Damage Diagnosis of Beam-like Structures Based on Sensitivities of Principal Component Analysis Results

Transcription

1 Damage Diagnoi of Beam-like Structure Baed on Senitivitie of Principal Component Analyi Reult NGUYEN Viet Ha, GOLNVAL Jean-Claude Univerity of Liege, Aeropace & Mechanical Engineering Department, Structural Dynamic Reearch Group, Chemin de chevreuil, 1 B52/3, B-4000 Liège 1, Belgium NOMENCLATURE K, M, C tiffne, ma and damping matrix X obervation (naphot) matrix p vector of parameter U,V,Σ matrice of left and right ingular vector and of ingular value k k α, β projection coefficient ji ji H ω Δ d, d FRF matrix angular frequency enitivity variation firt and econd derivative of the vector Δ ABSTRACT Thi paper addree the problem of damage detection and localization in linear-form tructure. Principal Component Analyi (PCA) i a popular technique for dynamic ytem invetigation. The aim of the paper i to preent a damage diagnoi method baed on enitivitie of PCA reult in the frequency domain. Starting from Frequency Repone Function (FRF) meaured at different location on the beam, PCA i performed to determine the main feature of the ignal. Senitivitie of principal direction obtained from PCA to beam parameter are then computed and inpected according to the location of enor; their variation from the healthy tate to the damaged tate indicate damage location. t i worth noting that damage localization i performed without the need of modal identification. Once the damage ha been localized, it evaluation may be quantified if a tructural model i available. Thi evaluation i baed on a model updating procedure uing previouly etimated enitivitie. The efficiency and limitation of the propoed method are illutrated uing numerical and experimental example. 1. ntroduction A dynamic tranformation, reulting from a variety of caue, e.g. tructural damage or nonlinearity onet, may diturb or threaten the normal working condition of a ytem. Hence, quetion of the detection, localization and everity etimation of thoe event have attracted the attention of countle engineering reearcher in recent time. The detection, localization and aement of the damage allow to reduce maintenance cot and to enure afety. n the lat decade, the problem of damage localization and aement ha been approached from many direction. Often baed on monitoring modal feature, thee procee can be achieved by uing an analytical model and/or promptly by meaurement. Damage can caue change in tructural parameter, involving the ma, damping and tiffne matrice of the tructure. Thu many method deal directly with thee ytem matrice. The Finite Element Method (FEM) i an efficient tool in thi proce [1]. The problem of detection may be reolved by thi method through model updating or enitivity analyi. For damage localization and evaluation, model updating i utilized to recontruct the tiffne perturbation matrix [2]. Thi may be combined with a genetic

