HAL Id: hal-00686194
https://hal-upec-upem.archives-ouvertes.fr/hal-00686194
Submitted on 8 Apr 2012
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Data and model uncertainties in complex aerospace engineering systems
M. Pellissitti, Evangéline Capiez-Lernout, H. Pradlwarter, G.I. Schueller, Christian Soize
To cite this version:
M. Pellissitti, Evangéline Capiez-Lernout, H. Pradlwarter, G.I. Schueller, Christian Soize. Data and
model uncertainties in complex aerospace engineering systems. 6th International Conference on Struc-
tural Dynamics, Université de Marne-la-Vallée, Sep 2005, Paris, France. pp.Pages: 677-682. �hal-
00686194�
$ATA AND MODEL UNCERTAINTIES IN COMPLEX AEROSPACE ENGINEERING SYSTEMS
- 0ELLISSETTI % #APIEZ,ERNOUT ( 0RADLWARTER ') 3CHUELLER # 3OIZE
!"342!#4 4HE DYNAMICAL ANALYSIS OF COMPLEX MECHANICAL SYSTEMS IS IN GENERAL VERY SENSITIVE TO RANDOM UNCERTAINTIES )N ORDER TO TREAT THE LATTER IN A RATIONAL WAY AND TO INCREASE THE ROBUSTNESS OF THE DYNAMICAL PRE DICTIONS THE RANDOM UNCERTAINTIES CAN BE REPRESENTED BY PROBABILISTIC MODELS 4HE STRUCTURAL COMPLEXITY OF THE DYNAMICAL SYSTEMS ARISING IN THESE lELDS RESULTS IN LARGE lNITE ELEMENT MODELS WITH SIGNIlCANT RANDOM UNCER TAINTIES 0ARAMETRIC PROBABILISTIC MODELS CAPTURE THE UNCERTAINTY IN THE PARAMETERS OF THE NUMERICAL MODEL OF THE STRUCTURE WHICH ARE OFTEN DIRECTLY RELATED TO PHYSICAL PARAMETERS IN THE ACTUAL STRUCTURE EG 9OUNGS MODU LUS -ODEL UNCERTAINTIES WOULD HAVE TO BE MODELED SEPARATELY /N THE OTHER HAND THE PROPOSED NONPARAMETRIC MODEL OF RANDOM UNCERTAINTIES REPRESENTS A GLOBAL PROBABILISTIC APPROACH WHICH IN ADDITION TAKES DIRECTLY INTO ACCOUNT MODEL UNCERTAINTY SUCH AS THAT RELATED TO THE CHOICE OF A PARTICULAR TYPE OF lNITE ELEMENT 4HE UNCERTAIN PARAMETERS OF THE STRUCTURE ARE NOT MODELED DIRECTLY BY RANDOM VARIABLES RVS INSTEAD THE PROBABILITY MODEL IS DIRECTLY INTRODUCED FROM THE GENERALIZED MATRICES OF A MEAN REDUCED MATRIX MODEL OF THE STRUCTURE BY USING THE MAXIMUM ENTROPY PRINCIPLE 3OIZE )N THIS FORMULATION THE GLOBAL SCATTER OF EACH RANDOM MATRIX IS CONTROLLED BY ONE REAL POSITIVE SCALAR CALLED DISPERSION PARAMETER
!N EXAMPLE PROBLEM FROM AEROSPACE ENGINEERING SPECIlCALLY THE &% MODEL OF THE SCIENTIlC SATELLITE ).4%
'2!, OF THE %UROPEAN 3PACE !GENCY %3! !LENIA IS USED TO ELUCIDATE THE TWO APPROACHES &IRST THE ANALYSIS BASED ON THE PARAMETRIC FORMULATION IS CARRIED OUT THE ASSOCIATED RESULTS ARE THEN USED TO CALIBRATE THE DISPERSION PARAMETERS AND TO CONSTRUCT THE REDUCED MATRICES OF THE NONPARAMETRIC MODEL
).42/$5#4)/.
