Advances in Structural Engineering

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1 Frequency-Doman Substructure Isolaton for Local Damage Identfcaton by Jln Hou, Lukasz Jankowsk and Jnpng Ou Reprnted from Advances n Structural Engneerng Volume 8 No. 5 MULTI-SCIENCE PUBLISHING CO. LTD. 6 Eldon Way, Hockley, Essex SS5 4AD, Unted Kngdom

2 Frequency-Doman Substructure Isolaton for Local Damage Identfcaton Jln Hou,*, Lukasz Jankowsk and Jnpng Ou, School of Cvl Engneerng, Dalan Unversty of Technology, Dalan 64, Chna Insttute of Fundamental Technologcal Research, Polsh Academy of Scences, Warsaw, Poland School of Cvl Engneerng, Harbn Insttute of Technology, Harbn 59, Chna (Receved: March 4; Receved revsed form: July 4; Accepted: August 4) Abstract: Ths paper proposes a frequency-doman method of substructure dentfcaton for local health montorng usng substructure solaton method (SIM). The frst key step of SIM s the numercal constructon of the solated substructure, whch s a vrtual and ndependent structure that has the same physcal parameters as the real substructure. Damage dentfcaton and local montorng can be then performed usng the responses of the smple solated substructure and any of the classcal methods amed orgnally at global structural analyss. Ths paper extends the SIM to frequency doman, whch allows the computatonal effcency of the method to be sgnfcantly ncreased n comparson to tme doman. The mass-sprng numercal model s used to ntroduce the method. Two alumnum beams wth the same substructure are then used n expermental verfcaton. In both cases the method performs effcently and accurately. Key words: structural health montorng (SHM), damage dentfcaton, substructurng, frequency doman, boundary.. INTRODUCTION Structural Health Montorng (SHM) has become a hot and ntensvely researched feld n cvl engneerng: montorng and damage dentfcaton play mportant roles n mantanng ntegrty and safety of structures (Y et al. ; Zhou, Yan, Wang and Ou ). Many effectve methods have been proposed for damage dentfcaton (Fan and Qao ; Moragasptya et al. ; Wang et al. ; Zhou, Yan and Ou ). However, accurate global dentfcaton of large realworld structures s not easy due to ther complex and often unknown boundary condtons, temperature effect (Xa et al. ), nonlnear components, small senstvty of global response to localzed damages, etc. Furthermore, global dentfcaton (parameter dentfcaton of the global structure) often nvolves large numbers of unknowns and sensors. Ths s costly, rarely feasble n practce, and usually yelds severely ll-condtoned dentfcaton problems. The substructurng approach seems to be a possble soluton. At present, there are two man knds of substructurng methods that can be appled to substructure dentfcaton: () Mechancal substructurng/couplng methods (MSC methods), whch assemble dynamc characterstcs of the global structure from known dynamc characterstcs of all ts substructures. In applcatons to substructural montorng, the global structure s frst separated nto several ndependent substructures, and then dynamc tests and dentfcaton are performed separately on each substructure. The substructures are then coupled back nto the global structure va dsplacement coordnaton on the nterface (Ewns, ; Maa and Slva 997, 65-). () Substructurng/decouplng methods (SD methods), whose purpose s to decouple an unknown substructure from an unknown global structure for the purpose of *Correspondng author. Emal address: houjln@dlut.edu.cn; Fax: ; Tel: Advances n Structural Engneerng Vol. 8 No. 5 7

3 Frequency-Doman Substructure Isolaton for Local Damage Identfcaton local substructural montorng or dentfcaton. To ths end, frst a local dynamc test s performed on the global structure, and then the local substructure s analyzed by takng nto account the nteracton between t and the global structure. Methods of ths knd do not separate the substructure from the global structure durng the test (Tee et al. 5). The MSC methods are usually used n mechancal or aerospace feld. For structures n cvl engneerng, only the response of a global structure can be obtaned durng ts servce perod and t s mpossble to mechancally separate ts substructures. Therefore most exstng substructural approaches for structures n cvl engneerng are substructurng/decouplng (SD) methods. The method proposed n ths paper belongs thus to SD methods. Such methods can focus on local small substructures; they need only a few sensors placed on the substructure and yeld smaller and numercally much more feasble dentfcaton problems. To detect and locate the substructural damage, one can compare locally senstve nformaton obtaned before and after damage. For example, Yun and Bahng () used natural frequences and mode shapes for local montorng of stffness modfcatons. Bao et al. () used the damage basc probablty assgnment (BPA) functon of substructures for prelmnary damage localzaton. An and Ou () presented a model updatng method that utlzes four cost functons nvolvng free vbraton acceleratons and local mode shapes to detect local damage of a truss structure. A substructure s a local part of the global structure, and so t s not ndependent of the global structure. In order to focus on the substructure only, most exstng SD methods separate the substructure from the global structure; the nterface forces are then used for couplng both structures and need to be dentfed together wth substructural parameters. The dentfcaton s performed mostly based on the equaton of moton of the substructure. Koh et al. () employed genetc algorthms (GA) as the search tool for ts advantages ncludng the ease of mplementaton and the desrable characterstcs of global search. Tee et al. (5) developed a dvde-and-conquer approach for dentfcaton at the substructural level of frst-order and second-order models. Law et al. () dentfed the couplng forces between substructures usng the damped least-squares method. Wang et al. () employed the concept of the quas-statc dsplacement vector to smplfy the nterface forces, and use a GA method to dentfy the substructure. A mult-feature GA method was used by Trnh and Koh () to estmate substructural mass, dampng and stffness parameters. Xng and Mta () confne each substructure of a mult-storey shear buldng to a few degrees of freedom only (Dofs) and use overlappng substructures, then they apply drectly the ARMAX method for dentfcaton. Zhu et al. () dentfy the structural damage, the external movng force and nterface forces of adjacent substructures smultaneously from the measured dynamc acceleraton responses. The above methods are formulated n tme doman. There are also many studes of substructure dentfcaton n frequency doman. In order to avod the need for complete nstrumentaton of the substructure, Yuen and Katafygots (6) presented an output-only bayesan frequency-doman approach for substructure dentfcaton and montorng n lnear MDOF systems. Tee et al. (9) use System Equvalent Reducton Expanson Process to condense the sub-model for substructural dentfcaton. Zhang et al. () ntroduced a control system to dentfy storey parameters n a shear structure usng Cross Power Spectral Densty (CPSD). Xa et al. () developed Kron s substructurng method to compute the frst-order dervatves of the egenvalues and egenvectors. The substructure and the global structure reman coupled by the unknown nterface forces that are exposed on the separated substructural nterface. As these forces nfluence substructural responses, the methods used for substructure dentfcaton are very dfferent from the methods of global dentfcaton, whch cannot be drectly appled to the substructure n all ther varety and flexblty. Damage of the global structure can be detected by drect sgnal processng of the constructed response usng tme seres methods (Nar et al. ) or wavelet analyss (Rucka and Wlde ), or be estmated by optmzng the fnte element (FE) model of the substructure usng ts flexblty matrx (Duan et al. 5), natural frequences and mode shapes (Hassots ), tme-doman response (Suwal / a and Jankowsk ) or frequency-doman response (Ln and Ewns 99), etc. Hou et al. () have proposed the Substructure Isolaton Method (SIM), where the solated substructure s an ndependent structure and all global dentfcaton methods can be appled to ts dentfcaton. The SIM proceeds n two stages. Frst, n the solaton stage, measured responses are drectly utlzed to construct the responses of the solated substructure, whch s an ndependent vrtual structure wth the same structural parameters as the real substructure, but solated from the global structure wth vrtual supports placed on the nterface. Then, n the dentfcaton stage, local dentfcaton of the substructure s performed based on the constructed responses of the solated substructure 8 Advances n Structural Engneerng Vol. 8 No. 5