2 algorithm [3] or baed on modal parameter enitivity [4]. n thee cae, a well fitted numerical model i eential to compare with the actual ytem. Method uing meaurement are alo widely ued becaue of their availability in practice. Yan and Golinval [5] achieved damage localization by analyzing flexibility and tiffne without ytem matrice, uing time data meaurement. Koo et al. [6] detected and localized low-level damage in beam-like tructure uing deflection obtained by modal flexibility matrice. Following localization, Kim and Stubb [7] etimated damage everity baed on the mode hape of a beam tructure. Rucka and Wilde [8] decompoed meaured FRF by continuou wavelet tranform in order to achieve damage localization. Cao and Qiao [9] recently ued a novel Laplacian cheme for damage localization. Other author have located damage by comparing identified mode hape [10] or their econd-order derivative [11] in varying level of damage. Sampaio et al. [12] extended the method propoed in [11] through the ue of meaured FRF. Natural frequency enitivity ha alo been ued extenively for the purpoe of damage localization. Ray and Tian [10] dicued the enitivity of natural frequencie with repect to the location of local damage. n that tudy, damage localization involved the conideration of mode hape change. Other author [13-15] have located damage by meauring natural frequency change both before and after the occurrence of damage. However, uch method, baed on frequency enitivity with repect to damage variable require an accurate analytical model. Jiang and Wang [16] extended the frequency enitivity approach by eliminating that requirement. However, an optimization cheme i till needed to etimate the unknown ytem matrice through an identified model uing input-output meaurement data. t i not only the iue of localization that ha become the ubject of recent tudy; the aement of damage i alo increaingly attracting the interet of reearcher [2-4, 7, 14]. Yang et al. [17] etimated damage everity by computing the current tiffne of each element. They ued Hilbert-Huang pectral analyi baed only on acceleration meaurement uing a known ma matrix aumption. Taking an alternative approach, other author have ued method involving the updating of a finite element model of the examined tructure and have ued enitivity analyi to dicover the effective parameter. For example, Meina et al. [13] etimated the ize of defect in a tructure baed on the enitivity of frequencie with repect to damage location where all the tructural element were conidered a potentially damaged ite. Teughel and De Roeck [18] identified damage in the highway bridge. They updated both Young modulu and the hear modulu uing an iterative enitivity baed FE model updating method. Thi tudy focue on the ue of enitivity analyi for reolving the problem of damage localization and evaluation. Natural frequencie are known to be ucceful in characterizing dynamical ytem. Mode hape, meanwhile, have been conidered effective in model updating, ince thee hape condene mot of the deformation databae of the tructure. Here, we ue not only enitivity of frequency, but alo of mode hape, a ubject which appear le developed in the literature. A modal identification i not neceary for the objective of localization. n monitoring the ditortion of a enitivity vector, the localization may be carried out in the firt tep. An analytical model i then needed for model updating, and thi enable the aement of the damage. 2. Senitivity analyi for Principal Component Analyi The behavior of a dynamical ytem depend on many parameter related to material, geometry and dimenion. The enitivity of a quantity to a parameter i decribed by the firt and higher order of it partial derivative with repect to the parameter. Senitivity analyi of modal parameter may be a ueful tool for uncovering and locating damaged or changed component of a tructure. On one hand, we know that the dynamic behavior of a ytem i fully characterized by it modal parameter which reult from the reolution of an eigenvalue problem baed on the ytem matrice (when a model i available). On the other hand, principal component analyi (PCA) of the repone matrix of the ytem i alo a way to extract modal feature (i.e. principal direction) which pan the ame ubpace a the eigenmode of the ytem [19]). The econd approach baed on PCA i ued in thi tudy to examine modal parameter enitivitie. X m N Let u conider the obervation matrix which contain the dynamic repone (naphot) of the ytem where m i the number of meaured co-ordinate and N i the number of time intant. We will aume that it

3 depend on a vector of parameter p. The obervation matrix X can be decompoed uing Singular Value Decompoition (SVD): X=Xp ( ) =UΣV T (1) where U and V are two orthogonal matrice, whoe column repreent repectively left and right ingular vector; Σ contain ingular value of decending importance: σ1 > σ2 >... > σ m. A enitivity analyi i performed here by taking the derivative of the obervation matrix with repect to p: X = U T + Σ T ΣV U V + UΣ V p p p p T (2) Through thi equation, the enitivity of the ytem dynamic repone how it dependence on the enitivity of each SVD term. So, the determination of U p, Σ p and V i neceary. Junkin and Kim [20] developed a p method to compute the partial derivative of SVD factor. The ingular value enitivity and the left and right ingular vector enitivity are imply given by the following equation: m m σ i T X Ui k Vi k = Ui Vi ; = α p p jiu j ; = β jivj (3) k k p k j = 1 p k j = 1 The partial derivative of the ingular vector are computed through multiplying them by projection coefficient. Thee coefficient are given by equation (4) for the off-diagonal cae and by equation (5) for the diagonal element. T T k 1 T X T X k 1 T X T X αji = σi Uj Vi + σ j Ui Vj ; βji = σ j Uj Vi + σi Ui V j σ p i σ j k pk p σi σ j k pk, j i (4) 1 σ 1 T k k T X σ i k k T X i αii βii = Ui Vi ; αii βii = +, σ σ Vi U i j = i (5) i pk pk i pk pk The enitivity analyi for PCA may alo be developed in the frequency domain, e.g. by conidering frequency repone function (FRF) [19]. A the dynamical ytem matrice depend on a vector of parameter p, the FRF matrix take the form: 1 2 H( ω, p) = ω ( ) + ω ( ) + ( ) M p i C p K p (6) where ω repreent the circular frequency. With regard to enitivity analyi, the partial derivative of equation (6) with repect to one parameter p k may be written [19], [24]: 2 H ( ω M+ iωc+ K) 2 M C K = H( ω, p) H( ω, p) = H( ω, p) ω iω H( ω, p ) (7) pk pk pk pk pk Equation (7) provide a way of determining the derivative of the FRF matrix needed for the enitivity analyi by mean of the partial derivative of the ytem matrice. Let u conider the FRF for a ingle input at location, and build a ubet of the FRF matrix (6): h1( ω1) h1( ω2)... h1( ωn ) h2( ω1) h2( ω2)... h2( ωn ) H ( ω) = (8) h ( ω1) ( ω2)... ( ω ) m hm hm N where m i the number of meaured co-ordinate and N i the number of frequency line. Thi matrix i the frequency domain analog of the obervation matrix X. The row in (8) repreent the repone at the meaured degree of freedom (DOF), while the column are naphot of the FRF at different frequencie. We conider that thi matrix depend on a given et of parameter. We can ae it principal

