Novel SHM method to locate damages in substructures based on VARX models

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1 Journal of Physcs: Conference Seres PAPER OPE ACCESS ovel SHM method to locate damages n substructures based on VARX models To cte ths artcle: U Ugalde et al 2015 J. Phys.: Conf. Ser Vew the artcle onlne for updates and enhancements. Related content - Structural health montorng by means of elastc wave propagaton Wesaw Ostachowc and Macej Radesk - Damage evaluaton by a guded wavehdden Markov model based method Hanfe Me, Shenfang Yuan, Le Qu et al. - Crack propagaton montorng n a fullscale arcraft fatgue test based on guded wave-gaussan mxture model Le Qu, Shenfang Yuan, Qao Bao et al. Ths content was downloaded from IP address on 11/02/2018 at 09:13

2 ovel SHM method to locate damages n substructures based on VARX models U Ugalde 1, J Anduaga 1, F Martíne 1, A Iturrospe 2 1 Department of Sensors, IK4-Ikerlan, Mondragon, Span 2 Department of Electroncs and Computer Scences, Mondragon Go Eskola Polteknkoa, Mondragon, Span E-mal: uugalde@kerlan.es Abstract. A novel damage localaton method s proposed, whch s based on a substructurng approach and makes use of Vector Auto-Regressve wth exogenous nput (VARX) models. The substructurng approach ams to dvde the montored structure nto several mult-dof solated substructures. Later, each ndvdual substructure s modelled as a VARX model, and the health of each substructure s determned analyng the varaton of the VARX model. The method allows to detect whether the solated substructure s damaged, and besdes allows to locate and quantfy the damage wthn the substructure. It s not necessary to have a theoretcal model of the structure and only the measured dsplacement data s requred to estmate the solated substructure s VARX model. The proposed method s valdated by smulatons of a two-dmensonal lattce structure. 1. Introducton Structural Health Montorng (SHM) s the process of mplementng a damage detecton and characteraton strategy for engneerng structures [1]. SHM s regarded as a very mportant engneerng feld n order to secure structural and operatonal safety; ssung early warnngs on damage or deteroraton, avodng costly repars or even catastrophc collapses [2]. Most of the exstng vbraton based SHM methods could be classfed nto two dfferent approaches: global approaches and local approaches [3]. In the global approaches, the goal s to montor the health of the entre structure. These global methods have been tested and mplemented n dfferent types of structures durng the last 30 years [4]. However, for many large systems, global montorng s not practcal due to the lack of senstvty of global features regardng local damages, naccuraces of developed models or the hgh cost of sensng, cablng and computatonal operatons [5]. On the other hand, local SHM methods are focused on evaluatng the state of reduced parts wthn the entre structures, based on substructurng methods. Ths approach ams to overcome global method s problems, dvdng the whole structure nto substructures and analyng each one ndvdually. Several research works have proposed substructurng methods for large-scale structures. Koh [6] presented a dvde and conquer strategy to montor large structures based on the dvson of the whole structure nto solated substructures. For each substructure, structural parameters are dentfed Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3.0 lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work, journal ctaton and DOI. Publshed under lcence by Ltd 1

