Structural Damage Assessment from Modal Data using a System Identification Algorithm with a Regularization Technique

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1 KEERC-MAE Jont Semnar on Rsk Mtgaton for Regons of Moderate Sesmcty Unversty of Illnos at Urbana-Champagn, August 5-8, MAE Center Structural Damage Assessment from Modal Data usng a System Identfcaton Algorthm wth a Regularzaton echnque Joo Sung KANG, Inho YEO and Hae Sung LEE 3 Graduate Student, Seoul Natonal Unversty School of Cvl, Urban and Geosystem Eng., Shllm-dong, Kwanak-gu, Seoul 5-74, Korea emal: skang94@plaza.snu.ac.kr Senor Researcher, Hghway Management Corporaton Research Insttute, Ichon , Korea emal: hyeo@hman.co.kr 3 Assocate Professor, Seoul Natonal Unversty School of Cvl, Urban and Geosystem Eng., Shllm-dong, Kwanak-gu, Seoul 5-74, Korea emal: chslee@plaza.snu.ac.kr ABSRAC: he parameter estmaton scheme for System Identfcaton (SI) has been developed by many researchers for more than three decades. It s acheved by comparng the measured responses of a real structure and the calculated responses of a mathematcal model. Generally the dynamc test s appled to a large structure for dagnoss of the system. Modal parameter estmaton s formulated as a mnmzaton problem the dscrepancy between the real measurement data and the predctons of a mathematcal model of the structure. It s usually dffcult to measure at all of the degrees of freedom and to collect data from all of the modes. Furthermore, the measurement data always nclude a consderable measurng errors. Inverse problem for parameter estmaton s severe ll-posedness problem because of the sparseness of measurement data and the measurng error. khonov norm for the change of the stffness s used as the regularzaton functon. Geometrc Mean Scheme (GMS) s employed to determne a regularzaton factor. KEYWORDS: System Identfcaton, modal data, ll-posedness, regularzaton, GMS

2 INRODUCION Structural damage often causes a loss of stffness n one or more elements of a structure that affects ts modal responses such as modal frequences and mode shapes. Many methods have been developed to detect the locaton and severty of damage based on these changes. In ths study, system dentfcaton (SI) based on the mnmzaton of least square errors between measured mode shape vector and calculated mode shape vector s employed. o solve nonlnear optmzaton problem, the recursve quadratc programmng (RQP) and the Fletcher actve set strategy are employed []. In RQP, senstvty of calculated mode shape vector wth respect to the system parameters s requred. Current proposed algorthms to calculate senstvty of mode shape vector, such as the modal method, the modfed modal method, and the Nelson s method [6] are vald only when the mode shape vector s normalzed wth respect to mass matr. Unless fully measured mode shape vector s avalable due to economc or physcal restrcton, normalzaton of measured mode shape vector cannot be obtaned. So an algorthm to calculate senstvty of the mode shape vector whch s normalzed wth respect to an arbtrary matr s developed. It s known that SI s typcally ll-posed nverse problem whch suffers from severe numercal nstabltes, such as non-estence, non-unqueness, and dscontnuty of soluton. A regularzaton technque [,3] s adopted to overcome such numercal nstabltes. As a regularzaton functon, khonov norm whch s dfference between a baselne stffness property and an assumed stffness property s used. A regularzaton factor plays the most mportant role for estmaton of both numercally and physcally meanngful soluton [,3]. GMS proposed by Park [8] s used to determne an approprate regularzaton factor. In real stuaton, measurement data suffer from the measurement noses. When the measurement data are polluted wth nose, t s very dffcult to dstngush whether the damage s caused ether by real damage or by nose n measurement data. Snce measurement nose s nevtable n the real stuaton, the estmated system parameters from SI may be easly meanngless n the damage detecton and assessment. o overcome ths drawbacks, data perturbaton scheme proposed by Helmstad and Shn [5] and statstcal approach proposed by Yeo [9] are ncorporated wth SI for damage detecton and assessment. PARAMEER ESIMAION In ths study damage s defned as the reducton of a system parameters from ts baselne value whch s assumed as a pror nformaton. System parameters are estmated by the output error estmator usng modal data such as Eq. (). Mnmze Π( ) = nmd = φ φˆ subect to R( ) ()