4 component through SVD by (1) where the left ingular vector give patial information, the diagonal matrix of ingular value how caling parameter and the right ingular vector repreent modulation function depending on frequency. n other word, thi SVD eparate information depending on pace and on frequency. The enitivity of the i th principal component can be computed by (3)-(5). Firt, we compute the SVD of the FRF matrix in (8) for the et of repone and the choen input location. Then, the partial derivative of (8) are determined uing equation (7). For a particular input, only a ubet of the derivative in (7) i needed. 3. Damage localization baed on enitivity analyi of the FRF matrix n the following, enitivity analyi i ued to reolve the problem of damage localization and evaluation. We preent now ome implification that may be carried out in experimental practice. Giving the FRF matrix H for a ingle input at location of the ytem and it SVD, the enitivity computation of the principal component (PC) require the partial derivative H / p k which are a ubet of H / p k. Thi quantity may be aeed by (7) requiring the partial derivative of the ytem matrice with repect to ytem parameter. f the parameter concerned i a coefficient k e of the tiffne matrix K, the partial derivative of the ytem matrice are uch that M / pk and C / pk equal zero and K/ pk = K / ke. Although only a ubet of H / p k i needed for a particular input, i.e. H / p k, which correpond to the th column of H / p k, the calculation of (7) demand the whole matrix H, which turn out to be cotly. However, we can compute H / by meauring only ome column of H, a explained below. p k We recall that our parameter of interet i a coefficient k e of the tiffne matrix K. Equation (6) how that FRF matrice are ymmetric if ytem matrice are ymmetric. n experiment, the number of degree of freedom (DOF) equal the number of repone enor. So, the FRF matrix ha the ame ize a the number of enor. Let u conider for intance a tructure intrumented with 4 enor. The FRF matrix take the ymmetrical form: a b c d ( ω) = b e f g H (9) c f h i d g i k Auming that k e accord to the econd DOF only, we have: M / pk = 0 ; C / pk = 0 and: K K = = (10) pk ke Equation (7) allow u to deduce the partial derivative of the FRF matrix: [ b e f g] b H K [ b e f g] e = H( ω, p) H( ω, p ) = [ ] (11) pk pk b e f g f [ ] b e f g g To compute the enitivity of H, only the th column of H / p k i needed, which i written in (12) in etting T H k = [ b e f g]. Thi relie entirely upon the column correponding to k e in the FRF matrix in equation (9). e H ke ke, p k = H H (12)

