Transmissibility of nonlinear output frequency response functions with application in detection and location of damage in MDOF structural systems

Transcription

1 Trasmssblty of olear output frequecy respose fuctos wth applcato detecto ad locato of damage MDOF structural systems Z.Q. Lag G. Par C.R. Farrar M.D. Todd Z. Mao L. Zhao K. Worde To cte ths verso: Z.Q. Lag G. Par C.R. Farrar M.D. Todd Z. Mao et al.. Trasmssblty of olear output frequecy respose fuctos wth applcato detecto ad locato of damage MDOF structural systems. Iteratoal Joural of No-Lear Mechacs lsever <.6/.olmec...9>. <hal-75958> HAL Id: hal Submtted o Nov HAL s a mult-dscplary ope access archve for the depost ad dssemato of scetfc research documets whether they are publshed or ot. The documets may come from teachg ad research sttutos Frace or abroad or from publc or prvate research ceters. L archve ouverte plurdscplare HAL est destée au dépôt et à la dffuso de documets scetfques de veau recherche publés ou o émaat des établssemets d esegemet et de recherche fraças ou étragers des laboratores publcs ou prvés.

2 Author s Accepted Mauscrpt Trasmssblty of olear output frequecy respose fuctos wth applcato detecto ad locato of damage MDOF structural systems Z.Q. Lag G. Par C.R. Farrar M.D. Todd Z. Mao L. Zhao K. Worde PII: S-746- DOI: do:.6/.olmec...9 Referece: M 85 To appear : Iteratoal Joural of No- Lear Mechacs Receved date: 9 Jue Accepted date: March Cte ths artcle as: Z.Q. Lag G. Par C.R. Farrar M.D. Todd Z. Mao L. Zhao ad K. Worde Trasmssblty of olear output frequecy respose fuctos wth applcato detecto ad locato of damage MDOF structural systems Iteratoal Joural of No-Lear Mechacs do:.6/.olmec...9 Ths s a PDF fle of a uedted mauscrpt that has bee accepted for publcato. As a servce to our customers we are provdg ths early verso of the mauscrpt. The mauscrpt wll udergo copyedtg typesettg ad revew of the resultg galley proof before t s publshed ts fal ctable form. Please ote that durg the producto process errors may be dscovered whch could affect the cotet ad all legal dsclamers that apply to the oural perta.

3 Trasmssblty of Nolear Output Frequecy Respose Fuctos wth Applcato Detecto ad Locato of Damage MDOF Structural Systems Z Q Lag G Par C R Farrar M D Todd Z Mao L Zhao ad K Worde 4 Departmet of Automatc Cotrol ad Systems geerg Uversty of Sheffeld Mapp Street Sheffeld S JD UK The geerg Isttute Los Alamos Natoal Laboratory MS T- Los Alamos NM USA Departmet of Structural geerg Uversty of Calfora Sa Dego La Jolla Calfora USA 4 Departmet of Mechacal geerg Uversty of Sheffeld Mapp Street Sheffeld S JD UK Abstract Trasmssblty s a well-ow lear system cocept that has bee wdely appled the dagoss of damage varous egeerg structural systems. However egeerg practce structural systems ca behave olearly due to certa ds of damage such as e.g. breathg cracs. I the preset study the cocept of trasmssblty s exteded to the olear case by troducg the Trasmssblty of Nolear Output Frequecy Respose Fuctos NOFRFs. The NOFRFs are a cocept recetly proposed by the authors for the aalyss of olear systems the frequecy doma. A NOFRF trasmssblty-based techque s the developed for the detecto ad locato of both lear ad olear damage MDOF structural systems. Numercal smulato results verfy the effectveess of the ew techque. xpermetal studes o a three-storey buldg structure demostrate the potetal to apply the developed techque to the detecto ad locato of damage practcal MDOF egeerg structures. Keywords: Trasmssblty; Nolear MDOF systems; Damage detecto ad locato.

4 . Itroducto A wealth of methods exsts for the detecto ad locato of damage structural systems. These methods clude tme ad frequecy doma techques parametrc ad oparametrc approaches ad emprcal ad model-based approaches for plates shells compostes ad other types of structures []. The core dea of most of these avalable techques s to compare some features evaluated from o-le measured structural resposes to the features evaluated from resposes measured uder the systems ormal worg codtos to assess whether damage has occurred ad f ths s the case where the damage s located. The frequecy-doma trasmssblty fucto s a sgfcat feature that ca be appled to detect locate ad quatfy damage mult-degree of freedom MDOF structural systems where structural dyamc sesor arrays ca be used to mae dfferetal dyamc trasmssblty measuremets []. For MDOF structures structural damage affects both the system poles ad zeros. But as aalysed [] ad [] zeros are much more sestve tha poles to localsed damage as zeros deped o the put ad output locatos whereas poles do ot. Trasmssblty fuctos are determed solely by the system zeros ad they are therefore potetally better dcators of localsed damage. The frequecy-doma trasmssblty fucto s essetally a lear system cocept. It s ormally defed as the rato of the spectra of two dfferet system outputs of terest ad s also equal to the rato of the system frequecy respose fuctos FRFs assocated wth the two outputs. Although as demostrated by umercal studes [] ths trasmssblty cocept ca sometmes be used for the detecto ad locato of damage olear structural systems the cocept s geerally putdepedet for olear systems. Cosequetly oe s geerally ot able to use the tradtoal trasmssblty fucto to dstgush the effect of system put from the effect of the chage of system propertes due to the occurrece of damage olear structural systems. I structural systems certa types of damage ofte mafest themselves as the troducto of a olearty to a otherwse lear system [4]. xamples clude post-bucled structures Duffg olearty rattlg ots mpactg system wth

