Probabilistic Investigation Of Sensitivities Of Advanced .
Probabilistic Investigation ofSensitivities of Advanced TestAnalysis Model CorrelationMethodsLiz Bergman, Matthew S. Allen, andDaniel C. KammerDept. of Engineering PhysicsUniversity of Wisconsin-MadisonRandall L. MayesSandia National Laboratories
Test Analysis Correlation7,146 DOFFEMfailsUpdate FEMTAM/TestCorrelationpassesFEMvalidated108 DOFTAMSelect sensor locationsand reduce FEM108 SensorTestfails TAM/FEMCorrelationpassesFEM mass matrix must be reduced to testdegrees of freedom (TAM) in order tocompute modal orthogonality.Orbital Sciences2
The Controversy Current FEM reduction algorithms Static TAM: fails for heavy, soft structures. May be difficult toachieve good TAM/FEM correlationFundamental FEM propellant mode (left) and fundamentalFEM propellant mode predicted by Static TAM (right) Improved Reduced Static (IRS) TAM: ill-conditioned under certaincircumstances Modal TAM: Trivial to achieve perfect TAM/FEM correlation,however it has a reputation of being highly sensitive to experimentalor modal-mismatch errors3
Purpose of ResearchStudy the sensitivity of various TAMs togain insight into factors that stronglyaffect sensitivity A probabilistic analysis will be used tocharacterize the effect of measurementerrors on TAM sensitivity 4
Relevant Literature GuyanFreed, AM and Flanigan, CC (1990): Modal TAM most sensitive, sensorsplaced using modal kinetic energyAvitabile, P, Pechinsky, F, and O’Callahan, J (1992): Sensor placement isvital to TAM performance, SEREP and Hybrid perform better than StaticTAM for small sensor setsChung, YT (1998): Sensor placement was not discussed and nosignificant difference could be seen between the TAMsModalLargeCross Orthogonality of Test Tower(Chung 1998)ReducedAvitabile, P, Pechinsky, F, andO’Callahan, J (1992)5
Relevant Literature Gordis, JH (1992), Blelloch, P and Vold, H (2005) : Notes ill-conditioning in dynamic reduction equation: Proposes that IRS TAM will be ill conditioned if the naturalfrequencies of the structure with the o-set DOF pinned are similar tothe frequencies of the structure of interest.Recently, this theory seems to have been applied to other TAMtechniques such as the Modal TAM. 6
Model Generic Satellite 7,146 DOFTarget modes: first 18 consecutive flexible modes (0.311.8 Hz)108 sensorsTarget Mode 5Target Mode 6Target Mode 7Target Mode 82.7 Hz2.8 Hz3.5 Hz3.7 Hz7
Test Analysis Models – Static TAMa sensor locationo omitted DOFEigenvalue problemLower partition equation M aa ω M oa2i[KoaM ao φia K aa M oo φio K oa][] ωi2 M oa {φia } K oo ωi2 M oo {φio } 00Neglect the mass of theo-set DOFK ao φia 0K oo φio {φio } K oo ωStatic Transformation Matrix (eachcolumn represents a constraint mode)2i0 1M oo K oa ωi2 M oa {φia }I [TS ] 1 K oo K oa 8
Test Analysis Models – IRS TAM 1{φio } K oo ωi2 M oo K oa ωi2 M oa {φia }Ill-conditioned when ωi2 is near any of theeigenvalues of the Koo, Moo systemApproximate thefrequency terms 1ω {φia } M S K S {φia }Calculate the IRStransformation matrix[TIRS ] [TS ] [Ti ]2i0 M aa 0[Ti ] 1 0Koo M oa M ao I 1 M S KS 1M oo K oo K oa 9
Test Analysis Models – IRS TAMO-set system Mode 1FEM Target Mode 1816.8 Hz11.8 Hz10
Test Analysis Models – Static and IRS TAM Mass weighted effective independence did not select thelumped masses (the lumped masses were essential to TAMFEM correlation)Modal kinetic energy applied to all 18 target modes was notsufficientA significant amount of hand selection and engineeringjudgment was used (modified modal kinetic energy method)5 core lumpedmasses11
Test Analysis Models – Modal TAMPhysical coordinates in terms ofmodal coordinatesPartitioned EquationsSolve for modal coordinates interms of the sensor DOFModal transformation matrix xa φa {q} xo φo {xa } [φa ]{q}{xo } [φo ]{q}{q} [φaT φa ] 1 [φaT ]{xa }I [TM ] T 1 T φo (φa φa ) φa 12
Test Analysis Models – Modal TAM Sensor placement achieved with EffectiveIndependenceMaximize the determinant ofthe Fisher information matrixEffective Independencemax Q max φ φaE Di φai Q 1φaiTTa0.0 E Di 1.013
Test Analysis Models – Modal TAMModal TAM o-set frequencies are similar to the FEM frequencies, sothe theory of Gordis suggests that this TAM will be sensitive.O-set system Mode 2FEM Target Mode 4O-set system Mode 5FEM Target Mode 71.2 Hz1.8 Hz3.2 Hz3.5 Hz14
Test Analysis Models – Modal usingCondition Number Sensor Placment{q} [φ φ] [φ ]{x } 1Ta aModal coordinates in termsof the sensor DOFTaaSolution is more sensitive if the conditionnumber of φa , is large.Begin with a visualization set, and add sensors that minimizethe condition number of aφ15
Test Analysis Correlation7,146 DOFFEMfailsUpdate FEMTAM/TestCorrelationpassesFEMvalidated108 DOFTAMSelect sensor locationsand reduce FEM108 SensorTestfailsTAM/FEMCorrelationpassesOrbital Sciences16
Correlation Metrics Orthogonality Criteria: 0 off diagonal term 0.1O [φ FEM ]Cross OrthogonalityT [] M TAM [φ FEM ]Criteria: 0 off diagonal term 0.10.95 diagonal term 1.0CO [φ FEM ]T [] M TAM [φTAM ]Frequency Comparison Criteria:f errorf FEM fTAM *100 3%f FEM17
TAM-FEM CorrelationMax off diagonal term: 0.05Max off diagonal term: 6e-4*Modal TAM always produces perfect orthogonality for TAM-FEM correlation18
Test Analysis Correlation7,146 DOFFEMfailsUpdate FEMTAM/TestCorrelationpassesFEMvalidated108 DOFTAMSelect sensor locationsand reduce FEM108 SensorTestfailsTAM/FEMCorrelationpassesOrbital Sciences19
