Bayesian Uncertainty Quantification and Propagation in Nonlinear Structural Dynamics

Transcription

1 Bayesan Uncertanty Quantfcaton and Propagaton n onlnear Structural Dynamcs Dmtros Gagopoulos, Dmtra-Chrstna Papadot 2, Costas Papadmtrou 2 and Sotros atsavas 3 Dept. of Mechancal Engneerng, Unversty of Western Macedona, Kozan, Greece 2 Dept. of Mechancal Engneerng, Unversty of Thessaly, Volos, Greece 3 Dept. of Mechancal Engneerng, Arstotle Unversty, Thessalon, Thessalon, Greece ABSTRACT A Bayesan uncertanty quantfcaton and propagaton (UQ&P) framewor s presented for dentfyng nonlnear models of dynamc systems usng vbraton measurements of ther components. The measurements are taen to be ether response tme hstores or frequency response functons of lnear and nonlnear components of the system. For such nonlnear models, stochastc smulaton algorthms are sutable Bayesan tools to be used for dentfyng system and uncertanty models as well as perform robust predcton analyses. The UQ&P framewor s appled to a small scale expermental model of a vehcle wth nonlnear wheel and suspenson components. Uncertanty models of the nonlnear wheel and suspenson components are dentfed usng the expermentally obtaned response spectra for each of the components tested separately. These uncertantes, ntegrated wth uncertantes n the body of the expermental vehcle, are propagated to estmate the uncertantes of output quanttes of nterest for the combned wheel-suspenson-frame system. The computatonal challenges are outlned and the effectveness of the Bayesan UQ&P framewor on the specfc example structure s demonstrated. Keywords: System Identfcaton, Uncertanty Identfcaton, Bayesan Inference, onlnear Dynamcs, Substructurng.. Introducton Structural model updatng methods (e.g. []) are used to reconcle lnear and nonlnear mathematcal models of a mechancal system wth avalable expermental data from component or system tests. For complex structural dynamcs models t s often the case that these models are usually dscretzed lnear fnte element (FE) models for a large part of the structure wth localzed nonlneartes n solated structural parts. Examples nclude vehcle models that consst of lnear structural components, such as the body of the vehcle, and nonlnear structural components, such as the suspenson and the wheel components, whch may exhbt strongly nonlnear behavour. To buld hgh fdelty models for such complex structures wth localzed nonlneartes, one should reconcle lnear and nonlnear structural models wth expermental data avalable at both the component and system level. Ths wor presents the challenges of Bayesan uncertanty quantfcaton and propagaton of complex nonlnear structural dynamcs models and results of dentfcaton of nonlnear models of a small scale expermental vehcle consstng of lnear and nonlnear components. The goal s to buld hgh fdelty models of the components to smulate the behavour of the combned system. Bayesan technques [2-3] have been proposed to quantfy the uncertanty n the parameters of a structural model, select the best model class from a famly of compettve model classes [4-5], as well as propagate uncertantes for robust response and relablty predctons [6]. Posteror probablty densty functons (PDFs) are derved that quantfy the uncertanty n the model parameters based on the data. For nonlnear structural models, the measurements are taen to be ether response tme hstores or frequency response functons of nonlnear systems. Computatonally ntensve stochastc smulaton algorthms (e.g., Transtonal MCMC [7]) are sutable tools for dentfyng system and uncertanty models as well as for performng robust predcton analyses. These algorthms requre a large number of system analyses to be performed over the space of uncertan parameters. However, for relatvely large order FE models nvolvng hundreds of thousands or even mllon degrees of freedom and localzed nonlnear actons actvated durng system operaton, such re-analyses may requre excessve