5 dscotutes or breathg cracs blear stffess. Therefore effectve techques are eeded whch ca relably detect ad locate both these types of olear damage as well as lear chages due to damage e.g. a stffess or mass chage such as wth corroso or loss of a elemet. May researchers have addressed dfferet aspects of ths mportat ssue usg dfferet approaches. These approaches clude for example mutual formato ad trasfer etropy-based statstcal olearty detecto methods [4-8] auto-bspectral aalyss [9-] olear system detfcato techques [] ad frequecy doma AR model based techque []. For MDOF structural systems whch clude beams rotor shafts mult-storey buldgs ad brdges etc. [4567] damage detecto has bee studed by Zhu ad Wu [8] where the structure wth damage s cosdered to be lear ad the locato ad magtude of the damage are estmated usg measured chages the system atural frequeces. Based o a oe-dmesoal structural model Saellarou ad Fassos [9] [6] have used a stochastc output-error vbrato-based methodology to detect damage structures where the damaged elemets are modeled as compoets of cubc stffess. Recetly usg the cocept of Nolear Output Frequecy Respose Fuctos NOFRFs [] several methods have bee developed whch use the respose sgals to determstc puts cludg susods to detect ad locate olear damage MDOF structural systems [7]. The basc prcple ad effectveess of these methods have bee verfed by theoretcal aalyss ad umercal smulato. The obectve of the preset study s to exted the frequecy-doma trasmssbltybased damage detecto ad locato techque to the olear case to systematcally develop a more geeral trasmssblty aalyss-based damage detecto ad locato approach for MDOF structural systems. The bass of ths study s a ew trasmssblty cocept ow as the trasmssblty of the NOFRFs. The NOFRFs are a ew cocept recetly proposed ad comprehesvely vestgated by the authors for the aalyss of olear systems the frequecy doma []. The cocept of the trasmssblty of the NOFRFs s defed as the rato of the maxmum order NOFRFs assocated wth two dfferet output resposes of terest a olear system. For the MDOF structural systems cosdered the preset study t s prove that the NOFRF trasmssblty s equal to the tradtoal trasmssblty uder certa

6 codtos but s geerally dfferet from the tradtoal trasmssblty. However the NOFRF trasmssblty oly depeds o the system lear characterstcs ad therefore does ot chage wth the system put. Ths excellet property of the NOFRF trasmssblty s exploted the preset study to develop a ew techque that ca detect ad locate both lear ad olear damage MDOF structural systems. I addto to theoretcal dervato ad aalyss both umercal smulato studes ad expermetal tests are coducted the preset study. The results verfy the effectveess of the ew techque ad demostrate that the techque has cosderable potetal to be appled damage detecto ad locato for real egeerg structures.. Trasmssblty of Nolear Output Frequecy Respose Fuctos Cosder the class of sgle-put multple-output SIMO olear systems whch are stable at zero equlbrum wth the outputs represeted a eghbourhood of the equlbrum by the Volterra seres [5] N... τ x t h τ u t τ dτ for.. I equato s the umber of the system outputs ut ad x t are the put ad the th output of the system h τ... τ s the th order Volterra erel assocated wth the th olearty. system output ad N deotes the maxmum order of the system The output frequecy resposes of ths class of olear systems to a geeral put ca be descrbed as [4] N π for H U dσ where s the spectrum of the th system output represets the th order frequecy respose of the system s th output U s the spectrum of the system put ad 4

7 H τ... τ h τ... τ e + + dτ... dτ s ow as the th order Geeralsed Frequecy Respose Fucto GFRF assocated wth the th system output whch s the exteso of the frequecy respose fuctos of SIMO lear systems to the th order olear case. The term H... U + + dσ equato deotes the tegrato of dmesoal hyper-plae H U over the - The Nolear Output Frequecy Respose Fuctos NOFRFs are a cocept recetly troduced by the authors []. For SIMO olear systems the NOFRFs are defed as G H... U dσ N 4 U dσ uder the codto that U U dσ The NOFRFs defed by equato 4 ca be regarded as a alteratve exteso of the frequecy respose fuctos of SIMO lear systems to the th order olear case. The most dstctve characterstc of the NOFRFs s ther oe dmesoal ature whch ca sgfcatly facltate the aalyss of olear systems the frequecy doma [8-]. From equatos ad 4 t ca be show that the output frequecy respose of SIMO olear systems ca be represeted usg the NOFRFs as N N G U 6 For SIMO lear systems N. So equato 6 reduces to 5

8 G U H U 7 where U U dσ U 8 G H U σ H 9 U dσ wth H deotg the frequecy respose fuctos FRFs of the system. Therefore equato 6 s a exteso of the well-ow lear system relatoshp 7 to the olear case. Cosderg that the trasmssblty of SIMO lear systems are ormally defed as T H G H G L where {... } ad < the cocept of the trasmssblty of the NOFRFs s proposed to exted the lear system trasmssblty cocept to the olear case as follows. Defto: For sgal put mult output olear systems the trasmssblty of the NOFRFs betwee the th ad th system outputs s defed as G N T G N where {... } ad <. Obvously whe the system s lear.e. N T G L T G 6

9 that s the trasmssblty of the NOFRFs defed above reduces to the tradtoal trasmssblty of lear systems. I addto as the NOFRFs are depedet of the chage of the system put ampltude the NOFRF trasmssblty also does ot chage wth the ampltude. Ths s the same as the property of the trasmssblty of lear systems.. The NOFRF Trasmssblty of MDOF Nolear Structural Systems. MDOF olear structural systems Cosder the mult-degree-of-freedom structural system show Fg. For the smplcty of troducg the ma deas assume that there s oly oe possble olear compoet located betwee the J- th ad J th masses the system wth J {... } ; the put force u t s appled at the th mass; ad the moto of all the masses s oe-dmesoal. I addto t s assumed that the restorg forces of the olear sprg ad damper deoted by FS Δ t ad FD Δ t ca be ay cotuous olear fuctos of the deformato Δ t x t x t ad ts dervatve Δ t respectvely. Thus by deotg J J NF t... FS Δ t + FD Δ t FS Δ t FD Δ t... ad ' u 4 F t t the MDOF olear structural system ca be descrbed as M x t + Cx t + Kx t NF t + F t 5 where x t [ x ] t x t s the dsplacemet vector ad J ' J M m m m 7