3 usng the Extended Kalman flter (EKF). However, the EKF usually requre knowledge of the system and ts dynamcs [7]. Yun and Lee [8] detected damage n structures combnng a substructurng method and expermental modellng [9]. Most recently, Xng [10] presented another damage detecton method based on a substructurng method and Auto-Regressve Movng Average wth exogenous nput (ARMAX) models. Damage ndcators were obtaned for each estmated substructural model calculatng the dfference between the squared natural frequences n the healthy state and the squared natural frequences durng the structure lfetme. All the natural frequences were computed from ther respectve estmated ARMAX models. Ths damage detecton method was valdated through smulatons and expermental test. The method proposed by Xng [10] doesn t requre any theoretcal model of the structure [9]. evertheless, as only one nternal DOF s measured n each substructure, s not possble to gve nformaton about the damage locaton wthn the substructures. In ths paper, a damage localaton method based on the combnaton of a substructurng method and expermental modellng s proposed. The substructurng method s used to solate a mult-dof substructure from the rest of the structure, and each solated substructure s modelled as a Vector Auto-Regressve wth exogenous nput (VARX) model. VARX models ncorporate data measured n dfferent nternal DOFs and ther coeffcent matrces descrbe the relatonshp between the measured nternal DOFs through some structural characterstcs (mass, stffness, dampng ). Therefore, the proposed method could potentally locate the damage wthn the substructure by analyng varatons on the VARX model over the tme. Furthermore, the proposed method doesn t requre any theoretcal model of the structure. The rest of the paper s organed as follows. Frst, the proposed method s presented n secton 2. Secondly, the proposed method s evaluated by seres of smulatons. In secton 3, smulaton results are dscussed and fnally the concludng remarks are presented n secton The proposed method The behavor of the structure s descrbed by a lumped parameter model, where we assume that all objects are rgd bodes and all nteractons between the rgd bodes take place va sprngs and dampers. We assume the structure conssts of bars that are connected together by rgd jonts. The pont n whch two or more bars are joned s called node and the number of structural nodes wll depend on the topology of the structure. The forces could only be transmtted along the axal drecton of the bars and the load could only be appled at the two ends of each bar. We assume that the mass of each bar s dstrbuted equally between ts two nodes, a half n the frst one and the other half n the second one. On the other hand, the structure could be subjected to arbtrary external loadng that s assumed to be known and could act on any node. The structure s dvded nto dfferent substructures and these substructures are solated from the remanng structure. The substructures could contan several nternal () and nterface (j) nodes. The nterface nodes are located n the border between the selected substructure and the remanng structure. On the other hand, nternal nodes are located wthn the substructure and they are not connected to the nodes of the remanng structure. 2

4 The dynamc equatons for an nternal node are formulated as follows: m ( f (,,, )) F x xk k k x k 1 m ( f (,,, )) F k 1 m ( f (,,, )) F k 1 yk k k k k where m s the lumped mass of the nternal node and x, and are the absolute acceleratons of the nternal node n x, y and axes respectvely. f xk, f yk and f k are lnear or non-lnear functons used to calculate the total nternal force appled n the node by the nodes connected to hm. These functons depend on the value of,,k and k, where and represent the absolute dsplacement and the velocty of the node and k and k are the absolute dsplacements and veloctes of the nodes that are connected to the node, all of them n x, y and axes. Furthermore, F x, F and F are the external forces that are actng n the nternal node. k (1) Expandng f xk, f yk and f k functons as Taylor seres [11] and selectng only the frst term, the dynamc equatons for an nternal node are stated as: m ( k ( ) k ( ) k ( ) c ( ) c ( ) c ( )) F x xxk x xk xyk yk xk k xxk x xk xyk yk xk k k 1 m ( k ( ) k ( ) k ( ) c ( ) c ( yxk x xk yyk yk yk k yxk x xk yyk k 1 (2) ) c ( )) F yk yk k m ( k ( ) k ( ) k ( ) c ( ) c ( ) c ( )) F xk x xk yk yk k k xk x xk yk yk k k 1 k x where x,, and x,, are the absolute dsplacements and veloctes of the node and xk, yk, k and xk, yk, k are the absolute dsplacements and veloctes of the nodes that are connected to the node, all of them n x, y and axes. On the other hand, k and c are coeffcents related to the stffness and dampng values of the bars that are n contact wth node. The fnte central dfference method [10] s used to obtan the approxmaton of the dsplacement s frst and second dervatves. Repeatng the explaned process (equaton 2) for the other nternal nodes and representng the expressons n matrx form, the substructural dynamc equaton s stated as: 3

