is determined bythe box-counting method We can approximately simulatethe sound propagation pathways by employing the fractalrandom walk model Figure 7 shows five random pathwaysfor the sound propagation in the PFMM when the soundwave is incident from left to right

The fractal software FractalFox was used to calculatethe tortuosity fractal dimension of the sound propagationpathways shown in Figure 7(a) By this method the crosssection is discretized by square boxes of size 119871 and thenumber of such 119871-sized boxes along the perimeter of the poreregions is recorded as119873(119871) This step is repeated for variousbox sizes and a ln-ln plot of119873(119871) versus 119871 is constructedTheslope of the best-fit straight line indicates the tortuosity fractaldimension As for the first sound propagation pathway inFigure 7(a) the calculated procedure of the tortuosity fractaldimension is as follows (1) transform the format of the graphfor the pathway to JPG format and import the new photo tothe FractalFox software (2) set the initial box size and thefinal box size as 5mm and 50mm respectively and the stepsize is set as 5mm (3) select the box-counting method andrun the ldquoL Current Fractalrdquo order Then the regression dataof the number of boxes and box sizes can be obtained Finally

the tortuosity fractal dimension is obtained by the slope of thebest-fit straight line of the ln-ln plot of119873(119871) versus 119871

Figure 7(b) illustrates the calculated result of the tortu-osity fractal dimension for the first pathway and 119863t = 105By repeating the same procedure the calculated results ofthe tortuosity fractal dimensions for other sound propagationpathways are 108 102 107 and 110 respectivelyThe averagevalue 119863t = 106 is very close to the theoretical calculatedvalue

322 Calculation of119863f The pore area fractal dimension canbe obtained through an analysis of microstructural imagesof the PFMMrsquos cross sections [28] The SEM photographs ofthe PFMM for the fractal analysis are shown in Figure 8 Thebox-countingmethod is used to calculate the pore area fractaldimension and the calculation procedure for the FractalFoxsoftware is presented The number of boxes119873(119871) is requiredto cover the pore area completely

Similarly the pore area fractal dimension is also deter-mined by the slope of linear fit through the data on a ln-lnplot of119873(119871) versus 119871The logarithmic plots of the cumulativenumber of pores 119873(119871 ge 119889) versus 119889 are shown in Figure 9(120601 = 081 086 and 090) It can be seen that the number ofthe cumulative pores decreases with the pore size

As shown in Figure 9 the pore area fractal dimensioncan be obtained as 119863f = 149 (Figure 9(a)) 119863f = 157(Figure 9(b)) and119863f = 159 (Figure 9(c))33 Fibrous Clusters Radius 119877 Based on the microstructurephotographs of the PFMM samples we measured the radiusof fibrous clusters by SEMAs shown in Figure 10 we obtainedthe SEM photographs (times50 times) of the PFMM samplesThen the particle size measurement software SmileViewwas employed to measure the radius of fibrous clustersThe procedure of the calculation of the radius of fibrousclusters is as follows (1) import the SEM photograph to thesoftware SmileView (2) set the scale of the measurement as1 1000 (120583m) (3) measure the radius of fibrous clusters

Figure 11 shows the photographs of the micrographicsurface and the radius of fibrous clusters for each sampleis measured five times As shown in Table 2 the measuredvalues of average radii are 359 355 and 346 120583m

4 Sound Absorption Verificationand Discussion

In this section the established fractal acoustic model is val-idated by the values of Biot-Allard model and experimentaldata We measured the values of the static flow resistivity forthree PFMM samples and the static flow resistivity formula isvalidated

41 Experiment and Verification of 120590 Two measuring meth-ods of the static flow resistivity are introduced by the ISO9053standard [29] one is steady airflow method and the otheris alternating airflow method In this section we used thesteady airflow method to measure the static flow resistivityFigure 12(a) shows a simplified sketch of the device for the


(1)

(2)

(3)

(4)

(5)

(a)

50

45

40

35

30

25

20

1515 20 25 30 35 40

ln[N

(L)]

ln (L)

DＮ = 105

(b)

Figure 7 The results of the tortuosity fractal dimension 119863t calculated by the box-counting method (a) Five possible random soundpropagation pathways (b) the result of119863t for the first pathway

(a) (b) (c)

Figure 8 The SEM photographs of the PFMMrsquos cross sections (a) sample 1 (b) sample 2 (c) sample 3

steady airflow resistivity measurement and Figure 12(b) isblanked since it is an in-house apparatus of the Instituteof Acoustics at the Chinese Academy of Sciences and waspublished without their permission

The setup is made up of differential gauge flow metersand specimen tube The inner diameter of the measuringtube is greater than 95mm and the minimum air velocity canreach 05 times 10minus3ms The measured values of the static flowresistivity for three PFMM samples are shown in Table 3

Furthermore we calculated the static flow resistivity ofthe PFMM in terms of the presented static flow resistivityformula considering the influences of the tortuous fractaldimension the pore area fractal dimension and the porosityFirstly in order to evaluate the average effect of the tortuosityfractal dimension on the static flow resistivity we calculatedthe values of the static flow resistivity and the deviationsbetween calculated results and experimental data for threePFMM samples with 119863t = 102 105 107 108 and 110respectively The calculated values and the deviations areshown in Table 4

It can be seen that the maximum absolute deviationsof the static flow resistivity (sample 1 to sample 3) are

1521 1914 and 1555 respectively the minimum abso-lute deviations are 167 008 and 241 respectivelyParticularly the deviations of the static flow resistivity calcu-lated by employing the average tortuosity fractal dimensionsare 195 363 and 396 respectively Therefore thecalculations for the static flow resistivity by using the averagevalue of 119863t can reduce the stochastic errors and improve theprediction accuracy

In addition to evaluate the average effect of fibrous radius119877 the static flow resistivities with different fibrous radii arecalculated The calculated values and the deviations of thestatic flow resistivity are shown in Table 5 The results showthat the maximum absolute deviations of the static flowresistivity are 1785 1723 and 2232 respectively theminimum absolute deviations are 083 675 and 477respectively The average errors are 195 193 and 241respectively It can be concluded that the calculations ofthe static flow resistivity by employing the average radii arereasonable

Next the tortuosity fractal dimension 119863t is set as 106and the pore area fractal dimensions 119863f are set as 149157 and 159 respectively The shape factor 119892q is set as 111


4

5

6

7

8

9

1510 20 25 30 35 40 45

ln (L)

ln[N

(L)]

D = 149

(a)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 157

(b)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 159

(c)

Figure 9 The calculated results of the pore area fractal dimension119863f for the PFMM samples

Table 2 The measuring results of R for fibrous clusters (unit 120583m)

Fiber clusters 1198771 1198772 1198773 1198774 1198775 Mean valueSample 1 354 358 363 347 375 359Sample 2 343 347 371 351 364 355Sample 3 336 340 366 353 334 346

and the fibrous cluster radii are 359 times 10minus4m 355 times 10minus4mand 346 times 10minus4m respectively And then the calculatedresults of the static flow resistivity of three PFMM samplesare compared with experimental data Figure 13(a) shows thecomparisons between the calculated results and experimentaldata at different porosities It can be seen that the theoreticalresults agree well with the experimental data

