A New Structural Damage Identification Method Based on Wavelet Packet Energy Entropy of Impulse Response

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1 Send Orders for Reprins o reprins@benhamscience.ae 570 The Open Civil Engineering Journal, 2015, 9, Open Access A New Srucural Damage Idenificaion Mehod Based on Wavele Pace Energy Enropy of Impulse Response Can He 1, Jianchun Xing 1,*, Juelong Li 2, Wei Qian 2 and Xun Zhang 1 1 College of Defense Engineering, PLA Universiy of Science and Technology, Nanjing, , China 2 Technical Managemen Office of Naval Defense Engineering, Beijing, , China Absrac: Exciaion maes a grea influence on he wavele energy disribuion of he response signal, his deficiency leads ha he radiional srucural damage idenificaion mehod based on wavele energy has a low precision. In order o solve his problem, a new srucural damage idenificaion mehod based on wavele pace energy enropy (WPEE) of impulse response is presened in his paper. Firsly, naural exciaion echnique (NExT) is adoped o exrac srucural impulse response. Then, WPEE of he impulse response is compued, and he change rae of WPEE is used o consruc he srucural damage index. An experimen of damage idenificaion on a pile srucure is provided o verify he effeciveness of he proposed mehod. Experimen resuls show ha his mehod can accuraely idenify he single damage and muli-damage. Keywords: Impulse response, srucural damage idenificaion, wavele pace energy enropy. 1. INTRODUCTION Due o srong wind force, orrenial rain, severe earhquae, explosion and oher abnormal loads, large-scale civil engineering may be damaged in is long service period [1]. Once srucural damage occurs, i will reduce he durabiliy of he srucure, even hrea he safey of he srucure. Therefore, srucural damage idenificaion has become an imporan research opic in he engineering proecion field [2]. In he pas few years, many echnologies have been developed o achieve srucural damage idenificaion, such as: idenificaion mehods based on ime domain analysis [3, 4], idenificaion mehods based on modal parameers [5, 6] and idenificaion mehods based on ime-frequency domain analysis [7, 8]. Among hese mehods, because of low enropy and muli-resoluion, wavele ransform has become a popular echnology in he field of srucural damage idenificaion. Kim e al. [9] found ha he occurrence and developmen of srucural damage could be accuraely idenified by monioring he changes of wavele coefficiens of he response signal. Hou e al. [10] proposed a new damage idenificaion mehod by deecing he singulariy of he signals based on wavele ransform, and he mehod had been successfully applied o he Benchmar model which was provided by he American Sociey of Civil Engineers (ASCE). Zio e al. [11] applied he coninuous wavele ransform o analyze he acousic emission signals from dynamic ess conduced on a reinforced concree slab wih a shaing able, he resuls showed his mehod can idenify he crac in he reinforced concree slab accuraely. Ding e al. [12] firsly achieved wavele pace energy specrum (WPES) by decomposing *Address correspondence o his auhor a he College of Defense Engineering, PLA Universiy of Science and Technology, Nanjing, , China; Tel: (86) ; Fax: (86) ; xjc893@163.com he wavele pace of he response signal, and hen compued he change rae of he WPES o deermine wheher he srucural damage occurs or no. This mehod has been applied in he Runyang Yangze River Bridge and achieved good resuls. According o he energy-damage heory, Li e al. [13] proposed a srucural damage idenificaion mehod based on wavele pace analysis and neural newor, which was esed on he Benchmar model and achieved saisfacory idenificaion effecs. As a meric of he uncerainy sae, informaion enropy is inroduced ino he field of srucural damage idenificaion. Xie e al. [14, 15] consruced srucural damage index based on he ransfer enropy and approximae enropy of he srucural response. By combining informaion enropy wih wavele analysis echnology, Ren e al. [16] proposed o use of he wavele ime enropy and he relaive wavele enropy as srucural damage index. Alhough he srucural damage idenificaion mehods have developed significanly, some deficiencies remain. The primary defec is ha mos exising srucural damage index varies wih he changes of exciaion, herefore, i is difficul o deermine ha he change of damage index is derived from he changes of exciaion or he real damage. In order o solve his problem, a new srucural damage idenificaion mehod based on wavele pace energy enropy (WPEE) of impulse response is presened. The mehod firsly exracs he impulse response by naural exciaion echnique (NExT). Then, a new damage index is consruced based on change rae of WPEE. The remaining pars of he paper are organized as follows: Secion 2 inroduces he wavele pace heory and informaion heory. Secion 3 proposes a new damage index based on change rae of WPEE. Secion 4 provides a simulaed experimen using a pile srucure o verify he effeciveness of he new mehod, and he conclusion is given in Secion / Benham Open