.UMERICAL MODELS HAVE BECOME A VITAL SOURCE OF IN FORMATION FOR THE MANUFACTURERS OF COMPLEX STRUCTURAL SYSTEMS 7ITH THESE MODELS THE DYNAMICAL BEHAVIOR OF THE MANUFACTURED STRUCTURAL SYSTEMS CAN BE PRE DICTED BEFOREHAND )N PRACTICE THE ACCURACY OF EVERY MANUFACTURING PROCESS IS LIMITED #ONSEQUENTLY THE MANUFACTURED SYSTEM IS DIFFERENT FROM THE DESIGNED SYSTEM 4HESE DIFFERENCES CAN HAVE SIGNIlCANT EFFECTS ON THE DYNAMICS OF THE STRUCTURE &OR THIS REASON A DE TERMINISTIC MODEL HEREAFTER REFERRED TO AS THE
IS USUALLY NOT SUFlCIENT FOR A ROBUST PREDICTION OF THE DYNAMIC RESPONSE OF THE STRUCTURE 4HE ROBUST NESS OF THE PREDICTIONS IS HOWEVER AN INDISPENSABLE PREREQUISITE FOR ITS PRACTICAL APPLICATION 4O INCREASE THE ROBUSTNESS OF THE PREDICTIONS THE MEAN MODEL CAN BE EXTENDED TO CONSTRUCT A PROBABILISTIC MODEL )N THIS PAPER TWO PROBABILISTIC APPROACHES FOR MODEL ING RANDOM UNCERTAINTIES ARE CONSIDERED NAMELY THE PARAMETRIC AND THE NONPARAMETRIC APPROACH 4HE PARAMETRIC PROBABILISTIC APPROACH ALLOWS DATA UN
CERTAINTIES TO BE MODELED BY CONSIDERING THE UNCER TAIN PHYSICAL PARAMETERS OF THE MECHANICALNUMERICAL MODEL AS RANDOM QUANTITIES 3UCH UNCERTAIN PARAME TERS ARE THE GEOMETRICAL PARAMETERS THE COMPONENTS OF THE ELASTICITY TENSOR OR THE BOUNDARY CONDITIONS 0ARAMETRIC APPROACHES HAVE BEEN SHOWN TO BE EFl CIENT FOR MODELING DATA UNCERTAINTIES AND ARE WIDELY USED IN COMPUTATIONAL MECHANICS SEE FOR INSTANCE )BRAHIM 3INGH AND ,EE ,IN AND #AI 3CHUELLER %D 3CHUELLER 3CHENK AND 3CHUELLER 7HILE SOME ATTEMPTS HAVE BEEN MADE TO INCORPORATE MODEL UNCERTAINTIES WITH THE PARAMETRIC APPROACH CF EG -ENEZES AND 3CHUELLER -ENEZES AND "RENNER IT IS TYPICALLY FO CUSED TO MODEL THE SCATTER IN THE PARAMETERS OF A GIVEN MODEL
/N THE OTHER HAND THE NONPARAMETRIC APPROACH AIMS TO TAKE INTO ACCOUNT MODEL UNCERTAINTIES TO BEGIN WITH )TS THEORETICAL CONCEPTS HAVE BEEN DEVELOPED IN 3OIZE 3OIZE AND EXPERIMENTAL VALIDATION HAS BEEN CARRIED OUT IN #HEBLI AND 3OIZE )N THE NONPARAMETRIC PROBABILISTIC APPROACH THE GENERALIZED