Due to the limitation of our experimental data the staticflow resistivity formula has been further verified by thecalculated values in the literature The physical parameters ofthe PFMM and the calculated data are shown in Table 6

Figure 13(b) displays the comparisons between the resultsevaluated by the static flow resistivity formula and the data in

[30] It can be shown that the static flow resistivity decreaseswith the porosity and the results calculated by the theoreticalformula coincide very well with the literaturersquos data whichproves the accuracy of the static flow resistivity formula

42 Sound Absorption Verification In order to validate thefractal acoustic model we measured the sound absorptioncoefficients of three PFMM samples The tested samples aremade of stainless steel fibers with the same diameter and thethickness is 25mm The sound absorption coefficients aremeasured by employing the transfer functionmethod and thetested setup is BampK 4206 impedance tube


Table 3 The physical parameters of the PFMM samples and experimental results of static flow resistivity

Sample parameters Porosity120601 ()Fiber diameter119889 (120583m)

Thicknessℎ (mm)Shape factor119892q Flow resistivity120590 (Nmminus4s)

Sample 1 81 50 25 111 98756Sample 2 86 50 25 111 34655Sample 3 90 50 25 111 19748

Fibrous cluster

Measuring scale

2R2

2R11 1000

1000 m

Figure 10 Typical porous morphology of fibrous clusters where119877 denotes the fibrous cluster radius and the measuring scale ofcalculation is 1 1000

Figure 14 plots the predictions evaluated by the frac-tal acoustic model and Biot-Allard model compared withexperimental data It can be seen that the theoretical resultsfor sound absorption agree well with experimental dataAs shown in Figure 14 the values calculated by the fractalacoustic model fit the measured ones better than Biot-Allard model at low frequencies The small deviations maybe attributed to the computational errors of 119863t and 119863f Insummary the sound absorption coefficient of the PFMMincreases with the frequency In frequency domain the soundabsorption coefficient increases monotonically from 100 to2000Hz but decreases from 2000 to 4000Hz When thefrequency is higher than 4000Hz the sound absorptioncoefficient is increased For the sound absorption of thePFMM the peak value of the curve is mainly formed bythe resonance between sound wave and the PFMM What ismore the PFMMhas superior capability of energy dissipationfor high-frequency sound wave and the sound absorptioncoefficient reaches a peak value when the energy dissipationability of the PFMM approaches its maximum value

43 Influences of Fractal Dimensions In this section theinfluences of the tortuosity fractal dimension and the porearea fractal dimension on the specific surface acousticimpedance 119885s1198850 the complex wave number 119896 and thesound absorption coefficient 120572 are presented and discussed

431 Influences of 119863t and 119863f on 119885s1198850 Figure 15 plots thereal and imaginary parts of 119885s1198850 for sample 1 with 119863t =106 108 and 110 respectively As shown in Figure 15(a)

the specific acoustic resistance (the real part of 119885s1198850)increases with the tortuosity fractal dimension in low-frequency range but decreases in high-frequency rangeFurthermore the peak value of the real part of 119885s1198850curve is increased with the tortuosity fractal dimension andmoves to the low frequency Figure 15(b) shows that thespecific acoustic reactance (the imaginary part of 119885s1198850)decreases slightly with the tortuosity fractal dimension inlow-frequency range but increases in high-frequency rangeIn other words the specific acoustic resistance changesobviously with the increase of 119863t but the specific acousticreactance changes slightly in the whole frequency range

On the other hand the influences of the pore area fractaldimension on the specific surface acoustic impedance 119885s1198850are illustrated in Figure 16

As shown in Figure 16(a) the specific acoustic resistancedecreases with the pore area fractal dimension in low-frequency range but increases in high-frequency range Thepeak value of the real part of 119885s1198850 curve increases with thepore area fractal dimension andmoves to the high-frequencyrange Figure 16(b) shows that the specific acoustic reactancedecreases slightly with the pore area fractal dimension in low-frequency range but increases in high-frequency range

432 Influences of 119863t and 119863f on 119896 The changing rulesbetween the complex wave number and the fractal dimen-sions are discussed in this section Figure 17 shows that boththe real and the imaginary parts of the complex wave numberincrease with the tortuosity fractal dimension The real partof 119896 increases with the frequency but the imaginary part of 119896decreases

Figure 18 illustrates the variation of the complex wavenumber with the pore area fractal dimension As shownin Figure 18(a) the real part of 119896 increases with the fre-quency and decreases with the pore area fractal dimensionFigure 18(b) shows that the imaginary part of 119896 decreaseswith the frequency and finally keeps constant Furthermorethe imaginary part of 119896 decreases with the pore area fractaldimension

433 Influences of119863t and119863f on120572 Theeffects of119863t and119863f onthe sound absorption coefficient 120572 are plotted in Figure 19 Asshown in Figure 19(a) the variations of the sound absorptioncoefficient with different tortuosity fractal dimensions (119863t =106 108 and 110) and 119863f = 149 are plotted The resultsshow that the sound absorption coefficient increases withthe tortuosity fractal dimension in 500ndash2500Hz and 4800ndash6400Hz ranges but decreases in 2500ndash4800Hz range Withthe increasing of the tortuosity fractal dimension the fluid


358 m363 m

347 m

375 m

354 m

1000 m

(a)

347 m

371 m

364 m

351 m343 m

1000 m

(b)

334 m

336 m

366 m353 m

340 m

1000 m

(c)

Figure 11 The photographs of the micrographic surface for 119877measuring (a) sample 1 (b) sample 2 (c) sample 3

Differentialgauge

AirflowFlow meters

Sample

(a) (b)

Figure 12 The static flow resistivity measurement (a) Schematic of the setup (b) This Figure is blanked since it is an in-house apparatus ofthe Institute of Acoustics at the Chinese Academy of Sciences and was published without their permission

0

The theoretical results of sample 1Measured data of sample 1The theoretical results of sample 2Measured data of sample 2The theoretical results of sample 3Measured data of sample 3

055 060 065 070 075 080 085 090

Porosity ()

1000000

800000

600000

400000

200000Flow

resis

tivity

(N

Ｇminus4s)

(a)

0

10000

20000

30000

40000

50000

60000

70000

The static flow resistivity data in [30]Fractal static flow resistivity model

065 070 075 080 085 090

Porosity ()

Flow

resis

tivity

(N

Ｇminus4s)

(b)

Figure 13 Comparisons between the calculated and experimental results of the static flow resistivity (a) compared with the test data (b)compared with the calculated results in [30]


Table4Th

ecalculated

andmeasuredstaticflo

wresis

tivities

ofdifferent

tortuo

sityfractald

imensio

ns

Num

ber

119863 tSample

1Sample

2Sample

3Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

110

283737

98756

1521

31233

34655

987

16677

19748

1555

210

593381

544

34684

008

18366

700

310

7100410

167

37190

731

19584

830

410

8104120

543

38510

1112

20223

241

5110

111960

1337

41289

1914

21564

920

Meanvalue

106

96834

195

35915

363

18965

396


Table5Th

ecalculated


wresis

tivity

ofdifferent

fibrous

radii

Num

ber

Sample

1Sample

2Sample

3

119877 1 (120583m)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 2 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 3 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