2 computatonal tme. Methods for drastcally reducng the computatonal demands at the system, algorthm and hardware levels nvolved n the mplementaton of Bayesan framewor have recently been developed. At the system level, effcent computng technques can be ntegrated wth the Bayesan framewor to handle large order models and localzed nonlnear acton. Specfcally, component mode synthess and multlevel substructurng technques can acheve substantal reductons n computatonal effort. At the system level, effcent computng technques are ntegrated wth Bayesan technques to effcently handle large order models of hundreds of thousands or mllons degrees of freedom (DOF) and localzed nonlnear actons actvated durng system operaton. Specfcally, fast and accurate component mode synthess (CMS) technques have recently been proposed [8], consstent wth the FE model parameterzaton, to acheve drastc reductons n computatonal effort. In addton, automated multlevel substructurng technques [9] are used to acheve substantal reductons n computatonal effort n the re-analyss of lnear substructures. At the level of the Transtonal MCMC (TMCMC) algorthm, surrogate models are adopted to drastcally reduce the number of computatonally expensve full model runs [0]. At the computer hardware level, parallel computng algorthms are proposed to effcently dstrbute the computatons n avalable mult-core CPUs [0]. In ths wor the Bayesan UQ&P framewor s appled to dentfy models of lnear and nonlnear components of a small scale expermental model of a vehcle. The dentfcaton of the uncertanty models of the nonlnear wheel and suspenson components s nvestgated usng the expermentally obtaned response spectra. The uncertanty models for the vehcle frame are also obtaned usng expermental data. The uncertanty s propagated to output quanttes of nterest for the combned wheel-suspenson-frame system. The computatonal challenges and effcency of the Bayesan UQ&P framewor are outlned. The effectveness of the framewor on the specfc example structure s dscussed. 2. Revew of Bayesan Formulaton for Parameter Estmaton and Model Class Selecton = Î = } Consder a parameterzed FE model class Μ of a nonlnear structure and let qm Î R be the structural model parameters to q be estmated usng a set of measured response quanttes. In nonlnear structural dynamcs, the measured quanttes may consst of full response tme hstores D { yˆ 0 R,,, at DOF and at dfferent tme nstants t t, where s 0 the tme ndex and s the number of sampled data wth samplng perod t, or response spectra 0 D = { yˆ Î R, =,,} at dfferent frequences, where s a frequency doman ndex. In addton, let { y ( q 0 ) Î R, =,, } be the model response predctons (response tme hstores or response spectra), correspondng m to the DOFs where measurements are avalable, gven the model class Μ and the parameter set the observaton data and the model predctons satsfy the predcton error equaton q Î R m q. It s assumed that yˆ = y ( q Μ ) + e m m () where the error term e ~ ( m, S( q e )) s a Gaussan vector wth mean zero and covarance S ( ). It s assumed that the error terms e,,, are ndependent. Ths assumpton may be reasonable for the case where the measured quanttes are the response spectra. However, for measured response tme hstores ths assumpton s expected to be volated for small samplng perods. The effect of correlaton n the predcton error models s not consdered n ths study. The notaton S ( q e ) s used to denote that a model s postulated for the predcton error covarance matrx that depends on the parameter set q. e Bayesan methods are used to quantfy the uncertanty n the model parameters as well as select the most probable FE model class among a famly of compettve model classes based on the measured data. The structural model class Μ s augmented to nclude the predcton error model class that postulates zero-mean Gaussan models. As a result, the parameter set s augmented to nclude the predcton error parameters e. Usng PDFs to quantfy uncertanty and followng the Bayesan formulaton (e.g. [2-3, ]), the posteror PDF p( D, Μ ) of the structural model and the predcton error parameters (, ) gven the data D and the model class Μ can be obtaned n the form m e [ pd ( Μ)] p( D, Μ) exp J( ) ( Μ) /2 2 (2) 0 2 det ( ) e q e