MATRICES ISSUED FROM A MEAN REDUCED MATRIX MODEL OF THE STRUCTURE ARE REPLACED BY RANDOM GENERALIZED MATRICES 4HE PROBABILISTIC DESCRIPTION OF THESE RAN DOM MATRICES IS CONSTRUCTED BY USING THE MAXIMUM ENTROPY PRINCIPLE UNDER CONSTRAINTS DElNED BY THE AVAILABLE INFORMATION AND YIELDS A NEW CLASS OF RAN DOM MATRICES CALLED THE vPOSITIVE DElNITE ENSEMBLEv 3OIZE A 3OIZE B 7ITH SUCH A FORMULATION THE GLOBAL DISPERSION LEVEL OF EACH RANDOM MATRIX IS CONTROLLED BY A UNIQUE POSITIVE SCALAR WHICH IS CALLED THE DISPERSION PARAMETER
)N THIS PAPER THE PARAMETRIC APPROACH IS USED TO CON STRUCT A REFERENCE SOLUTION OF THE PROBABILISTIC RE SPONSE 4HIS SOLUTION IS USED TO CALIBRATE THE NON PARAMETRIC MODEL FROM WHICH RESPONSE PREDICTIONS ARE DERIVED ! TEST EXAMPLE FROM AEROSPACE ENGINEER ING INVOLVES THE FREQUENCY RESPONSE ANALYSIS OF THE ).4%'2!, SATELLITE OF THE %UROPEAN 3PACE !GENCY 0!2!-%42)# 02/"!"),)34)# !002/!#(
&/2 $9.!-)#!, 3934%- 7)4( 2!.$/- 5.#%24!).4)%3
"ASED ON THE THEORY OF THE LINEAR VISCOELASTICITY WITH OUT MEMORY THE MEAN lNITE ELEMENT MATRIX EQUATION OF THE STRUCTURE IS WRITTEN FOR ALL/IN BAND
/ /
U/ F/ $ IN WHICHU/ ANDF/ ARE THE VECTORS OF THE
$/&S AND OF THE EXTERNAL FORCES RESPECTIVELY )N THE ABOVE EQUATION AND IN THE SEQUEL AN UNDERSCORE DE NOTES MEAN MATRICES AND VECTORS 3INCE THE STRUC TURE HAS A FREE BOUNDARY THE MEAN MASS MATRIX IS A POSITIVEDElNITE SYMMETRIC !! REAL MA TRIX AND THE MEAN DAMPING AND STIFFNESS MATRICES ARE POSITIVE SEMIDElNITE SYMMETRIC!!REAL MATRI CES &URTHERMORE IT IS ASSUMED THAT THE KERNEL OF THE MEAN MATRICESANDIS IDENTICAL CONSTITUTED OF RIGIDBODY MODES WITH DENOTED AS
$ % % % $
,ET8 $ % % % $ BE THE RANDOM VALUED VEC TOR WHOSE COMPONENTS ARE INDEPENDENT 'AUSSIAN RVS AND DESCRIBE MECHANICAL PARAMETERS SUCH AS GEOMET RICAL PARAMETERS OF THE STRUCTURE COEFlCIENTS OF THE ELASTICITY TENSOR MASS DENSITY ETC #LEARLY THE RAN DOMNESS PROPAGATES TO THE lNITE ELEMENT MASS DAMP ING AND STIFFNESS MATRICES 4HE RANDOM lNITE ELEMENT MODEL IS THEN WRITTEN AS
/-PAR /$PAR +PAR
5PAR/ F/ $
IN WHICH5PAR/IS THE VALUED VECTOR OF THE $/&S AND WHERE-PAR 8 AND$PAR 8 +PAR 8ARE THE RANDOM lNITE ELEMENT MASS AND DAMPING STIFFNESS MATRICES WITH VALUES IN THE SET OF THE POSITIVEDElNITE SYMMETRIC!!REAL MA TRICES AND IN THE SET OF THE POSITIVE SEMIDElNITE SYM METRIC!!REAL MATRICES
./.0!2!-%42)# !002/!#( &/2 $9 .!-)#!, 3934%-3 7)4( 2!.$/- 5.