1354

102510

98756

380

343

40625

34655

1723

336

21708

19748

993

2358

97937

083

347

38757

1184

340

20690

477

3363

92574

626

371

29541

1476

366

15340

2232

4347

11117

01257

351

36995

675

353

17766

1004

5375

81123

1785

364

31916

790

334

22241

1262

Meanvalue

359

96834

195

355

35331

193

346

19272

241


0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

Frequency (Hz)Sample 1

Experimental dataFractal acoustic modelBiot-Allard model

(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Sample 2Experimental dataFractal acoustic modelBiot-Allard model

(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)

Figure 14 Comparisons between the calculated andmeasured sound absorption coefficients of the PFMM samples (a) sample 1 (b) sample2 and (c) sample 3

Table 6 The physical parameters of the porous metallic samples and calculated static flow resistivities

Samplenumber

Porosity120601 ()Fiber radius119903 (120583m)

Thicknessℎ (mm)Tortuosity120572infin

Static flowresistivity120590 (Nmminus4s)

1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)

Figure 15 Predictions of specific surface acoustic impedance 119885s1198850 with different tortuosity fractal dimensions (a) The real part of 119885s1198850(b) The imaginary part of 119885s1198850

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)

Figure 16 Predictions of specific surface acoustic impedance 119885s1198850 with different pore area fractal dimensions (a) The real part of 119885s1198850(b) The imaginary part of 119885s1198850

pathways for sound propagation become more tortuous andthe static flow resistivity is increased At 500ndash2500Hz rangethe number of sound waves in the PFMM is increased withthe tortuosity fractal dimension and more sound energy canbe dissipated sufficiently Therefore the sound absorptioncoefficient increases with the tortuosity fractal dimensionWhen the frequency is increased and reaches the 2500ndash4800Hz range the wavelength of sound wave is reduced andapproaches the porous diameter Then it becomes difficult

for the sound wave to be incident and the sound absorptioncoefficient decreases with the tortuosity fractal dimensionFinally from 4800 to 6400Hz the wavelength of sound waveis reduced further and the number of incident sound waves isincreased So the sound absorption coefficient increases againwith the tortuosity fractal dimension

On the other hand Figure 19(b) plots the changingrules of the sound absorption coefficients with the frequencyfor different pore area fractal dimensions (119863f = 12 14


5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)

Figure 17 Predictions of the complex wave number 119896 with different tortuosity fractal dimensions (a) The real part of 119896 (b) The imaginarypart of 119896

5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)

Figure 18 Predictions of the complex wave number 119896 with different pore area fractal dimensions (a) The real part of 119896 (b) The imaginarypart of 119896

and 16) and 119863t = 106 The results show that the soundabsorption coefficient decreases with the pore area fractaldimension in 500ndash2500Hz and 5000ndash6400Hz range butincreases in 2500ndash5000Hz range From 500 to 2500Hz theporosity is increased with the pore area fractal dimensionand the acoustic resistance plays a major role Furthermorethe larger the porosity the smaller the acoustic resistanceand the acoustic reactance Therefore the sound absorptioncoefficient decreases with the pore area fractal dimensionwhen the frequency is lower than 2500Hz However when

the frequency is higher than 2500Hz but lower than 5000Hzthe acoustic reactance plays a major role Then the porosityis increased and the larger the spaces among fibers themore absorbed the acoustic energy So the sound absorptioncoefficient is increased At 5000ndash6400Hz range due to thereduction of sound energy attenuation the sound absorptioncoefficient decreases againwith the increasing of the pore areafractal dimension

In summary the sound absorption coefficient is asso-ciated with the tortuosity fractal dimension the pore area


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)

Figure 19 The Sound absorption coefficient versus the fractal dimensions (a) 119863f = 149 and 119863t = 106 108 and 110 respectively (b)119863t = 106 and119863f = 12 14 and 16 respectively

fractal dimension and the porosity Furthermore the geo-metrical parameters such as the pore diameter the thicknessand the arrangement of fibers are related to the sound absorp-tion performance of the PFMM Therefore the optimizationof sound absorption capability for the PFMMis dependent onthe design of geometrical parameters which are determinedby the fractal acoustic model

5 Conclusions

A quantitative fractal acoustic model of the PFMM consid-ering the geometrical microstructure is derived in this paperThe fractal acoustic model is mainly related to the tortuosityfractal dimension and the pore area fractal dimension Basedon the presented acoustic theoretical model the acousticperformance including sound absorption the specific sur-face acoustic impedance and the complex wave numberare investigated both theoretically and experimentally Theexperimental results and the values of Biot-Allard model areadopted to validate the theoretical model predictions and agood agreement is achieved The fractal acoustic theory isalso applicable to other materials having a microstructuresimilar to the PFMMsThemain conclusions are summarizedas follows

(1) In frequency domain the sound absorption coeffi-cient increases in 100ndash2000Hz and 4000ndash6400Hzrange but decreases in 2000ndash4000Hz range At a fixedvalue of119863f the sound absorption coefficient increaseswith the tortuosity fractal dimension in 500ndash2500Hzand 4800ndash6400Hz ranges but decreases from 2500to 4800Hz In addition the sound absorption coeffi-cient decreaseswith the pore area fractal dimension in

500ndash2500Hz and 5000ndash6400Hz ranges but increasesfrom 2500 to 5000Hz

(2) The specific acoustic resistance changes obviouslywith the increase of the tortuosity fractal dimensionbut the specific acoustic reactance changes slightly inthewhole frequency rangeThe specific acoustic resis-tance decreases with the pore area fractal dimensionin low frequencies but increases in high frequenciesthe specific acoustic reactance decreases slightly withthe pore area fractal dimension in low frequencies butincreases in high frequencies

(3) Both the real and imaginary parts of the complexwave number increase with the tortuosity fractaldimensionThe real part increases with the frequencyand decreases with the pore area fractal dimensionthe imaginary part decreases with the frequency andthe pore area fractal dimension and finally keeps aconstant

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this article

Acknowledgments

This work is supported by the National Basic Research Pro-gramof China (Grant no 2011CB610306) and theNatural Sci-ence Foundation of Shaanxi Province of China (2015JM5154)The authors are also grateful to the Northwest Institute forNonferrous Metal Research for providing them with thePFMM samples


References

[1] J H Wu Z P Hu and H Zhou ldquoSound absorbing property ofporous metal materials with high temperature and high soundpressure by turbulence analogyrdquo Journal of Applied Physics vol113 no 19 Article ID 194905 2013

[2] F G Sun H Chen J Wu and K Feng ldquoSound absorbingcharacteristics of fibrous metal materials at high temperaturesrdquoApplied Acoustics vol 71 no 3 pp 221ndash235 2010

[3] J P Arenas and M J Crocker ldquoRecent trends in porous sound-absorbingmaterialsrdquo Sound and Vibration vol 44 no 7 pp 12ndash17 2010

[4] X L Wang F Peng and B J Chang ldquoSound absorption ofporous metals at high sound pressure levelsrdquo Journal of theAcoustical Society of America vol 126 no 2 pp EL55ndashEL612009