5 ... A... ( n) ( n 1) nx nx ( n) ( n 1) ( n 2) 1x 1x 1x ( n) ( n 1) ( n 2) 1y 1y 1y ( n) ( n 1) ( n 2) ( n) ( n 1) ny ny ( n) ( n 1) n n A j1x j1x j1y j1y j1 j1... ( n 2) ( n 2) ( n 2) ( n) ( n 1) ( n) ( n 1) ( n) ( n 1) B10... B11... B B ( n) ( n 1) jnx ( n) jny jn ( n) nx ny n j1x 1x j1y 1y j1 1 jnx jnx ( n 2) F ( n 1) ( n 2) F ( n 1) ( n 2) F ( n 1) ( n 2) F ( n 1) ( n 1) ( n 2) F ( n 1) jny jny ny ( n 1 ) ( n 2) F jn jn ( n 1) n nx (3) where 1x,..., n and j1x,..., jn are the absolute dsplacements of all nternal and nterface nodes n x, y and axes. Equaton (3) could be regarded as a VARX model [12], where 1x,..., n corresponds to the endogenous varables and j1x,..., jn and F 1x,..., F n correspond to the exogenous varables. A k s a n x n endogenous coeffcent matrx, where n denotes the number of nternal DOFs of the substructure. The elements of A k matrx are related to the physcal propertes of the bars that connect the nternal nodes between them. On the other hand, B k s a n x m exogenous coeffcent matrx, where m denotes the number of nterface DOFs of the substructure. The elements of B k are related to the physcal propertes of the bars that connect one nternal node to another nterface node. In the proposed method, the substructures are modelled as a VARX models. The state of each substructure s evaluated analyng devatons n ts estmated coeffcent matrces (A k and B k ) respect to ts estmated coeffcent matrces (A k and B k ) n the healthy condton. Frstly substructural damages are detected and secondly the damages are located wthn the substructure. 3. umercal results A lnear and tme nvarant two-dmensonal lattce structure s studed n ths secton. The structure conssts of stanless steel bars that are connected together by rgd jonts and we assume that the forces could only be transmtted along the axal drecton of the bars and the load could only be appled at the two ends of each bar. The structural behavour s descrbed by a lumped parameter model, where we assume that all object are rgd bodes and all nteractons between the rgd bodes take place va sprngs. In the studed case, the structure s modelled as a sxteen DOF mass-sprng model (see fgure 1). Furthermore, a ten DOF substructure s solated from the general structure, where absolute dsplacement 6x, 6y, 7x, 7y, 8x and 8y correspond to nternal DOFs and absolute dsplacement 4x, 4

6 4y, 5x and 5y correspond to nterface DOFs. As shown n fgure 1, none external force s appled wthn the substructure. Fgure 1. Isolated substructure n the structural model Below, the dynamc equatons for the nternal node are formulated: 2 x k, k, x xk k, k, k, yk k1 2 ( k, cos k, n k, ( x xk k, k, ( yk k1 m ( k cos ( ) k cos sn ( )) (4) m k s ) k sn )) where m represent the lumped mass of the nternal node and x and are the absolute acceleratons of the nternal node n x and y axes. The nternal node s supportng nternal forces, one force for each node connected to hm. These forces depend on the stffness and the angle respect to the x axs of the sprngs located between the nternal node and the nodes that are connected to hm, as well as the dsplacements of these nodes n x and y axes ( x,, xk, yk ). Followng the procedure descrbed n secton 2, we get the VARX model. Equaton (5) could be regarded as a four exogenous and sx endogenous varables VARX model [12]. The exogenous varables are the measured absolute dsplacements n 4x, 4y, 5x and 5y and the endogenous varables are the measured absolute dsplacement n 6x, 6y, 7x, 7y, 8x and 8y. ( n) ( 1) ( 2) 6 x n 6 x n 6 x ( n) ( n 1) ( n 2) ( n1) 7 x 7 x 7 x 4 x ( n) ( n 1) ( n 2) ( n1) 8x 8x 8x 5x A A 1 2 B 1 ( n) ( n 1) ( n 2) 6 y 6 y 6 y ( 1) 4 y n ( n) ( n 1) ( n 2) 7 y 7 y 7 y ( n1) 5 y ( n) ( n 1) ( n 8y 8 y 2) 8y (5) 5