3 where m T - T ( q) = [ ( q )- ˆ] S ( q) [ ( q )- ˆ] m e m J å y y y y (3) r= s the weghted measure of ft between the measured and model predcted quanttes, ( Μ ) s the pror PDF of the model parameters and pd ( Μ) s the evdence of the model class Μ. For a large enough number of expermental data, and assumng for smplcty a sngle domnant most probable model, the posteror dstrbuton of the model parameters can be asymptotcally approxmated by the mult-dmensonal Gaussan dstrbuton [2, ] centered at the most probable value ˆq of the model parameters that mnmzes the functon g( ; Μ) ln p( D, Μ) wth covarance equal to the nverse of the Hessan h( ) of the functon g( ; Μ ) evaluated at the most probable value. For a unform pror dstrbuton, the most probable value of the FE model parameters concdes wth the estmate obtaned by mnmzng the weghted resduals n (3). An asymptotc approxmaton based on Laplace s method s also avalable to gve an estmate of the model evdence pd ( Μ) []. The estmate s also based on the most probable value of the model parameters and the value of the Hessan h( q ) evaluated at the most probable value. The Bayesan probablstc framewor s also used to compare two or more competng model classes and select the optmal model class based on the avalable data. Consder a famly Μ ={Μ,,, }, of alternatve, competng, q parameterzed FE and predcton error model classes and let q Î R be the free parameters of the model class Μ. The posteror probabltes P( Μ D) of the varous model classes gven the data D s [4] P( Μ D) = pd ( Μ ) P( Μ ) pd ( Μ ) Fam (4) where P( Μ ) s the pror probablty and pd ( Μ ) s the evdence of the model class Μ. The optmal model class s selected as the one that maxmzes P ( M D) gven by (4). For the case where no pror nformaton s avalable, the pror probabltes are assumed to be P ( M ) = / m, so the model class selecton s based solely on the evdence values. The asymptotc approxmatons may fal to gve a good representaton of the posteror PDF n the case of multmodal dstrbutons or for undentfable cases manfested for relatvely large number of model parameters n relaton to the nformaton contaned n the data. For more accurate estmates, one should use SSA to generate samples that populate the posteror PDF n (2). Among the SSA avalable, the TMCMC algorthm [7] s one of the most promsng algorthms for selectng the most probable model class among compettve ones, as well as fndng and populatng wth samples the mportance regon of nterest of the posteror PDF, even n the undentfable cases and mult-modal posteror probablty ( ) dstrbutons. In addton, the TMCMC samples,,, drawn from the posteror dstrbuton can be used to yeld s an estmate of the evdence pd ( Μ ) requred for model class selecton [7, 2-3]. The TMCMC samples can further be used for estmatng the probablty ntegrals encountered n robust predcton of varous performance quanttes of nterest [6]. In partcular, f q( ) s an output quantty of nterest condtonal on the value of the parameter set, the posteror robust measure of q gven the data and tang nto account the uncertanty n s obtaned from the sample estmate M best Μ = å m q () Eq ( D, ) ( ; Μ) q s = (5) where m q () ( ; Μ) s the condtonal mean value of q( ) gven the model class. q

4 3. Applcaton to a Small Scale Laboratory Vehcle Model In order to smulate the response of a ground vehcle an expermental devce was selected and set up [4]. More specfcally, the selected frame structure comprses a frame substructure wth predomnantly lnear response and hgh modal densty plus four supportng substructures wth strongly nonlnear acton. Frst, Fgure a shows a pcture wth an overvew of the expermental set up. In partcular, the mechancal system tested conssts of a frame substructure (parts wth red, gray and blac color), smulatng the frame of a vehcle, supported on four dentcal substructures. These supportng substructures consst of a lower set of dscrete sprng and damper unts, connected to a concentrated (yellow color) mass, smulatng the wheel subsystems, as well as of an upper set of a dscrete sprng and damper unts connected to the frame and smulatng the acton of the vehcle suspenson. Also, Fgure b presents more detals and the geometrcal dmensons of the frame subsystem. Moreover, the measurement ponts ndcated by -4 correspond to connecton ponts between the frame and ts supportng structures, whle the other measurement ponts shown concde wth characterstc ponts of the frame. Fnally, pont E denotes the pont where the electromagnetc shaer s appled. (a) (b) Fg. (a) Expermental set up of the structure tested, (b) Dmensons of the frame substructure and measurement ponts. In order to dentfy the parameters of the four supportng subsystems, whch exhbt strongly nonlnear