Optimization Algorithms for System Integration

Transcription

1 Optmzaton Algothms fo System Integaton Costas Papadmtou 1, a and Evaggelos totsos 1,b 1 Unvesty of hessaly, Depatment of Mechancal and Industal Engneeng, Volos 38334, Geece a costasp@uth.g, b entotso@uth.g Keywods: Stuctual Dynamcs, Stuctual Identfcaton, Damage Detecton, Optmal Senso Locaton, Bayesan nfeence, Infomaton Entopy. Abstact. hs wok outlnes the optmzaton algothms nvolved n ntegatng system analyss and measued data collected fom a netwok of sensos. he ntegaton s equed fo stuctual health montong poblems asng n stuctual dynamcs and elated to (1) model paamete estmaton used fo fnte element model updatng, () model-based damage detecton n stuctues and (3) optmal senso locaton fo paamete estmaton and damage detecton. hese poblems ae fomulated as sngle- and mult-obectve optmzaton poblems of contnuous o dscete-valued vaables. Gadent-based, evolutonay, hybd and heustc algothms ae pesented that effectvely addess ssues elated to the estmaton of multple local/global solutons and computatonal complexty asng n sngle and mult-obectve optmzaton nvolvng contnuous and dscete vaables. Intoducton Successful health montong and dagnoss of stuctual systems depends to a lage extent on the ntegaton of cost-effectve ntellgent sensng technques, accuate physcs-based computatonal models smulatng stuctual behavo, effectve system dentfcaton methods, sophstcated health dagnoss algothms, as well as decson-makng expet systems to gude management n plannng optmal cost-effectve stateges fo system mantenance, nspecton and epa/eplacement. A stuctual health montong system ncopoates algothms elated to (1) fnte element model paamete estmaton (updatng), () stuctual damage detecton based on fnte element model updatng, and (3) optmal senso locaton. Fnte element model updatng methods based on vbaton data ae often used to develop hgh fdelty models so that model pedctons ae consstent wth measued data. he need fo model updatng ases because thee ae always assumptons and numecal eos assocated wth the pocess of constuctng a theoetcal model of a stuctue and pedctng ts esponse usng the undelned model. Revews of model updatng methodologes based on modal data can be found n [1]. Moeove, model updatng methodologes ae useful n pedctng the stuctual damage by contnually updatng the stuctual model usng vbaton data [-8]. Such updated models obtaned peodcally thoughout the lfetme of the stuctue can be futhe used to update the esponse pedctons and lfetme stuctual elablty based on avalable data [9]. Optmal senso locaton methods efe to algothms fo optmzng the locaton and numbe of sensos n the stuctue such that the measue data contan the most mpotant nfomaton fo stuctual dentfcaton puposes. Algothms based on nfomaton theoy and usng a nomnal fnte element model of the stuctue, have been poposed to addess ths poblem [10-13]. Effectve heustc optmzaton tools [14,15] have also been developed fo effcently solvng the esultng nonlnea sngle- and mult-obectve optmzaton poblems nvolvng dscete-valued vaables. hs wok fst pesents a Bayesan methodology fo the paamete estmaton and damage detecton used n the stuctual health montong system. It s shown that the fomulatons esult n optmzaton poblems wth espect to contnuous vaables. Computatonal algothms fo solvng these optmzaton poblems ae poposed to ovecome convegence poblems and pematue convegence to local optma. In patcula, hybd algothms based on evoluton stateges and