[5] P Nordin S Sarin and E Rademaker ldquoDevelopment of newliner technology for application in hot stream areas of aero-enginesrdquo in Proceedings of the 10th AIAACEAS AeroacousticsConference AIAA Paper 2004-3033 2004

[6] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoThe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956

[7] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquo Journal ofAcoustical Society of America vol 28 pp 179ndash191 1956

[8] T Cox and P DrsquoAntonio Acoustic Absorbers and DiffusersTheory Design andApplication TaylorampFrancis NewYorkNYUSA 2009

[9] YKawasima ldquoSoundpropagation in a fibre block as a compositemediumrdquo Acta Acustica United with Acustica vol 10 no 4 pp208ndash217 1960

[10] V Tarnow ldquoCompressibility of air in fibrous materialsrdquo TheJournal of the Acoustical Society of America vol 99 no 5 pp3010ndash3017 1996

[11] V Tarnow ldquoCalculation of the dynamic air flow resistivity offiber materialsrdquo Journal of the Acoustical Society of America vol102 no 3 pp 1680ndash1688 1997

[12] J-F Allard and Y Champoux ldquoNew empirical equations forsound propagation in rigid frame fibrous materialsrdquo Journal ofthe Acoustical Society of America vol 91 no 6 pp 3346ndash33531992

[13] M E Delany and E N Bazley ldquoAcoustical properties of fibrousabsorbentmaterialsrdquoApplied Acoustics vol 3 no 2 pp 105ndash1161970

[14] N Sebaa Z E A FellahM FellahW Lauriks andCDepollierldquoMeasuring flow resistivity of porous material via acousticreflected wavesrdquo Journal of Applied Physics vol 98 no 8 ArticleID 084901 2005

[15] Z E A Fellah M Fellah F G Mitri N Sebaa C DepollierandW Lauriks ldquoMeasuring permeability of porous materials atlow frequency range via acoustic transmitted wavesrdquo Review ofScientific Instruments vol 78 no 11 Article ID 114902 2007

[16] M Sadouki Z E A Fellah A Berbiche et al ldquoMeasuringstatic viscous permeability of porous absorbing materialsrdquo TheJournal of the Acoustical Society of America vol 135 no 6 pp3163ndash3171 2014

[17] C Van der Kelen Characterization of Anisotropic AcousticProperties of Porous Materials-Inverse Estimation of Static FlowResistivity Digitala Vetenskapliga Arkivet Sweden Stockholm2011

[18] J-ZWang Z-P Xi H-P TangW-DHuang J-L Zhu andQ-B Ao ldquoFractal dimension for porous metal materials of FeCrAlfiberrdquo Transactions of Nonferrous Metals Society of China vol23 no 4 pp 1046ndash1051 2013

[19] J F Allard and N Atalla Propagation of Sound in PorousMediaModelling Sound Absorbing Materials Elsevier Applied ScienceLondon UK 1993

[20] A J Katz and A H Thompson ldquoFractal sandstone poresimplications for conductivity and pore formationrdquo PhysicalReview Letters vol 54 no 12 pp 1325ndash1328 1985

[21] B B Mandelbrot and W H Freeman The Fractal Geometry ofNature 1982

[22] FM Borodich andA BMosolov ldquoFractal roughness in contactproblemsrdquo Journal of Applied Mathematics and Mechanics vol56 no 5 pp 681ndash690 1992

[23] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of surfacesrdquoJournal of Tribology vol 112 no 2 pp 205ndash216 1990

[24] R Pitchumani andB Ramakrishnan ldquoA fractal geometrymodelfor evaluating permeabilities of porous preforms used in liquidcomposite moldingrdquo International Journal of Heat and MassTransfer vol 42 no 12 pp 2219ndash2232 1999

[25] S W Wheatcraft and S W Tyler ldquoAn explanation of scale-dependent dispersivity in heterogeneous aquifers using con-cepts of fractal geometryrdquoWater Resources Research vol 24 no4 pp 566ndash578 1988

[26] M M Denn Process Fluid Mechanics Prentice-Hall Engle-wood Cliffs NJ USA 1980

[27] B M Yu and P Cheng ldquoA fractal permeability model for bi-dispersed porousmediardquo International Journal ofHeat andMassTransfer vol 45 no 14 pp 2983ndash2993 2002

[28] T Ai R Zhang H W Zhou and J L Pei ldquoBox-countingmethods to directly estimate the fractal dimension of a rocksurfacerdquo Applied Surface Science vol 314 pp 610ndash621 2014

[29] ISO ldquoAcousticsmdashmaterials for acoustical applicationsmdashdetermination of airflow resistancerdquo International StandardISO 90531991 International Organization for StandardizationGeneva Switzerland 1991

[30] S T Liu W J Chen and Y C Zhang ldquoDesign optimization ofporous fibrous material for maximizing absorption of soundsunder set frequency bandsrdquo Applied Acoustics vol 76 pp 319ndash328 2014

RoboticsJournal of

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Journal of


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RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



(a) (b)

Figure 1 The PFMM samples (a) photograph of the samples (b) SEM photo of micrographic surface

wave to measure the static flow resistivity of porous materialand showed that the reflected wave is much more sensitiveto the static flow resistivity than to porosity Fellah et al [15]introduced an acoustical transmission method to measurepermeability of porous materials at low-frequency rangeOn the basis of a temporal model of the direct and inversescattering problem the proposed method is simple rapidand efficient Experimental and numerical validation resultsof this method were presented Sadouki et al [16] establishedan acoustical method to measure static viscous permeabilityof porous absorbing materials This method was derivedwithout a priori estimation of the porosity but by simplyusing the transmitted waves Van der Kelen [17] investigatedthe static flow resistivity of two porousmaterials (fibrous glasswool and melamine foam) and the measurement methodand inverse estimation were employed to determine theanisotropic flow resistivity The research results showed thatthe homogeneities of the density and the static flow resistivityfor the two materials are different but seem to have nocorrelation with each other However the sound propagationpathways in the PFMM are tortuous because of the irregularporous structure and there have been no theoretical modelsproperly demonstrating the variation rules of the static flowresistivity with porous structures and pathways Thereforeit is essential to present a quantitative description of thedisordered pores in the PFMMs for evaluating the soundabsorption performance

Wang et al [18] studied the microstructure features ofFeCrAl fiber materials and obtained the fractal dimensionsby the scanning electron microscope (SEM) photographsThe research results showed that the porous structure of thePFMM demonstrated obvious fractal features Figure 1(a)displays the photographs of the PFMM samples and theirmicrostructures (fibrous clusters (at macrolevel) and fibers(at microlevel)) are shown in Figure 1(b)

Based on the foregoing discussion it can be found that aquantitative description of the disordered PFMM structureis crucial to the evaluation of the static flow resistivityAllard and Atalla [19] established the equivalent fluid modelconsidering the impacts of heat conduction air viscosityand the structure factor on sound propagation Howeverthese parameters are difficult to measure directly The static

flow resistivity is a key parameter which is a function ofthe complex porous structure and porosity Based on thesimilarity of aperiodic characters between the PFMM andsandstone or islands on the earth they are analogous forporous structure and the arrangement of pores [20 21]Fractal techniques have been used in diverse engineeringapplications involving physical phenomenon in disorderedstructures andmultiple scales [22 23]Therefore it is possibleto obtain the static flow resistivity formula of the PFMM by afractal analysis on the porous microstructures