characterstcs, a seres of tests was performed. To nvestgate ths further, the elements of the supportng unts were dsassembled and tested separately. Frst, Fgure 2a shows a pcture of the expermental setup and presents graphcally the necessary detals of the expermental devce that was set up for measurng the stffness and dampng propertes of the supports, whle Fgure 2b shows the equvalent mechancal model. (a) (b) Fg. 2 (a) Expermental set up for measurng the support stffness and dampng parameters, (b) Equvalent model. The expermental process was appled separately to both the lower and the upper sprng and damper unts of the supportng substructures and can be brefly descrbed as follows. Frst, the system shown n Fgure 2 s excted by harmonc forcng through the electromagnetc shaer up untl t reaches a perodc steady state response. When ths happens, both the hstory of the acceleraton and the forcng sgnals are recorded at each forcng frequency. Some characterstc results obtaned n ths manner are presented n the followng sequence of graphs. ext, Fgure 3a presents the transmssblty functon of the system tested, obtaned expermentally for three dfferent forcng levels. Specfcally, ths functon s defned as the rato of

5 the root mean square value of the acceleraton to the root mean square value of the forcng sgnal measured at each forcng frequency. The contnuous, dashed and dotted lnes correspond to the smallest, ntermedate and largest forcng ampltude, respectvely. Clearly, the devatons observed between the forcng levels ndcate that the system examned possesses nonlnear propertes. Moreover, nether the appled forcng s harmonc, especally wthn the frequency range below 0Hz. To llustrate ths, Fgure 3b shows two perods of the actual exctaton force appled for the same three exctaton levels n obtanng the results of Fgure 3a, whch were recorder at a fundamental forcng frequency of 4Hz. Fg. 3 (a) Transmssblty functon of the support system, for three dfferent forcng levels, (b) Hstory of the external force appled wth a fundamental harmonc frequency 4Hz A number of models of the restorng and dampng forces, say f and f, respectvely, were tred for modelng the acton of r d the supports and compared wth the expermental results. The classc lnear dependence of the restorng force on the dsplacement and of the dampng forces on the velocty of the support unt was frst assumed. However, crtcal comparson wth the expermental results usng the Bayesan model selecton framewor demonstrated that the outcome was unacceptable n terms of accuracy. Eventually t was found that an acceptable form of the restorng forces s the one where they reman vrtually n a lnear relaton wth the extenson of the sprng, namely fr ( x) x (6) whle the dampng force was best approxmated by the followng formula c2 x fd ( x ) c x c x 3 (7) As usual, the lnear term n the last expresson s related to nternal frcton at the support, whle the nonlnear part s related to the exstence and actvaton of dry frcton. More specfcally, n the lmt 0, the second term n the rght hand sde of Equaton (7) represents energy dsspaton acton correspondng to dry frcton. On the other sde, n the lmt term represents classcal vscous acton and can actually be absorbed n the frst term. 4. Results c 3 c 3 The value of the parameters appearng n the assumed models of the restorng and dampng forces of the supports, le the coeffcents, c, c and c n Equatons (6) and (7), are determned by applyng the Bayesan uncertanty quantfcaton 2 3 and calbraton methodology. Results are obtaned based on expermental response spectra values for both the dsplacement and acceleraton of ether the wheel or the suspenson component. It s assumed that the predcton errors n the Bayesan formulaton are uncorrelated wth predcton error varance dag(, ) dag( I, I ), where and 2 I I are the covarance matrces for the predcton errors correspondng to the dsplacements and acceleratons,, ths