10 c c c c c c c c c c c c c C K are the system mass dampg ad stffess matrces respectvely.. The NOFRF trasmssblty ad propertes The MDOF system descrbed by equato 5 ca clearly be regarded as a oe-put - output system. The system put s the force appled to the th mass ut ad outputs are the dsplacemets of the masses. Therefore deote the output spectra of the th ad th masses the system as ad the ad ca be descrbed usg the NOFRF cocepts as N U G 6 ad N U G 7 respectvely where G ad G are the th order NOFRFs assocated wth the dsplacemets of the th ad th masses of the system ad U s defed equato 5 wth U represetg the spectrum of the put force t u. I ths case the NOFRF trasmssblty T defed equato refers to the trasmssblty betwee the dsplacemets of masses ad the MDOF structural

11 9 system; ad T L defed equato represets the rato betwee the st order NOFRFs assocated wth the dsplacemets of the th ad th masses. Based o these deftos a seres of propertes regardg the NOFRF trasmssblty of the MDOF structural system 5 ca be obtaed. These are summarzed the followg two propostos. Proposto For system 5 T ad T L are completely determed by the system lear characterstc parameters M C ad K as follows + + z J z J z J z J z Q Q Q Q T 8 + z z z L Q Q T 9 where + + K C M Q Q Q Q I addto N where G G T ad there exst the followg relatoshps betwee T ad T L < otherwse f T T J T T L L Proof: The coclusos 8- ca be reached by followg the dervato for the NOFRF propertes of MDOF systems [6] ad tag to accout the deftos of T ad T L troduced above. Proposto

12 For system 5 the NOFRF trasmssblty T ad a fucto of frequecy defed by [ G T G ] ca be determed from the system put ad output spectra as follows; T U U f the verse of matrx U U 4 exsts. I U ad U are the spectra of two dfferet put forces separately appled to the system ad are the spectra of the dsplacemet resposes of masses ad the system to the put force wth spectrum U ad ad are the spectra of the dsplacemet resposes of masses ad the system to the put force wth spectrum U. Proof: See the Appedx. Proposto shows that for system 5 the NOFRF trasmssblty T ad the rato of the st order NOFRFs T L are all depedet of the system put ad completely determed by the system lear characterstc parameters. Ths property mples that ust le the tradtoal trasmssblty cocept for lear SIMO systems the NOFRF trasmssblty T ca be used to evaluate chages the lear characterstc parameters of the olear MDOF system 5. I addto Proposto shows that T equals to the rato of the hgher-tha-oe order NOFRFs assocated wth mass ad equato ad reveals a mportat relatoshp betwee T ad T L equato.

13 Proposto dcates that uder the vertblty codto of matrx 4 the NOFRF trasmssblty T of system 5 ca be determed usg the system put ad output spectra obtaed from two separate tests. Proposto also shows that defed ca be obtaed usg the same system put ad output spectra. The sgfcace of s that t ca be used to evaluate the dfferece betwee T ad T L L. Ths s because f T T otherwse [ G G T G L [ T T ] ] 5 6 geeral. Ths together wth the relatoshp betwee T ad T L revealed Proposto ca be exploted to determe the locato of the olear compoet the system. 4. A NOFRF Trasmssblty-based Techque for the Detecto ad Locato of Damage MDOF Structural Systems The troducto of the cocept of the NOFRF trasmssblty Secto s to address the damage detecto ad locato ssue for the MDOF structural system 5. quato 5 descrbes a class of MDOF olear structural systems where the olear compoet ca represet damage the system such as for example a crac a beam. The M C ad K matrces represet the lear characterstcs of the system ad damage the system ca also chage the values of these matrces. Therefore the obectves are to detect the exstece of the olear compoet ad locate ts posto ad to detfy ay chages the M C ad K matrces. These obectves are to be acheved based o the put force ad correspodg resposes of all the masses the system whch ca for example be the outputs of a array of sesors ftted to the structure for health motorg purposes.

14 The tradtoal trasmssblty cocept has already bee used [] as a dfferetal dcator to detect ad locate structural damage. Although some olear damage was also cosdered ths prevous study the coclusos reached are based o a specfc case study ad ca ot be exteded to geeral olear MDOF structural systems. Ths s smply because the tradtoal trasmssblty s a lear system cocept; ad the result s put-depedet the olear case. I order to resolve the problem wth tradtoal trasmssblty cocept-based approaches a NOFRF trasmssblty based techque for the detecto ad locato of damage the MDOF structural system 5 s proposed ths secto based o the system NOFRF trasmssblty ad propertes that have bee derved Secto.. The basc deas of the damage detecto ad locato techque are to apply two dfferet forces to excte the system to get two sets of correspodg system resposes. The evaluate the spectra of the force exctatos ad correspodg resposes ad detect the exstece of olear compoet/damage the system by checg the vertblty of matrx 4. Ths system olearty detecto method s based o the fact that f there s o olear compoet/damage system 5 the system s lear so that 7 U U ad the verse of matrx 4 does ot exst. Cosequetly a vertble codto of matrx 4 such as e.g. whether the determat of matrx 4 s zero ca be used to detect whether there exsts a olear compoet/damage the system. After that f the matrx-vertblty-chec dcates that there s o olear compoet the system the the system s lear ad the tradtoal trasmssblty based techque wll be appled for the detecto ad locato of possble lear damage the system. Otherwse there s a olear damage/compoet the system. quato wll be appled to evaluate T + ad + for each par of cosecutve masses the system that s for -. The the NOFRF

15 trasmssblty results T + - wll be used to evaluate the chages the system lear characterstc parameters to detect ad locate possble lear damage the system ad + - wll be used to determe the locato of the system olear damage/compoet. The theoretcal bass of determg the locato of the olear compoet s equatos ad whch mply that T T + + T T L + L G G + + [ T [ T L + L + T T + ] f J + ] Otherwse 8 Therefore theoretcally the system olear compoet s located betwee masses * ad * + where * s determed by the values of + - such that + + > whe * Otherwse < 9 Based o these deas ad cosderg varous ssues assocated wth real data aalyss observed comprehesve expermetal studes that have bee coducted by the authors o MDOF test rgs [4] the damage detecto ad locato techque for MDOF structural system 5 s proposed as follows. xcte the system twce usg two dfferet puts wth spectra U ad U respectvely; Measure the resposes of all the masses the system to each put exctato ad calculate the spectra of the measured resposes to obta ad. valuate dex IND as defed by where IND Meda{ D qδ ; q... Q}