#%24!).4)%3
4HE MAIN IDEA OF THE NONPARAMETRIC APPROACH 3OIZE 3OIZE CONSISTS IN REPLACING THE GENERAL IZED MATRICES OF A MEAN REDUCED MATRIX MODEL OF THE STRUCTURE BY RANDOM MATRICES
3INCE WE ARE INTERESTED IN THE ELASTIC MOTION OF THE STRUCTURE WE THEN INTRODUCE THE!REAL MATRIX WHOSE COLUMNS ARE THE!EIGENVECTORS RELATED TO THESTRICTLY POSITIVE LOWEST EIGENFREQUEN CIES ( / 4HE MEAN REDUCED MATRIX MODEL IS WRITTEN AS
U/ Q/ $
IN WHICHQ/IS THE VECTOR OF THE GENERALIZED CO ORDINATES SOLUTION OF THE MEAN REDUCED EQUATION /RED /RED RED
Q/ / $ IN WHICH / F/ IS THE VECTOR OF THE GENERALIZED FORCES AND WHERE THE MEAN RE DUCED MASS DAMPING AND STIFFNESS MATRICESRED RED AND RED ARE POSITIVEDElNITE SYMMETRIC REAL MATRICES
&OR LINEAR ELASTODYNAMICS IN THE LOW FREQUENCY RANGE THE NONPARAMETRIC MODEL OF RANDOM UNCERTAINTIES YIELDS THE RANDOM MATRIX EQUATION 3OIZE 3OIZE /-NPARRED /$NPARRED +NPARRED
1/ / $ IN WHICH -NPARRED $NPARRED AND +NPARRED ARE POSITIVE DElNITE SYMMETRICREALVALUED MATRICES COR RESPONDING TO THE RANDOM REDUCED MASS DAMPING AND STIFFNESS MATRICES AND WHERE1/IS THE VALUED RANDOM VECTOR OF THE RANDOM GENERALIZED COORDINATES 4HE VALUED RANDOM VECTOR5NPAR/IS THUS RECON STRUCTED BY
5NPAR/ 1/ %
4HE NONPARAMETRIC PROBABILISTIC APPROACH REQUIRES THE NORMALIZATION OF THE MEAN REDUCED MATRICES SUCH THATRED RED AND RED IN WHICHAND ARE DIAGONALREAL MATRICES %ACH RANDOM MA TRIX IS WRITTEN AS
-NPARRED '
$NPARRED '
+NPARRED ' %
4HE PROBABILITY DISTRIBUTION OF RANDOM MATRICES' ' AND ' IS DERIVED FROM THE MAXIMUM EN TROPY PRINCIPLE ISSUED FROM THE INFORMATION THEORY 3HANNON WITH THE AVAILABLE INFORMATION 3OIZE )T CAN BE SHOWN THAT RANDOM MATRICES' 'AND'ARE INDEPENDENT RVS WHOSE DISPER SION LEVEL CAN BE CONTROLLED BY THE POSITIVE REAL PA RAMETERS,,AND,WHICH ARE INDEPENDENT OF THE DIMENSION
4HE FORMAL PROBABILISTIC DESCRIPTION OF THE RAN DOM POSITIVEDElNITE SYMMETRIC REAL MATRIX'IS DE SCRIBED IN 3OIZE 3OIZE &OR NUMERICAL CALCULATIONS IE -ONTE #ARLO SIMULATION THE FOLLOW ING PROCEDURE HAS BEEN PROPOSED TO GENERATE REALIZA TIONS OF THE RANDOM MATRIX'
' , , $
)N THE ABOVE EQUATION,IS ANUPPER TRI ANGULAR RANDOM MATRIX RESULTING FROM THE #HOLESKY FACTORIZATION SUCH THAT
RVS,$ ARE INDEPENDENT FOR # REALVALUED RV, CAN BE WRIT TEN AS, .IN WHICH., AND WHEREIS A REALVALUED 'AUSSIAN RV WITH ZERO MEAN AND VARIANCE EQUAL TO
FOR POSITIVEVALUED RV , CAN BE WRITTEN AS , .