6 respectvely. The parameter space s sx dmensonal and ncludes (, c, c, c,, ). Parameter estmaton results are obtaned usng the parallelzed and surrogate-based verson [0] of the TMCMC algorthm [7] wth 500 samples per stage. Eght computer worers were used to perform n parallel the computatons nvolved n the TMCMC algorthm. The computatonal tme requred to run all 5500 samples for the TMCMC stages, wthout surrogate approxmaton, for the SDOF model s approxmately 7 hours. Surrogate modelng [0] reduces further ths tme by approxmately one order of magntude. For llustraton purposes, results for the TMCMC samples projected n the two-dmensonal parameter spaces (, c ) and ( c, c ) are shown n Fgure 4 for the SDOF system, shown n Fgure 2(b), correspondng to the suspenson 2 component. It s clear that the uncertantes n the dampng parameters c and c are relatvely hgh and c and c are hghly 2 2 correlated along certan drectons n the parameter space. Fg. 4 Model parameter uncertanty: projecton of TMCMC samples n the two dmensonal parameter spaces The parameter uncertantes are propagated through the SDOF model to estmate the uncertantes n the dsplacement and acceleraton response spectra. The results are shown n Fgures 5 and 6 for the dsplacement and acceleraton response spectra, respectvely and are compared to the expermental values of the response spectra. An adequate ft s observed. Dscrepances between the model predctons and the expermental measurements are manly due to the model errors related to the selecton of the partcular forms of the restorng force curves n (6) and (7). The Bayesan model selecton strategy based on equaton (5) can be used to select among alternatve restorng force models n an effort to mprove the observed ft. Fg. 5 Uncertanty propagaton: dsplacement response spectra uncertanty along wth comparsons wth the expermental data for the suspenson component. (a) moderate exctaton level, (b) strong exctaton level The above procedure has been repeated for the wheel component to dentfy the uncertantes n the lnear stffness and nonlnear dampng model. In addton, the uncertantes n nne stffness-related parameters of the frame component were also

7 estmated usng the Bayesan methodology and the expermental values the frst ten modal frequences and the mode shape components at 72 locatons of the frame [5]. The lnear fnte element model has DOFs. Due to excessve computatonal cost arsng n stochastc smulaton algorthms, the model was reduced usng a recently developed CMS method for FE model updatng [8]. The reduced model has 30 DOFs, resultng n substantal computatonal savngs of more than two orders of magntude. Due to space lmtatons, results of the parameter estmaton are not shown here. Fg. 6 Uncertanty propagaton: acceleraton response spectra uncertanty along wth comparsons wth the expermental data for the suspenson component. (a) moderate exctaton level, (b) strong exctaton level The estmates of the model parameter values and ther uncertantes for each component are used to buld the model for the combned wheel-suspenson-frame structure. The number of DOFs of the nonlnear model of the combned structure s The parametrc uncertantes are then propagated to uncertantes n the response of the combned structure. The CMS was agan used to reduce the number of DOFs to 34 and thus drastcally reduce the computatonal effort that arses from the re-analyses due to the large number of TMCMC samples and the nonlnearty of the combned system. Selected uncertanty propagaton results are next presented. Specfcally, the parameters of the wheel model and the model of the frame structure are ept to ther mean values and only the uncertantes n the model parameters of the suspenson components are consdered. Such uncertantes are propagated to uncertantes for the acceleraton transmssblty functon at a pont on the wheel, the connecton of the wheel wth the frame and an nternal pont on the frame as shown n Fgure 7. It s observed that a large uncertanty n the response spectra s obtaned at the resonance regon close to 3.4 Hz, whch s domnated by local wheel body deflectons. The response n the resonance regons close to 58 Hz and 68 Hz s manly domnated by deflecton of the frame structure. It s observed that the uncertantes n the suspenson parameters do not sgnfcantly affect the response spectra at the resonance regons. As a result, response spectra obtaned expermentally n these resonance regons for the complete vehcle model are not expected to be adequate to dentfy uncertantes n the parameters of the suspenson model. Fg. 7 Uncertanty propagaton: acceleraton transmssblty functon uncertanty for combned system. (a) wheel DOF, (b) DOF at connecton between suspenson and frame, (c) frame DOF.