On the basis of Biot-Allardmodel and the proposed staticflow resistivity formula a fractal acoustic model was pre-sented to investigate the acoustic performance of the PFMMWe focus our attention on the derivation of a fractal acousticmodel for the PFMM The sound propagation pathwaysand porous microstructure are characterized by two fractaldimensions one describing the tortuosity of the capillarypathways and the other relating the size of the capillary flowpathways to their quantity in the PFMM [24] The fractalflow resistivity formula yields an analytical expression forthe static flow resistivity which is a function of the twofractal dimensions and porosity of the PFMM Moreoverthe relationship between sound absorption performanceand geometrical parameters can be obtained by the fractalacoustic model The organization of the paper is as followsIn Section 2 the derivation of the fractal acoustic modeland the static flow resistivity formula are presented andthe calculation of the tortuosity fractal dimension the porearea fractal dimension the maximum pore diameter and thefibrous cluster radius is carried out in Section 3 In Section 4the sound absorption is validated by experimental data andthe influences of the fractal dimensions on the specific surfaceacoustic impedance the complex wave number and thesound absorption coefficient are discussed The conclusionsare drawn in Section 5

2 Acoustic Models

21 Introduction of Biot-Allard Model For the Biot-Allardmodel a layer of porous material is assumed to be a layer of



120588 = 120572infin1205880 (1 + 1205901206011198951205961205880120572infin119866119888 (119904)) (1)



119870 = 1205741198750120574 minus (120574 minus 1) 119865 (1198612120596) (2)




(3)


120572 = 1 minus 10038161003816100381610038161003816100381610038161003816119885s minus 12058801198880119885s + 12058801198880

100381610038161003816100381610038161003816100381610038162 (4)





119872(120576) prop 120576119863 (5)




119871 t (119889) = 1198891minus119863t119871119863t0 (6)



Incident sound wave

d

L0

LＮ

(a)

d

LＮ

(b)




119873(119871 ge 119889) = (119889max119889 )119863f (7)




120590 = 1199012 minus 1199011119881ℎ (8)


ℎV

p1p2




119902 (119889) = 119892q Δ119875119871 t (119889)1198894120583 (9)


119876 = 119892q Δ119875120583119871119863t0119863f3 + 119863t minus 119863f

1198893+119863tmax (10)





119876 = 119860119881 (11)




120572infin = 1cos2120593 = (119871 t (119889)1198710 )

2 (13)


119871 t (119889)1198710 = (1198710119889 )119863tminus1 (14)


120572infin = (1198710119889 )2(119863tminus1) (15)


120588 = 12058801198792(119863tminus1) (1 + 12059012060111989512059612058801198792(119863tminus1)119866119888 (119904)) (16)



Incident sound u

L0

LＮ(d)










0

L0

x

y

LＮ




119863t = ln 119871 t minus ln 119889ln (1198710119889) (18)














(1)

(2)

(3)

(4)

(5)

(a)

50

45

40

35

30

25

20

1515 20 25 30 35 40

ln[N

(L)]

ln (L)

DＮ = 105

(b)


(a) (b) (c)










4

5

6

7

8

9

1510 20 25 30 35 40 45

ln (L)

ln[N

(L)]

D = 149

(a)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 157

(b)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 159

(c)














Fibrous cluster

Measuring scale

2R2

2R11 1000

1000 m












358 m363 m

347 m

375 m

354 m

1000 m

(a)

347 m

371 m

364 m

351 m343 m

1000 m

(b)

334 m

336 m

366 m353 m

340 m

1000 m

(c)


Differentialgauge

AirflowFlow meters

Sample

(a) (b)


0


055 060 065 070 075 080 085 090

Porosity ()

1000000

800000

600000

400000

200000Flow

resis

tivity

(N

Ｇminus4s)

(a)

0

10000

20000

30000

40000

50000

60000

70000


065 070 075 080 085 090

Porosity ()

Flow

resis

tivity

(N

Ｇminus4s)

(b)



Table4Th

ecalculated


wresis

tivities

ofdifferent

tortuo

sityfractald

imensio

ns

Num

ber

119863 tSample

1Sample

2Sample

3Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

110

283737

98756

1521

31233

34655

987

16677

19748

1555

210

593381

544

34684

008

18366

700

310

7100410

167

37190

731

19584

830

410

8104120

543

38510

1112

20223

241

5110

111960

1337

41289

1914

21564

920

Meanvalue

106

96834

195

35915

363

18965

396


Table5Th

ecalculated


wresis

tivity

ofdifferent

fibrous

radii

Num

ber

Sample

1Sample

2Sample

3

119877 1 (120583m)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 2 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 3 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

1354

102510

98756

380

343

40625

34655

1723

336

21708

19748

993

2358

97937

083

347

38757

1184

340

20690

477

3363

92574

626

371

29541

1476

366

15340

2232

4347

11117

01257

351

36995

675

353

17766

1004

5375

81123

1785

364

31916

790

334

22241

1262

Meanvalue

359

96834

195

355

35331

193

346

19272

241


0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t



(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)


(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)



Samplenumber




1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120588 = 120572infin1205880 (1 + 1205901206011198951205961205880120572infin119866119888 (119904)) (1)



119870 = 1205741198750120574 minus (120574 minus 1) 119865 (1198612120596) (2)




(3)


120572 = 1 minus 10038161003816100381610038161003816100381610038161003816119885s minus 12058801198880119885s + 12058801198880

100381610038161003816100381610038161003816100381610038162 (4)





119872(120576) prop 120576119863 (5)




119871 t (119889) = 1198891minus119863t119871119863t0 (6)



Incident sound wave

d

L0

LＮ

(a)

d

LＮ

(b)




119873(119871 ge 119889) = (119889max119889 )119863f (7)




120590 = 1199012 minus 1199011119881ℎ (8)


ℎV

p1p2




119902 (119889) = 119892q Δ119875119871 t (119889)1198894120583 (9)


119876 = 119892q Δ119875120583119871119863t0119863f3 + 119863t minus 119863f

1198893+119863tmax (10)





119876 = 119860119881 (11)




120572infin = 1cos2120593 = (119871 t (119889)1198710 )

2 (13)


119871 t (119889)1198710 = (1198710119889 )119863tminus1 (14)


120572infin = (1198710119889 )2(119863tminus1) (15)


120588 = 12058801198792(119863tminus1) (1 + 12059012060111989512059612058801198792(119863tminus1)119866119888 (119904)) (16)



Incident sound u

L0

LＮ(d)










0

L0

x

y

LＮ




119863t = ln 119871 t minus ln 119889ln (1198710119889) (18)














(1)

(2)

(3)

(4)

(5)

(a)

50

45

40

35

30

25

20

1515 20 25 30 35 40

ln[N

(L)]

ln (L)

DＮ = 105

(b)


(a) (b) (c)










4

5

6

7

8

9

1510 20 25 30 35 40 45

ln (L)

ln[N

(L)]