IN WHICH . IS DE lNED ABOVE AND WHEREIS A POSITIVEVALUED GAMMA RANDOM VARIABLE WHOSE PROBABILITY DENSITY FUNCTION WITH RESPECT TOIS WRITTEN AS
) $ WHERE)
)$%.4)&)#!4)/. /& 4(% $)30%23)/. 0!
2!-%4%23 &/2 4(% ./.0!2!-%42)#
!002/!#(
)N THIS SECTION THE IDENTIlCATION OF THE DISPERSION PA RAMETERS , , AND , IS OUTLINED BASED ON THE
PARAMETRIC PROBABILISTIC MODEL ,ETPAR ANDNPAR BE THE NON ZERO lRST RANDOM EIGENVALUES RELATED TO THE REAL RANDOM GENERALIZED EIGENVALUE OBTAINED WITH THE PARAMETRIC AND THE NONPARAMETRIC MODEL OF RANDOM UNCERTAINTIES 4HE PROBABILITY DENSITY FUNCTIONS OF RANDOM EIGENVALUESPAR ANDNPAR DENOTED ASPAR
(
ANDNPAR (ARE THEN COMPARED IN THE LEAST SQUARE SENSE 4HE FUNCTION,$ ,IS INTRODUCED
,$ ,
NPAR
,$ ,PAR
PAR
$
IN WHICH THE NORMIS GIVEN BY
XX
%
4HE IDENTIlCATION IS THEN CARRIED OUT SUCH THAT PARAM ETERS,AND,ARE SOLUTION OF THE EQUATION
,$ , %
$ISPERSION PARAMETER ,IS IDENTIlED SEPARATELY BY USING THE IDENTIlCATION METHOD PROPOSED IN #APIEZ ,ERNOUT AND 3OIZE ,ET$PARREDBE THE RANDOM RE DUCED DISSIPATION MATRIX FROM THE PARAMETRIC PROBA BILISTIC MODEL 4HE DISPERSION PARAMETER,IS WRITTEN AS
,
"PAR
TRREDTRRED $ WHERE"PAR $PARREDRED$
AND
-%4(/$/,/'9 /& 2%3/,54)/. &/2
!.!,9:).' 4(% 2!.$/- 2%30/.3%
! STOCHASTIC CONVERGENCE ANALYSIS ALLOWS TO SPECIFY THE NUMBEROF MODES RELATED TO THE MEAN lNITE EL EMENT MODEL OF THE SATELLITE TO BE KEPT AND TO SPECIFY THE NUMBEROF REALIZATIONS USED IN THE -ONTE #ARLO NUMERICAL SIMULATION 4HE CONVERGENCE IS MONITORED BY DElNING THE FOLLOWING SEQUENCE5NPARSUCH THAT
5NPAR
5NPAR//
$ IN WHICH5NPAR/DENOTES THE (ERMITIAN NORM OF RANDOM VECTOR5NPAR/ 4HE NORM5NPARIS ESTI MATED WITH THE FUNCTION$ $ SUCH THAT
$
NPAR/$ -/ %
4HE CONlDENCE REGIONS OF THE RANDOM RESPONSE RELATED TO A GIVEN PROBABILITY LEVEL ) ARE CON SIDERED 4HE EXPONENTS NPAR OR PAR ARE OMITTED IN THIS SECTION SINCE THE EXPRESSIONS APPLY IN GEN ERAL ,ET OBS BE THE OBSERVATION NODE ,ET UOBS/ BE THE DETERMINISTIC VECTOR AND LET 5OBS/ BE THE RANDOM VECTOR RELATED TO THE THREE TRANSLATIONAL
$/&S OF NODE OBS 7E THEN INTRODUCE THE SCALAR OBS/ UOBS/AND THE RANDOM VARI ABLE "OBS/ 5OBS/ 4HE MEAN VALUE"OBS/OF THE RANDOM RESPONSE IS INTRODUCED SUCH THAT
"OBS/ 5OBS/ % ,ET/lXED IN 4HE QUANTILE FUNCTIONOBS)/
OF RANDOM VARIABLE"OBS/IS DElNED SUCH THAT OBS)/
OBS/) $