2 gadent methods ae necessay and well-suted optmzaton tools fo solvng the esultng nonconvex sngle-obectve optmzaton poblem and dentfyng the global optmum fom multple local ones. Fo gadent-based algothms, computatonally effcent schemes fo estmatng the gadents and Hessans of the obectve functons ae poposed and shown to sgnfcantly educe the computatonal effot and the numbe of teatons equed fo convegence. ext, theoetcal and computatonal ssues asng n the selecton of the optmal senso locaton fo paamete estmaton and damage detecton ae addessed. he nfomaton entopy s used as the pefomance measue of a senso confguaton. he optmal senso locaton poblem s fomulated as sngle- and mult-obectve optmzaton poblems nvolvng dscete-valued vaables. Accuate and computatonally effcent heustc algothms fo solvng these poblems ae outlned. Bayesan Methodology fo Model Paamete Estmaton Consde a paametezed class of stuctual models (e.g. a class of fnte element models o a class of modal models) chosen to descbe the nput-output behavo of a stuctue. Let R be the vecto of fee paametes (physcal o modal paametes) n the model class. A Bayesan statstcal system dentfcaton methodology [16,17] s used to estmate the values of the paamete set and the assocated uncetantes usng the nfomaton povded fom dynamc test data. Fo ths, the uncetantes n the values of the stuctual model paametes ae quantfed by pobablty densty functons (PDF) that ae updated usng the dynamc test data. Accodng to the Bayesan stuctual dentfcaton methodology, assumng ndependent and zeo-mean Gaussan pedcton eos e() k wth vaance σ, the updatng PDF p(, σ D) of the paamete sets and σ = ( σ1,, σ o ), gven the measued data D and the class of models, takes the fom [18]: c D p( σ, D) = exp J( σ ; ) π ( ) πσ ( σ) (1) ( π ) D0 ρ( σ ) whee J( σ ; ) s a measue of ft of the measued esponse chaactestcs and the coespondng esponse chaactestcs pedcted by a patcula model n the model class, and ρ( σ ) s a functon of the pedcton eo paametes σ, π ( ) and πσ ( σ ) ae the po dstbuton fo the paamete sets and σ, espectvely, = 0, 0 s the numbe of esponse chaactestcs, D s the numbe of measued data sets, and c s a nomalzng constant chosen such that the PDF n (1) ntegates to one. me Hstoy Data. Fo the case fo whch the esponse chaactestcs consst of the esponse tme 0 hstoes data D = { x ( k t) R, = 1,, 0, k = 1,, D} at 0 measued DOFs, whee s the numbe of the sampled data usng a samplng ate t, then D D J( σ ; ) = J ( ), J ( ) = x ( k) x ( k; ) () 0 = 1σ D k= 1 0 D ( ) = σ = 1 ρ σ, and x (; k ), = 1,, d, ae the pedctons of the sampled esponse tme hstoes obtaned fom a patcula model n the model class coespondng to a specfc value of the paamete set. Modal Data. Fo the case whee the esponse chaactestcs consst of modal data ( k) ( k) 0 ( ) D = { ω, φ R, = 1,, m}, whee k ( ) ae the modal fequences and φ k ae modeshape ω

3 components at 0 measued DOFs, m s the numbe of obseved modes and D s the numbe of modal data sets avalable, then m J( σ ; ) = ε ( ω, ) (, ) ω ε D + Lφφ = 1 σω σ (3) φ whee ω ω β Lφ φ εω (, ) and (, ω = εω = εlφ ) φ = ε = (4) φ ω φ ω ( ) and ( ) d φ R, = 1,, m, ae the pedctons of the modal fequences and modeshapes obtaned fo a patcula value of the model paamete set, = m s the ( ) numbe of measued data pe modal set, D = 1 and ( )/( ) β = φ Lφ Lφ ( Lφ ) s a nomalzaton constant that accounts fo the dffeent scalng between the measued and the pedcted modeshape. Most Pobable Model. ote that the pobablty densty functon n (1) quantfes the uncetanty n the values of the paametes and σ gven the data D. Fo a non-nfomatve po dstbutons π ( ) and πσ ( σ ), the optmal value of the paamete set gven the data s obtaned by [19] mnmzng the measue of ft J( σ ; ). Bayesan Methodology fo Damage Detecton Damage detecton s accomplshed by ntoducng a famly of µ model classes 1 µ and assocatng each model class to a damage patten n the stuctue, ndcatve of the locaton of damage. Each model class s assumed to be paametezed by a numbe of stuctual model paametes scalng the stffness contbutons of a possbly damaged substuctue, whle all othe substuctues ae assumed to have fxed stffness contbutons equal to those coespondng to the undamaged stuctue. Usng a Bayesan model selecton famewok, the pobable damage locatons ae anked accodng to the posteo pobabltes of the coespondng model classes. he most pobable model class wll be ndcatve of the substuctue that s damaged, whle the pobablty dstbuton of the model paametes of the coespondng most pobable model class wll be ndcatve of the sevety of damage n the dentfed damaged substuctue. Usng Bayes theoem, the posteo pobabltes of the vaous model classes gven the data D s pd ( ) P( ) P( D) = d (5) whee pd ( ) s the pobablty of obsevng the data fom the model class, P( ) s the po pobablty of the model class, whle d s selected so that the sum of all model pobabltes equals to one. Assumng thee s no po pefeence as to what class of models we choose, we may set that P ( ) = 1/ µ n (5). he followng asymptotc appoxmaton has been ntoduced to gve a useful and nsghtful estmate of the ntegal nvolved n pd ( ) n (5) [19-0].