D = 149

(a)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 157

(b)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 159

(c)














Fibrous cluster

Measuring scale

2R2

2R11 1000

1000 m












358 m363 m

347 m

375 m

354 m

1000 m

(a)

347 m

371 m

364 m

351 m343 m

1000 m

(b)

334 m

336 m

366 m353 m

340 m

1000 m

(c)


Differentialgauge

AirflowFlow meters

Sample

(a) (b)


0


055 060 065 070 075 080 085 090

Porosity ()

1000000

800000

600000

400000

200000Flow

resis

tivity

(N

Ｇminus4s)

(a)

0

10000

20000

30000

40000

50000

60000

70000


065 070 075 080 085 090

Porosity ()

Flow

resis

tivity

(N

Ｇminus4s)

(b)



Table4Th

ecalculated


wresis

tivities

ofdifferent

tortuo

sityfractald

imensio

ns

Num

ber

119863 tSample

1Sample

2Sample

3Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

110

283737

98756

1521

31233

34655

987

16677

19748

1555

210

593381

544

34684

008

18366

700

310

7100410

167

37190

731

19584

830

410

8104120

543

38510

1112

20223

241

5110

111960

1337

41289

1914

21564

920

Meanvalue

106

96834

195

35915

363

18965

396


Table5Th

ecalculated


wresis

tivity

ofdifferent

fibrous

radii

Num

ber

Sample

1Sample

2Sample

3

119877 1 (120583m)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 2 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 3 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

1354

102510

98756

380

343

40625

34655

1723

336

21708

19748

993

2358

97937

083

347

38757

1184

340

20690

477

3363

92574

626

371

29541

1476

366

15340

2232

4347

11117

01257

351

36995

675

353

17766

1004

5375

81123

1785

364

31916

790

334

22241

1262

Meanvalue

359

96834

195

355

35331

193

346

19272

241


0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t



(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)


(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)



Samplenumber




1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Incident sound wave

d

L0

LＮ

(a)

d

LＮ

(b)




119873(119871 ge 119889) = (119889max119889 )119863f (7)




120590 = 1199012 minus 1199011119881ℎ (8)


ℎV

p1p2




119902 (119889) = 119892q Δ119875119871 t (119889)1198894120583 (9)


119876 = 119892q Δ119875120583119871119863t0119863f3 + 119863t minus 119863f

1198893+119863tmax (10)





119876 = 119860119881 (11)




120572infin = 1cos2120593 = (119871 t (119889)1198710 )

2 (13)


119871 t (119889)1198710 = (1198710119889 )119863tminus1 (14)


120572infin = (1198710119889 )2(119863tminus1) (15)


120588 = 12058801198792(119863tminus1) (1 + 12059012060111989512059612058801198792(119863tminus1)119866119888 (119904)) (16)



Incident sound u

L0

LＮ(d)










0

L0

x

y

LＮ




119863t = ln 119871 t minus ln 119889ln (1198710119889) (18)














(1)

(2)

(3)

(4)

(5)

(a)

50

45

40

35

30

25

20

1515 20 25 30 35 40

ln[N

(L)]

ln (L)

DＮ = 105

(b)


(a) (b) (c)










4

5

6

7

8

9

1510 20 25 30 35 40 45

ln (L)

ln[N

(L)]

D = 149

(a)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 157

(b)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 159

(c)














Fibrous cluster

Measuring scale

2R2

2R11 1000

1000 m












358 m363 m

347 m

375 m

354 m

1000 m

(a)

347 m

371 m

364 m

351 m343 m

1000 m

(b)

334 m

336 m

366 m353 m

340 m

1000 m

(c)


Differentialgauge

AirflowFlow meters

Sample

(a) (b)


0


055 060 065 070 075 080 085 090

Porosity ()

1000000

800000

600000

400000

200000Flow

resis

tivity

(N

Ｇminus4s)

(a)

0

10000

20000

30000

40000

50000

60000

70000


065 070 075 080 085 090

Porosity ()

Flow

resis

tivity

(N

Ｇminus4s)

(b)



Table4Th

ecalculated


wresis

tivities

ofdifferent

tortuo

sityfractald

imensio

ns

Num

ber

119863 tSample

1Sample

2Sample

3Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

110

283737

98756

1521

31233

34655

987

16677

19748

1555

210

593381

544

34684

008

18366

700

310

7100410

167

37190

731

19584

830

410

8104120

543

38510

1112

20223

241

5110

111960

1337

41289

1914

21564

920

Meanvalue

106

96834

195

35915

363

18965

396


Table5Th

ecalculated


wresis

tivity

ofdifferent

fibrous

radii

Num

ber

Sample

1Sample

2Sample

3

119877 1 (120583m)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 2 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 3 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

1354

102510

98756

380

343

40625

34655

1723

336

21708

19748

993

2358

97937

083

347

38757

1184

340

20690

477

3363

92574

626

371

29541

1476

366

15340

2232

4347

11117

01257

351

36995

675

353

17766

1004

5375

81123

1785

364

31916

790

334

22241

1262

Meanvalue

359

96834

195

355

35331

193

346

19272

241


0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t



(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)


(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)



Samplenumber




1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119876 = 119860119881 (11)




120572infin = 1cos2120593 = (119871 t (119889)1198710 )

2 (13)


119871 t (119889)1198710 = (1198710119889 )119863tminus1 (14)


120572infin = (1198710119889 )2(119863tminus1) (15)


120588 = 12058801198792(119863tminus1) (1 + 12059012060111989512059612058801198792(119863tminus1)119866119888 (119904)) (16)



Incident sound u

L0

LＮ(d)










0

L0

x

y

LＮ




119863t = ln 119871 t minus ln 119889ln (1198710119889) (18)














(1)

(2)

(3)

(4)

(5)

(a)

50

45

40

35

30

25

20

1515 20 25 30 35 40

ln[N

(L)]

ln (L)

DＮ = 105

(b)


(a) (b) (c)










4

5

6

7

8

9

1510 20 25 30 35 40 45

ln (L)

ln[N

(L)]

D = 149

(a)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 157

(b)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 159

(c)














Fibrous cluster

Measuring scale

2R2

2R11 1000

1000 m












358 m363 m

347 m

375 m

354 m

1000 m

(a)

347 m

371 m

364 m

351 m343 m

1000 m

(b)

334 m

336 m

366 m353 m

340 m

1000 m

(c)


Differentialgauge

AirflowFlow meters

Sample

(a) (b)


0


055 060 065 070 075 080 085 090

Porosity ()

1000000

800000

600000

400000

200000Flow

resis

tivity

(N

Ｇminus4s)

(a)

0

10000

20000

30000

40000

50000

60000

70000


065 070 075 080 085 090

Porosity ()

Flow

resis

tivity

(N

Ｇminus4s)

(b)



Table4Th

ecalculated


wresis

tivities

ofdifferent

tortuo

sityfractald

imensio

ns

Num

ber

119863 tSample

1Sample

2Sample

3Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

110

283737

98756

1521

31233

34655

987

16677

19748

1555

210

593381

544

34684

008

18366

700

310

7100410

167

37190

731

19584

830

410

8104120

543

38510

1112

20223

241

5110

111960

1337

41289

1914

21564

920

Meanvalue

106

96834

195

35915

363

18965

396


Table5Th

ecalculated


wresis

tivity

ofdifferent

fibrous

radii

Num

ber

Sample

1Sample

2Sample

3

119877 1 (120583m)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 2 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 3 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