4 Jln Hou, Lukasz Jankowsk, and Jnpng Ou and any of the standard methods amed orgnally at global dentfcaton. The selecton of the dentfcaton method usually depends on the characterstcs of the constructed responses of the solated substructure. Ths paper uses natural frequences to optmze the substructure. In the orgnal formulaton of the SIM (Hou et al. ), the solated substructure and ts responses were constructed n tme doman. It nvolved computng a soluton to a very large and extremely ll-condtoned dscrete lnear system, whch orgnated from a dscretzaton of a system of Volterra ntegral equatons of the frst knd and had the dmensons proportonal to the number of the consdered tme steps. Soluton of such a system s tme-consumng, whch sgnfcantly lmted the manageable measurement tme nterval. Ths paper formulates the SIM n frequency doman, whch dramatcally mproves ts computatonal effcency: even though sgnfcantly longer measurement tme ntervals are used, the method performs much faster. The next secton derves the SIM usng frequencydoman responses. Secton makes use of a smple mass-sprng system to ntroduce the applcaton of the method. The last secton verfes expermentally the proposed approach usng an alumnum cantlever beam.. SUBSTRUCTURE ISOLATION METHOD IN FREQUENCY DOMAIN (SIM-FD).. Constructon of the Isolated Substructure Denote by x(t) the vector of dsplacements n all Dofs of the global structure, and by M, C and K the correspondng mass, stffness, and dampng matrces. It s assumed that the Eqn of moton n tme doman can be wrtten as: Mx() t + Cx() t + Kx() t = f () t () where the vector f(t) collects all the external exctatons. The Fourer transform F, appled to both sdes of (), yelds the frequency-doman quas-statc form of the equaton of moton: ( ω M + j ω C+K ) X ( ω) = F ( ω) () where X(ω) = (Fx)(ω) and (F)(ω) = (Ff)(ω) Let the subscrpt s denote the Dofs nternal to the substructure, b the Dofs of ts nterface and r all the Dofs outsde the substructure. Henceforth, these subscrpts wll be used to mark the correspondng blocks of system matrces and the correspondng parts of response and exctaton vectors. Eqn can be thus stated n the equvalent form, Mss Msb ω M M M + bs bb br Mrb Mrr Css Csb Kss Ksb jω Cbs Cbb C br + Kbs Kbb K br Crb Crr Krb Krr Xs ( ω) Fs ( ω) X ( ) b ω = Fb ( ω) Xr ( ω) Fr ( ω) Assume that n+ external exctatons F (ω), F (ω),, F n (ω) are appled to the structure and let X (ω), X (ω),, X n (ω) be the correspondng responses. Denote by P(ω) the followng lnear combnaton of the exctatons: (4) where Z (ω) are arbtrary complex combnaton coeffcents. Let Y(ω) denote the correspondng structural response, whch s a smlar combnaton that can be stated separately for nternal and nterface Dofs as Y s = Y n b n = P( ω) = F ( ω) + Z ( ω) F ( ω), ( ω) = X ( ω) + s ( ) ( ) = ( ) + D ( ) ( ω) s s Z ω X ω X ω ω Z ( ω) = X ( ω) + Z b ( ω) X ( ω) = = ( ) + ( ) ( ) b Xb ω B ω Z ω, where the matrces D(ω) and B(ω) are composed of the respectve responses of the nternal and nterface Dofs and the vector Z(ω) collects the combnaton coeffcents, ( ) = ( ) ( ) Z ω Z ω L Zn ω, D( ω) = n Xs ( ω) L Xs ( ω), n B( ω) = Xb ( ω) L Xb ( ω ). n (6) For the reasons descrbed n Secton., F (ω)s called the basc exctaton, and the correspondng response X (ω) s the basc response; F (ω),, F n (ω), are the constranng exctatons, and the correspondng T, () (5) Advances n Structural Engneerng Vol. 8 No. 5 9