4 ( π ) pd ( ) ~ / π ( ) [ (, J σ ))] det h (, σ ) D (6) whee, fo unfom po dstbuton of the paametes n a model class, s the value that mnmzes J (, σ ), and h (, σ ) s defned by D h (, σ ) = ln[ J (,σ )] (7) = n whch = [ /,, / ] s the usual gadent vecto wth espect to the paamete set, 1 and σ s the optmal pedcton eo vaance fo the model class. he appoxmate estmate s unelable when the optmal s outsde the egon Θ { : u = } of vaaton of, whee u ae the values of at the undamaged condton. Altenatvely, one can use mpotance samplng method to compute the ntegal nvolved n estmatng pd ( ) n (5) [19]. Unde the assumpton that the po dstbutons π ( ) ae non-nfomatve unfom dstbutons ove the ange of vaaton of, the pobablty of the model class s gven by [0] log P( M ) log (, ; ) ( D = J J σ + β ; ) + P( M) + d (8) whee d s constant ndependent of the model class M, J = D /, and the facto β( ; ) n (8), known as the Ockham facto, smplfes fo lage numbe of data J to [1, ] β( ; ) = β = logj (9) whee t s evdent that t depends fom the numbe model class M. he optmal model class M best P( M D) measue of ft J (, σ ; ) by the optmal model of a model class pobablty of a model class M J (, ; ) Based on the Ockham facto β depends on the numbe of the model paametes nvolved n the s selected as the model class that maxmzes the pobablty gven by (8). It s evdent that the selecton of the optmal model class depends on the between the measued chaactestcs and the chaactestcs pedcted M. hus, the fst tem n (8) gves the dependence of the fom how well the model class pedcts the measuements. he smalle the value of σ, the hghe the pobablty P( M D) of the model class M. smplfed n (9), the odeng of the model classes n (8) also of the stuctual model paametes that ae nvolved n each model class. Specfcally, model classes wth lage numbe of paametes ae penalzed n the selecton of the optmal model class. It should be ponted out that damage detecton usng the methodology eques the soluton of µ optmzaton poblems wth the obectve functon fo each model class M to be the measue of ft functon J (, σ ).

5 Optmzaton Issues Related to Model Updatng and Damage Detecton he esultng optmzaton poblems ae solved usng avalable numecal algothms. he optmzaton of J (, ) σ wth espect to fo gven σ can eadly be caed out numecally usng any avalable algothm fo optmzng a nonlnea functon of seveal vaables. hese sngle obectve optmzaton poblems may nvolve multple local/global optma. Conventonal gadentbased local optmzaton algothms lack elablty n dealng wth the estmaton of multple local/global optma obseved n stuctual dentfcaton poblems [18, 3], snce convegence to the global optmum s not guaanteed. Evoluton stateges [4] ae moe appopate and effectve to use n such cases. Evoluton stateges ae andom seach algothms that exploe bette the paamete space fo detectng the neghbohood of the global optmum, avodng pematue convegence to a local optmum. A dsadvantage of evoluton stateges s the slow convegence at the neghbohood of an optmum snce they do not explot the gadent nfomaton. A hybd optmzaton algothm [19] should be used that explots the advantages of evoluton stateges and gadent-based methods. Specfcally, an evoluton stategy s used to exploe the paamete space and detect the neghbohood of the global optmum. hen the method swtches to a gadent-based algothm statng wth the best estmate obtaned fom the evoluton stategy and usng gadent nfomaton to acceleate convegence to the global optmum. Gadent and Hessan Computatons. In ode to guaantee the convegence of the gadent-based optmzaton methods fo stuctual models nvolvng a lage numbe of DOFs wth seveal contbutng modes, the gadent of the obectve functon wth espect to the paamete set has to be estmated accuately. It has been obseved that numecal algothms such as fnte dffeence methods fo gadent evaluaton does not guaantee convegence. Moeove, gadent computatons wth espect to the paamete set usng the fnte dffeence method eques the soluton of as many egenvalue poblems as the numbe of paametes. Analytcal expessons fo the gadent of the obectve functons can be used to ovecome the convegence poblems. hese analytcal expessons ae next gven fo the case of stuctual dentfcaton based on modal data. In patcula, elson s method [5] s used fo computng analytcally the fst devatves of the egenvalues and the egenvectos. he advantage of the elson s method compaed to othe methods s that the gadent of egenvalue and the egenvecto of one mode ae computed fom the egenvalue and the egenvecto of the same mode and thee s no need to know the egenvalues and the egenvectos fom othe modes. Fo each paamete n the set ths computaton s pefomed by solvng a lnea system of the same sze as the ognal system mass and stffness matces. elson s method has also been extended n ths wok to compute the second devatves of the egenvalues and the egenvectos. Fnally, the computaton of the gadents and the Hessan of the obectve functons s shown to nvolve the soluton of a sngle lnea system, nstead of lnea systems equed n usual computatons of the gadent and ( + 1) lnea systems equed n the computaton of the Hessan. hs educes consdeably the computatonal tme, especally as the numbe of paametes n the set ncease. he expessons fo the fst and second devatves of the obectve functons ae next pesented. Due to space lmtatons detals of the devatons ae not shown. he gadent of squae eos ε ( ) and ε ( ) nvolved n obectves (see equatons (3) and (4)) ae gven by ε ω ( ) ε ( ) ( ) * ( ) and K M ε ω ω φ = φ ω φ ω φ = x F, (10) whee F, ( = I Mφφ )( K ωm ) φ and x s gven by the soluton of the lnea system