3 where, φ, φˆ, nmd, R() are system parameter vector, calculated mode shape vector of -th mode, measured mode shape vector of -th mode, the total number of the measured modes and constrants of system parameters, respectvely. REGULARIZAION he parameter estmaton wth the output error estmator s typcally ll-posed nverse problem. Ill-posed problems suffer from three nstabltes: nonestence of soluton, nonunqueness of soluton and/or dscontnuty of soluton when measured data s polluted by nose []. So far, many authors have attempted to overcome an nstablty problem by mposng upper and lower constrants on the system parameters. However, Neuman [6] and Helmstad [4] have shown that constrants are not suffcent to guarantee a meanngful soluton. In ths study t s utlzed regularzaton technque n order to overcome ll-posedness n optmzaton processng. he followng regularzaton functon s used for the current dentfcaton of a structure. Π R β = () where β and denote the regularzaton factor and the system parameters representng baselne stffness propertes of a structure, respectvely. s the khonov norm of a matr. he regularzaton effect n parameter estmaton process s determned by the regularzaton factor. Some rgorous methods to fnd an optmal regularzaton factor have been proposed for lnear nverse problems. he geometrc mean scheme (GMS) proposed by Park s adopted to determne the optmal regularzaton factor [8]. In the GMS, the optmal regularzaton factor s defned as the geometrc mean between the mamum sngular value and the mnmum sngular value of the Gauss-Newton hessan matr of the error functon gven n Eq. (). β = S (3) ma S mn where β, S ma, S mn denote regularzaton factor, mamum sngular value and mnmum sngular value whch s not zero, respectvely. By addng the regularzaton functon to the error functon, the regularzed output error estmator s defned as follows :

4 nmd Mnmze ˆ β Π = φ ( ) φ + subect to R( ) (4) = SENSIYVIY o solve the constraned nonlnear optmzaton problem epressed by Eq. (4), the recursve quadratc programmng (RQP) and the Fletcher actve set strategy are employed []. In RQP we need the senstvty of the Eq. () wth respect to system parameters. he senstvty s shown n Eq. (5) Π nmd ˆ, = φ φ φ, = (5) where the subscrpt ( ), denotes the partal dervatve wth respect to a system parameter. In case where the mode shape vector s normalzed by mass matr of the structural system, several methods to calculate the senstvty are already proposed, such as the modal method, the modfed modal method, and the Nelson s method [6] and so on. When the mode shape vector s normalzed by mass matr, the senstvty matr of the vector s as follows: nmd φ K, φ φ, = ( ) (6) φ ( λ λ ) φ Mφ where λ, M, and K, denote egen value, the mass matr, and the senstvty matr of the stffness matr of a structural system wth respect to a system parameter, respectvely. However, n the system dentfcaton where the partally measured mode shape s used to determne desgn varables we cannot use the senstvty of the normalzed mode shape by the mass matr. If we assume that φ s normalzed by arbtrary matr C, then we can epress ts senstvty φ, as follows: φ Cφ, φ =, φ, φ Cφ φ (7) φ Cφ φ Cφ If we substtute m c for φ C φ, the Eq. (5) can be represented as the followng equaton. ( φ φ ) φ φ,, C, 3 = φ (8) mc mc

5 MEASURED DAA PERURBAION SCHEME If a suffcent number of measured data sets are avalable for the same measurement condton, the effect of measurement nose on dentfcaton results may be reduced by averagng the measured data. In real stuatons, however, only lmted sets of nosy measurement data are avalable. herefore, t s usually dffcult to determne whether the changes of the system parameters are caused by measurement nose or by actual damage. he data perturbaton method, whch has been proposed by Helmstad and Shn [5] for a numercal smulaton study, s employed. In the data perturbaton method, a seres of the system dentfcaton s performed wth generated data sets around a gven set of measured dsplacements by perturbng the gven data wth a small magntude. As a result, the dentfed system parameters are nterpreted statstcally wth ther dstrbutons. o obtan samples of a system parameter for ts statstcal dstrbuton, the measured data perturbaton teraton s performed wth the followng perturbed measurement data. ( u ) k k = ( u ) ( + η ) (9) k k where ( u ) and η are a perturbed dsplacement for load case and a random number, respectvely, for the -th component of the measured dsplacement at the k-th teraton. DAMAGE SEVERIY After mean and standard devaton are evaluated usng the measured data perturbaton scheme, t s the net progresson to determne whch member s damaged. Because baselne value of the system parameter s a pror knowledge, t s possble to defne damage by comparng the estmated mean value of a member wth ts baselne value. When any estmated system parameter value s less than ts baselne value, t s regarded that the system parameter encounters damage. Accordng to the statstcal nterpretaton proposed by Yeo [9], we can calculate the damage severty as follows : S D = (%) () where and are the baselne value and the estmated mean value of a desgn system parameter. EXAMPLE he numercal eample structure s -story frame structure as shown n Fg.. Snce the aal rgdty can be neglected n a frame structure, the only fleural rgdty EI of each member s selected as the system parameter. he damage s assumed that the rotatonal rgdty of