IN WHICHOBS/IS THE CUMULATIVE DENSITY FUNC TION OF RANDOM VARIABLE"OBS/ ,ET"OBS-/ #
% % % # "OBS-/ BE THE ORDERED STATISTIC ASSOCI ATED WITH "OBS-/ # % % % # "OBS-/ 4HE UNBIASED ESTIMATION OF CUMULATIVE DENSITY FUNCTION OBS/IS DElNED AS
OBS/
"OBS-/ $ IN WHICH IS SUCH THAT IF AND IF NOT ,ETOBS ANDOBS BE THE) QUANTILE AND THE)QUANTILE 5SING %QS AND YIELDS
OBS/ "OBS-/$ lXNS)$ OBS/ "OBS-/$ lXNS)$ IN WHICH lXXIS THE INTEGER PART OF REAL
.5-%2)#!, %8!-0,%
4HE MEAN lNITE ELEMENT MODEL OF THE SATELLITE IS A THREE DIMENSIONAL MESH WITH $/&S SEE &IG URE ,ETEXCBE THE XTRANSLATIONAL $/& OF THE SATEL LITE STRUCTURE SUBJECTED TO A DETERMINISTIC LOAD 4HE FORCE VECTOR IS THEN WRITTEN AS F/ /G IN WHICH THE VECTOR G $ % % % $ IS SUCH THAT ,EXC$ $ % % % $ ! WHERE THE CONCENTRATED
&IGURE &INITE ELEMENT MODEL OF THE SATELLITE PROVIDED BY
%3!%34%#
MASS % AND WHERE/IS A PRE SCRIBED ACCELERATION SUCH THAT / %!%
IF / #%AND/ %!%IF /
% -EAN REDUCED DAMPING MATRIXREDIS SUCH THATRED
(', IN WHICH'IS THE MODAL DAMPING RATIO RELATED TO EIGENMODE SUCH THAT ' % IF /#%% AND ' % IF/%%
4HE UNCERTAIN PARAMETERS OF THE SATELLITE ARE MODELED BY INDEPENDENT 'AUSSIAN RVS WITH COEFlCIENTS OF VARIATION BETWEEN AND 4HE RANDOMNESS OF THE DISSIPATION IS INTRODUCED FROM THE MEAN RE DUCED DAMPING MATRIXRED -ODAL DAMPING RATIOS ARE MODELED BY INDEPENDENT RVS 2ANDOM MATRIX
$PARREDIS SUCH THAT$PARRED
PAR , IN WHICH IS THE RANDOM MODAL DAMPING RATIO WHOSE PROBA BILITY DISTRIBUTION IS ,OGNORMAL WITH OF STANDARD DEVIATION AROUND ITS MEAN VALUE'
4HE PROBABILITY DISTRIBUTION(PAR
(IS ESTIMATED WITH REALIZATIONS CF &IGURE 4HE MINI MIZATION OF FUNCTION,$ ,YIELDS, % AND , % 4HE APPROXIMATE PDFS FROM THE TWO APPROACHES MATCH REASONABLY WELL &IGURE
&IGURE )DENTIl CATION OF DISPERSION PARAMETERS AND 'RAPH OF THE PROBABILITY DISTRIBUTIONSPARFOR THE PARA METRIC APPROACH THIN LINE ANDNPAR FOR THE NONPARA METRIC APPROACH THICK LINE WITH DISPERSION PARAMETERS SUCH THAT
AND
SHOWS THE ESTIMATION OF,WITH RESPECT TO THE NUM BEROF REALIZATIONS USED FOR THE -ONTE #ARLO SIMU LATION )T IS SEEN THAT A GOOD CONVERGENCE IS OBSERVED FOR AND YIELDS, %
&IGURE )DENTIl CATION OF DISPERSION PARAMETER 'RAPH OF
! -ONTE#ARLO SIMULATION HAS BEEN PERFORMED WITH REALIZATIONS AND OBSERVING THE FUNCTION $ FOR AND 4HE CONVERGENCE ANALYSIS SHOWED THAT WITH AND THE RESULTING APPROXIMATION IS ADEQUATE