6 A X = D (11) * wth D ( L β βl ) = φ φ / φ and X eplaced by. Fo notatonal convenence, the dependence of seveal vaables on the paamete set has been dopped. Fo an n n matx A = K ω M, A s used to denote the modfed matx deved fom by eplacng the * elements of the k -th column and the k -th ow by zeoes and the ( k, k ) element of A by one, whee k denotes the element of the modeshape vecto φ wth the hghest absolute value. Also, the * n vecto b s used to denote the modfed vecto deved fom b by eplacng the k -th element of the vecto b by zeo. Also, K and M n the fomulaton denote the quanttes K and M that can be obtaned ethe analytcally o numecally usng fnte element methods. Smlaly, t can be shown that the (, ) element of the Hessan of adequately appoxmated n the fom (assumng that M = 0 ) and ε ω ( ) 4 [ ( ) ][ ( = K ω M K ωm ) ω φ φ φ φ ] εφ ( ) * * * * = (, )(, ) β Lφ, XX, φ Lφ x A ε ( ) and ω ε φ ( ) can be (1) z F z F F F (13) whee z s gven by the soluton of the lnea system (11) wth D = I M φ φ L β Lφ φ D = I M φφ L. ( ) ( ) and X s gven by (11) wth ( ) It s clea that the computaton of the fst and second devatves of the squae eos fo the modal popetes of the -th mode wth espect to the paametes n eques only the solutons of the lnea system (11), ndependent of the numbe of paametes n. Fo a lage numbe of paametes n the set, the above fomulaton fo the gadents and Hessan of the mean eos n modal fequences and n the modeshape components n (4) ae computatonally vey effcent and nfomatve. Optmal Senso Locaton Methodology fo Model Paamete Estmaton he nfomaton entopy H ( δ, D) [13], ntoduced as a unque scala measue of the uncetanty n the estmate of the stuctual paametes, s used fo optmzng the senso confguaton n the stuctue fo dentfyng the paametes n a model class. he nfomaton entopy depends on the avalable data D and the senso confguaton vecto δ. It has been shown [14] that fo a lage numbe of measued data,.e. as D, the followng asymptotc esult holds fo the nfomaton entopy fo a model class 1 1 H ( δ, D ) H ( δ ;, σ ) = ln( π ) ln[det h (, σ ; δ )] (14) whee ( δ, D) = agmn J( ; D) s the optmal value of the paamete set that mnmzes the measue of ft J( ; σ ) gven n () o (3) fo a model class, and h (, σ) h (, σ; δ) s an postve defnte matx defned by (7) and asymptotcally appoxmated by [14]

