Probabilistic Investigation of Sensitivities of Advanced Test- Analysis Model Correlation Methods

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1 Probbilistic Investigtion of Sensitivities of Advnced Test- Anlysis Model Correltion Methods Liz Bergmn, Mtthew S. Allen, nd Dniel C. Kmmer Dept. of Engineering Physics University of Wisconsin-Mdison Rndll L. Myes Sndi Ntionl Lbortories

Sensor Test fils TAM/FEM Correltion psses FEM mss mtrix must be reduced to

2 Test Anlysis Correltion 7,146 DOF FEM Updte FEM fils TAM/Test Correltion psses FEM vlidted Select sensor loctions nd reduce FEM 108 DOF TAM 108 Sensor Test fils TAM/FEM Correltion psses FEM mss mtrix must be reduced to test degrees of freedom (TAM) in order to compute modl orthogonlity. Orbitl Sciences 2

The Controversy Current FEM reduction lgorithms Sttic

My be difficult to chieve good TAM/FEM correltion

propellnt mode predicted by Sttic TAM (right) Improved

circumstnces Modl TAM: Trivil to chieve perfect TAM/FEM

3 The Controversy Current FEM reduction lgorithms Sttic TAM: fils for hevy, soft structures. My be difficult to chieve good TAM/FEM correltion Fundmentl FEM propellnt mode (left) nd fundmentl FEM propellnt mode predicted by Sttic TAM (right) Improved Reduced Sttic (IRS) TAM: ill-conditioned under certin circumstnces Modl TAM: Trivil to chieve perfect TAM/FEM correltion, however it hs reputtion of being highly sensitive to experimentl or modl-mismtch errors 3

4 Purpose of Reserch Study the sensitivity of vrious TAMs to gin insight into fctors tht strongly ffect sensitivity A probbilistic nlysis will be used to chrcterize the effect of mesurement errors on TAM sensitivity 4

5 Relevnt Literture Guyn Freed, AM nd Flnign, CC (1990): Modl TAM most sensitive, sensors plced using modl kinetic energy Avitbile, P, Pechinsky, F, nd O Cllhn, J (1992): Sensor plcement is vitl to TAM performnce, SEREP nd Hybrid perform better thn Sttic TAM for smll sensor sets Chung, YT (1998): Sensor plcement ws not discussed nd no significnt difference could be seen between the TAMs Modl Lrge Reduced Cross Orthogonlity of Test Tower (Chung 1998) Avitbile, P, Pechinsky, F, nd O Cllhn, J (1992) 5

6 Relevnt Literture Gordis, JH (1992), Blelloch, P nd Vold, H (2005) : Notes ill-conditioning in dynmic reduction eqution: Proposes tht IRS TAM will be ill conditioned if the nturl frequencies of the structure with the o-set DOF pinned re similr to the frequencies of the structure of interest. Recently, this theory seems to hve been pplied to other TAM techniques such s the Modl TAM. 6

8 Hz) 108 sensors Trget Mode 5 Trget Mode 6

7 Model Generic Stellite 7,146 DOF Trget modes: first 18 consecutive flexible modes ( Hz) 108 sensors Trget Mode 5 Trget Mode 6 Trget Mode 7 Trget Mode Hz 2.8 Hz 3.5 Hz 3.7 Hz 7

8 Test Anlysis Models Sttic TAM = sensor loction o = omitted DOF Eigenvlue problem 2 M ωi M o M M o oo φ i + φio K K o K K o oo φ i = φio Lower prtition eqution [ Ko ωi M o ]{ φi} + [ Koo ωi M oo ]{ φio} = 0 Neglect the mss of the o-set DOF 0 0 { } 2 1 φ = K ω M K ω 2 M { φ } io oo i oo o i o i Sttic Trnsformtion Mtrix (ech column represents constrint mode) I K K [ ] TS = 1 oo o 8

9 Test Anlysis Models IRS TAM { } 2 1 φ = K ω M K ω 2 M { φ } io oo i oo o i o i Ill-conditioned when ω i2 is ner ny of the eigenvlues of the Koo, Moo system Approximte the frequency terms Clculte the IRS trnsformtion mtrix ω 2 ~ ~ i i S S 1 { φ } = M K { φ } [ T ] = [ T ] + [ T ] IRS S i i M 0 0 ~ K oo M o M oo Koo K o o [ Ti ] = M S K S M I ~ 9

10 Test Anlysis Models IRS TAM O-set system Mode Hz FEM Trget Mode Hz 10

11 Test Anlysis Models Sttic nd IRS TAM Mss weighted effective independence did not select the lumped msses (the lumped msses were essentil to TAM- FEM correltion) Modl kinetic energy pplied to ll 18 trget modes ws not sufficient A significnt mount of hnd selection nd engineering judgment ws used (modified modl kinetic energy method) 5 core lumped msses 11

12 Test Anlysis Models Modl TAM Physicl coordintes in terms of modl coordintes Prtitioned Equtions Solve for modl coordintes in terms of the sensor DOF Modl trnsformtion mtrix x x o φ = φo {} q { x } = [ φ ]{ q} { x } = [ φ ]{ q} o {} [ ] [ T 1 T q = φ φ φ ]{ x } [ T ] M o I T φo ( φ φ ) = 1 T φ 12

of the Fisher informtion mtrix mx Q = mx φ T φ

13 Test Anlysis Models Modl TAM Sensor plcement chieved with Effective Independence Mximize the determinnt of the Fisher informtion mtrix mx Q = mx φ T φ Effective Independence E Di 1 = φ Q φ 0.0 E 1. 0 i it Di 13