4O SIMPLIFY THE NOTATIONS INDICIAL EXPONENTSNPARAND
PAR ARE OMITTED 7E CONSIDER THE RANDOM RESPONSE OF THE FREE SATELLITE AT NODE OBS SEE lGURE IN LOW FREQUENCY BAND 4HE NUMERICAL CALCULATIONS ARE CAR RIED OUT WITH AND &IGURES
&IGURE #ONl DENCE REGION OF RANDOM DISPLACEMENT RELATED TO NODEOBSIN D" OVER A LOWFREQUENCY BAND AND OBTAINED WITH THENONPARAMETRICPROBABILISTIC APPROACH DETERMINISTIC RESPONSE OF THE MEAN MODEL THICK DASHEDDOTTED LINE MEAN OF THE RANDOM RESPONSE FOR THE STOCHASTIC MODEL THIN DOTTED LINE LOWER AND UPPER ENVELOPES OF THE CONl DENCE RE GION CORRESPONDING TO A PROBABILITY LEVEL EQUAL TODARK GRAY l LLED ZONE
&IGURE #ONl DENCE REGION OF RANDOM DISPLACEMENT RELATED TO NODEOBSIN D" OVER A LOWFREQUENCY BAND AND OBTAINED WITH THEPARAMETRICPROBABILISTIC APPROACH DETER MINISTIC RESPONSE OF THE MEAN MODEL THICK DASHEDDOTTED LINE MEAN OF THE RANDOM RESPONSE FOR THE STOCHASTIC MODEL MID THIN DOTTED LINE LOWER AND UPPER ENVELOPES OF THE CONl DENCE RE GION CORRESPONDING TO A PROBABILITY LEVEL EQUAL TODARK GRAY l LLED ZONE
AND DISPLAY THE GRAPHS RELATED TO THE CONlDENCE RE GION OF THE RANDOM DISPLACEMENTS OF NODE OBS OB TAINED FOR A PROBABILITY LEVEL EQUAL TO %AND CON
STRUCTED WITH THE QUANTILE METHOD 4HE THICK DASHED DOTTED LINE SHOWS THE GRAPH* OBS* IN WHICH
* /&+ 4HE THIN DOTTED LINE CORRESPONDS TO
*"OBS* 4HE CONlDENCE REGION CORRESPONDS TO THE GRAY lLLED ZONE WHOSE ENVELOPES ARE DELIMITED BY THE MAPPINGS*OBS*AND*OBS* &IGURE REFERS TO THE NONPARAMETRIC lGURE TO THE PARAMET RIC APPROACH #LEARLY THE RESPECTIVE CONlDENCE RE GIONS MAY BE COMPARED FOR FREQUENCIES LOWER THAN WHICH JUSTIlES THE RELEVANCE OF THE IDENTIlCA TION PROCEDURE OF THE DISPERSION PARAMETERS "UT lG URE SHOWS THAT FOR FREQUENCIES GREATER THAN THE MEAN OF THE RANDOM RESPONSE OBTAINED WITH THE NONPARAMETRIC APPROACH IS VERY DIFFERENT FROM THE RE SPONSE OF THE MEAN MODEL &URTHERMORE THERE EXIST FREQUENCIES FOR WHICH THE RESPONSE OF THE MEAN MODEL IS OUTSIDE FROM THE CONlDENCE REGION /N THE CON TRARY THIS PHENOMENON IS NOT PRESENT FOR THE RANDOM RESPONSE OBTAINED WITH THE PARAMETRIC PROBABILISTIC MODEL DESCRIBED BY lGURE 4HESE DIFFERENCES ARE EXPLAINED BY THE ABILITY OF THE NONPARAMETRIC PROB ABILISTIC MODEL TO REPRESENT MODEL UNCERTAINTIES
#/.#,53)/.