2 A New Srucural Damage Idenificaion Mehod The Open Civil Engineering Journal, 2015, Volume BACKGROUND 2.1. Wavele Pace Theory Wavele ransform is considered as an effecive signal processing echnology in ime-frequency domain. However, wavele scale funcion is changed as a binary sysem, which leads o a poor frequency resoluion in he high frequency band. Wavele pace ransform provides a more precise mehod on signal processing. Furher decomposiion of high frequency par is carried ou by wavele pace ransform. According o he characerisic of he signal, wavele pace ransform can adapively selec he appropriae frequency band o mach wih he signal specrum. Therefore, wavele pace ransform can increase he ime-frequency resoluion. Wavele pace is a subdivision of wavele subspace in accordance wih a binary sysem. In order o achieve wavele n pace, i firsly requires consrucing a new space U j by combining scale subspace V j wih wavele subspace W j. Then, i needs o compue he orhonormal basis of he space, he orhonormal basis is he wavele pace. If U 0 j and U 1 j can be represened as: U j 0 = V j, U j 1 = W j, j Z (1) 0 U j+1 Then, he decomposiion relaions may be expressed as: = U j 0 +U j 1, j Z (2) I is assumed ha ( x) and ( x) are he orhogonal scale funcion and wavele funcion, respecively. They are defined as follows: μ 0 (x) = (x) μ 1 (x) = (x) A funcion cluser can be defined in accordance wih he following recursive clusering mehods: u 2n () = 2 h()u n (2 ) (4) Z u 2n+1 () = 2 g()u n (2 ) (5) Z where g() = (1) h(1 ). When n=0, Equaion (4) and (5) can be changed as: u 0 () = h u 0 (2 ) (6) Z u 1 () = g u 0 (2 ) (7) Z In he muli-resoluion analysis, ( x) and ( x) mee he wo-scale equaions: () = h (2 ) {h } z l 2 (8) z () = g (2 ) {g } z l 2 (9) z (3) u () and 0 u () 1 are he scale funcion and wavele funcion, respecively. The orhogonal wavele pace can be consruced by{ u ( )}( n Z + ). n If n is a parameer of refinemen muliplier and n = 2 l + m, hen he wavele pace can be simplified as: j,,n () = 2 j/2 n (2 j ) (10) l/2 l where n() = 2 un(2 ), jn,,() is considered as he wavele pace wih a scale indicaor j, posiion indicaor and frequency indicaor n. If g j n () U j n, hen g j n can be represened as: g n j () = d j,n l u n (2 j l) (11) l Wavele pace decomposiion algorihm can be represened as: d l j,2n = d l j,2n+1 = j+1,n a 2l d j+1,n b 2l d (12) (13) As he scale increases, he orhogonal wavele basis funcions has a lower frequency resoluion, which is he defec of he orhogonal wavele. Wavele pace can mae furher segmenaion of frequency band, hereby increasing he frequency resoluion Informaion Enropy Informaion enropy was proposed by C. Shannon in 1948, which is considered as a measure of he uncerainy of a random even and described by probabiliy disribuion funcion. I is assumed ha he probabiliy of each resul is he vecor p=( p 1, p 2,, p m ), which saisfies he following equaion: m p i i=1 = 1 (14) where 0 p i 1, i = 1, 2, m. Funcion H(.) was inroduced o define he informaion enropy. H m ( X ) = H m (p 1, p 2,, p m ) = p i log p i (15) where 0 is a consan. H(m) is he informaion enropy, which represens he uncerainy of a random even by probabiliy disribuion funcion. 3. A NEW DAMAGE IDENTIFICATION INDEX BASED ON WPEE 3.1. Impulse Response Exracion Based on NExT NExT is firsly proposed by JAMES and CARNE in 1995 [17], whose basic idea is ha he cross-correlaion m i=1