5 Frequency-Doman Substructure Isolaton for Local Damage Identfcaton responses X (ω),, X n (ω), are the constranng responses. Correspondngly, the matrces and B(ω), D(ω) are called the constranng matrces. Snce the substructure s assumed to be lnear, the lnearly combned exctaton P(ω) and the lnearly combned response Y(ω) satsfy together the equaton of moton (see Eqn ). The frst row of ths equaton can be then wrtten as ( ω M ss + jωc ss + K ss ) Ys ( ω ) = Ps ( ω ) + P c ( ω ), where P (7) (8) s the vector of the nterface forces that couple the substructure to the outsde structure va the nterface Dofs. If the combned responses of the substructure nterface vansh, Y b (ω) =, then the couplng nterface forces P c (ω) also vansh, and Eqn 7 s smplfed nto ( ω M ss + jωc ss + K ss ) Ys ( ω ) = Ps ( ω ) (9) Accordng to the second equaton of Eqn, the combned nterface responses Y b (ω) vansh and Eqn (9) holds, f the combnaton coeffcents Z(ω) satsfy ( ) ( ) ( ) = + + c sb sb sb b ω ω M jωc K Y ω = X b ( ω) + B( ω) Z ( ω) () Eqn () s a lnear equaton that s unquely solvable f the matrx B(ω) has full column rank, whch s possble only f the number n of the constranng exctatons/responses s not smaller than the number of the nterface Dofs. The soluton s gven by + ( ) = ( ) X ( ) Z ω B ω h ω () where the superscrpt + denotes the pseudo-nverse of a matrx. The correspondng combned response of the substructure can be then stated n the explct form as [ ] ( ) + Y s X s D B X b ( ω) = ( ω) ( ω) ( ω) ω () In the above analyss, the combnaton coeffcents are chosen n such a way that the combned response of the substructure nterface vansh. Ths s equvalent to addng fxed supports n all Dofs of the nterface and thus to full solaton of the substructure from the global structure. In other words: gven the properly lnearly combned exctaton P s (ω), the couplng nterface forces P c (ω) vansh and the substructure responds wth Y s (ω) as an ndependent structure, see Eqn 9. Notce that the exctaton P c (ω) and the response Y s (ω) are not drectly measured, but rather artfcally constructed from measured data. Hence, the constructed system s not physcal but vrtual. Such an ndependent vrtual substructure s called the solated substructure. It s solated from the outsde structure by addng fxed vrtual supports n all Dofs of ther nterface. The above approach requres the responses to be measured n all Dofs of substructure nterface. Therefore, t s hardly applcable to substructures wth a large number of nterface Dofs. The proposed method needs thus a proper selecton of the substructure, so that t has a smple nterface. However, the dentfcaton of the solated substructure can be performed usng standard methods of global dentfcaton, and so there s no smlar lmtaton mposed on the nteror of the substructure, whch can be complex wth a large number of Dofs... The Vrtual Support and Other Types of Sensors In the prevous secton, all the responses are assumed to be dsplacement n substructural and nterface Dofs. Consequently, the frequency response of the solated substructure s derved based on fxed vrtual supports. In fact, other knds of vrtual supports can be also used to emulate other types of boundary condtons, dependng on the type of the substructure and sensors placed on ts nterface. In the followng, solaton of a plane beam s used as an llustratve example. Fgure shows the exposed nterface of the substructure. For the sake of smplcty, the axal dsplacement of the neutral axs and the axal force are gnored for the moment. The followng physcal quanttes are of nterest on the nterface: the nternal shear force p(t) and bendng moment M(t), the vertcal dsplacement v(t), the rotaton θ(t) and the axal stran ε(t) of the beam surface. Notce that, n the absence of axal dsplacement and force, the stran ε(t) s proportonal to the bendng moment, but much easer to measure. Four dfferent types of boundary condtons can be now formulated, Interface Substructure p (t), n (t) e (t) M (t), q (t) The rest of global substructure Fgure. A beam substructure 4 Advances n Structural Engneerng Vol. 8 No. 5

6 Jln Hou, Lukasz Jankowsk, and Jnpng Ou 4 Fgure. Four types of vrtual supports vt ( ) = vt ( ) = pt ( ) = pt ( ) = 4 θ( t) = ε( t) = ε( t) = θ( t) = () whch defne four knds of vrtual supports that can be appled n the nterface to solate the substructure, see Fgure. In practce, t s usually mpossble to place physcal supports or apply the proper loads to make the nterface responses satsfy one of the four condtons lsted n Eqn. However, Secton. shows that a number of nonzero nterface responses can be lnearly combned to zero, so that the combned response satsfes the boundary condtons. As a result, the type of the vrtual support depends on the types of nterface sensors. For nstance, f v(t) and θ(t) are measured, then the vrtual fxed support can be constructed by a lnear combnaton of the measured responses. Or, f responses v(t) and ε(t) are measured, the vrtual pned support can be constructed. In real applcatons, the shear force p(t) s hard to measure, so that the rd and 4th knd of vrtual supports wll be usually not used. As n Secton., assume that there are n+ exctatons and denote the correspondng responses by Xs ( ω), Xs( ω),, Xs n ( ω) for the nterface sensors (that need to be complant n type wth one of ) and by X X X n b ( ω), b( ω),, b ( ω) for the sensors placed nsde the substructure. The substructure and all the sensors are lnear, so that Eqns 4 to 6 hold. As a result, Eqn () can be solved to fnd the combnaton coeffcents that make the combned response satsfy the boundary condtons. Fnally, the correspondng combned response of the solated substructure s gven by Eqn. Notce that for a complete solaton, Eqn needs to be exactly satsfed, whch means that B(ω) must be of full row rank. Rows of B(ω) correspond to nterface sensors and ts columns to combnaton coeffcents Z (ω), and hence the number n of constranng exctatons/responses must not be smaller than the number of the nterface sensors. If axal dsplacement of the neutral axs of the beam and axal force are to be consdered, the stran ε(t) of the surface of the beam s no longer a drect substtute for the nternal bendng moment. In such a case, two stran sensors can be placed on the opposte faces of the beam n the same dstance from ts neutral axs: the axal stress and the bendng moment wll be proportonal to the sum and to the dfference of ther measurements, respectvely... Local Damage Identfcaton Local dentfcaton of substructure damages s equvalent to damage dentfcaton of the solated substructure, whch s an ndependent system that has ts own natural frequences and mode shapes that can be found by nvestgaton of the constructed response Y s (ω) and the correspondng combned exctaton P s (ω) of the solated substructure, n s s s = ( ) = ( ) + ( ) ( ) P ω F ω Z ω F ω (4) where F s (ω) s the part of the vector F (ω) that corresponds to the nternal Dofs of the substructure, see Eqn 4, and the combnaton coeffcents are gven by Eqn to ensure proper solaton. As a result, local damage dentfcaton can be performed by optmzng a vector µ of certan parameters n the FE model of the solated substructure that are assumed to quantfy the damage. Specfc meanng of the parameters s applcaton-dependent (e.g., stffness modfcaton ratos). The optmzaton can be performed by any of the classcal methods that have been orgnally amed at global dentfcaton. Selecton of a partcular method depends on characterstcs of the constructed response, whch s dependent on the combned exctaton P s (ω) of the solated substructure. The combned exctaton s partcularly easy to fnd, f F s (ω) = for =,,n, that s f all the constranng exctatons are appled on the nterface or outsde the substructure. In such a case, the combned exctaton of the substructure equals ts basc exctaton F s (ω). The combned response Y s (ω) s then the response of the solated substructure to the basc exctaton only. In ths paper, a modal hammer s used as the exctaton n experment. Ths s a quas-mpulsve exctaton wth a broad spectrum that exctes many natural frequences of the solated substructure. The vector µ of damage parameters s dentfed by mnmzng the followng objectve functon ( ) = µ F ( ) ω µ ω m ω m (5) Advances n Structural Engneerng Vol. 8 No. 5 4