16 D U qδ det U qδ qδ Δ Q Q s a teger ad [ ] qδ qδ s a terval wth the frequecy rage of the system put. The result s used to determe whether or ot there s a olear compoet the system as follows: If IND ε the the system olearty s eglgble; otherwse there s a olear compoet the system. Here ε s zero theory a ose-free evromet but practce t should be a small umber assocated wth a ose threshold a case where the system bascally behaves learly. If the result Step dcates that there s eglgble olearty the evaluate the tradtoal lear trasmssblty betwee the resposes of all cosecutve masses to yeld L T over the frequecy rage [ ] ad assess the chages the system lear characterstcs by comparg the evaluated - wth the results T L + evaluated the system ormal operatg codtos to detect ad locate damage that ca be deduced from chages the system lear characterstc parameters v If the result Step dcates that the system s behavg olearly determe the NOFRF trasmssblty T + ad + from equato wth + over the frequecy rage [ ] for -. The locate the damage that maes the system behave olearly ad assess the state codto of the system lear compoets usg the evaluated NOFRF trasmssblty results T - to detect ad locate possble + damage that duces chages the system lear compoets. Locatg olear damage ca be acheved as follows. 4

17 a valuate + + d - b Fd a ormalzed result for + as max + {... } c xame +..- to fd a î such that + + ε > ε < ˆ ˆ The t ca be cocluded that the olear damage s located betwee mass î ad mass ˆ +. I practcal applcatos ε s a specfc case-depedet threshold whch ca be determed from expermetal data usg statstc aalyses. Fg shows a flow chart that llustrates how to follow the four-step procedure to mplemet ths NOFRF trasmssblty-based MDOF system damage detecto ad locato techque. Step of ths techque dcates that two tests are eeded for the system specto ad the force exctatos used the two tests should be dfferet. However t s worth potg out that ths does ot mea the techque eeds the data from two specally desged spectg tests. I fact the data collected from two spectg codtos where the put forces from ambet evromet are dfferet should also be suffcet for the mplemetato of the techque. Step s a practcal mplemetato of the dea of detectg the exstece of olearty the system based o the vertble codto of matrx 4. It s worth otg that the meda fucto s used ths mplemetato to avod outlyg values of the determat at some frequeces caused by ose or other dsturbaces affectg the aalyss results practce. Step s a tradtoal trasmssblty aalyss based procedure whch detects ad locates the system damage by evaluatg the chages the system lear characterstc parameters. I Step v + - whch are the ormalzed results of + 5

18 - are used to mplemet the equato 9 based dea of determg the locato of olear damage. A threshold of ε s troduced ths step to chage the purely theoretcal codto of to the more practcal codto of + ε +. Ths allows the effects of ose u-modeled dyamcs ad heret but less sgfcat system oleartes to be omtted the detecto of the relatvely cosderable system olearty so that the proposed method ca be appled egeerg practce. I addto by explotg the fact that NOFRF trasmssblty T + - are completely determed by the characterstcs of the system lear compoets Step v also assesses the state codto of the system lear compoets usg the evaluated NOFRF trasmssblty to detect ad locate possble lear damage. I the authors prevous studes a seres of techques [7] have bee proposed for the locato of olear damage MDOF structural systems. Most of these prevous techques [ 7] requre to test a spected structure usg specfc puts. These ca be acheved well cotrolled test codtos but are dffcult for may practcal stuatos. We realzed the problems whe processg expermetal data usg these techques ad ths motvated us to develop the ew techque troduced above. The ew techque oly eeds the system exctato ad respose data collected uder two dfferet loadg codtos ad ths requremet ca be satsfed eve for the cases where oly ambet exctatos to the system are avalable. I addto va the troducto of the NOFRF trasmssblty cocept the ew techque sgfcatly exteds the olearty locato method for MDOF systems []. Ths s because the ew techque provdes a comprehesve procedure whch ca ot oly be used to detect the exstece of olear damage ad fd ts locato but s also able to detect ad locate damage that ca duce chages the system lear characterstc parameters. > 6

19 5. Smulato Studes o a DOF Structural System I ths secto umercal smulato studes o a DOF system.e. Fg where the characterstc of the thrd sprg ca be olear.e. J are coducted to demostrate the effectveess of the techque proposed the last Secto. Cosder the DOF system four dfferet specfc cases as follows: Case I: m m m g ; 4.6 N/m; 4 c c c..6 Ns/m ; There s o olear compoet the system. Case II: 4 m m m g ;.6 N/m; 4 c c c..6 Ns/m ; The thrd sprg s olear.e. J wth ts restorg force gve by l x rl x x x x +. x l FS x x Case III: Case IV 4 m m m g ; N/m;.6 4 c c c..6 Ns/m ; 4. N/m The thrd sprg s olear.e. J wth ts restorg force gve by l x rl x x x x +. x l FS x x 4 m m m g ;.5.6 N/m;.6 4 c c c..6 Ns/m ;. 4 N/m 7

20 The thrd sprg s olear.e. J wth ts restorg force gve by l x rl x x x x +. x l FS x x The techque proposed Secto 4 was appled to the DOF system the four cases respectvely. I all these cases the appled force exctatos were the same whch are t + π 6 + s t u t s π ad u t u π. t For each case the dsplacemets of the three masses the system x t were obtaed by umercal smulato ad the appled puts ad umercally smulated system dsplacemet resposes were used to determe IND D π where D U π det U π π π π π ad π max + { } max + { } T T π T T π. order to detect the exstece ad fd the locato of the olear compoet the system from IND ad ad evaluate the characterstcs of the system lear compoets from T ad T. 8

