Bolton, J Stuart

Gardner, Bryce K.

Jung, In Hwa

Kang, Yeon June

English

Purdue E-Pubs

Purdue UniversityPurdue e-PubsPublications of the Ray W. Herrick Laboratories School of Mechanical Engineering1998Development of Axisymmetric Finite Elements forPoroelastic MaterialsJ Stuart BoltonPurdue University, bolton@purdue.eduYeon June KangSeoul National University, KoreaIn Hwa JungSeoul National University, KoreaBryce K. GardnerAutomated Analysis Corp.Follow this and additional works at: http://docs.lib.purdue.edu/herrickThis document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu foradditional information.Bolton, J Stuart; Kang, Yeon June; Jung, In Hwa; and Gardner, Bryce K., "Development of Axisymmetric Finite Elements forPoroelastic Materials" (1998). Publications of the Ray W. Herrick Laboratories. Paper 37.http://docs.lib.purdue.edu/herrick/37DEVELOPMENT OF AXISYMMETRIC FINITE ELEMENTSFOR POROELASTIC MATERIALSYeon June Kang and In Hwa JungSchool of Mechanical & Aerospace EngineeringSeoul National University, KoreaJ. Stuart BoltonRay W. Herrick Laboratories, Purdue University, U.S.A.Bryce K. GardnerAutomated Analysis Corporation, Ann Arbor, MI, U.S.A.OUTLINE• Introduction• Axisymmetrical Foam Finite Elements• Sound Transmission through Cylindrical & Conical Foam Plug- validation with 3-D Cartesian solutioni ith i t l lt- compar son w  exper men a  resu s- effect of finite termination impedance• Sound Attenuation in Foam-Lined Duct- comparison with Morse’s solution- comparison with experiment results- effect of circumferential boundary conditionAXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National UniversityINTRODUCTION• Cartesian Finite Elements of Poroelastic Materials- Normal incidence absorbtion coefficient( Y. J. Kang and J. S. Bolton, J. Acoust. Soc. Am. 98, 1995 )Normal incidence sound transmission loss-     ( Y. J. Kang and J. S. Bolton, J. Acoust. Soc. Am. 99, 1996 )( J. P. Coyette and H. Wynendaele, Inter-Noise 95 )( N. Attala and R. Panneton, Inter-Noise 95 )- Sound transmission through poroelastic wedges( Y. J. Kang and J. S. Bolton, J. Acoust. Soc. Am. 102, 1997 )• Sound Propagation along Lined Ducts- Axisymmetric circular ducts, Locally reacting( Y. Kagawa et al., J. Sound & Vib. 53, 1977 )- Rectangular ducts, Extended & Locally reacting( R. J. Astley and A. Cummings, J. Sound & Vib. 116, 1987 )AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University            Elastic Porous Material Theory based on BiotAXISYMMETRIC FOAM FINITE ELEMENTS      3D (           )                    2D (        )zθ,r, zr,St St i R l tiD i R l ti ress- ra n e a onsynam c e a onsTwo Wave EquationsWeak Forms Galerkin’s ApproximationsFoam Finite ElementsAXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National UniversityAXISYMMETRIC FOAM FINITE ELEMENTSU l d S t E ti aΚ  p    zazrari dvnvnrπjω Γ0 Γ)(2• ncoup e  ys em qua onsruu  e zrzrridnnrdnnrΓΓ)(Γ)( fK rzU  e rie zzzrridsnrΓΓΓ   zU   e zi dsnrΓ Γneed to be coupled using appropriate boundary conditions at the interface of two systemsAXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National UniversityAXISYMMETRIC FOAM FINITE ELEMENTSB d C diti• oun ary on onsp, vσUusUuv hjhja   )(1- Velocity continuity :fa shp nn - Force equilibrium ( fluid part ) :kin )()()( fzzfrzrfzzrfrra nnrnnrph  1- Force equilibrium ( frame part ) :AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National UniversityAXISYMMETRIC FOAM FINITE ELEMENTSC l d A ti F S t E ti• oup e  cous c- oam ys em qua ons  aΚ rup aΚ f 1FQfK zUuafΚ 32FF   zrU 4FAXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University• Axisymmetric vs. 3-D CartesianSOUND TRANSMISSION THROUGH CYLINDRICAL FOAM PLUG49 foam elements3.3 cm 2.8 cm 3.3 cm98 air elements192 nodesR = 1.45 cmair IIair I foam1456 foam elements2912 air elements5082 nodesair II* It t k 5500 ti l l ti ti t h f !air IfoamAXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University  a es  mes onger so u on me a  eac  requency SOUND TRANSMISSION THROUGH CYLINDRICAL FOAM PLUG• Validation with 3-D Cartesian Solution20oss (dB)15Axisymmetric Model3-D Modelnsmission Lo10Tra5Frequency (Hz)0102 104103AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National UniversitySOUND TRANSMISSION THROUGH FOAM PLUG• Experimental Setup • Anechoic TerminationDual ChannelSignal AnalyzerComputer2 B & K Type 2032Impedance1.51Re(z)Signal AmplifierCH BCH A SignalGeneratorNormalized 00.5Im(z)Microphones-0.5102 104103Two-MicrophoneImpedance Measurement TubeB & K Type 4206NewSampleHolderAnechoicTerminationFrequency (Hz)* Note:Measured impedance data was phase - corrected AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National Universitywhen it was applied to the model.SOUND TRANSMISSION THROUGH CYLINDRICAL FOAM PLUG• Effect of Circumferential Boundary Conditionsfree edge glued edgemeasuredFEM20measuredFEM20Loss (dB)1510 Loss (dB)1510Transmission 5Transmission L5Frequency (Hz)0102 104103Frequency (Hz)0102 