5 H i the th element of the vector k e, H k e entire matrix H ; only the column relating to k e i needed. 4. Localization indicator. Thu, the enitivity of H with repect to k e doe not involve the When H / p k ha been computed, the enitivity of principal component can be determined next uing equation (3)-(5). The enitivitie of the left ingular vector are good candidate for reolving localization problem of linear-form tructure, e.g. chain-like or beam-like tructure. n each working condition of the R ytem, we can compute the enitivity U i / pk. The reference tate i denoted by U i / pk, and the deviation of the current condition may be aeed a follow: R Δ=Δ( Ui / pk) = Ui / pk U i / pk (13) Other indicator may be utilized to better locate dynamic change, uch a: 1 1 d j = ( Δ Δ j j 1 ) and d j = 2 2 Δ Δ +Δ j+ 1 j (14) r r j 1 where r i the average ditance between meaurement point. The indicator d and d, effectively comparable with the firt and econd derivative of vector Δ, may allow the maximization of ueful information for damage localization. The formula d i widely identified in the literature of damage localization, e.g. in [11, 12]. However, the previou method compared mode hape vector or FRF data. n thi tudy, the enitivity of ingular vector i the ubject under examination. 5. Damage evaluation Once the damage ha been localized, it may then be aeed by a technique of model parameter updating. We firt aemble the modal feature of the ytem coming from the SVD of the FRF matrix (8) into a vector v called the model vector. Principal component (PC) in U or their energie in Σ may be conidered to contruct the model vector. n the literature, PC vector have been often conidered a more convenient for damage detection o they will be ued here. The model vector v of a ytem i uually a nonlinear function of the parameter T p = [ p1... p n ] where n p p i the number of parameter. The Taylor erie expanion (limited to the firt two term) of thi vector in term of the parameter i given by: n p v vp ( ) = va + pk = va + SΔp (15) p k = 1 k where va = v p= p repreent the model vector evaluated at the linearization point p = p a a. S i the enitivity matrix of which the column are the enitivity vector. The change in parameter are repreented by Δ p = p p. The reidual vector meaure the difference between analytical and meaured tructural behavior [21]: rw = Wvr = Wv( vm v( p )) (16) where v m repreent meaured quantitie. The weighting matrix W v (according to a weighted leat quare approach) take care of the relative importance of each term in the reidual vector r. Subtituting (15) into (16) and etting r a = v m v a lead to the linearized reidual vector: r = W r = W ( v v SΔ p) = W ( r SΔp ) (17) w v v m a v a Let u define the penalty (objective) function to be minimized a the weighted quared um of the reidual vector: J = ( r T T min wrw) = min ( r Wr), W = W T v Wv (18) Equation (17) may be olved from the objective function derivative J/ Δ p = 0, which produce the linear equation: WSΔ p= Wr, for which the olution i: v v a T 1 T Δ p = ( S WS) S Wra a (19)

6 n our approach, we will aume that the number of meaurement i larger than the number of updating parameter, which yield an overdetermined ytem of equation. Note that the conditioning of the enitivity matrix S play an important role in the accuracy and the uniquene of the olution. The olution may be implemented uing QR decompoition or SVD, which allow to check the conditioning of S [21]. The updating proce i hown in Figure 1 where the i th PC i conidered a the model vector. The two eparated branche correpond to: 1) the analytical model which will be updated; 2) experimental damage repone, input poition and correction parameter. The choice of correction parameter may be baed on the damage localization reult. Analytical model KMC,, ( ω 2 iω ) 1 2 ( ω ω ) H= M+ C+ K H/ p = H M/ p i C/ p K/ p H k k k k Experiment: H input, parameter p H, H / p k [ U, Σ, V ] = vd( H ) [ U, Σ, V] = vd( H ) U / p S, W i k i r = U U a i i T 1 T Δ p = ( S WS) S Wr pn = pn 1 +Δp a Kn = Kn 1 +Δp p n p n 1 (min J) + p Figure 1: Updating diagram 5.1. Choice of weighting matrix The poitive definite weighting matrix i uually a diagonal matrix whoe element are given by the reciprocal of the variance of the correponding meaurement [22]. Thi matrix may be baed on etimated tandard deviation to take into account the relative uncertainty in the parameter and meaurement: W = diag( w1, w2,..., wi,..., wm) ; W = Var with Var = diag( σ1, σ2,..., σi,..., σm) (20) where m i the number of meaurement and σ i i the tandard deviation of the i th meaurement. The relationhip between W and the variance matrix Var i aumed to be reciprocal becaue a correct data ha a mall variance but preent a ignificant weight in the etimate. Similarly, in our problem of damage evaluation, the weight may be intenified according to damaged location in order to improve the efficiency of the technique, i.e. to accelerate the convergence and raie the accuracy.