7 Frequency-Doman Substructure Isolaton for Local Damage Identfcaton where ω m s the th dentfed natural frequency of the solated substructure and ω F (µ) denotes the natural frequences of ts FE model;.4. Fast Fourer Transform (FFT) of the Response In real applcatons the measured response s dscrete, so the Fast Fourer Transform (FFT) needs to be used to compute the frequency response. When the tme-doman sgnal s of a fnte length and does not tend to zero n the ntegraton tme, spectral leakage s nevtable (Harrs 979). It can sgnfcantly affect the accuracy of the frequency response constructed n Eqn. To mnmze ts effects, the wndowng process s employed durng the FFT. Ths paper tests the exponental wndow w e (t;η) and the Hannng wndow w h (t), w h (6) (7) where η denotes the decay rate of the exponental wndow and T denotes the measurement tme nterval. The Fourer transform wth the exponental wndows s equvalent to the Laplace transform, F e ηt x( t) ( ω ) = L [ x ( t )]( s ), where L s the Laplace operator and s = jω + η. If the exponental wndow s used, Eqn () should be stated as (8) A useful feature of the exponental wndow can be consdered to support ts feasblty. Compare the free response of an n-dofs structure wth the same response after wndowng, ( ) = + x t Aφ e sn ω t ϕ, x t w ( t) = ( ) ( ) = e w ( η) = e t; πt + cos for t T T, elsewhere, s s + b ( ) = ( ) ( ) ( ) ωξτ ( ) t; η Aφe ω sn ωdt ϕ, (9) ηt e for t T, elsewhere, where φ s the shape of the th mode, ω s the th natural frequency, ξ s the dampng rato, and d [ ] ( ) Y s X s D s B s X s, η ω ξ + τ + ( ). The exponental wndow ncreases the dampng rato of the free response by and does not change ts frequency content, whch s unlke other wndows ncludng the Hannng wndow..5. Isolaton n Frequency Doman vs. Isolaton n Tme Doman Isolaton n tme doman (Hou et al. ) (SIM-TD) and n frequency doman (SIM-FD) are both based on the same concept: the substructure s solated from the global structure nto an ndependent structure by addng vrtual supports on ts nterface. The tme-doman counterpart of Eqn () and the correspondng formula for the response of the solated substructure are ( ) 5 d ω = ω ξ () ( ) ( ) = x t + x t τ z τ dτ, b = n t s s s = n t () () ( ) ( ) Y t = x t + x t τ z τ dτ. () Durng the solaton process the frst equaton s solved to fnd the functons z (t), whch are then substtuted nto the second equaton to fnd the response of the solated substructure. The frst equaton s a system of Volterra ntegral equatons of the frst knd (Kress 989). The advantages and dsadvantages of the tme- and frequency-doman approaches can be summarzed as follows: ) The SIM-TD drectly uses the measured tmedoman responses, so that t avods the errors related to the Fourer transform. However, the computatons extensvely employ tme-consumng convolutons and requre solvng a dscretzed verson of a system of Volterra ntegral equatons of the frst knd, whch s an extremely ll-condtoned problem that requres numercal regularzaton technques. ) In the SIM-FD, the computatonal effort s sgnfcantly reduced, as all the convolutons are effcently performed n frequency doman. The computatons can be focused on a selected frequency range only, whch further reduces the computatonal tme. As a result, a larger number of responses measured n a longer tme nterval can be used. However, the SIM- FD approach may ntroduce naccuraces related to the FFT and spectral leakage. b ηω 4 Advances n Structural Engneerng Vol. 8 No. 5