7 h (, ; δ) Q ( δ, σ, σ ) (15) as D. me Hstoy Data. Fo esponse tme hstoy data, substtutng () nto (7) and consdeng the lmtng case D, the esultng matx Q( δ, ) appeang n (15) smplfes to a postve sem-defnte matx of the fom 1 Q δ P (16) d D ( ) (,, σ ) = ( δ ) = 1 σ that contans the nfomaton about the values of the paametes based on the data fom all measued postons specfed n δ, whle the optmal pedcton eo vaances σ ae gven by σ = J ( ). he matx D ( ) P = k = 1 ( ) x ( k; ) x ( k; ) (17) s a postve sem-defnte matx contanng the nfomaton about the values of the paametes based on the data fom one senso placed at the -th DOF. he pedcton eo σ = J ( ) n (16) s computed fom σ = s1 + sg ( ), whee the fst tem accounts fo constant measuement eo and the second tem accounts fo model eo that depends on the stength g ( ) of the esponse chaactestcs wth the values of s 1 and s gvng the elatve sze of measuement and model eos. Modal Data. Fo modal data, the esultng matx Q matx gven by ( δ, ) smplfes to a postve sem-defnte m d 0 ( ) 0 ( ) D ω( ) ω ( ) Lφ Lφ Q (, ) δ δ = + (18) = 1 s1 + sω ( ) = 1 s1 + s L0 ( ) / φ 0 contanng the nfomaton about the values of the model paametes based on the modal data fom all sensos placed n the stuctue. he asymptotc appoxmaton of the nfomaton entopy s useful n the expemental stage of desgnng an optmal senso confguaton. Specfcally, the nfomaton entopy fo a model class s completely defned by the optmal value of the model paametes and the optmal pedcton eo σ = J ( ), = 1,, 0, expected fo a set of test data, whle the tme hstoy detals of the measued data do not ente explctly the fomulaton. he optmal senso confguaton s selected as the one that mnmzes the nfomaton entopy [13] wth espect to the set of 0 measuable DOFs. Howeve, n the ntal stage of desgnng the expement the data ae not avalable, and thus an estmate of the optmal model paametes and σ cannot be obtaned fom analyss. In pactce, useful desgns can be obtaned by takng the optmal model paametes and σ to have some nomnal values chosen by the desgne to be epesentatve of the system. Optmal Senso Locaton Methodology fo Damage Detecton he desgn of optmal senso confguatons fo povdng nfomatve measuements fo multple model classes,..., 1 µ s next addessed. Let I() δ IEI() δ be the effectveness of a senso

8 confguaton δ fo the th model class, whee IEI ( δ ) s the nfomaton entopy ndex gven by H( δ) H( δ, best) IEI ( δ) = (19) H ( δ ) H ( δ ) wth H, wost, best ( δ) H ( ;, δ σ ). he optmal senso confguaton fo the model class s selected as the one that mnmzes the nfomaton entopy ndex I ( δ ). In (19), δ best, s the optmal senso confguaton and δ wost, s the wost senso confguaton fo the th model class. he values of IEI ( δ ) ange fom zeo to one. he most effectve confguaton coesponds to value of IEI ( δ ) equal to zeo, whle the least effectve confguaton coesponds to value of IEI ( δ ) equal to one. he poblem of dentfyng the optmal senso locatons that mnmze the nfomaton entopy ndces fo all µ model classes s fomulated as a mult-obectve optmzaton poblem stated as follows. Fnd the values of the dscete-valued paamete set δ that smultaneously mnmzes the obectves [15]. I = I1 I µ ( δ) ( ( δ),..., ( δ)) Fo conflctng obectves I 1 (),..., δ I µ () δ, thee s no sngle optmal soluton, but athe a set of altenatve solutons, whch ae optmal n the sense that no othe solutons n the seach space ae supeo to them when all obectves ae consdeed. Such altenatve solutons, tadng-off the nfomaton entopy values fo dffeent model classes, ae known n mult-obectve optmzaton as Paeto optmal solutons. An advantage of the mult-obectve dentfcaton methodology s that all admssble solutons ae obtaned whch consttute model tade-offs n educng the nfomaton entopes fo each model class. hese solutons ae consdeed optmal n the sense that the coespondng nfomaton entopy fo one model class cannot be mpoved wthout deteoatng the nfomaton entopy fo anothe model class. he optmal ponts along the Paeto tade-off font povde detaled nfomaton about the effectveness of the senso confguaton fo each model class. Optmzaton Issues fo Optmal Senso Locaton Poblems Model Paamete Estmaton. An exhaustve seach ove all senso confguatons fo the computaton of the optmal senso locatons s extemely tme consumng and n most cases pohbtve, even fo models wth a small numbe of degees of feedom. Altenatve technques must be used to solve the dscete-vaable optmzaton poblem. Genetc algothms [16] ae well suted fo solvng geneal optmzaton poblems nvolvng dscete-valued vaables. A moe systematc and computatonally vey effcent appoach fo obtanng a good appoxmaton of the optmal senso confguatons fo a fxed numbe of 0 sensos s to use a sequental senso placement (SSP) appoach [14]. he computatons nvolved n the SSP algothm ae an nfntesmal facton of the ones nvolved n exhaustve seach method and can be done n ealstc tme, ndependent of the numbe of sensos and the numbe of model degees of feedom. Fo essentally the same accuacy, GA algothms eque sgnfcantly moe computatonal tme than the heustc SSP algothms. In all example cases consdeed, the SSP algothms outpefom, n tems of accuacy and comutatonal tme, the GA algothms. Damage Detecton. Genetc algothms ae well suted fo pefomng the mult-obectve optmzaton nvolvng dscete vaables. In patcula, the stength Paeto evoluton algothm [7] based on genetc algothms s most sutable fo solvng the esultng dscete mult-obectve optmzaton poblem and povdng nea optmal solutons [15]. A moe systematc and computatonally vey effcent appoach fo obtanng a good appoxmaton of the Paeto optmal (0)