Test Anlysis Models Modl TAM Modl TAM o-set

O-set system Mode 2 FEM Trget Mode 4 O-set

14 Test Anlysis Models Modl TAM Modl TAM o-set frequencies re similr to the FEM frequencies, so the theory of Gordis suggests tht this TAM will be sensitive. O-set system Mode 2 FEM Trget Mode 4 O-set system Mode 5 FEM Trget Mode Hz 1.8 Hz 3.2 Hz 3.5 Hz 14

15 Test Anlysis Models Modl using Condition Number Sensor Plcment Modl coordintes in terms of the sensor DOF {} [ ] [ T T q = φ φ φ ]{ x } 1 Solution is more sensitive if the condition number of, is lrge. φ Begin with visuliztion set, nd dd sensors tht minimize the condition number of φ 15

16 Test Anlysis Correltion 7,146 DOF FEM Updte FEM fils TAM/Test Correltion psses FEM vlidted Select sensor loctions nd reduce FEM 108 DOF TAM 108 Sensor Test fils TAM/FEM Correltion psses Orbitl Sciences 16

17 Orthogonlity Criteri: O = φ Cross Orthogonlity Criteri: CO = φfem Frequency Comprison Criteri: Correltion Metrics 0 off digonl term 0.1 [ ] [ T M ][ φ ] FEM ~ TAM 0 off digonl term digonl term 1.0 f error ~ FEM [ ] [ T M ][ φ ] = f FEM f FEM f TAM TAM TAM * 100 3% 17

18 TAM-FEM Correltion Mx off digonl term: 0.05 Mx off digonl term: 6e-4 *Modl TAM lwys produces perfect orthogonlity for TAM-FEM correltion 18

19 Test Anlysis Correltion 7,146 DOF FEM Updte FEM fils TAM/Test Correltion psses FEM vlidted Select sensor loctions nd reduce FEM 108 DOF TAM 108 Sensor Test fils TAM/FEM Correltion psses Orbitl Sciences 19

20 {} i φ FEM { U}= Noise Model nd Simulted Test Mode Shpes i φ1 i φ2 = i φ n i φn 1 Mx vlue *2%* {} i φ {} { } i Test = φ noise + φ FEM { U} = { φ} noise column vector of uniformly distributed rndom numbers between -1 nd 1 FEM trget mode 15 Test trget mode 15 20

21 Noise Model FEM ssumed to be perfect Noise contminted mode shpe Noise vector models the net effect of ll errors tht cuse the FEM mode shpes to disgree with the test mode shpes. Noise Distribution: Uniform no ssumption is mde bout the distribution of noise Noise Amplitude: Sensors with the smllest motion hve the lrgest noise to signl rtio Noise is smll on verge: ± 2% t sensor loctions with the lrgest motion. 21

22 TAM-Test Test Correltion Results (1 cse of Rndom Noise) Mx off digonl term: 0.27 Mx off digonl term:

23 TAM-Test Test Correltion Results (1 cse of Rndom Noise) Mx off digonl term: 0.05 Mx off digonl term:

24 Monte Crlo Simultion Thus fr, TAM-Test correltion hs been studied using only one noise profile Rndom noise dded in 10,000 itertions Orthogonlity computed for ech itertion Mximum off-digonl term of orthogonlity ws stored 24

25 TAM-Test Test Correltion Results Pss Fil Despite its low o-set frequencies, Modl TAM does not show high sensitivity! 25

26 TAM-Test Test Correltion Results If Orthogonlity > 0.1 one might Refine FEM before exiting test Repet test nd/or look for errors Updte the FEM In this cse, the FEM ws perfect (errors in test modes were purely rndom) Note: The specific rnking of different TAM methods my depend on: The structure of interest The chrcteristics of the noise Systemtic errors between the test nd FEM 26

27 TAM-Test Test Correltion Results Pss Fil Sensor selection is criticl to the performnce of ech TAM Most previous studies used the sme sensor set, usully optimized for the Sttic TAM 27

28 Predicting Stndrd Devition Recently, we hve developed formuls to nlyticlly predict sensitivity of TAM bsed on simple metrics For exmple, for the noise model used in this study: T Oij = [ φi + ni ] M TAM [ φi + ni ] ni = noise ( 2 T ) ( 2 ) ( ) σ ( O ) = M φ σ + φ M σ + M σσ ij TAM j i i TAM j TAM mn i j m m m m m n Sttic Modl EFI Modl C#TAM Modl with Sttic DOF Mximum Orthogonlity Off-Digonl Predicted STD Actul STD

29 Conclusions nd Future Work Conclusions IRS TAM ws ill-conditioned, s predicted by Gordis Modl TAM did not show high sensitivity even though its o-set frequencies were ner those of the trget modes Probbilistic nlysis more fully explins TAM sensitivity One cn even predict the sensitivity of the TAMs nlyticlly given the TAM Mss mtrix, mode shpes nd noise model. Future Work Develop more ccurte noise models Study the effect of systemtic mismtch between FEM nd test due to modeling errors. My need the Hybrid TAM in these cses Apply these methods to other physicl systems, nlyticlly nd experimentlly. Investigte systems with non-consecutive trget modes 29

Multivariate problems and matrix algebra

Multivariate problems and matrix algebra University of Ferrr Stefno Bonnini Multivrite problems nd mtrix lgebr Multivrite problems Multivrite sttisticl nlysis dels with dt contining observtions on two or more chrcteristics (vribles) ech mesured