Inverse Characterization of Poro-Elastic Materials Based on Acoustical Input Data

Transcription

1 Purdue University Purdue e-pubs Publications of the Ray W. Herrick Laboratories School of Mechanical Engineering Inverse Characterization of Poro-Elastic Materials Based on Acoustical Input Data J Stuart Bolton Purdue University, bolton@purdue.edu Kwanwoo Hong Follow this and additional works at: Bolton, J Stuart and Hong, Kwanwoo, "Inverse Characterization of Poro-Elastic Materials Based on Acoustical Input Data" (2009). Publications of the Ray W. Herrick Laboratories. Paper This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information.

2 Inverse Characterization of Poro Elastic Materials based on Acoustical Input Data J. Stuart Bolton and Kwanwoo Hong Ray W. Herrick Laboratories School of Mechanical Engineering Purdue University ASA San Antonio 29 October 2009

3 Introduction Brief history of standing wave tube Four-microphone standing wave tube Eti Estimation of fbit Biot parameters based on acoustical measurements 2

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6 Standing Wave Tube Standing wave method for measuring normal incidence absorption coefficients more than 00 years old Method is credited by a number of authors to J. Tuma (902) Subsequent experiments conducted by Weisbach (90) and Taylor (93) 5

7 PHYSICAL REVIEW, 2(4): OCT 93 6

8 J. Acoust. Soc. Am. 9, , May 947 7

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11 J. Acoust. Soc. Am. 90(4), , Oct 99 0

12 Four Microphone Method Four-microphone tube for silencer testing Munjal (Duct Acoustics) Two-load method Two-source method Four-microphone tube for material testing Suggested by Joseph Pope Yun and Bolton (997 SAE) Song and Bolton (2000 JASA) introduce transfer matrix approach Many articles since then

13 Transfer Matrix Method P jkx jkx ( Ae Be ) e jt Mic Mic 2 Mic 3 Mic 4 P jkx3 jkx3 3 ( Ce De ) e jt P jkx2 jkx2 2 ( Ae Be A B jkx2 j( Pe P2 e 2sin k ( x x s 2 ) ) jkx jkx j( P2 e Pe 2sin k( x x ) 2 ) e jkx2 jt ) Speaker Sound pressure and velocity relationship A P V x x0 TL 20log0 x 2 B T T Symmetric sample T =T 22, T T 22 -T 2 T 2 = Transmission loss Transfer matrix / T 2 d T T x 2 22 T T cosk d 2 p T T j sin k p d / p c 2 22 Property of material x 3 x 4 C P V p xd where T T 2 D P Anechoic termination (not required) P jkx4 jkx4 4 ( Ce De C D jkx4 j( P3 e P4 e 2sin k( x x 3 4 ) e 3 ) ) jkx jkx3 j( P4 e Pe 3 2sin k( x x ) 3 jkx4 2 2 V P V P P xd xd x0 x0 x0 xd T2 P V P V P V P V x0 xd xd x0 x0 xd xd x0 2 2 V V P V P V x0 xd xd xd x0 x0 T22 P V P V P V P V x0 xd xd x0 x0 xd xd x0 T T j pcp sin k pd cos k d p Limp or rigid porous material ( T2 0 ) jkd 2e2 e / c ct 0 2 k p T p 22 cos d T T c p T jt )

14 Anechoic Transmission Loss Aviation grade glass fiber (density=9.6 kg/m 3, flow resistivity= 3000 Rayls/m) cm tube Experiment Prediction using FEM (with edge constraint) Prediction without edge constraint 25 TL (db) 20 5 Increase in TL due to edge constraint 0 5 Shearing Resonance Frequency (Hz) Above shearing resonance finite size sample represents infinite sample Below shearing resonance all properties affected by edge-constraint 3