7 6. Application to damage localization 6.1. Numerical example of a cantilever beam Let u examine a teel cantilever beam with a length of 700 mm, and a quare ection of dimenion 14 mm. The beam i modeled by twenty finite element a illutrated in Figure 2. The input location i choen at node 7 and the naphot matrix i aembled from FRF correponding to the vertical diplacement at node 1 to 20. Figure 2: Cantilever beam We model the damage by a tiffne reduction of a beam element. Four tate are examined: the reference (healthy) tate, and 3 level of damage (L 1, L 2, L 3 ) induced by a reduction of tiffne of repectively 10%, 20% and 40%. The damage i aumed to occur in element 12. Note that the maximum deviation on the firt three frequencie from the reference tate are 0.70%, 1.41% and 2.81% for the 3 level repectively. For illutration, enitivity analyi reult are hown in Figure 3 according to the parameter p k = k 15. Note that imilar reult were obtained for other tiffne parameter in variou poition. The FRF were conidered in the frequency range from 0 Hz to 165Hz at interval of 1 Hz. Δ U 1 / k 15 of the firt left ingular vector with repect to the coefficient Figure 3 how the enitivity difference ( ) aociated to the 15th DOF in the tiffne matrix and it derivative d, d. t i oberved that the Δ( / k ) Figure 3: Δ( / k ) U 1 15, d, d for 3 level with pk = k 15 d how a dicontinuity d allow u to dicover explicitly the poition of the U 1 15 curve are dicontinuou at DOF 11 and 12; index with large variation around element 12 and finally, index damaged element. The method reveal itelf robut when damage develop in everal element. Localization reult are hown in Figure 4 in two different cae of damage. We note that index d doe not indicate the ame level of damage in the damaged element. However, the damage location are accurately indicated. The difference in magnitude i due to unequal enitivitie for variou damage location, a dicued in [10]. Figure 4: d for 3 level, damage at element 12, 13, 14 (left, pk = k 8 ); 7 and 16 (right, pk = k 18 )

8 Damage evaluation n thi ection, the beam i examined with the model of 10 element. We conider firtly the damage occurring in a ingle element. For example, the tiffne of element 5 i ubjected to reduction. FRF matrice are conidered for an excitation at node 9. Frequency line are elected from 0 to 200 rad/ec. Becaue the damage localization i readily identified, the firt PC and it derivative with repect to the damaged element are conidered. For the ake of efficiency and accuracy, only the tiffne of the damaged element i taken into account in the parameter vector p. The diagonal weighting matrix defined by (20), a dicued above, i intenified in entrie according to damaged element in order to increae the calculation efficiency. For thi purpoe, the tandard deviation of the principal component (PC) vector element i preliminarily aumed to be 10% of the correponding element in the conidered PC. And in particular, to take advantage of enitivity vector, the entrie matched to the damaged element are aigned a lower variance o a to give a larger weighting in the algorithm. A fitting variance for meaurement in the damaged element can improve ignificantly the convergence peed. n the tet below, the tandard deviation value according to the damaged element were multiplied by a factor between 1% and 3%. The evaluation of ome of the damage i hown in Figure 5. Several level are conidered o that the tiffne of element 5 i reduced of 5%, 10%, 20%, 30% and 50%. The data ha alo been perturbed with 5% of noie, but it till how atifactory reult. Figure 5: Evaluation of damage, noie free ( ) and noie at 5% ( ) Now let u examine the problem of ome element being damaged; thoe element may be ide by ide or ditant from each other. The reult of variou occurrence of damage are given in Figure 6 for the ame or for different level, repectively. 5% of noie wa taken into account. Figure 6: Evaluation of damage when tiffne reduction occurred in ome element