Noise Model and Simulated Test ModeShapes{φ }iFEM φ1i i Max value* 2% * {U } {φ }noise φ2 # φ i n 1 φni {φ } {φ }noise {φ }column vector of uniformly distributed{U } random numbers between -1 and 1iTestiFEMFEM targetmode 15Test targetmode 1520
Noise Model FEM assumed to be perfectNoise vector models the net effect of allerrors that cause the FEM mode shapesto disagree with the test mode shapes. Noise contaminatedmode shapeNoise Distribution: Uniform – noassumption is made about the distribution ofnoiseNoise Amplitude: Sensors with the smallestmotion have the largest noise to signal ratioNoise is small on average: 2% at sensorlocations with the largest motion.21
TAM-Test Correlation Results(1 case of Random Noise)Max off diagonal term: 0.27Max off diagonal term: 0.7922
TAM-Test Correlation Results(1 case of Random Noise)Max off diagonal term: 0.05Max off diagonal term: 0.0423
Monte Carlo Simulation Thus far, TAM-Test correlation has beenstudied using only one noise profile Random noise added in 10,000 iterations Orthogonality computed for each iteration Maximum off-diagonal term of orthogonalitywas stored24
TAM-Test Correlation ResultsPass FailDespite its low o-set frequencies, Modal TAMdoes not show high sensitivity!25
TAM-Test Correlation Results If Orthogonality 0.1 one might Refine FEM before exiting testRepeat test and/or look for errorsUpdate the FEMIn this case, the FEM was perfect (errors in testmodes were purely random)Note: The specific ranking of different TAMmethods may depend on: The structure of interestThe characteristics of the noiseSystematic errors between the test and FEM26
TAM-Test Correlation ResultsPass Fail Sensor selection is critical to theperformance of each TAM Most previous studies used the same sensorset, usually optimized for the Static TAM27
Predicting Standard Deviation Recently, we have developed formulas to analytically predictsensitivity of a TAM based on simple metricsFor example, for the noise model used in this study:Oij [φi ni ] M TAM [φi ni ]ni noiseTσ (Oij ) (mM TAM φ j)σi (22mmφi M TAMT)σ22j mm(M TAMn)2σ iσ jmnMaximum OrthogonalityOff-DiagonalPredicted STDActual STDStatic0.030.03Modal EFI0.0090.01Modal C#TAM0.0060.006Modal withStatic DOF0.020.0228
Conclusions and Future Work Conclusions IRS TAM was ill-conditioned, as predicted by GordisModal TAM did not show high sensitivity even though itso-set frequencies were near those of the target modesProbabilistic analysis more fully explains TAM sensitivity One can even predict the sensitivity of the TAMs analyticallygiven the TAM Mass matrix, mode shapes and noise model.Future Work Develop more accurate noise modelsStudy the effect of systematic mismatch between FEM andtest due to modeling errors. May need the Hybrid TAM in these casesApply these methods to other physical systems,analytically and experimentally. Investigate systems with non-consecutive target modes29
6 Relevant Literature Gordis, JH (1992), Blelloch, P and Vold, H (2005) : Notes ill-conditioning in dynamic reduction equation: Proposes that IRS TAM will be ill conditioned if the natural frequencies of the structure with the o-set DOF pinned are similar to the frequencies of the structure of interest. Rece
Tests for Paired Sensitivities Introduction This procedure gives power or required sample size for comparing two diagnostic test s when the outcome is sensitivity. In the design covered in this chapter, the sensitivities of two diagnostic tests are each performed on the same subject.
Several numerical techniques have been developed for re-gional transport models to evaluate source sensitivities and apportionments (Dunker et al., 2002; Cohan and Napelenok, 2011). A simple technique for evaluating source sensitivities is the brute-force method (BFM). Differences in the simu-lated pollutant concentrations between two .
The calculation of these sensitivities involve two terms: analysis terms that are encountered during the usual nite element analysis and isogeometric analy-sis and geometric sensitivities that are represented as the derivatives of positions and weights over design variables. Mesh re nement is often required for accu-rate nite element analysis.
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non-Bayesian approach, called Additive Regularization of Topic Models. ARTM is free of redundant probabilistic assumptions and provides a simple inference for many combined and multi-objective topic models. Keywords: Probabilistic topic modeling · Regularization of ill-posed inverse problems · Stochastic matrix factorization · Probabilistic .
A Model for Uncertainties Data is probabilistic Queries formulated in a standard language Answers are annotated with probabilities This talk: Probabilistic Databases 9. 10 Probabilistic databases: Long History Cavallo&Pitarelli:1987 Barbara,Garcia-Molina, Porter:1992 Lakshmanan,Leone,Ross&Subrahmanian:1997
characteristics of fracture response and reliability of welded structures. Hence, a probabilistic study on weld cracks and an investigation on how the weld and material parameters a ect structural reliability is timely and exciting. This paper presents a probabilistic methodology for fracture-mechanics analysis of weld cracks in a CCT specimen.
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