3 572 The Open Civil Engineering Journal, 2015, Volume 9 He e al. funcion of response signals and impulse response funcion have a similar form under he whie noise exciaion. Therefore, he cross-correlaion funcion of response signals can be considered as impulse response o achieve srucural damage idenificaion. The equaion of moion for he n DOF dynamic sysem is represened as: Mx() + Cx() + Kx() = f () (16) where M is he srucural mass marix, C is he srucural damping marix, K is he srucural siffness marix, x() is he displacemen vecor and f ()is he inpu exciaion. When poin is subjeced o exernal exciaion f (), he measured response of poin i x () can be compued as: 2 N x i = ir e r ( p) f ( p)dp (17) r=1 where ir is he rh mode of poin i, m r is a consan associaed wih exciaion poin and modal order r. If he response of poin j is xj (), hen he crosscorrelaion funcion of xi () and xj () can be compued as: = E[x i ( + )x j ()] = + e r (+ p) 2 N 2 N r=1 S=1 i ir js m s e r ( ) E[ f ( p) f (q)]dpdq (18) If he exciaion poin is he uni impulse force, hen he impulse response of poin i can be achieved as: 2 N h i = ir e r (19) r=1 I is assumed ha f () is whie Gauss noise, we can ge he following equaion based on he definiion of he crosscorrelaion funcion. E[ f ( p) f (q)] = m ( p q) (20) where () is he impulse funcion, m is a consan associaed wih he exciaion poin. Combined Equaion (18) wih Equaion (20), we can ge Equaion (21). 2 N 2 N ( ) = ir js m s m e r (+ p) e r ( ) dp (21) r=1 s=1 Inegral par of he Equaion (21) can be simplified as: = e r (+ p) e r ( ) dp (22) r + s e r So, Equaion (21) can be wrien as: 2 N 2 N ( ) = ir js m s m ( e r ) (23) r + s r=1 s=1 If n 2N jr = jsmr msm s= 1 r + s forms can be obained: 2 N 1 ( ), hen he following ( ) = n jr ir e r (24) r=1 Compared Equaion (19) wih Equaion (24), i can be seen ha he cross-correlaion funcion of response signals and impulse response can be boh expressed as he summaion of complex exponenial funcions. Therefore, he cross-correlaion funcion of response signals can be considered as impulse response o achieve srucural damage idenificaion A New Srucural Damage Idenificaion Index According o he wavele heory, when wavele funcions are a se of orhogonal basis funcions, he wavele ransform has he propery of energy conservaion. Therefore, Equaion (25) can be provided as: N x(), j, () 2 = x 2 (25) j=1 The definiion of wavele energy is provided as: E j = C j () 2 (26) where Cj ( ) is he wavele coefficiens of scale j. Wavele energy represens he quadraic sum of wavele coefficiens under a single scale. So, he oal wavele energy can be expressed as: E = E j = C j () 2 (27) j j The definiion of he relaive wavele energy can be provided as: p j = E j (28) E p j can express he energy disribuion of differen scales. I is a good ool o deec he abnormal poins of signals. Based on he informaion enropy heory and he definiion of he relaive wavele energy, he WPEE can be compued as: S w = p j log p j (29) j=1 where log is he logarihm using base 2. In order o accuraely describe he changes of he srucural sae propery, he change rae of WPEE is proposed as damage index in his paper, which is defined as follows: CR() = S w () S w (0) S w (0) (30)

4 A New Srucural Damage Idenificaion Mehod The Open Civil Engineering Journal, 2015, Volume where S w (0) is WPEE of he inac srucure. Sw () is WPEE of he srucural sae a ime. CR is he change rae of WPEE. CR expresses he changes of WPEE under differen srucure sae. Firsly, WPEE of he inac srucure is considered as a meric. If he srucures don damage in a period of ime, he dynamic characerisic of he srucure is relaively sable, he change rae of WPEE is very small. When srucural damage occurs, he response signal will change significanly. A his ime, WPEE will increase dramaically. Due o his feaure, change rae of WPEE can be used o idenify he srucural damage. The process of srucural damage idenificaion using CR is provided as Fig. (1). Sar Fig. (2). FE model of he pile srucure. Exracing cross-correlaion funcion using NexT Compuing he CR using Equaion (30) Comparing he CRs of differen elemens o idenify damage End Fig. (1). The flowchar of he proposed mehod. Fig. (3). Whie Gauss noise exciaion. 4. DEMONSTRATION CASE 4.1. Finie Elemen Model In order o verify he effeciveness of he proposed mehod, a case sudy of srucural damage idenificaion on a numerically simulaed pile srucure is provided. The oal lengh of pile is 10m, he cross-secional area of he pile is 0.4*0.4m 2. The finie elemen (FE) model of he pile is esablished using commercial sofware ANSYS, he pile is modeled by he elemen of Beam 4. The pile is divided ino 15 elemens, and 16 nodes are generaed in oal. Maerial parameers of he pile are se as: he elasic modulus E= N/m 2, Poisson raio μ=0.17, densiy =2500g/m 2. The FE model is shown in Fig. (2). The whie Gauss noise load is generaed as inpu exciaion. The duraion ime of load is se as 5s, he sep is se as 0.01s. The whie Gauss noise load is shown in Fig. (3). Node 1 is se as a reference node, and he response signal of node 1 is shown in Fig. (4). Fig. (4). Response signal of node A Single Damage Idenificaion Damage of he elemen is simulaed by reducing is siffness. In his case sudy, i is assumed ha he damaged