8 5. Conclusons A Bayesan UQ&P framewor was presented for dentfyng nonlnear models of dynamc systems usng vbraton measurements of ther components. The use of Bayesan tools, such as stochastc smulaton algorthms (e.g., TMCMC algorthm), may often result n excessve computatonal demands. Drastc reducton n computatonal effort to manageable levels s acheved usng component mode synthess, surrogate models and parallel computng algorthms. The framewor was demonstrated by dentfyng the lnear and nonlnear components of a small-scale laboratory vehcle model usng expermental response spectra avalable separately for each component. Such model uncertanty analyses for each component resulted n buldng a hgh fdelty model for the combned system to be used for performng relable robust response predctons that properly tae nto account model uncertantes. The theoretcal and computatonal developments n ths wor can be used to dentfy and propagate uncertantes n large order nonlnear dynamc systems that consst of a number of lnear and nonlnear components. Acnowledgements Ths research has been co-fnanced by the European Unon (European Socal Fund ESF) and Gree natonal funds through the Operatonal Program Educaton and Lfelong Learnng of the atonal Strategc Reference Framewor (SRF) Research Fundng Program: Heracltus II. Investng n nowledge socety through the European Socal Fund. REFERECES [] Yuen, K.V., Kuo S.C., Bayesan methods for updatng dynamc models, Appled Mechancs Revews 64(), 20. [2] Bec, J.L., Katafygots, L.S., Updatng models and ther uncertantes- I: Bayesan statstcal framewor, ASCE Journal of Engneerng Mechancs 24 (4), , 998. [3] Yuen, K.V., Bayesan Methods for Structural Dynamcs and Cvl Engneerng, John Wley & Sons, 200. [4] Bec, J.L., Yuen, K.V., Model selecton usng response measurements: Bayesan probablstc approach, ASCE Journal of Engneerng Mechancs 30(2), , [5] Yuen, K.V., Recent developments of Bayesan model class selecton and applcatons n cvl engneerng, Structural Safety 32(5), , 200. [6] Papadmtrou, C., Bec, J.L., Katafygots, L.S., Updatng robust relablty usng structural test data, Probablstc Engneerng Mechancs 6, 03-3, 200. [7] Chng J., Chen, Y.C., Transtonal Marov Chan Monte Carlo method for Bayesan updatng, model class selecton, and model averagng, ASCE Journal of Engneerng Mechancs 33, , [8] Papadmtrou, C., Papadot, D.C., Component Mode Synthess Technques for Fnte Element Model Updatng, Computers and Structures, DOI: 0.06/j.compstruc , 202. [9] Papaluopoulos, C., atsavas, S., Dynamcs of Large Scale Mechancal Models usng Mult-level Substructurng, ASME Journal of Computatonal and onlnear Dynamcs, 2, 40-5, [0] Angelopoulos, P., Papadmtrou, C., Koumoutsaos, P., Bayesan Uncertanty Quantfcaton and Propagaton n Molecular Dynamcs Smulatons: A Hgh Performance Computng Framewor, The Journal of Chemcal Physcs, 37(4). DOI: 0.063/ [] Chrstodoulou, K., Papadmtrou, C., Structural dentfcaton based on optmally weghted modal resduals, Mechancal Systems and Sgnal Processng 2, 4-23, [2] Muto, M., Bec, J.L., Bayesan updatng and model class selecton usng stochastc smulaton, Journal of Vbraton and Control 4, 7 34, [3] Bec, J.L., Au, S.K., Bayesan updatng of structural models and relablty usng Marov chan Monte Carlo smulaton, ASCE Journal of Engneerng Mechancs 28(4), , [4] Gagopoulos, D., atsavas, S., Hybrd (umercal-expermental) Modelng of Complex Structures wth Lnear and onlnear Components, onlnear Dynamcs, 47, 93-27, [5] Papadmtrou, C., totsos, E., Gagopoulos, D., atsavas, S., Varablty of Updated Fnte Element Models and ther Predctons Consstent wth Vbraton Measurements, Structural Control and Health Montorng, DOI: 0.002/stc.453, 20.