A procedure to track vibration modes under changing external factors : application to a pedestrian bridge

A p ocedu e o ack ib a ion modes unde changing ex e nal ac o s: applica ion o a pedes ian b idge So ia, José M.1; Díaz, I án M.2; Ga cía-Palacios, Jaime H.3; Ibán, No be o4; Lo enzana, An olín5 ABSTRACT A low-cos ib a ion moni o ing sys em has been de eloped and ins alled on an u ban s eel- pla ed s ess- ibbon oo b idge. The sys em con inuously measu es: he accele a ion (using 18 iaxial MEMS accele ome e s dis ibu ed along he s uc u e), he ambien empe a u e and he wind eloci y and di ec ion. Au oma ed ou pu -only modal pa ame e es ima ion based on he S ochas ic Subspace Iden i ica ion (SSI) is ca ied ou in o de o ex ac he modal pa ame e s, i.e., he na u al equencies, damping a ios and modal shapes. Thus, his pape analyzes he ime e olu ion o he modal pa ame e s o e a whole-yea da a moni o ing. Fi s ly, o simila en i onmen al/ope a ional ac o s, he unce ain ies associa ed o he ime window size used a e s udied and quan i ied. Secondly, a me hodology o ack he ib a ion modes has been es ablished since se e al o hem wi h closely-spaced na u al equencies a e iden i ied. Thi dly, he modal pa ame e s ha e been co ela ed agains ex e nal ac o s. I has been shown ha his s ess- ibbon s uc u e is highly sensi i e o empe a u e a ia ion ( equency changes o mo e han 20%) wi h s ongly seasonal and daily ends. Keywo ds: Modal analysis; Foo b idge; Con inuous dynamic moni o ing; En i onmen al e ec s. 1. INTRODUCTION The long- e m ib a ion moni o ing o ci il enginee ing s uc u es is inc easingly used o moni o bo h, he s uc u e in eg i y and he ib a ion se iceabili y. A numbe o examples o b idges and oo b idges equipped wi h a moni o ing sys em can be ound [7, 9, 10, 16]. In hese moni o ing sys ems, he ib a ions due o a ic o wind a e measu ed and eco ded. These da a can be used o con inuously ex ac he modal pa ame e s (na u al equencies, damping a ios and modal shapes) which may be employed o assess he s uc u al in eg i y since s uc u al damages lead o changes in he modal pa ame e s. Thus, ib a ion-based S uc u al Heal h Moni o ing (SHM) sys ems using au oma ed ou pu -only modal iden i ica ion (also known as Ope a ional Modal Analysis, OMA) ha e been ex ensi ely p oposed, o ins ance [8, 13]. One o he main p oblems o hese sys ems is ha hey ha e 1 Dp o. CMM and Theo y o S uc u es. ETSICCP, UPM (Spain). [email p o ec ed] (Co esponding au ho ) 2 Dp o. CMM and Theo y o S uc u es. ETSICCP, UPM (Spain). i an.m[email p o ec ed] 3 Dp o. Ci il Enginee ing: Hid aulic and Ene gy. ETSICCP, UPM (Spain). jaime.ga cí[email protected] 4 Mechanical Enginee ing Di ision. Ca i Resea ch Cen e (Spain). no iba@ca i .es 5 EII/ITAP, Uni e sidad de Valladolid (Spain). [email p o ec ed] 41 o cope wi h changing en i onmen al/ope a ional condi ions ha o en signi ican ly a ec he modal es ima ion. The main obs acle o a mo e widesp ead adop ion o hese moni o ing sys ems is he cos associa ed o hei ins alla ion. Recen ly, a success ul a emp o educe he cos o a moni o ing sys em has been ca ied ou h ough he use o low-cos senso s o measu e s uc u al ib a ions on a oo b idge. Thus, a low-cos con inuous ib a ion moni o ing sys em using small MEMS (Mic o Elec o-mechanical Sys ems) accele ome e s has been de eloped and ins alled on Ped o Gómez Bosque (PGB) oo b idge (Valladolid, Spain) in o de o ack i s long- e m dynamic pe o mance. The sys em con inuously measu es: he accele a ion (using 18 iaxial MEMS accele ome e s dis ibu ed along he s uc u e), he ambien empe a