7 A 1 and A 2 are 6 x 6 endogenous coeffcent matrces and B 1 s a 6 x 4 exogenous coeffcent matrx. These matrces are related to the physcal propertes of the substructural bars (mass, stffness and angle) and also depend on the used samplng perod. In addton to ths, equaton 6 shows the dependence between each matrx element and the substructural stffness values. A1 f ( k, k, k, k ) f ( k ) f ( k ) f ( k, k, k, k ) f ( k ) f ( k ) 11 4,6 5,6 6,7 6,8 12 6,7 13 6,8 14 4,6 5,6 6,7 6,8 15 6,7 16 6,8 f ( k ) f ( k, k, k ) f ( k ) f ( k ) f ( k, k, k ) f ( k ) 21 6,7 22 4,7 6,7 7,8 23 7,8 24 6,7 25 4,7 6,7 7,8 26 7,8 f ( k ) f ( k ) f ( k, k, k ) f ( k ) f ( ) f ( k, k, k ) 31 6,8 32 7,8 33 5,8 6,8 7,8 34 6,8 35 7,8 36 5,8 6,8 7,8 f ( k, k, k, k ) f ( k ) f ( k ) f ( k, k, k, k ) f ( k ) f ( k ) 41 4,6 5,6 6,7 6,8 42 6,7 43 6,8 44 4,6 5,6 6,7 6,8 45 6,7 46 6,8 f ( k ) f ( k, k, k ) f ( k ) f ( k ) f ( k, k, k ) f ( k ) 51 6,7 52 4,7 6,7 7,8 53 7,8 54 6,7 55 4,7 6,7 7,8 56 7,8 f ( k ) f ( k ) f ( k, k, k ) f ( k ) f ( k ) f ( k, k, k ) 61 6, ,8 63 5,8 6,8 7,8 64 6,8 65 7,8 66 5,8 6,8 7,8 A 2 I k (6) g ( k ) g ( k ) g ( k ) g ( k ) 11 4,6 12 5,6 13 4,6 14 5,6 g ( k ) 0 g ( k ) ,7 23 4,7 0 g ( k ) 0 g ( k ) 32 5,8 34 5,8 B1 g ( k ) g ( k ) g ( k ) g ( k ) 41 4,6 42 5,6 43 4,6 44 5,6 g ( k ) 0 g ( k ) ,7 53 4,7 0 g ( k ) 0 g ( k ) 62 5,8 64 5,8 In ths work, the structure s excted n the thrd mass (outsde the substructure) by a Gaussan whte nose and the dsplacements are recorded for each substructural DOF usng a data samplng frequency of 1000 H. Later, the substructural VARX model s estmated by the Multvarable Least-Square estmator (MLS) method [12] for a healthy state and for the damaged scenaros. All consdered damages are stffness losses of a specfc sprng wthn the structure. Three dfferent damage severtes (5%, 10% and 20%) and sx dfferent damage locatons are evaluated. In some of them, the damaged sprngs are wthn the substructure (k 4,6, k 4,7, k 6,7, k 6,8 ) and n the others, they correspond to external sprng (k 1,3, k 2,5 ). As we could see n equaton (6), the elements of matrces A 1 and B 1 are functon, among other thngs, of the stffness of the substructural sprngs. Matrx A 1 depends on the state of the sprngs that are located between the nternal nodes (k 6,7, k 6,8, k 7,8 ) and matrx B 1 depends on those sprngs located between the nternal and nterface nodes (k 4,6, k 4,7, k 5,6, k 5,8 ). In ths work we evaluate the state of the whole substructure. For ths purpose, frstly the substructural VARX model of the healthy state s estmated. Later, the substructural VARX model s reestmated n each new scenaro and the state of the substructure s evaluated n these new scenaros comparng these updated A 1 and B 1 matrces and the healthy ones. In the case of substructural damages, the damages wll be frstly detected and later located n one of the substructural sprng dependng on the vared elements wthn A 1 and B 1 matrces. On the other hand, f external sprngs are damaged, the elements of matrces A 1 and B 1 wll not change, so these external damages wll not be detected and located wthn the substructure. 6