6 support s lost perfectly by any severe load. It s assumed that the damage s reflected as the reducton of the fleural stffness of the member connected to the support. Fg. and Fg. show the geometry condton and boundary condtons and the fnte element model of the structure, respectvely. It s assumed that the structure s made of steel. herefore Young s modulus of each element s assumed unformly 6Gpa. A cross-secton of each element s. m. m rectangle. Moment of nerta from the neutral as s.33-8 m 4. 5% proportonal random nose for each mode shape s added to the nose-free mode shape vector. he baselne values of all the system parameters are N m, and whch s used as the ntal values for desgn parameters. A posteror nformaton used as the measurement s modal dsplacement ncludng horzontal and vertcal dsplacement, and rotatonal dsplacement at nodes 3,5,3 and 5. he total number of degrees of freedom of the structure s 48, and the number of measured degrees of freedom s. Frst three modes are used n the ROEE n Eq. (4). Fg. 3 represents the estmaton result as the normalzed values of system parameters. he fgure shows that system parameters of several members s reduced than ther baselne values. However only the three members (5, 9, ) are assessed as damaged members usng the statstcal approach of Yeo [9]. her damage severtes are shown n Fg. 4. he two members are ust ones connected wth the hnged support and the other member s the sde bay member. Maybe the reason that the member 5 s assessed to a damaged member s the sparseness of measurement data n parameter estmaton..3m Observaton Pont (7) (4) (3) 9 5 (8) (9) () (8) 9 4 (7) () ().3m Damage () (3) (4) 7 (6) (5) (6) () () 6 (5) (9).3m.3m Fg. Geometry and boundary condton Fg. FEM modelng

7 stffness property 5 5 Baselne stffness property Estmated stffness property Standard Devaton Damage Severty element number element number Fg. 3 Estmated result Fg. 4 Damage severty CONCLUSION he damage detecton and assessment algorthm usng mode shape vector s proposed. he khonov regularzaton technque s employed to allevate the ll-posedness of the nverse problem n SI. he GMS s utlzed to determne the optmal regularzaton factor. he method to calculate the senstvty matr of the mode shape vector, whch s normalzed by an arbtrary matr, s presented. Data perturbaton scheme s used to assess the structural damage statstcally. Statstcal approach s used to determne damaged members and assess the damage severty. In spte of successful detecton of the deteroraton of the frame s assessed as the reducton of fleural rgdty of the member connected to the support but the proposed algorthm stll has many problems. Because the system dentfcaton scheme usng modal data contans drawbacks caused by nsenstveness of lower mode shape to changes of structural propertes, proposed algorthm needs to hgher mode shape to dstngush the mode shape by changes of structural propertes but t can t be obtaned n real stuaton. So t s needed to update proposed algorthm to overcome these problems. REFERENCE. Banan, M.R. and Helmstad, K.D. (993), Identfcaton of structural systems from measured response, Structural Research Seres NO. 579, Dept. of Cvl Eng. Unv. of Illnos at Urbana-Champagn, May. Bu, H.D. (994), Inverse problems n the mechancs of materals: An ntroducton,

8 CRC Press, Boca Raton 3. Groetsch, C.W. (984). he theory of khonov regularzaton for Fredholm equatons of the frst knd, Ptman Advanced Publshng, Boston 4. Helmstad, K.D. (996). On the unqueness of modal parameter estmaton. J. of Sound and Vbraton 9(), Helmstad, K.D., and Shn, S.B. (997). Damage Detecton and Assessment of Structures from statc response. J. of Engneerng Mechancs, ASCE, 3(6), Nelson, R.B. (976). Smplfed Calculaton of Egenvector Dervatves, AIAA Journal, Vol. 4, Sept. 976, pp Neuman, S. P., and Yakowtz S. (979). A statstcal approach to the nverse problem of aqufer Hydrology. Water Resources Research, 5(4), Park, H.W., Shn, S.B. and Lee, H.S., Determnaton of Optmal Regularzaton Factor for System Identfcaton of Lnear Elastc Contnua wth the khonov Functon, Internatonal Journal for Numercal Methods n Engneerng, Vol. 5, No.,, pp Yeo, I.H., Shn, S.B., Lee, H.S., Chang, S.P., "Statstcal Damage Assessment of Framed Structures from Statc Responses," Journal of Engneerng Mechancs, ASCE, Vol. 6, No. 4, pp. 44-4,