5 574 The Open Civil Engineering Journal, 2015, Volume 9 He e al. Fig. (5). A single damage locaion. Fig. (7). MSECR index when elemen 4 is damaged. Fig. (8). WPES index when elemen 4 is damaged. Fig. (6). CRs of all he elemens when elemen 4 is damaged. elemen is elemen 4, he siffness of elemen 4 reduced by 20%. The damage locaion is shown in Fig. (5). Node 6 is seleced as he response node. The crosscorrelaion funcion of node 1 and node 6 is compued by Equaion (24). The CRs of all he elemens are compued in accordance wih he mehod in Fig. (1). The resul is given in Fig. (6). In order o prove he validiy and superioriy of our mehod, MSECR index proposed in [6] and WPES index proposed in [12] are seleced as comparaive experimens. The resuls are shown in Figs. (7 and 8) Muli-Damage Idenificaion A new experimen is provided o show he muli-damage idenificaion effec of he proposed mehod. I is assumed ha elemen 4 and 9 are boh damaged (see Fig. 9), he siffness of elemen 4 and 9 are reduced by 20%. The experimen resul is shown in Fig. (10). Fig. (9). Muli-damage locaions.

6 A New Srucural Damage Idenificaion Mehod The Open Civil Engineering Journal, 2015, Volume Resul Analysis As seen in Fig. (6), he CR of elemen 4 is 12.1%, which is much bigger ha he oher elemens. Therefore, we can obviously idenify ha he elemen 4 is damaged. Fig. (9) shows ha he CRs of elemens 4 and 9 are 9.7% and 8.4% respecively, which are boh bigger ha he oher elemens. So, he wo damages can be boh idenified by comparing CR. As shown in Figs. (7-9), our mehod can more clearly idenify he single damage. From Figs. (10-12), we can see our index is more sensiive for muli-damage idenificaion. Fig. (10). CRs of all he elemens when elemen 4 and elemen 9 damages. MSECR index and WPES index are seleced as comparaive experimens. The resuls are shown in Figs. (11 and 12). Fig. (11). MSECR index when elemen 4 and elemen 9 damages. Fig. (12). WPES index when elemen 4 and elemen 9 damages. 5. CONCLUSION The radiional idenificaion mehod has a low idenificaion precision. In order o solve his problem, a new srucural damage idenificaion mehod based on WPEE of impulse response is presened. The mehod firsly achieves he srucural impulse response by calculaing he crosscorrelaion funcion of he differen signals. Then, a new damage index is consruced based on change rae of WPEE. Wih he case analysis, some conclusions are summarized as follows: (1) The cross-correlaion funcion of response signals has he similar form wih impulse response. Therefore, crosscorrelaion funcion can be considered as impulse response o idenify he srucural damage. (2) When srucural damage occurs, he WPEE of damaged elemen will change significanly. By comparing he CR of differen elemens, he damaged elemen can be idenified accuraely. The experimen resuls show ha CR can idenify he single damage and muli-damage accuraely, which proves CR is an effeciveness srucural damage idenificaion index. CONFLICT OF INTEREST The auhors confirm ha his aricle conen has no conflic of ineres. ACKNOWLEDGEMENTS Declared none. REFERENCES [1] J. E. Padge, and C. Tapia, Susainabiliy of naural hazard ris miigaion: Life cycle analysis of environmenal indicaors for bridge infrasrucure, Journal of Infrasrucure Sysems, vol. 19, no. 4, pp , [2] D. Balageas, C. P. Frizen, and A. Güemes, Eds., Srucural Healh Monioring, ISTE, London, [3] S. F. Jiang, S. Y. Wu, and L. Q. Dong, A ime-domain srucural damage deecion mehod based on improved muliparicle swarm coevoluion opimizaion algorihm, Mahemaical Problems in Engineering, vol. 2014, Aricle ID , [4] J. Zhang, Y. L. Xu, J. Li, Y. Xia, and J. C. Li, Saisical momenbased srucural damage deecion mehod in ime domain, Earhquae Engineering and Engineering Vibraion, vol. 12, no. 1, pp , [5] W. Fan, and P. Qiao, Vibraion-based damage idenificaion mehods: a review and comparaive sudy, Srucural Healh Monioring, vol. 10, no. 1, pp , 2011.

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