u e and he wind eloci y and di ec ion [18]. PGB oo b idge is a singula s eel-pla ed s ess- ibbon oo b idge wi h a single span o 85 m. The dynamic beha io shows se e al low- equency ib a ion modes wi h closely-spaced na u al equencies and low damping a ios. A p ocedu e o ack ib a ion modes has been p og ammed o ca y ou an au oma ic modal pa ame e es ima ion. This p ocedu e has o cope wi h ib a ion modes wi h closely-spaced na u al equencies. In his case, so ing ib a ion modes using equency c i e ia and emo ing ou lie s ha lay ou o in e als o equency a ia ion migh no be adequa e [14, 16]. This pape desc ibes he de ailed p ocedu e ca ied ou o ack ib a ion modes o PGB oo b idge o e ime. The p ocedu e uses h ee ole ances based on he Modal Assu ance C i e ion (MAC). This app oach allows eliable moni o ing o ib a ion modes ha a e main ained o e ime. Rega ding he use o modal pa ame e a ia ions as possible damage de ec o in s uc u es, i is necessa y o obse e he modal a ia ions o e ime due o he in luence o some ex e nal agen s ac ing on he undamaged s uc u e [12, 14]. I is impo an o e alua e he magni ude o his modal a ia ion, as well as he main ac o s ha in luence he ela ionship be ween hem. The empe a u e is usually he mos signi ican en i onmen al ac o [20]. Howe e , o he ac o s, such as ope a ional o bounda y condi ions, may signi ican ly a ec he modal es ima es [1, 12]. Fini e Elemen (FE) models a e usually upda ed using he modal pa ame e s [4, 15]. Howe e , when he modal pa ame e s show signi ican a iabili y wi h changing en i onmen al ac o s, he upda ed model migh no be as accu a e as equi ed [2]. Hence, i is impo an o es ima e unce ain ies in he modal pa ame e iden i ica ion in o de o es ablish he deg ee o con idence o he upda ed model. This upda ed model can be use ul o gua an ee a success ul s uc u al in e en ion du ing s uc u al ehabili a ion [6, 11]. Tha is, non- des uc i e es s, such as an OMA, oge he wi h calib a ed models g ea ly bene i eliable s uc u al e alua ions. The pape con inues wi h he desc ip ion o he s uc u e and i s moni o ing sys em. Sec ion 3 analyzes he unce ain ies associa ed o a single ime his o y eco d. The ou pu -only modal pa ame e es ima ion using di e en da a blocks is ca ied ou . Thus, unce ain ies due o he selec ion o he da a block a e quan i ied. The esul s o con inuous dynamic analysis a e desc ibed in Sec ion 4. A acking me hod o ollow he e olu ion o he pe sis en ib a ion modes o e ime is p oposed. The in luence o he en i onmen al/ope a ional ac o s on he modal es ima es is s udied. Finally, some conclusions a e d awn and sugges ions o u u e wo k a e gi en. 42 2. THE FOOTBRIDGE AND ITS VIBRATION MONITORING 2.1. S uc u e desc ip ion PGB oo b idge, si ed in Valladolid (Spain), is a slende and ligh weigh s uc u e ha c ea es a pedes ian link o e he Pisue ga Ri e be ween a spo complex and he ci y cen e (see Figu e 1). This b idge, buil in 2011, is a singula s ess- ibbon oo b idge bo n by a p e ensioned ca ena y-shape s eel band wi h a single span o 85 m ha p o ides minimal impac on he su oundings. The s uc u e mainly consis s o a Co en s eel band o 94 m long, 3.6 m wide and only 30 mm hick which is p e- ensioned and ancho ed o he wo abu men s. The comple e s eel band is ab ica ed by welding 8-me e long shee s and a numbe o 110 p ecas conc e e slabs lay on he s eel band [17]. These slabs do no ha e bea ing capaci y in such a way ha he only s uc u al elemen is he band. The s uc u e is comple ed by ubbe loo ing and a s ainless s eel glass hand ail. Figu e 1. Ped o Gómez Bosque oo b