!LTHOUGH THE DISPERSION LEVEL RELATED TO THE RANDOM UNCERTAINTIES OF THE SATELLITE IS THE SAME FOR BOTH PROB ABILISTIC APPROACHES THE RANDOM FORCED RESPONSES DO NOT LOOK SIMILAR 3INCE THE TWO APPROACHES FOCUS ON DIFFERENT FACETS OF THE PROBLEM A DIRECT COMPARI SON IS NOT MEANINGFUL 7ITHIN THE FREQUENCY BAND
$ FOR WHICH THE MODEL UNCERTAINTIES ARE VERY SMALL THE DATA UNCERTAINTIES BEING PREPONDER ANT BOTH PARAMETRIC AND NONPARAMETRIC APPROACHES WHICH ALLOWS DATA UNCERTAINTIES TO BE MODELED YIELD SIMILAR RESULTS SEE &IGURESAND &OR HIGHER FRE QUENCIES HOWEVER lGURES SHOWS THAT THE CON lDENCE REGIONS CANNOT BE COMPARED 4HE PARAMETRIC APPROACH AS APPLIED TO THIS PROBLEM MODELS DATA UN CERTAINTIES WHILE THE NONPARAMETRIC APPROACH MOD ELS BOTH DATA AND MODEL UNCERTAINTIES 7ITHIN THE FRE QUENCY BAND $ FOR THE FREE SATELLITE THE RESULTS SHOW THAT THE SATELLITE STRUCTURE IS MORE SENSI TIVE TO MODEL UNCERTAINTIES THAN TO DATA UNCERTAINTIES
!#+./7,%$'-%.43
4HIS STUDY WAS PARTIALLY SUPPORTED BY THE h!-!$%53v 0ROJECT .R !USTRIA ,!- &RANCE WHICH IS GRATEFULLY ACKNOWLEDGED 4HE AUTHORS THANK %3!%34%# FOR PROVIDING THE ).4%'2!, SATELLITE lNITE ELEMENT MODELS
2%&%2%.#%3
!LENIA 4ECHNICAL NOTE )NTEGRAL STRUCTURAL MATHEMAT ICAL MODEL DESCRIPTION AND DYNAMIC ANALYSIS RESULTS INTTNAL 4ECHNICAL 2EPORT ND EDITION !LENIA
!EROSPAZIO 3PACE $IVISION 4URIN )TALY
#APIEZ,ERNOUT % 3OIZE # .ONPARAMETRIC MODEL ING OF RANDOM UNCERTAINTIES FOR DYNAMIC RESPONSE OF MIS TUNED BLADEDDISKS
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#HEBLI ( 3OIZE # %XPERIMENTAL VALIDATION OF A NONPARAMETRIC PROBABILISTIC MODEL OF NONHOMOGE NEOUS UNCERTAINTIES FOR DYNAMICAL SYSTEMS
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3INGH 2 ,EE # &REQUENCY RESPONSE OF LINEAR SYS TEMS WITH PARAMETER UNCERTAINTIES
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3OIZE # ! NONPARAMETRIC MODEL OF RANDOM UNCER TAINTIES FOR REDUCED MATRIX MODELS IN STRUCTURAL DYNAM
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3OIZE # -AXIMUM ENTROPY APPROACH FOR MODELING RANDOM UNCERTAINTIES IN TRANSIENT ELASTODYNAMICS
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3OIZE # A ! COMPREHENSIVE OVERVIEW OF A NON PARAMETRIC PROBABILISTIC APPROACH OF MODEL UNCERTAINTIES FOR PREDICTIVE MODELS IN STRUCTURAL DYNAMICS IN PRESS
3OIZE # B 2ANDOM MATRIX THEORY FOR MODELING UNCER TAINTIES IN COMPUTATIONAL MECHANICS
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