21 Tables -4 show the umercal aalyss results obtaed together wth the theoretcal values of these results. From Table t s ow that the value of IND s a very small umber dcatg that the system Case I was behavg learly whch s obvously correct. Because of the very small determat value the NOFRF trasmssblty results were determed usg equato Step of the techque ad o results for + - were evaluated for ths case. A comparso of the IND value Table wth that Table dcates that the system Case II was behavg cosderably more olearly. The cocluso that there exsts a olear compoet the system ca readly be reached for ths case f choosg ε 9 ad comparg IND wth ths threshold. Moreover t s ow from the results Table that f choosg > ε.445 ε < ε 4 the. Ths as descrbed the proposed techque dcates that ˆ that s the olear compoet s located betwee mass ˆ ad mass ˆ + whch s obvously correct for Case II. I addto comparg the results of T T evaluated usg the techque ad ther theoretcal values shows a excellet agreemet whch verfes the effectveess of the system put/output data-based aalyss. A comparso of the results of the NOFRF trasmssblty Tables ad shows that the T results are the same but the T results are dfferet the two tables. Ths s because there s a olear compoet located betwee mass ad mass Case II the system s completely lear Case I ad apart from ths dfferece the systems the two cases are the same. Therefore f the system s ormal Case I but addto to olear damage there also exst lear damage Case II whch duces chages the system parameters that affect T the the damage ca be effectvely detected by comparg the T results the two cases. 9

22 The aalyss results for Cases III ad IV Tables ad 4 further cofrm the effectveess of the proposed techque. I addto a comparso of the results of T T Tables ad shows that the chage of the system parameter from N/m Case II to N/m Case III causes the chages the evaluated values for T T. Ths demostrates the depedece of these results evaluated by the proposed techque o the system characterstc parameter ad dcates that the proposed techque ca be used to detect ad evaluate the chage the system parameter. However a comparso of the results of T T Tables ad 4 show that the chage of the system parameter from 4.6 N/m Case II to N/m Case IV duces o chage the evaluated values for T T. Because s a characterstc parameter of the system olear compoet ths result mples that the NOFRF trasmssblty results are completely determed by the characterstcs of the system s lear compoets. Therefore the NOFRF trasmssblty should oly be used to evaluate the chages the system lear compoets. 6. xpermetal setup 6. xpermetal studes The three-storey buldg structure [] show Fg was used the study to demostrate the potetal of the proposed NOFRF trasmssblty-based techque for the detecto ad locato of damage practcal MDOF structural systems. The structure cossts of alumum colums ad plates assembled usg bolted ots wth a rgd base. The structure sldes o rals that allow movemet oly oe drecto. At each floor four colums 7.7x.5x.6cm are coected to the top ad bottom alumum plates.5x.5x.5 cm whch form a four degree-of-freedom system. Addtoally a ceter colum 5.x.5x.5 cm ca be suspeded from the top of each floor Fgs ad 4 show the case where the colum s suspeded from the top of the secod floor whch s used to duce olear behavours whe the colum cotacts a

23 bumper mouted o the ext floor. The posto of the bumper ca be adusted to vary the extet of the olearty. Ths source of olearty ca for example smulate the fatgue cracs that subsequetly ope ad close uder operatoal ad evrometal codtos. A electromagetc shaer provdes the exctato to the groud floor of the structure. A force trasducer was attached at the ed of a stger to measure the put force from the shaer. Four accelerometers are attached to each floor at the opposte sde from the exctato source to measure the respose from each floor. Fg. 4 shows the sprg-damper model of the three-storey buldg structure whch s clearly a specfc case of the DOF model Fg.. 6. xpermets ad obectves of expermetal data aalyss The data collected from twelve expermets coducted o the structure were aalyzed for the preset study to evaluate the performace of the proposed damage detecto ad locato techque dfferet practcal stuatos. The detals of the expermets are summarzed Table 5. Sx dfferet state codtos of the structure were vestgated. These are the structural state codtos uder xpermets # ad # xpermets # ad #4 xpermets #5 ad #6 xpermets #7 ad #8 xpermets #9 ad # ad xpermets # ad # respectvely. The obectves of the expermetal data aalyss are to apply the proposed techque to two sets of sgals measured by the force trasducer ad four accelerometers from xpermet # ad xpermet # respectvely to evaluate state codto of the structural system for 456. It s worth potg out that the descrpto for the proposed techque Secto 4 the dsplacemet of each degree of freedom system 5 s cosdered to be the system output respose t ca be show that the techque wll be the same f stead of dsplacemet the accelerato s regarded as the system output. Therefore the proposed techque ca drectly be appled to process the expermetal data from the three storey buldg structure for the purposes of damage detecto ad locato. The aalyss results for structural state codto wll be compared wth the aalyss results for structural state codto to exame the dfferece betwee the results to see how the troducto of olearty to the system chages the outcome of

24 aalyss. Ths s to smulate the stuato of detectg the exstece of olear damage ad determg ts locato. The the aalyss results for structural state codto wll be compared wth the aalyss results for structural state codto 4 to demostrate how the chages the characterstc parameters of lear compoets a olear structure ca be detfed by the aalyss. Ths s to smulate the stuato where the proposed techque s appled to a olear structural system ad the chages structural lear compoets are to be detected. Fally the aalyss results for structural state codto 5 wll be compared wth the aalyss results for structural state codto 6 to demostrate the capablty of the proposed techque locatg damage that oly duces chages the characterstcs of the system s lear compoets. It s worth otg that the appled susodal put exctato wth a gve ampltude each expermet as show Table 5 s the sgal geerated by a computer whch was used to cotrol the shaer s force output. However real expermetal data show that the force lterally measured by the force trasducer s ot oly dfferet from the susodal sgal ampltude due to scalg etc effects but also cotas harmocs due to the heret olear ature of shaer dyamcs. Therefore the preset study whe the proposed techque was appled to aalyze the expermetal data the rage of frequeces of the put force exctato that was actually used the aalyss covers ot oly the frequecy of the appled susodal put but also ts harmocs. 6. The results of expermetal data aalyss Tables 6 7 ad 8 show the comparso of the expermetal data aalyss results for structural state codtos ad ad 4 ad 5 ad 6 respectvely. Because the test rg s a 4DOF structure 4. was evaluated to + determe the locato of the system olearty; ad - NOFRF trasmssblty results.e. T T T were evaluated over the frequeces of 4 terest. Due to the olear effects of shaer dyamcs as metoed above the rage of frequeces of the exctato force lterally appled to the structure covers ot oly the frequecy of the susods geerated by the shaer cotrol computer but also ts