8 Table 1 shows whch elements of A 1 and B 1 matrces depend exclusvely on the propertes of the substructural sprngs. In ths work, only the varatons of these elements (see table 1) are analyed and the state of each substructural sprng s evaluated dependng on ths analyss. For example, the stffness of the sprng that jons nodes 5 and 6 (k 5,6 ) affects exclusvely n four elements wthn the matrx B 1 (B 1(1,2), B 1(1,4), B 1(4,2) and B 1(4,4) ). For ths reason, the varatons of these four elements are analyed to determne f the sprng k 5,6 s damaged or not. In the present study, the damage severty for each sprng s gven by the mean varaton of the elements that should be analyed (see table 1). Table 1. Analyed elements wthn A 1 and B 1 matrces Internal sprng A 1 elements B 1 elements k 4,6 - (1,1),(1,3).(4,1),(4,3) k 4,7 - (2,1),(2,3),(5,1),(5,3) k 5,6 - (1,2),(1,4),(4,2),(4,4) k 5,8 - (3,2),(3,4),(6,2),(6,4) k 6,7 k 6,8 k 7,8 (1,2),(1,5),(2,1),(2,4) (4,2),(4,5),(5,1),(5,4) (1,3),(1,6),(3,1),(3,4) (4,3),(4,6),(6,1),(6,4) (2,3),(2,6),(3,2),(3,5) (5,3),(5,6),(6,2),(6,5) Regardng to the results, for external damages (reducng k 1,3 and k 2,5 values), the estmated stffness modfcaton for all substructural sprngs are almost ero, so the method determnes that the substructure s healthy. For nternal damages (reducng k 4,6, k 4,7, k 6,7 and k 6,8 values), the estmated stffness modfcatons are shown n fgure 2. In these four scenaros, the substructural damages are frstly detected and later located n the proper sprng. Furthermore, the results show that the method estmates the severty of the damage. Fgure 2. Estmated stffness modfcaton for each substructural sprng (nternal damages) 7

9 4. Conclusons Ths paper proposes a novel SHM method to locate damages n multdmensonal structures. A substructure of nterest s solated by a substructurng method and a VARX model of the solated substructure s obtaned. The analyss of the estmated VARX model s carred out n order to evaluate the health of the solated substructure. It s not necessary to have any theoretcal model and only the measured dsplacement data s requred to estmate the solated substructure s VARX model. A lnear and tme nvarant model of a two-dmensonal lattce structure s smulated to evaluate the proposed method. The results show that the method not only allows detectng damages wthn the substructure, because t also allows estmatng ther locaton and ther severty. The proposed method s also suted for three dmensonal lattce structures, where the number of element s connectons ncreases. Our research group s already applng ths method n a laboratory lattce structure and the results wll be publshed soon. References [1] J. Ko and Y., "Technology developments n structural health montorng of large-scale brdges," Engneerng structures, vol. 27, pp , [2] A. Mta, "Structural dynamcs for health montorng, Sankesha Co., Ltd, agoya, vol. 114, [3] L. Jankowsk, "Dynamc load dentfcaton for structural health montorng," [4] S. W. Doeblng, C. R. Farrar, M. B. Prme, and others, "A summary revew of vbraton-based damage dentfcaton methods," Shock and vbraton dgest, vol. 30, pp , [5] J. Hou, L. Jankowsk, and J. Ou, "Structural damage dentfcaton by addng vrtual masses," Structural and Multdscplnary Optmaton, vol. 48, pp , [6] C. G. Koh, L. M. See, and T. Balendra, "Estmaton of structural parameters n tme doman: a substructure approach," Earthquake Engneerng \& Structural Dynamcs, vol. 20, pp , [7] A. W. Oreta and T.-a. Tanabe, "Element dentfcaton of member propertes of framed structures," Journal of Structural Engneerng, vol. 120, pp , [8] C. Yun and H. Lee, "Substructural dentfcaton for damage estmaton of structures," Structural Safety, vol. 19, pp , [9] R. Isermann and M. Munchhof, "Identfcaton of dynamc systems: an ntroducton wth applcatons", Sprnger Scence \& Busness Meda, [10] Z. Xng and A. Mta, "A substructure approach to local damage detecton of shear structure," Structural Control and Health Montorng, vol. 19, pp , [11] L. B. Zhao, J. Y. Zhang, and Y. X. Zhao, "Taylor Seres umercal Method n Structural Dynamcs," Advanced Materals Research, vol. 33, pp , [12] H. Lutkepohl, "ew ntroducton to multple tme seres analyss: Sprnger, 2007". 8