8 Jln Hou, Lukasz Jankowsk, and Jnpng Ou. NUMERICAL EXAMPLE Ths secton uses a mass-sprng system for ntal verfcaton of the proposed solaton method n frequency doman... Sx-Dof Mass-Sprng System The mass-sprng system s a 6-Dof structure shown n Fgure (a). The stffness of each floor s k = kn/m; all the lumped masses are m = 4 kg. The st and nd order dampng ratos are %. The substructure conssts of the 4th, 5th and 6th mass, see Fgure (b)... Exctatons, Sensors and the FRF (Frequency Response Functon) Assume that two acceleraton sensors are used, X and X5, placed respectvely on the rd and 5th mass. Assume also that two mpulse exctatons, F and F6, can be appled respectvely to the rd and 6th mass. Altogether four responses (denoted X-Fj) are smulated; they correspond to the four terms used n Eqn to construct the response of the solated substructure, see Table. Fgure 4 plots the smulated FRFs of the global system. They are drectly used n Eqn to construct the FRF of the solated substructure. The result s compared n Fgure 5 to the accurate frequency response computed usng the FE model of the substructure. The response obtaned usng the SIM s the same as that drectly computed, whch confrms that the solated substructure can be constructed successfully. The stars n Fgure 5, as m, c, k ( =,,,,6) (a) Global structure (b) lsolated substructure Fgure. A 6-Dof mass-sprng system Table. Exctatons and responses Exctatons Sensors Constranng F Basc F6 Interface X X-F: B(ω) X-F6: X b(ω) Internal X5 X5-F: B(ω) X5-F6: X s(ω) well as n Fgures 8 to, mark the accurate natural frequences computed usng the FE model of the solated substructure... Wndowng and the FFT In order to smulate a real applcaton wth tme-doman exctatons, a smulated hammer exctaton s appled at F and F6, see Fgure 6. The responses of the global structure are shown n Fgure 7. The samplng frequency s Hz and the tme nterval s T =.56s. Three wndowng functons are used to decrease the spectral leakage: no wndowng, the exponental wndow (Eqn 6) and the Hannng wndow (Eqn 7). In order to ncrease the frequency resoluton of the FFT, several zeros are added n front of the wndowed responses (the measurement tme nterval s ncreased eghtfold and flled wth zeros at the begnnng). The results of the FFT are used n Eqn to construct the FRF of the solated substructure. Fgure 8 to Fgure compare the FRF constructed usng the SIM to the actual FRF computed usng the FE model. The theoretcal FRF s smooth, and peaks appear only around structural frequences. But due to truncaton of the measured response n tme doman, there s spectral leakage n ts computed fast Fourer transform, so that the computed FRF s not smooth and has many small peaks, see Fgure 8 (No-FEM). For ths reason, the solated substructure constructed usng such a frequency response cannot provde relable modal nformaton, see Fgure 8 (No-SIM). It can be seen that only the exponental wndow (Fgure 9) can lead to consstent results, provded the wndow decay rate s properly selected. Even f the Hannng wndow (Fgure ) yelds almost the same postons of the peaks, the ampltudes devate sgnfcantly. As mentoned n Secton.4, the exponental wndow only ncreases the dampng rato of the free response and does not change ts frequency content, whch s unlke other wndows ncludng the Hannng wndow. Therefore, the exponental wndow s selected n the method. The exponental wndow ncreases the effectve system dampng. Thus, the constructed FRF depends on the wndow decay rate η or, alternatvely, on the attenuaton rato r at the end of tme nterval T, whch are related to each other by η = In r T. () Four exemplary values of the parameter r are tested to construct the FRF of the solated substructure, see Advances n Structural Engneerng Vol. 8 No. 5 4

9 Frequency-Doman Substructure Isolaton for Local Damage Identfcaton Ampltude X-F X5-F X-F6 X5-F Fgure 4. FRFs of the global structure Magntude 4 SIM FEM ω Fgure 5. FRF of the solated substructure Exctaton (N) t (s) Fgure 6. Smulated hammer exctaton Fgure. For r = e-, there are two peaks near the frst natural frequency, whch mght suggest that e- s not small enough to reduce the spectral leakage. On the other hand, too small r ntroduces too much dampng nto the system, whch s apparent n Fgure. For ths numercal example, t s thus decded to use a compromse value of r = e-. The man reason of spectral leakage s that the ampltudes of the truncated structural responses are not low. The exponental wndow can be used to decay the responses. In order to determne the attenuaton rato r, structural dampng s assumed to be zero, and then there s no attenuaton of the smulated structural response. In ths way, the attenuaton s determned only by the attenuaton rato r. Hammer exctaton n Fgure 6 s appled n Dof and DOf 6, and the acceleratons of Dof and DOf 5 a (m/s )... X-F X5-F X-F6 X5-F t (s) Fgure 7. Responses of the global structure 44 Advances n Structural Engneerng Vol. 8 No. 5

10 Jln Hou, Lukasz Jankowsk, and Jnpng Ou Ampltude No-SIM No-FEM ω Fgure 8. FRFs of the solated substructure, no wndowng: (No-SIM) constructed by the SIM; (No-FEM) computed from the FE model Ampltude Exp-SIM Exp-FEM ω Fgure 9. FRFs of the solated substructure, exponental wndow: (Exp-SIM) constructed by the SIM (wth r = e-); (Exp -FEM) computed from the FE model Ampltude 9 6 Hann-SIM Hann-FEM ω Fgure. FRFs of the solated substructure, Hann wndow: (Hann-SIM) constructed by the SIM; (Hann -FEM) computed from the FE model Ampltude r = e r = e r = e r = e 4 ω Fgure. FRFs constructed by the SIM wth the exponental wndow and dfferent attenuaton ratos r are measured and used to construct the structural response of the solated substructure. The response X5-F6 s shown n Fgure. Frst, the nfluence of the attenuaton rato r on the constructed solated substructure s studed. Let r= α, α [,], and compute the nephogram of the constructed frequency response of the solated substructure n dependence on the frequency and the attenuaton rato r (Fgure ). The dark color represents large ampltudes, that s the locaton of the dark color marks the egen frequency locaton of the solated substructure. The three dark lnes reflect the relaton between the frst three frequences of the solated Advances n Structural Engneerng Vol. 8 No. 5 45