25 harmocs. Because of ths the preset study dex IND was determed from equato wth 4 Q9 Ω ad Ω. Here Ω s the frequecy of the susodal sgal geerated by the shaer cotrol computer. I xpermets # ad # for example Ω π 6 rad/s. I addto the values of the NOFRF trasmssblty over the frequeces up to the ffth harmoc were evaluated for each case. Comparso of the aalyss results for structural state codtos ad The results of IND Table 6 show that the test rg uder structural state codto behaves cosderably more olearly tha structural state codto ad f 4 choosg ε 5 t ca be cocluded from Step of the proposed techque that the test rg s a lear system uder structural state codto but a olear oe uder structural state codto. The cocluso s obvously correct as uder structural state codto a olear effect was troduced to the 4DOF structural system va fttg a ceter colum ad bumper o the groud floor wth a.mm gap betwee the colum ad bumper. Ths dcates that the exstece of a system olearty ca effectvely be detected by the proposed techque. I addto f ε Step of the proposed techque s chose as ε. the from + evaluated uder structural state codto t ca readly be cocluded that the system olear compoet was located betwee masses ad.e. o the groud floor whch s aga a correct aalyss result. It s worth otg that the expermetal studes reported ths paper ths ε was selected by observg the obtaed data aalyss results. I practce a prcpled method based o the statstcs of may eed to be used to set ths threshold. + A comparso of the NOFRF trasmssblty results evaluated for state codtos ad shows that overall the two sets of results are dfferet. Ths s beleved to be due to the effects of heret but ot very sgfcat olearty the three-storey buldg structure; otherwse the T. ad T. results should be very smlar uder the two state codtos as they are theoretcally the same. vdece of the heret structural olearty s the value of IND uder state codto whch s much smaller tha the IND uder state codto but ca obvously ot be cosdered to be

26 zero le the IND Table the result obtaed from a smulato study. However t ca be observed that the results of the NOFRF trasmssblty at the base frequecy that s T π 6 ad T π 6 uder the two dfferet state codtos are bascally smlar. Ths cofrms the theoretcal aalyss to a certa extet ad mples that smlar practcal stuatos the NOFRF trasmssblty results evaluated at the base frequecy should be used to assess ay chages system lear compoets to detect ad locate system lear damage. Other expermetal data aalyses also cofrm ths observato. Comparso of the aalyss results for structural state codtos ad 4 4 The two values of IND Table 7 are all cosderably larger tha ε 5. Therefore based o the proposed techque the 4DOF structural system was olear uder both structural state codtos ad 4. I addto because s less tha ε. ad ad 4 are all larger tha ths ε Step of the proposed techque dcates that the system olear compoet was located betwee masses ad.e. o the frst floor. These were as descrbed Table 5 exactly the stuatos of the 4DOF structure uder structural state codtos ad 4. A comparso of the values of the NOFRF trasmssblty evaluated for state codto ad state codto 4 as gve Table 7 shows a cosderable dfferece betwee the codtos of the structural lear compoets. The relatve dfferece betwee the NOFRF trasmssblty values uder the two dfferet state codtos averaged over the results evaluated at the fve harmoc frequeces s 5.75% for T. 8.4% for T. ad.4% for T. respectvely. Ths s as expected because 4 compared wth state codto 4 four mass blocs were added o the top of the buldg structure uder state codto whch creases the value of parameter m the system model. It ca be aalytcally show that the chage of the system parameter m ca be reflected by the values of all NOFRF trasmssbltes the system. Therefore addto to a olear damage a chage the lear characterstc of the system s also effectvely detected. 4

27 Comparso of the aalyss results for structural state codtos 5 ad 6 From the results of IND ad Table 8 the same coclusos as those + reached above about the lear/olear ature of the system behavors ad the locato of the system olear compoet ca be reached for the 4DOF structural system uder structural state codtos 5 ad 6 whch are also correct. A comparso of the values of the NOFRF trasmssblty evaluated for state codto 5 ad state codto 6 as gve Table 8 however shows a dfferet stuato. The relatve dfferece betwee the NOFRF trasmssblty values uder the two dfferet state codtos aga averaged over the results evaluated at the fve harmoc frequeces s 7.99% for T. 7.5% for T. ad.6% for T. respectvely. Compared wth the 5.75% dfferece betwee the two sets of T. results Table 7 the 7.99% dfferece betwee the T. results uder state codtos 5 ad 6 s much less sgfcat. I addto apart from T π 4 the T. results uder the two dfferet state codtos have very lttle dfferece. Because the observed dfferece betwee the T π results uder state codtos 5 ad 6 ca be explaed to be due to ose ad other factors the expermetal results Table 8 bascally show that compared wth the data aalyss results for state codto 5 the aalyss result T. for state codto 6 almost has o chage but the aalyss result T. for state codto 6 has a relatvely 4 sgfcat chage. These observatos ca be cofrmed by theoretcal aalyses. Compared wth state codto 5 three mass blocs were added o the groud floor of the structure uder state codto 6 ad ths fact creases the value of parameter m the system model. It ca be show that the NOFRF trasmssblty T. oly 4 depeds o c c ad m so that theory the values of T. ca ot be affected by the crease of m 4. It ca also be proved that for the 4DOF system Fg.4 whe the olear compoet s located betwee masses ad m 4 wll affect 5