15 Estimation of Biot Parameters Software available to estimate Biot parameters by performing optimal fit to measured acoustical data (flow resistivity, it porosity, tortuosity, t viscous characteristic ti length, thermal characteristic length, bulk density, Young s modulus, loss factor, Poisson ratio) ESI-FOAM-X (rigid, limp) COMET/Trim (rigid, id limp, elastic) Original software based on transversely infinite layered representation: i.e., edge constraint effects are not included 4

16 Infinite Panel Model: COMET/TRIM 5

17 Infinite Panel Model: Limitation ficient Absorption Coeff Transmission Los ss [db] Experiment Trim Frequency [Hz] 0 Experiment Trim Frequency [Hz] Note that this model does not simulate the low frequency transmission loss fluctuation caused by shearing resonance of the sample 6

18 Finite Element Models: COMET/SAFE The software COMET/SAFE is used to model and compute the absorption and transmission loss having a finite depth and finite size layer of porous material. Afi finite it element tbased program that tallows for the analysis of sound traveling through various media including fluids, solids and foam-like substances. Finite element implementation is based on u-u and p-u versions of Biot theory. All models used in this work involved axisymmetric i elements. The new version of TRIM supports automated inverse characterization capability based on SAFE. 7

19 Finite Element Model Note that finite model can simulate the low frequency transmission loss fluctuation caused by shearing resonance of the sample 8

20 Inverse Characterization Questions: Is it possible to determine the Biot parameters from acoustical measurements? Do parameters act independently? How many parameters can be estimated? To help answer these questions, introduce a procedure based on Singular Value Decomposition Singular Value Decomposition is widely used linear algebraic method to identify the principal components in the field of image processing and signal processing. 9

21 Sensitivity Matrix Analysis Procedures. Linearize absorption and transmission coefficient close to a certain parameter set 2. Use absorption and/or transmission coefficient values for certain number of frequencies to construct a sensitivity matrix 3. Perform singular value decomposition on the sensitivity matrix and extract t singular values to determine effective rank (number of independent parameters) 4. Calculate condition number (the smaller the better) 20

22 Sensitivity Matrix Analysis Linearize the expression for the absorption and transmission coefficient in the vicinity of a certain parameter set For frequency f ( x) f ( x Real Solution 9 xi 0 ) ( ) i xi dx i Calculate by using central difference scheme ± % difference of material properties xi Approximate Solution 2x i xi x i 2

23 Sensitivity Matrix Analysis For n frequencies the equation can be combined as a matrix y y For n frequencies, the equation can be combined as a matrix ) ( ) ( 9 2 dx x x x x x x f x f x f f f ) (... ) ( dx x x x x x x f x f x f f f n n n n n Sensitivity Matrix Perform singular value decomposition: g p M=UΣV * The rank of the matrix M equals the number of non-zero singular values which is the same as the number of non zero elements in the matrix Σ.

24 Rigid Foam Sensitivity Matrix Analysis Use COMET/TRIM rigid foam type material that has 5 material properties. E.g., Porosity, flow resistivity, tortuosity, viscous and thermal characteristic length. The nominal values of the material properties are ient Absorption Coeffici Transmission Loss [d db] Frequency [Hz] Frequency [Hz] Porosity Flow Tortuosity VCL TCL Resistivity , * *0-5 23

25 Rigid Foam Sensitivity Matrix Analysis Effect of adding frequency data for absorption coefficient Effect of adding frequency Effect of adding frequency Singular Value Condition Number 0-4 st SV 2nd SV 3rd SV 4th SV 5th SV Number of Frequency Number of Frequency Adding additional frequency data reduces the condition number, but the condition number is too big to consider that the sensitivity matrix is well-posed. 24

26 Rigid Foam Sensitivity Matrix Analysis Effect of adding frequency data for transmission coefficient Effect of adding frequency Effect of adding frequency Singular Value st SV 2nd SV 3rd SV 4th SV 5th SV Number of Frequency Condition Number Number of Frequency 58 Adding additional frequency data reduces the condition number, but the condition number is too big to consider that the sensitivity matrix is well-posed. 25