9 t i worth noting that the accuracy of the reult depend on everal factor when many element are damaged, namely, the relative poition of the damaged element and the difference between their level of damage. nput poition alo play an important role on the number of iteration of the updating proce. Generally, the etimate i more effective if the degree of damage of different element are not too different. Data recorded from an excitation cloe to the damaged location often accelerate the convergence peed Experiment involving a ma-pring ytem The next example involve the ytem of eight DOF hown in Figure 7 and for which data are available in [23]. The ytem comprie 8 tranlating mae connected by pring. n the undamaged configuration, all the pring have the ame contant: 56.7 kn/m. Each ma weigh gram; the weight i gram for the ma located at the end which i attached to the haker. Figure 7: Eight degree of freedom ytem The acceleration repone and alo the FRF of all the mae are meaured with the excitation force applied to ma 1 the firt ma at the right-hand end (Figure 7). The FRF are aembled o a to localize the damage by the propoed method. Frequency line are elected from 0 to 55 Hz at interval of Hz. Two type of excitation are produced: hammer impact and random excitation uing a haker. Firt, everal experiment were implemented with the ytem in the healthy tate (denoted H ) and in the damage tate (denoted D ). Then damage wa imulated by a 14% tiffne reduction in pring 5 (between mae 5 and 6). A the excitation wa applied only on ma 1, the partial derivative wa taken with repect to the firt DOF (equation (9)-(12)). The damage index d i hown in Figure 8, where the healthy tate are denoted H and the damaged tate D. The damage index mark a clear ditinction between the two group healthy and damaged. Healthy tate how regular indexe in all poition, o they do not diplay any abnormality. By contrat, all the damaged curve reveal a high peak in point 6 or 5 where the lope i the mot noticeable. Damage evaluation n order to evaluate the damage, it i neceary to build an analytical model. The firt PC and it enitivity are aeed in all condition for the damage evaluation. So, for the ake of conitent data, a frequency line i elected from 6 to 29 Hz, which cover only the firt phyical mode and remove noiy low frequencie. The variation in tiffne of pring 5 are hown in Figure 9 for impact and random excitation repectively. n both type of excitation, the reult are very atifactory. All fale-poitive indicate mall tiffne variation in the element concerned, and they how any detection of damage. By contrat, all damaged tate give an evaluation very cloe to the exact damage a tiffne reduction of 14%.

10 Figure 8: d by impact (left) and random excitation (right) 7. Concluion Figure 9: Evaluation of damage in cae of impact (left) and random (right) excitation The enitivity computation of principal component by analytical method ha been verified in [19] in both the time domain and the frequency domain. The contribution of the preent tudy i it application of enitivity analyi in the frequency domain to the problem of damage localization and evaluation. Damage localization i achieved in the firt tep a a reult of the difference in principal component enitivity between the reference and the damaged tate. The method ha proved efficient in damage localization in circumtance where either only one or where everal element are involved. A enitivity computation from FRF i eay, the technique hould be uitable for online monitoring. The analytical model i not required in damage localization but it i neceary in carrying out damage evaluation. 8. Reference [1]. Huynh D., He J. and Tran D., Damage location vector: A non-detructive tructural damage detection technique, Computer and Structure 83, pp , [2]. Koh B. H. and Ray L. R., Localiation of damage in mart tructure through enitivity enhancing feedback control, Mechanical Sytem and Signal Proceing, 17(4), pp , [3]. Gome H. M. and Silva N. R. S., Some comparion for damage detection on tructure uing genetic algorithm and modal enitivity method, Applied Mathematical Modelling, pp , 2008.

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