1354

102510

98756

380

343

40625

34655

1723

336

21708

19748

993

2358

97937

083

347

38757

1184

340

20690

477

3363

92574

626

371

29541

1476

366

15340

2232

4347

11117

01257

351

36995

675

353

17766

1004

5375

81123

1785

364

31916

790

334

22241

1262

Meanvalue

359

96834

195

355

35331

193

346

19272

241


0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t



(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)


(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)



Samplenumber




1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

L0

x

y

LＮ




119863t = ln 119871 t minus ln 119889ln (1198710119889) (18)














(1)

(2)

(3)

(4)

(5)

(a)

50

45

40

35

30

25

20

1515 20 25 30 35 40

ln[N

(L)]

ln (L)

DＮ = 105

(b)


(a) (b) (c)










4

5

6

7

8

9

1510 20 25 30 35 40 45

ln (L)

ln[N

(L)]

D = 149

(a)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 157

(b)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 159

(c)














Fibrous cluster

Measuring scale

2R2

2R11 1000

1000 m












358 m363 m

347 m

375 m

354 m

1000 m

(a)

347 m

371 m

364 m

351 m343 m

1000 m

(b)

334 m

336 m

366 m353 m

340 m

1000 m

(c)


Differentialgauge

AirflowFlow meters

Sample

(a) (b)


0


055 060 065 070 075 080 085 090

Porosity ()

1000000

800000

600000

400000

200000Flow

resis

tivity

(N

Ｇminus4s)

(a)

0

10000

20000

30000

40000

50000

60000

70000


065 070 075 080 085 090

Porosity ()

Flow

resis

tivity

(N

Ｇminus4s)

(b)



Table4Th

ecalculated


wresis

tivities

ofdifferent

tortuo

sityfractald

imensio

ns

Num

ber

119863 tSample

1Sample

2Sample

3Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

110

283737

98756

1521

31233

34655

987

16677

19748

1555

210

593381

544

34684

008

18366

700

310

7100410

167

37190

731

19584

830

410

8104120

543

38510

1112

20223

241

5110

111960

1337

41289

1914

21564

920

Meanvalue

106

96834

195

35915

363

18965

396


Table5Th

ecalculated


wresis

tivity

ofdifferent

fibrous

radii

Num

ber

Sample

1Sample

2Sample

3

119877 1 (120583m)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 2 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 3 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

1354

102510

98756

380

343

40625

34655

1723

336

21708

19748

993

2358

97937

083

347

38757

1184

340

20690

477

3363

92574

626

371

29541

1476

366

15340

2232

4347

11117

01257

351

36995

675

353

17766

1004

5375

81123

1785

364

31916

790

334

22241

1262

Meanvalue

359

96834

195

355

35331

193

346

19272

241


0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t



(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)


(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)



Samplenumber




1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(1)

(2)

(3)

(4)

(5)

(a)

50

45

40

35

30

25

20

1515 20 25 30 35 40

ln[N

(L)]

ln (L)

DＮ = 105

(b)


(a) (b) (c)










4

5

6

7

8

9

1510 20 25 30 35 40 45

ln (L)

ln[N

(L)]

D = 149

(a)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 157

(b)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 159

(c)














Fibrous cluster

Measuring scale

2R2

2R11 1000

1000 m












358 m363 m

347 m

375 m

354 m

1000 m

(a)

347 m

371 m

364 m

351 m343 m

1000 m

(b)

334 m

336 m

366 m353 m

340 m

1000 m

(c)


Differentialgauge

AirflowFlow meters

Sample

(a) (b)


0


055 060 065 070 075 080 085 090

Porosity ()

1000000

800000

600000

400000

200000Flow

resis

tivity

(N

Ｇminus4s)

(a)

0

10000

20000

30000

40000

50000

60000

70000


065 070 075 080 085 090

Porosity ()

Flow

resis

tivity

(N

Ｇminus4s)

(b)



Table4Th

ecalculated


wresis

tivities

ofdifferent

tortuo

sityfractald

imensio

ns

Num

ber

119863 tSample

1Sample

2Sample

3Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

110

283737

98756

1521

31233

34655

987

16677

19748

1555

210

593381

544

34684

008

18366

700

310

7100410

167

37190

731

19584

830

410

8104120

543

38510

1112

20223

241

5110

111960

1337

41289

1914

21564

920

Meanvalue

106

96834

195

35915

363

18965

396


Table5Th

ecalculated


wresis

tivity

ofdifferent

fibrous

radii

Num

ber

Sample

1Sample

2Sample

3

119877 1 (120583m)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 2 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 3 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

1354

102510

98756

380

343

40625

34655

1723

336

21708

19748

993

2358

97937

083

347

38757

1184

340

20690

477

3363

92574

626

371

29541

1476

366

15340

2232

4347

11117

01257

351

36995

675

353

17766

1004

5375

81123

1785

364

31916

790

334

22241

1262

Meanvalue

359

96834

195

355

35331

193

346

19272

241


0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t



(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)


(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)



Samplenumber




1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










4

5

6

7

8

9

1510 20 25 30 35 40 45

ln (L)

ln[N

(L)]

D = 149

(a)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 157

(b)

1510 20 25 30 35 40 45

ln (L)

4

5

6

7

8

9

ln[N

(L)]

D = 159

(c)














Fibrous cluster

Measuring scale

2R2

2R11 1000

1000 m












358 m363 m

347 m

375 m

354 m

1000 m

(a)

347 m

371 m

364 m

351 m343 m

1000 m

(b)

334 m

336 m

366 m353 m

340 m

1000 m

(c)


Differentialgauge

AirflowFlow meters

Sample

(a) (b)


0


055 060 065 070 075 080 085 090

Porosity ()

1000000

800000

600000

400000

200000Flow

resis

tivity

(N

Ｇminus4s)

(a)

0

10000

20000

30000

40000

50000

60000

70000


065 070 075 080 085 090

Porosity ()

Flow

resis

tivity

(N

Ｇminus4s)

(b)



Table4Th

ecalculated


wresis

tivities

ofdifferent

tortuo

sityfractald

imensio

ns

Num

ber

119863 tSample

1Sample

2Sample

3Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

110

283737

98756

1521

31233

34655

987

16677

19748

1555

210

593381

544

34684

008

18366

700

310

7100410

167

37190

731

19584

830

410

8104120

543

38510

1112

20223

241

5110

111960

1337

41289

1914

21564

920

Meanvalue

106

96834

195

35915

363

18965

396


Table5Th

ecalculated


wresis

tivity

ofdifferent

fibrous

radii

Num

ber

Sample

1Sample

2Sample

3

119877 1 (120583m)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 2 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 3 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