28 4 the value of T.. Therefore the data aalyss results Table 8 ca be wellustfed. Further dscussos for the data aalyses The NOFRF trasmssblty aalyss results Table 7 show that compared to state codto 4 more masses are added o the top floor state codto ; The results Table 8 dcate that compared to state codto 5 more masses are added o the groud floor state codto 6. Therefore f the mass chage represets a lear damage the proposed techque has these cases ot oly detected the damage but also foud ts locato. All the results show Tables 6-8 ad dscussed above show that the expermetal data aalyss coducted usg the proposed techque ca effectvely dstgush the dfferet state codtos of the 4DOF expermetal structural system ad eglectg the effects of uavodable ose u-modeled dyamcs ad partcularly the heret but relatvely sgfcat system olearty the data aalyss results are also cosstet wth theoretcal aalyses. The expermetal studes therefore demostrate the effectveess of the proposed techque detectg ad locatg system olearty ad evaluatg chages the characterstc parameters of the system lear compoets usg the NOFRF trasmssblty cocept. Because the system olearty ad chages system lear compoets ca represet dfferet ds of damage practcal MDOF structural systems the proposed techque has cosderable potetal to be appled damage detecto ad locato o real egeerg structures. 7. Coclusos I the preset study the cocept of the trasmssblty of the NOFRFs has bee proposed for a class of olear MDOF structural systems. The NOFRF trasmssblty s based o the recetly proposed NOFRF cocept ad exteds the trasmssblty cocept for lear systems to the olear case. A NOFRF trasmssblty based techque s the developed for the detecto ad locato of damage MDOF structural systems. Both smulato ad expermetal studes have bee coducted. The results verfy the effectveess of proposed techque ad demostrate that the techque has the potetal to be appled practce to detect ad 6

29 locate damage a wde rage of MDOF egeerg structural systems. Although the MDOF system model cosdered the preset study s of a relatvely smple form the basc deas ca be exteded to much more complcated cases cludg systems where there are more tha oe olear compoets ad/or the moto of each mass s of a mult-dmesoal ature. Further research studes wll focus o addressg these ssues. More comprehesve expermetal studes wll also be coducted to vestgate how to effectvely apply the basc dea of the proposed techque to the health motorg of a wde rage of real egeerg compoets ad structures cludg beams plates shafts ad brdges. Appedx Proof of Proposto : Rewrte equato 6 as follows N G U + T T A. Substtutg equato 7 to A. for the sde the thrd term o the rght had sde of ths equato ad tag to accout equato yeld [ G T U + T G ] U + T A. Because T G G are all depedet of the system put equato A. mples that f system 5 s excted by two dfferet force puts wth ther spectra beg U ad U respectvely the the correspodg spectra q q of the dsplacemet resposes of masses ad deoted by ad q ca be descrbed as U U + T + T A. Therefore T U U A.4 7

30 f matrx U U s vertble. Refereces. S W Doeblg C R Farrar M B Prme D W Shevtz Damage detfcato ad health motorg of structural ad mechacal systems from chages ther vbrato characterstcs: a lterature revew. Los Alamos Natoal Laboratory report LA-7- MS T J Johso D Adams Trasmssblty as a dfferetal dcator of structural damage Tras. of the ASM Joural of Vbrato ad Acoustcs Vol.4 pp J Mottershead O the zeros of structural frequecy respose fuctos ad ther sestvtes Mechacal Systems ad Sgal Processg Vol. 998 pp J M Nchols M Seaver S T Trcey M D Todd C Olso L Overbey Detectg olearty structural systems usg the trasfer etropy. Physcal Revew M. Paluš Testg for olearty usg redudaces: quattatve ad qualtatve aspects. Physca D pp M. Paluš V. Komare Z. Hrcr ad K. Sterbova Sychrozato as adustmet of formato rates: Detecto from bvarate tme seres. Physcal Revew T. Schreber Measurg Iformato Trasfer Physcal Revew Letter Vol. 85 pp A. Kaser ad T. Schreber Iformato Trasfer cotuous processes. Physca D 66 pp J.M. Nchols P. Marzocca ad A. Mlaese O the use of the auto-bspectral desty for detectg quadratc olearty structural systems Joural of Soud ad Vbrato Vol. 8 pp A. Rvola P.R. Whte Bspectral aalyss of the blear oscllator wth applcato to the detecto of cracs Joural of Soud ad Vbrato Vol pp C R Farrar K Worde M D Todd G Par J Nchols D Adams M T Bemet K Farholt Nolear system detfcato for damage detecto Los Alamos Natoal Laboratory report LA D Adams C R Farrar Classfyg lear ad olear structural damage usg frequecy doma AR models Structure Health Motorg Vol. pp85. 8

31 . A. Neld P.D. Mcfadde ad M.S Wllams A dscrete model of vbratg beam usg tme-steppg approach Joural of Soud ad Vbrato Vol. 9 pp T G Chodros A D Dmarogoas J Yao Vbrato of a beam wth breathg crac Joural of Soud ad Vbrato Vol.9 pp Ja Kcs Rotor dyamcs IFFM Publshers S.D. Fassos J.S. Saellarou Stochastc output error vbrato-based damage detecto ad assessmet structures uder earthquae exctato Joural of Soud ad Vbrato Vol pp C.R. Farrar ad S.W. Doeblg Lessos Leared from Applcatos of Vbrato Based Damage Idetfcato Methods to Large Brdge Structures Proc. of the Iteratoal Worshop o Structural Health Motorg Staford CA Sept 997 pp H. Zhu M. Wu The Characterstc receptace method for damage detecto large Moo-coupled Perodc structures Joural of Soud ad Vbrato Vol.5 pp S. D. Fassos J.S. Saellarou Tme-seres methods for fault detecto ad detfcato vbratg structures Phlosophcal Trasactos of the Royal Socety A: Mathematcal Physcal ad geerg Sceces 65 7 pp Z.Q. Lag S.A. Bllgs ergy trasfer propertes of olear systems the frequecy doma Iteratoal Joural of Cotrol. Vol.78 5 pp Z Q Lag Z K Peg A ovel approach for olearty detecto vbratg systems Joural of Soud ad Vbrato Vol. 4 8 pp6-65. Z K Peg Z Q Lag S A Bllgs Novel method for detectg the o-lear compoets perodc structures Proc. IMech Vol. 8 Part C: J. Mechacal geerg Scece pp9-9.. Z K Peg Z Q Lag Detectg the posto of o-lear compoet perodc structures from the system resposes to dual susodal exctatos. Iteratoal Joural of No-Lear Mechacs Vol. 4 7 pp Z. Q. Lag S. A. Bllgs Output frequecy characterstcs of olear systems Iteratoal Joural of Cotrol Vol pp I W Sadberg A perspectve o System Theory I Trasactos o Crcuts ad Systems Vol. CAS- 984 pp Z.K Peg Z.Q Lag S.A Bllgs Nolear Output Frequecy Respose Fuctos of MDOF Systems wth Multple Nolear Compoets Iteratoal Joural of No-lear Mechacs Vol