27 Sensitivity matrix Rigid Foam Sensitivity Matrix Analysis 0.4 a T Difference Difference Porosity Flow resistivity Tortuosity Viscous CL Thermal CL Frequency [Hz] Porosity Flow resistivity Tortuosity Viscous CL Thermal CL Frequency [Hz] Sensitivities to porosity and flow resistivity are quite close to each other for both absorption and transmission coefficients 26

28 Singular Vector Ab ti C ffi i t Absorption Coefficient Note: Effect of material property (porosity) and 2(flow resistivity) is almost the same on all five singular vectors. 27

29 Rigid Foam Sensitivity Matrix Analysis Fixed porosity case result for absorption coefficient Effect of adding frequency 0 4 Effect of adding frequency Singular Value Co ondition Number st SV 2nd SV 3rd SV 4th SV Number of Frequency Number of Frequency 8 Fixing gporosity reduces the condition number significantly and makes the sensitivity matrix well-posed. 28

30 Rigid Foam Sensitivity Matrix Analysis Combine both absorption and transmission coefficient sensitivity matrix Singular Value Condition Number Adding other acoustical measurements reduces the condition number further 29

31 Rigid Foam Sensitivity Matrix Analysis To verify the effect of low and high condition number during the automatic inverse characterization in COMET/TRIM, two different cases were studied Solution Initial value Found value Found value 2 Porosity Flow resistivity 50,000 25,000 65,000 5,050 Tortuosity Viscous C.L 3.0* *0-5.77* *0-5 Thermal C.L 9.0* * * *0-5 30

32 O ti l Inverse I Ch t i ti Optimal Characterization 4 parameter search gives near optimal result 3

33 Inverse Characterization based on FEM Use COMET/SAFE (FEM) elastic foam type material that has 9 material properties listed below. Layer thickness = 5 cm, Sample is fixed around circumferential edge Porosity Flow R i i i Resistivity Tortuosity VCL TCL Density Young s modulus d l Poisson s ratio i Loss f factor , * * ,

34 Singular Vectors for Absorption and T i i Transmission Higher order singular vectors for absorption & transmission coefficient case Even higher E hi h (6th, 7th, 8th, and d 9th) order d singular i l vectors t have wide range of values all parameters independent 33

35 Inverse Characterization Results Solution Initial guess Unfixed Porosity Flow resistivity 50,000 45,000 5,203 Tortuosity Viscous C.L 3.0* * *0 5 Thermal C.L 9.0*0 5.05* *0 5 Density Young s modulus 50,000 45,000 53,445 Poisson s ratio Loss factor parameters estimated with reasonable accuracy 34

36 I Ch t i ti Results R lt Inverse Characterization 35

37 Inverse Procedure based on FEM The finite element model s condition number is significantly smaller than the condition number based on the plane wave model. Absorption coefficient: Transmission coefficient: This result is due to the fact that the finite element model can simulate finite sample size effects such as low frequency shearing resonance of the sample inside the tube. Therefore, the inverse characterizations based on the finite element model have better chance to extract correct material properties. 36

38 Conclusions Standing wave tubes can provide both absorption and transmission loss data for estimation of Biot parameters by inverse methods, but edge constraint effects have a significant impact on the results By using a linearization and SVD procedure, the stability of the inverse process can be improved by removing material properties that makes the sensitivity matrix ill-conditioned. Inverse procedures based on finite element models of edge- constrained samples may offer improved performance by making the effect of input parameters more independent 37

39 Acknowledgments P. E. Doak Joe Pope L&L Products United Technologies Research Center 3M Corporation (Jon Alexander) Bruel & Kjaer (Oliviero Olivieri, Jason Kunio, Jorgen Hald) NASA (Richard Silcox) Richard Yun, Heuk Jin (Bryan) Song, Jinho Song, Jeong-woo Kim, Taewook Yoo, Kwanwoo Hong, Kang Hou Tanya Wulf 38