11 Frequency-Doman Substructure Isolaton for Local Damage Identfcaton. A (m/s ) t (s) Fgure. Responses X5-F6 of the global structure wthout dampng 5, (r = ) α α r =. T (s) 5 ω ω ω 5 ω (Hz) 5 ω ω ω ω (Hz) Fgure. Nephogram of the constructed frequency response of the solated substructure n dependence on the frequency and the attenuaton rato r Fgure 4. In case of r =., nephogram of the constructed frequency response of the solated substructure n dependence on the frequency and tme length T substructure and the attenuaton rato r. It can be seen that the smaller value of α,.e. the bgger value of r, corresponds to larger errors between the constructed frequency and the actual values (the dotted lnes n Fgure ) of the solated substructure. The error can be neglected when α s greater than,.e. r s less than.. Then the nfluence of the tme length T of response on the constructed solated substructure s studed. Let r =., and the range of tme length T be (, 5s]. The frequency response of the solated substructure s constructed usng the measured response durng tme [,T]. Fgure 4 shows the nephogram of the constructed frequency response of the solated substructure n dependence on the frequency and tme length T. It can be seen that the constructed values have good accuracy. Therefore, when r =., the frequency response of solated substructure can be constructed accurately no matter the length of the measured tme. Besdes, the attenuaton rato r ams to decay the structural response, and ts value can be.. Therefore, n the comng experment verfcaton, let r =.. 4. EXPERIMENTAL VERIFICATION 4.. Expermental Setup An alumnum cantlever beam s used for expermental verfcaton. The beam s vertcally suspended on a stable frame, see Fgure 5(a). Its upper end s fxed to the frame, and the bottom end s ether free or fxed usng a sponge support (Fgure 5(b)). The dmensons of beam s shown n Fgure 6. The crosssecton s.7 cm. cm. Young s modulus of the beam s 7 GPa, and the densty s 7 kg/m. The beam s slender, so that the axal stran can be neglected; the gravty s consdered: when any of the beam segments s tlted off the vertcal equlbrum poston, the gravty affords a certan restorng force, whch s consdered n the equaton of moton. The upper part wth the length of 79.4 cm s the substructure to be solated and locally dentfed, see Fgure 6. In the experment, a segment wth the length of. cm s cut evenly along ts wdth on two sdes as shown n Fgure 5(c) and the dmenson of the damage s shown n Fgure 7. There are 5 notches on each sde. The depth of each notch s.78 cm, and the average dstance between two adjacent notches s./(5 ) =.7 46 Advances n Structural Engneerng Vol. 8 No. 5

12 Jln Hou, Lukasz Jankowsk, and Jnpng Ou Stran Stran Stran } Velocty (c) Damage segment (d) Velocty measurement Substructure Velocty stran Stran Damage Stran (a) Beam (b) Optonal sponge Fgure 5. Cantlever beam cm. Because notches are very narrow, the mass of the damaged secton s assumed to reman unchanged. The beam wdth s.7 cm, therefore the damage factor of the damaged secton s calculated as.78 /.7 4%. Then the damaged secton s taken as one of the segments to be dentfed. Snce the beam s slender, t s easy to excte ts hgh natural frequences. The FEM model of the substructure s thus densely dvded nto 48 elements (95 Dofs). In most other SD methods of local substructural montorng, the full state of the substructure (n all ts Dofs and at each tme step) need to be computed, so that the substructure cannot have too many Dofs. In the method proposed here, the substructure can be as complex as requred: only the complexty of ts nterface s lmted by the necessty of placng sensors n the nterface Dofs. As a result, applcaton to large and complex substructures wth a large number of nternal Dofs s possble. A relatvely dense dvson of the substructure nto 48 elements s used to verfy ths pont. Three PVDF pezoelectrc flm sensors are placed on the substructure to measure the stran; a laser vbrometer (Fgure 5(d)) s used to measure the velocty n the transverse drecton. They are numbered Sensors to 4, respectvely. Sensor ( stran n Fgure 5(a)) and Sensor 4 ( Velocty n Fgure 5(d)) are located on the substructure nterface to measure ts responses..4 Fgure 6. The dmensons of beam Fgure 7. The dmensons of damage Two versons of the beam are tested: the orgnal beam wth a free bottom end (beam ) and the same beam wth an addtonal sponge support on the free end (beam ). These beams share the same substructure, hence they can be used to verfy the robustness of the solaton process wth respect to unknown changes of the outsde structure ncludng potental nonlneartes of beam ntroduced by the sponge support. All the measurements are separately performed on each of the beams..7 Advances n Structural Engneerng Vol. 8 No. 5 47

13 Frequency-Doman Substructure Isolaton for Local Damage Identfcaton A modal hammer s used to apply exctaton n the transverse drecton. Two sensors are placed on the nterface, so at least three exctatons need to be separately appled, ncludng one basc exctaton nsde the substructure and two constranng exctatons outsde t. In order to make the measured responses ndependent of each other, the poston of each exctaton s lmted to a dfferent part of the beam, see Table. The exctaton s appled by hand: note that a more accurate postonng s not requred, whch s a practcal advantage of the method. Snce the beam s slender and lght, the orgnal FE model s bult consderng the gravty of the cantlever beam and the weght of the stran sensors, as well as the stffness of the sensors. Hammer exctaton knocks the beam randomly and frequences of the beam are dentfed usng the measured responses. Table (Identfed) lsts the frst seven dentfed frequences of the ntact beam and the damaged beam. The orgnal FE model s smple, Young s modulus and the geometrcal parameters are accurate, as well as the fxed end constrant. The masses and stffnesses of the stran sensors are updated usng the measured response, whch s a smpler procedure not ntroduced here n detal. The frequences of the updated FE model are lsted n Table (FEM) and compared to the dentfed values. Errors are computed usng Eqn. It can be seen that the updated FE model s very accurate. F ω ω ε = m ω m () The solated substructure has vrtual supports, so t cannot be excted drectly, and ts natural frequences cannot be obtaned expermentally. But the FE model of the global structure s accurate and so the correspondng damaged substructure FE model s accurate, and t can be used drectly for comparson wth the constructed frequences of solated substructure. 4.. Measured Responses In order to obtan the necessary dynamc nformaton of the structure, the samplng frequency of khz s used, and the measured tme nterval s T = 4 s. Two beams and three exctatons are used, there are thus sx groups of measured responses, see Fgure 8. The legend X(,j,k) denotes the response of the th sensor to the jth hammer exctaton measured n the kth beam. 4.. Substructure Isolaton The measured nterface responses (sensors and 4) are used to construct a sngle vrtual pnned support on the nterface, see Secton.. The correspondng FRFs of sensors and (as f placed n the solated substructure) are constructed by Eqn. In order to confrm that the numercally solated substructure s ndependent of the global structure (ts constructed response s not nfluenced by modfcatons of the outsde structure), responses of beam are combned wth responses of beam. Such an approach mght be also convenent n practce: f the substructure s unchanged, then the measured responses can be combned wth those measured n another tme, no matter whether the components outsde the substructure are changed or not. Table. Exctatons and ther postons Number Type Poston Basc exctaton Insde the substructure Constranng exctaton Outsde the substructure, near the nterface Constranng exctaton Outsde the substructure, far from the nterface Table. The comparson of the natural frequences Intact Damaged Order Identfed FEM Error Identfed FEM Error % % % % % % % % % % % % % % 48 Advances n Structural Engneerng Vol. 8 No. 5