104103AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University  SOUND TRANSMISSION THROUGH CONICAL FOAM PLUGfree edge20measuredFEM15B)10sion Loss (dB5Transmiss0102 104103AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National UniversityFrequency (Hz)SOUND TRANSMISSION THROUGH CYLINDRICAL & CONICAL FOAM PLUG• Effect Finite Termination Impedancemeasured termination ( cone; glue; free)20ideal termination ( cone; glue; free)15ss (dB)10smission Los5Trans10000 2000 3000 4000 5000Frequency (Hz)0AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National UniversitySOUND ATTENUATION IN FOAM-LINED CIRCULAR DUCT• Macroscopic Physical Properties of Foams Obtained byMeasurement and OptimizationFoam Type Foam A Foam B Foam CParameter (polyester) (polyether) (polyester)Flow resistivity (mks Rayls/m) 13666 30814 46417 measuredTortuosity 3 58 4 28 6 13 optimized(Structure factor) . . .Porosity 0.96 0.96 0.96 assumedBulk density)3(k / 32 29 32 measuredg mBulk Young’sModulus (Pa) 30400 25200 85800 optimizedLoss factor 0.3 0.3 0.3 assumedPoisson’s ratio 0.4 0.4 0.4 assumedAXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University• Measured and Predicted Absorption CoefficientSOUND ATTENUATION IN FOAM-LINED CIRCULAR DUCTFoam A Foam C0.810.810.40.60.40.600.20 1000 2000MeasuredOptimized by analytical approachFrequency (Hz)00.20 1000 2000MeasuredOptimized by analytical approachOptimized by FEMFrequency (Hz)AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University• Experimental SetupSOUND ATTENUATION IN FOAM-LINED CIRCULAR DUCTImpedance Tube Foam liningMicrophone AnechoicTermination1 m 0.6 mB&K 4206ConditioningAmplifierB&K type 2690Amplifier FrequencyAnalyzerHP35670A• Open Area Fraction (radius of airway / radius of tube)0.5 0.80.70.6AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University• Performance of Anechoic TerminationSOUND ATTENUATION IN FOAM-LINED CIRCULAR DUCT1 1.5Anechoic termination0 60.80 51Impedance real partNormal0.4.0.Impedance imaginary partnized Impedanc00.2-1-0.5ce0 500 1000 1500 2000Frequency (Hz)AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University• Comparison with Experimental Results (Foam A)SOUND ATTENUATION IN FOAM-LINED CIRCULAR DUCTOPEN AREA FRACTION1000FEM,FEM,measuredmeasuredOpen area fraction 0.5:Open area fraction 0.6:0.50 6100FEM,FEM,measuredmeasured(dB/m)   Open area fraction 0.7:Open area fraction 0.8:0.7.10Attenuation (0.81100 1000 104Frequency (Hz)AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University• Comparison with Experimental Results (Foam B)SOUND ATTENUATION IN FOAM-LINED CIRCULAR DUCTOPEN AREA FRACTION1000FEM,FEM,measuredmeasuredOpen area fraction 0.5:Open area fraction 0.6:0.50 6100FEM,FEM,measuredmeasured(dB/m)Open area fraction 0.7:Open area fraction 0.8:0.7.10Attenuation (0.81100 1000 104Frequency (Hz)AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University• Comparison with Experimental Results (Foam C)SOUND ATTENUATION IN FOAM-LINED CIRCULAR DUCTOPEN AREA FRACTION1000FEM,FEM,measuredmeasuredOpen area fraction 0.5:Open area fraction 0.6:0.50 6100FEM,FEM,measuredmeasured(dB/m)Open area fraction 0.7:Open area fraction 0.8:0.7.10Attenuation (0.81100 1000 104Frequency (Hz)AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University• Bulk Reacting Vs. Locally Reacting LinerSOUND ATTENUATION IN FOAM-LINED CIRCULAR DUCTFoam A Foam C0.5, 0.80.5, 0.8FEM:Morse's sol:0.5, 0.80.5, 0.8FEM:Morse's sol:10000.5, 0.8/m)Measured:10000.5, 0.8/m)Measured:0 5 0 8 0 5 0 810100ttenuation (dB/10100ttenuation (dB/. . . .1100 1000 104At1100 1000 104AtFrequency (Hz) Frequency (Hz)AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University• Effect of Boundary ConditionSOUND ATTENUATION IN FOAM-LINED CIRCULAR DUCTFoam A Foam C0.5,0.5,0.80.80 5 0 8FEM (lubricated):FEM (constrained): 0.5,0.5,0.80.80 5 0 8FEM (lubricated):FEM (constrained):M d1000. , ./m)Measured:1000. , ./m)easure :0 5 0 8 0 5 0 810100ttenuation (dB/10100ttenuation (dB/. . . .1100 1000 104At1100 1000 104AtFrequency (Hz) Frequency (Hz)AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National UniversityCONCLUSIONThe AXISYMMETRICAL FOAM FINITE ELEMENTS has been formulated and•          validated for its accuracy and efficiency.• It has many applications such as sound transmission and attenuation in           axisymmetric configurations.• Constraining the circumference of the foam plugs decreased the transmission loss           at high frequencies.• Finite termination impedance had rippling effect on sound transmission loss at low            frequencies.• Thicker liner does not always guarantee high sound attenuation        .• Locally reacting assumption is valid for some limited cases.AXISYMMETRIC POROELASTIC FINITE ELEMENTS Seoul National University

Development of Axisymmetric Finite Elements for Poroelastic Materials

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Development of Axisymmetric Finite Elements for Poroelastic Materials

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