9 font and the coespondng Paeto optmal senso confguatons fo a fxed numbe of 0 sensos s to use a sequental senso placement (SSP) appoach [15], extendng the SSP algothm [14] to handle Paeto optmal solutons. he total numbe of vecto functon evaluatons usng the extended SSP algothm s nfntesmally small compaed to the numbe of vecto functon evaluatons equed n an exhaustve seach method. umecal applcatons [14, 15] show that the Paeto font constucted by ths heustc algothm, n most cases examned, concdes wth, o s vey close to, the exact Paeto font. In all cases, the extended SSP algothm outpefoms, n tems of accuacy and computatonal tme, avalable dscete mult-obectve optmzaton algothms such as the stength Paeto evoluton algothm based [7] on genetc algothms. Summay Optmzaton poblems encounteed n ntegatng system analyss and measued data (esponse tme hstoes o modal data) collected o dentfed usng a senso netwok, ae evewed. he pesent study concentates on stuctual dentfcaton poblems elated to (1) model paamete estmaton used fo fnte element model updatng, () model-based damage detecton n stuctues and (3) optmal senso locaton fo paamete estmaton and damage detecton. A Bayesan methodology was pesented to addess the fst two poblems and an nfomaton entopy method was used to addess the thd poblem. hese poblems ae fnally fomulated as sngle and multobectve optmzaton poblems nvolvng contnuous o dscete-valued vaables. Gadent-based, evolutonay, hybd and heustc algothms wee poposed to effcently solve these poblems as well as effectvely addess ssues elated to multple local/global solutons and computatonal complexty. Fo the fst two poblems, computatonally effcent methods fo estmatng the gadents and Hessans of the obectve functons wth espect to the model paametes ae shown to sgnfcantly educe the computatonal effot fo solvng the optmzaton poblems. Heustc algothms ae effectve fo solvng the sngle- and mult-obectve optmzaton poblems asng n optmal senso locaton fomulatons. hese algothms ae based on sequental senso placement schemes and outpefom, n tems of accuacy and computatonal effcency, avalable GA-based algothms. Acknowledgements hs eseach was co-funded 75% fom the Euopean Unon (Euopean Socal Fund), 5% fom the Geek Mnsty of Development (Geneal Secetaat of Reseach and echnology) and fom the pvate secto, n the context of measue 8.3 of the Opeatonal Pogam Compettveness (3 d Communty Suppot Famewok Pogam) unde gant 03-Ε -54 (PEED 003). hs suppot s gatefully acknowledged. Refeences [1] J.E. Motteshead and M.I. Fswell: Jounal of Sound and Vbaton Vol. 167() (1993), p. 347 [] H. Sohn, K.H. Law: Eathquake Engneeng and Stuctual Dynamcs Vol. 6 (1997), p.159 [3] C.P. Ftzen, D. Jennewen and. Kefe: Mechancal Systems and Sgnal Pocessng Vol. 1(1) (1998), p. 163 [4] M.W. Vank, J.L. Beck and S.K. Au: Jounal of Engneeng Mechancs (ASCE) Vol. 16(7) (000), p. 738 [5] C. Papadmtou: 9th ASCE Jont Specalty Confeence on Pobablstc Mechancs and Stuctual Relablty (004), Albuqueque, ew Mexco. [6] H.F. Lam, C.. g and M. Vedt: Jounal of Sound and Vbaton Vol. 305(1-) (006), p. 34

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