14 Jln Hou, Lukasz Jankowsk, and Jnpng Ou Voltage (V) X(,,) X(,,) X(,,) X(4,,) Voltage (V) X(,,) X(,,) X(,,) X(4,,) 4 Tme (s) (a) Beam, basc response (exctaton ) 4 Tme (s) (b) Beam, basc response (exctaton ) Voltage (V) X(,,) X(,,) X(,,) X(,4,) 4 Tme (s) (c) Beam, constranng response (exctaton ) Voltage (V) X(,,) X(,,) X(,,) X(4,,) 4 Tme (s) (d) Beam, constranng response (exctaton ) Voltage (V) X(,,) X(,,) X(,,) X(4,,) 4 Tme (s) (e) Beam, constranng response (exctaton ) Voltage (V) X(,,) X(,,) X(,,) X(4,,) 4 Tme (s) (f) Beam, constranng response (exctaton ) Fgure 8. The measured responses of beams and To ths end, the basc and the constranng responses measured ether n beam or n beam can be used. There are four combnatons and, consequently, the four correspondng FRFs of the solated substructure are constructed. Ther ampltudes are plotted n Fgure 9, where the legend X-B(j,k) denotes the FRF of the th sensor n the solated substructure constructed by usng the basc responses of the jth beam and the constranng responses of the kth beam. For comparson purposes, the stars mark the natural frequences computed usng the FE model of the damaged substructure. The attenuaton rato r n the numercal example above s adjusted by constructng frequency responses wth dfferent values of r, and the value s chosen that makes the constructed frequency response close to the actual one. Such data are not avalable n expermental practce, hence the value determned n the numercal example s accepted also here, that s r =. (decay rate η =.5). The peaks of the constructed FRFs (Fgure 9) are obvous, and the frst seven natural frequences of the solated substructure can be thus obtaned by peak-pckng. They are n good agreement wth the natural frequences computed usng the FE model of the damaged substructure, see Table 4. It shows that the bggest error s %, and n all four cases, the constructed frequences of the damaged solated substructure are Advances n Structural Engneerng Vol. 8 No. 5 49

15 Frequency-Doman Substructure Isolaton for Local Damage Identfcaton ( ) ( ) Ampltude of FFT 5 4 X-B(,) X-B(,) FEM Ampltude of FFT 5 4 X-B(,) X-B(,) FEM ( ) ( ) (a) B-B (b) B-B ( ) ( ) Ampltude of FFT 5 4 X-B(,) X-B(,) FEM Ampltude of FFT 5 4 X-B(,) X-B(,) FEM ( ) ( ) (c) B-B (d) B-B Fgure 9. The four constructed FRFs of the solated substructure Table 4. Natural frequences of the solated substructure (Hz) Expermental dentfcaton Theoretcal FEM B-B B-B B-B B-B Order Intact Damaged ω error ω error ω error ω error % % 7..85% 7..85% % % 5.6.% 5.6.% %.8.%.5.4%.5.% % 95.7.% 9..% % % 89.6% 89..5% 89..9% % 46.5% % % % 55.5% % % close to those of ts FE model, whch proves that the solated substructure s constructed accurately. In all four combnaton cases, the dentfed natural frequences are smlar, whch confrms that the solaton process s ndependent of the outsde structure: f the substructures are the same, the constructed solated substructures are also the same, no matter the outsde that can be unknown, nonlnear or changng Identfcaton of the Isolated Substructure The substructure s dvded nto fve segments, see Fgure, where each segment contans 9 or fnte elements. The second segment s actually damaged, so that the 4 5 Fgure. Dvson of the substructure nto fve segments actual damage extents of the fve segments are [.4 ] T. The damages of the substructure are dentfed by mnmzng the square dstance (Eqn 5) between the constructed natural frequences of the solated substructure and the natural frequences computed usng ts FE model. The damages are dentfed wth a good accuracy n all four combnaton cases, see Fgure and 5 Advances n Structural Engneerng Vol. 8 No. 5

16 Jln Hou, Lukasz Jankowsk, and Jnpng Ou Table 5. The proposed method performs well wth a substructure wth 95 Dofs, whch makes t a substructure sgnfcantly more complex than the substructures used n most other SD methods of local substructural montorng. 5. DISCUSSION 5.. Tme-Doman and Frequency-Doman Method Ths paper proposes a frequency-doman method of substructure solaton. In comparson to the tmedoman method (Hou et al. ), t s computatonally sgnfcantly more effcent, whch s one of ts man advantages. Let n be the number of the constranng exctatons (nterface sensors), and denote by n t the number of the tme steps. Snce the number of spectral lnes after the FFT s proportonal to the number of the tme steps, and for each spectral lne a pseudonverse of an n n matrx needs to be computed, the tme complexty of the frequency-doman method, ncludng the cost of the FFT, s Onn ( t + nntlognt). The solaton n tme-doman requres a sngle computaton Damage extent Actual B-B B-B B-B B-B 4 5 Segment number Fgure. The dentfed damage extents of the substructure of the pseudonverse of a large n t n n t n matrx, whch yelds a sgnfcantly hgher tme complexty of Onn ( t ). For example, the measurement tme nterval consdered n ths experment s 4 s, whle for the tme-doman method t s only.4 s (at the same samplng rate). Despte the ten tmes longer tme nterval, the frequency-doman method s approxmately sx orders of magntude faster. In terms of accuracy, there s no sgnfcant dfference between the damage extents dentfed here (Fgure 6) and the damage extents dentfed usng the tmedoman approach and the same objectve functon (Fgure 8, Hou et al. ). 5.. Constranng Exctatons The matrx B(ω) n Eqn () conssts of structural responses to constranng exctatons. To avod excessve ll-condtonng, the correlaton between ts columns should be as close to zero as possble. However, f closely-spaced, smlar constranng exctatons are used, then the correspondng responses are also smlar and the correlaton s hgh. Therefore, essentally dfferent constranng exctatons should be used; n ths example, they are appled n dfferent postons wth a certan dstance from each other. 5.. Substructure Interface and Constranng Sensors Two factors need to be consdered n practcal applcaton of the method. The frst s the selecton of the substructure, whch should have a smple nterface wth the outsde global structure. The second s the selecton of the constranng sensors and ther locaton on the nterface: t depends on the ntended type of the vrtual supports, whch should ensure that the constructed response of the solated substructure s senstve to the local nformaton beng dentfed. Table 5. Identfed damage extents of the substructure and ther absolute errors Segment number 4 5 Actual damage B-B Identfed Error.6% 4.5% 5.%.%.% B-B Identfed Error.85% 4.%.44%.%.% B-B Identfed Error 9.9% 9.4%.54%.% 4.4% B-B Identfed Error 5.%.5% 7.7%.% 9.8% Advances n Structural Engneerng Vol. 8 No. 5 5