1354

102510

98756

380

343

40625

34655

1723

336

21708

19748

993

2358

97937

083

347

38757

1184

340

20690

477

3363

92574

626

371

29541

1476

366

15340

2232

4347

11117

01257

351

36995

675

353

17766

1004

5375

81123

1785

364

31916

790

334

22241

1262

Meanvalue

359

96834

195

355

35331

193

346

19272

241


0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t



(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)


(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)



Samplenumber




1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Fibrous cluster

Measuring scale

2R2

2R11 1000

1000 m












358 m363 m

347 m

375 m

354 m

1000 m

(a)

347 m

371 m

364 m

351 m343 m

1000 m

(b)

334 m

336 m

366 m353 m

340 m

1000 m

(c)


Differentialgauge

AirflowFlow meters

Sample

(a) (b)


0


055 060 065 070 075 080 085 090

Porosity ()

1000000

800000

600000

400000

200000Flow

resis

tivity

(N

Ｇminus4s)

(a)

0

10000

20000

30000

40000

50000

60000

70000


065 070 075 080 085 090

Porosity ()

Flow

resis

tivity

(N

Ｇminus4s)

(b)



Table4Th

ecalculated


wresis

tivities

ofdifferent

tortuo

sityfractald

imensio

ns

Num

ber

119863 tSample

1Sample

2Sample

3Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

110

283737

98756

1521

31233

34655

987

16677

19748

1555

210

593381

544

34684

008

18366

700

310

7100410

167

37190

731

19584

830

410

8104120

543

38510

1112

20223

241

5110

111960

1337

41289

1914

21564

920

Meanvalue

106

96834

195

35915

363

18965

396


Table5Th

ecalculated


wresis

tivity

ofdifferent

fibrous

radii

Num

ber

Sample

1Sample

2Sample

3

119877 1 (120583m)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 2 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 3 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

1354

102510

98756

380

343

40625

34655

1723

336

21708

19748

993

2358

97937

083

347

38757

1184

340

20690

477

3363

92574

626

371

29541

1476

366

15340

2232

4347

11117

01257

351

36995

675

353

17766

1004

5375

81123

1785

364

31916

790

334

22241

1262

Meanvalue

359

96834

195

355

35331

193

346

19272

241


0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t



(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)


(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)



Samplenumber




1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










358 m363 m

347 m

375 m

354 m

1000 m

(a)

347 m

371 m

364 m

351 m343 m

1000 m

(b)

334 m

336 m

366 m353 m

340 m

1000 m

(c)


Differentialgauge

AirflowFlow meters

Sample

(a) (b)


0


055 060 065 070 075 080 085 090

Porosity ()

1000000

800000

600000

400000

200000Flow

resis

tivity

(N

Ｇminus4s)

(a)

0

10000

20000

30000

40000

50000

60000

70000


065 070 075 080 085 090

Porosity ()

Flow

resis

tivity

(N

Ｇminus4s)

(b)



Table4Th

ecalculated


wresis

tivities

ofdifferent

tortuo

sityfractald

imensio

ns

Num

ber

119863 tSample

1Sample

2Sample

3Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

110

283737

98756

1521

31233

34655

987

16677

19748

1555

210

593381

544

34684

008

18366

700

310

7100410

167

37190

731

19584

830

410

8104120

543

38510

1112

20223

241

5110

111960

1337

41289

1914

21564

920

Meanvalue

106

96834

195

35915

363

18965

396


Table5Th

ecalculated


wresis

tivity

ofdifferent

fibrous

radii

Num

ber

Sample

1Sample

2Sample

3

119877 1 (120583m)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 2 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 3 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

1354

102510

98756

380

343

40625

34655

1723

336

21708

19748

993

2358

97937

083

347

38757

1184

340

20690

477

3363

92574

626

371

29541

1476

366

15340

2232

4347

11117

01257

351

36995

675

353

17766

1004

5375

81123

1785

364

31916

790

334

22241

1262

Meanvalue

359

96834

195

355

35331

193

346

19272

241


0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t



(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)


(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)



Samplenumber




1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table4Th

ecalculated


wresis

tivities

ofdifferent

tortuo

sityfractald

imensio

ns

Num

ber

119863 tSample

1Sample

2Sample

3Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

110

283737

98756

1521

31233

34655

987

16677

19748

1555

210

593381

544

34684

008

18366

700

310

7100410

167

37190

731

19584

830

410

8104120

543

38510

1112

20223

241

5110

111960

1337

41289

1914

21564

920

Meanvalue

106

96834

195

35915

363

18965

396


Table5Th

ecalculated


wresis

tivity

ofdifferent

fibrous

radii

Num

ber

Sample

1Sample

2Sample

3

119877 1 (120583m)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 2 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 3 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

1354

102510

98756

380

343

40625

34655

1723

336

21708

19748

993

2358

97937

083

347

38757

1184

340

20690

477

3363

92574

626

371

29541

1476

366

15340

2232

4347

11117

01257

351

36995

675

353

17766

1004

5375

81123

1785

364

31916

790

334

22241

1262

Meanvalue

359

96834

195

355

35331

193

346

19272

241


0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t



(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)


(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)



Samplenumber




1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table5Th

ecalculated


wresis

tivity

ofdifferent

fibrous

radii

Num

ber

Sample

1Sample

2Sample

3

119877 1 (120583m)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 2 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)119877 3 (120583m

)

Staticflo

wresistiv

ity120590(N

mminus4s)

Measured

value

120590(Nmminus4s)

Deviatio

n(

)

1354

102510

98756

380

343

40625

34655

1723

336

21708

19748

993

2358

97937

083

347

38757

1184

340

20690

477

3363

92574

626

371

29541

1476

366

15340

2232

4347

11117

01257

351

36995

675

353

17766

1004

5375

81123

1785

364

31916

790

334

22241

1262

Meanvalue

359

96834

195

355

35331

193

346

19272

241


0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t



(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)


(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)



Samplenumber




1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1000 2000 3000 4000 5000 6000 700000

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t



(a)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)


(b)

00

02

04

06

08

10

Soun

d ab

sorp

tion

coeffi

cien

t


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

(c)



Samplenumber




1 7638 50 25 112 403232 7908 50 25 111 280573 8134 50 25 110 205234 8325 50 25 108 155945 8489 50 25 108 122096 8751 50 25 106 80097 9033 50 25 104 4797


00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00

05

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0

5

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(b)


0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

05

10

15

20

25

Re(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

0

5

minus5

minus10

minus15

minus20

Im(Z

s)Z

0

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

D = 149DＮ1 = 102

DＮ2 = 106

DＮ3 = 110

(b)


5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

Re(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

00

minus02

minus04

minus06

minus08

minus10

Im(k

)

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)






0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10So

und

abso

rptio

n co

effici

ent

D = 149DＮ1 = 106

DＮ2 = 108

DＮ3 = 110

(a)

0 1000 2000 3000 4000 5000 6000 7000Frequency (Hz)

01

02

03

04

05

06

07

08

09

10

Soun

d ab

sorp

tion

coeffi

cien

t

DＮ = 106D1 = 12

D2 = 14D3 = 16

(b)



5 Conclusions






Competing Interests


Acknowledgments



References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Fractal Model for Acoustic Absorbing of Porous Fibrous...

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Transcript of Fractal Model for Acoustic Absorbing of Porous Fibrous...