32 7. Z.K Peg Z.Q Lag F L Chu G Meg Locatg Nolear Compoets Perodc Structures Usg the Nolear ffects Structure Health Motorg Press. 8. Peg Z K Lag Z Q Bllgs S A Noler output frequecy respose fuctos for mult-put olear Volterra systems. Iteratoal Joural of Cotrol Vol.8 7 pp Peg Z K Lag. Z Q Bllgs S A Aalyss of Locally Nolear MDOF Systems Usg Nolear Output Frequecy Respose Fuctos. Joural of Vbrato ad Acoustcs I press.. Peg Z K Lag Z Q Chu F L Numercal aalyss of craced beams usg olear output frequecy respose fuctos Computers & Structure Vol.86 8 pp m J- m J- J m J J+ m J+ m - m ut c c x J- c J c x x J- x J+ c J x J+ x - Fgure The MDOF structural system cosdered the preset study

33 Step xcte system 5 twce usg two dfferet put forces measure the resposes of all the masses to each exctato ad evaluate the spectra of the appled put forces ad the measured resposes. Step - Calculate dex IND from equato usg the system put ad output spectra evaluated Step Step - Chec f IND ε? YS Step I ths case o olear damage exsts the system. So valuate the tradtoal trasmssblty usg equato ; ad Assess the chages the system lear characterstcs from the evaluated trasmssblty results. NO Step v I ths case there exsts olear damage the system. So valuate the NOFRF trasmssblty T + ad + from equato ; Locate the olearty followg steps a-c from the evaluated values for + ; ad Assess the chages the system lear characterstcs from the evaluated NOFRF trasmssblty T + Fgure A flow chart of the NOFRF trasmssblty-based MDOF system damage detecto ad locato techque

34 Fgure Schematc represetato of a three-storey buldg structure used for the expermetal studes m d Floor m st Floor m Groud Floor m 4 u t x t x t x t x 4 t Fgure 4 The 4DOF system model of the three storey buldg structure

35 Table Aalyss Results for Case I of Smulato Studes System Characterstcs Aalyss Results Theoretcal Values IND.e- π π max NA NA max T T π T T π Table Aalyss Results for Case II of Smulato Studes System Aalyss Theoretcal Values Characterstcs Results IND.e-4 > π.9 π.4 4.e-5 max.445 max. T T π T T π.68.68

36 Table Aalyss Results for Case III of Smulato Studes System Aalyss Theoretcal Values Characterstcs Results IND.9e-4 > π.4 π e-5 max.578 max. T T π T T π.6.6 Table 4 Aalyss Results for Case IV of Smulato Studes System Aalyss Theoretcal Values Characterstcs Results IND 6.49e-4 > π.56 π.4 4.e-5 max.6 max. T T π T T π

37 xpermet xpermet # xpermet # Table 5 Detals of the xpermets Iput exctato appled by shaer cotrol computer 6Hz susodal wth ampltude 6Hz susodal wth ampltude 6 Structural state codto uder whch expermet was coducted State Codto The ceter colum ad bumper were ftted o the groud floor but Gap was so large that o olearty was troduced to the structure. xpermet # xpermet #4 xpermet #5 xpermet #6 xpermet #7 xpermet #8 xpermet #9 xpermet # 6Hz susodal wth ampltude 4 6Hz susodal wth ampltude 6 4Hz susodal wth ampltude 4Hz susodal wth ampltude.5 4Hz susodal wth ampltude 4Hz susodal wth ampltude.5 Hz susodal wth ampltude Hz susodal wth ampltude 4 State Codto The ceter colum ad bumper were ftted o the groud floor ad Gap.mm. State Codto * The ceter colum ad bumper were ftted o the st floor ad Gap.mm; * 5% stffess reducto the two colums located betwee the groud ad frst floors ad the tersectos of plae B wth plaes C ad D; * 4 mass blocs were added to the top floor. State Codto 4 * The ceter colum ad bumper were ftted o the st floor ad Gap.mm; * 5% stffess reducto the two colums located betwee the base ad frst floor ad the tersectos of plae B wth plaes C ad D; State Codto 5 The ceter colum ad bumper were ftted o the st floor ad Gap.mm; xpermet # xpermet # Hz susodal wth ampltude Hz susodal wth ampltude 4 State Codto 6 * The ceter colum ad bumper were ftted o the st floor ad Gap.mm; * mass blocs were added to the groud floor. 5

38 Table 6 Comparso of the Real Data Aalyss Results for Structural State Codtos ad The expermetal data aalyss results for the test rg uder State Codto The expermetal data aalyss results for the test rg uder State Codto IND 9.e-5 5.8e N/A T π T π T π T π T π T π T π T π T π T π T π T π T π T π T π

39 Table 7 Comparso of the Real Data Aalyss Results for Structural State Codtos ad 4 The expermetal data aalyss results for the test rg uder Status State codto The expermetal data aalyss results for the test rg uder Status State codto 4 IND 5.6e- 9.9e T π T π T π T π T π T π T π T π T π T π T π T π T π T π T π

40 Table 8 Comparso of the Real Data Aalyss Results for Structural State Codtos 5 ad 6 The expermetal data aalyss results for the test rg uder Status State Codto 5 The expermetal data aalyss results for the test rg uder Status State Codto 6 IND T π T π T π T π.4.49 T π T π T π T π T π T π T π T π T π T π T π