17 Frequency-Doman Substructure Isolaton for Local Damage Identfcaton 6. CONCLUSIONS Ths paper extends the substructure solaton method (SIM) nto frequency doman (SIM-FD). The effcency and accuracy of the approach are verfed usng a masssprng numercal model and a beam experment. The method focuses on local damage dentfcaton of the substructure, so that, n contrast to other substructurng methods, unknown nterface forces (dampng coeffcents, state vectors, etc.) do not need to be dentfed. Therefore, damage parameters are the only unknowns, so that local dentfcaton can be generally easer and numercally more stable. It s an advantage n applcatons to substructures that feature a larger number of nternal Dofs. The proposed SIM-FD transfers the solaton process from tme doman nto frequency doman. The solated substructure can be constructed separately for each frequency of nterest, whch sgnfcantly decreases the numercal costs of solaton. Durng the solaton process, selecton of the wndowng functon for the FFT s mportant. The exponental wndow s suggested, whch avods the spectral leakage and preserves the frequency nformaton of the solated substructure. The SIM requres the nterface responses to be measured n order to construct the vrtual supports. In complex boundary condtons, t may restrct the applcablty of the method. Ths lmtaton s a subject of an ongong research. ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of the Major State Basc Research Development Program of Chna (97 Program)(CB65), of Natonal Scence Foundaton of Chna (NSFC) (5857, 5866), of the Fundamental Research Funds for the Central Unverstes(Chna) (DUTLK), of Specal Fnancal Grant from the Chna Postdoctoral Scence Foundaton (T555), and of the Project of Natonal Key Technology R&D Program (Chna) (BAKB, BAKB, 6BAJB5). Fnancal support of the Polsh Natonal Scence Centre Project AIA (DEC-/5/B/ST8/97) and the FP7 EU project Smart-Nest (PIAPP-GA--8499) s gratefully acknowledged. REFERENCES An, Y. and Ou, J. (). Expermental and numercal studes on model updatng method of damage severty dentfcaton utlzng four cost functons, Structural Control and Health Montorng, Vol., No., pp. 7. Bao, Y., L, H., An, Y. and Ou, J. (). Dempster Shafer evdence theory approach to structural damage detecton, Structural Health Montorng, Vol., No., pp. 6. Duan, Z., Yan, G., Ou, J. and Spencer, B.F. (5). Damage localzaton n ambent vbraton by constructng proportonal flexblty matrx, Journal of Sound and Vbraton, Vol. 84, No. -, pp Ewns, D.J. (). Modal Testng, Theory, Practce, and Applcaton, nd ed, Research Studes Press, England. Fan, W. and Qao, P. (). Vbraton-based damage dentfcaton methods: A revew and comparatve study, Structural Health Montorng, Vol., No., pp. 8-. Hassots, S. (). Identfcaton of damage usng natural frequences and markov parameters, Computers and Structures, No. 74, pp Harrs, F.J. (978). On the use of wndows for harmonc analyss wth the dscrete Fourer transform, Proceedngs of the IEEE, Vol. 66, No., pp Hou, J., Jankowsk, L. and Ou, J. (). Expermental study of the substructure solaton method for local health montorng, Structural Control and Health Montorng, Vol. 9, No. 4, pp Km, H. and Melhem, H. (). Fourer and wavelet analyses for fatgue assessment of concrete beams, Expermental Mechancs, Vol. 4, No., pp. 4. Koh, C.G. Hong, B. and Law, C.Y. (). Substructural and progressve structural dentfcaton methods, Engneerng Structures, Vol. 5, No., pp Kress R. (989). Lnear Integral Equatons. In: Appled Mathematcal Scences, Vol 8, Sprnger, New York. Law, S.S. and K. Zhang, Duan, Z.D. (). Structural damage detecton from couplng forces between substructures under support exctaton, Engneerng Structures, Vol., No. 8, pp. 8. Ln, R.M. and Ewns, D.J. (99). Model updatng usng FRF data, 5th Int. Semnar on Modal Analyss, Leuven, Belgum, pp Maa, N. and Slva, J. (997). Theoretcal and Expermental Modal Analyss, Research Studes Press, England. Moragasptya, H., Thambratnam, D., Perera, N. and Chan, T. (). Health montorng of buldngs durng constructon and servce stages usng vbraton characterstcs, Advances n Structural Engneerng. Vol. 5, No. 5, pp Nar, K.K., Kremdjan, A.S., Le, Y., Lynch, J.P. and Law, K.H. (). Applcaton of tme seres analyss n structural damage evaluaton, Internatonal Conference on Structural Health Montorng, Tokyo, Japan. Rucka, M. and Wlde, K. (). Neuro-wavelet damage detecton technque n beam, plate and shell structures wth expermental valdaton, Journal of Theoretcal and Appled Mechancs, Vol. 48, No., pp Suwal / a, G. and Jankowsk, L. (). A model-free method for dentfcaton of mass mod_catons, Structural Control and Health Montorng, Vol. 9, No., pp. 6. Tee, K.F., Koh, C.G. and Quek, S.T. (5). Substructural frst- and second-order model dentfcaton for structural damage 5 Advances n Structural Engneerng Vol. 8 No. 5