idge. Landscape iew. 2.2. Moni o ing sys em A s uc u al ib a ion moni o ing sys em was de ised in o de o con inuously es ima e he modal pa ame e s o he s uc u e and o assess hei changes unde a ying en i onmen al condi ions. The e o e, apa om he accele ome e s needed o pe o m a modal analysis, senso s o he wind and en i onmen al empe a u e condi ions we e ins alled. The moni o ing sys em comp ises 18 iaxial accele ome e s, 9 a each side o he deck, a empe a u e senso and an anemome e wi h a ane. Wi es and accele a ion senso s we e ins alled inside he hand ail so he s uc u e aes he ic was no modi ied in any way. This ac in oduced addi ional complica ions: (i) he ins alla ion p ocess was a labo ious ask, and (ii) addi ional angula ans o ma ions a e equi ed o ob ain he accele a ion ec o in he s uc u e axes o each accele ome e . The ib a ion senso used o he moni o ing sys em was he low-cos MEMS accele ome e ADXL327 (ANALOG DEVICES) able o measu e he s a ic accele a ion o g a i y. The ADXL327 is a e y small, low powe , 3-axis accele ome e wi h signal condi ion ol age ou pu . The key p ope ies o his senso a e: measu emen ange up o ±2.5g, sensi i i y up o 500 mV/g and bandwid h up o 550 Hz. Howe e , his senso is no designed o ansmi he signal o e long dis ances. To o e come his p oblem, an ad-hoc condi ioning ci cui was designed o enhance i s long-dis ance pe o mance. Fi s , h ee 43 capaci o s, one o each channel, we e placed o ix he equency bandwid h o 100 Hz. Second, since he accele ome e has o be supplied by 3.6 V o ge i s maximum nominal sensi i i y o 500 mV/g, he powe supply uni o 12 V and a ol age egula o o 3.6 V we e in eg a ed in o each ci cui boa d in o de o a oid powe losses by he long dis ance wi es. Thi dly, an ope a ional ampli ie was used o educe signi ican ly he ou pu impedance, achie ing hus a good signal- o-noise a io o accele a ion pa ame e (cons an sensi i i y and low noise). The achie ed signal- o-noise a io was 25 μg/√Hz, which was conside ed o be enough o moni o ing he s uc u al ib a ions. Finally, each ci cui boa d wi h all i s componen s was co e ed wi h a plas ic coa ing o p o ec i om en i onmen al condi ions. Then, he accele ome e s we e eady o be ins alled inside he ube o he hand ail. The sensing sys em was comple ed by a empe a u e senso (model T0110 ansmi e o Come ) and a wind sen y (model 03002L o R.M. Young Company) o measu e he speed and di ec ion o he wind. The empe a u e senso and wind sen y we e ins alled on he public ligh owe si ed closed o he s uc u e (see Figu e 2). Figu e 2. Moni o ing sys em. Dis ibu ion o senso s. The moni o ing sys em hen comp ises 57 ol age channels ha a e p ocessed con inuously. The da a logge Compac RIO 9076 (Na ional Ins umen s) wi h wo NI 9205 wi h 32 analog inpu channels is used o eal- ime da a acquisi ion. The equency sampling o each channel was chosen o be 200 Hz, enough o iden i y he modal pa ame e s o he s uc u e and o a oid aliasing p oblems du ing he pos -p ocessing. The ac ual o ien a ion o he accele ome e s ins alled inside he hand ail is unknown. Howe e , he Eule angles be ween he accele ome e coo dina e sys em and he s uc u e coo dina e sys em can be de i ed aking in o accoun he ollowing: (i) he longi udinal axis o each accele ome e ma ches wi h he longi udinal axis o he oo b idge, and (ii) he accele ome e s a e able o measu e accele a ion due o g a i y. The e o e, he ans o ma ion ma ix be ween bo h coo dina e sys ems is ob ained, and hen, he accele a ion in he global axes can be inally calcula ed [18]. Be o e i s inal ins alla ion, a labo a o y alida ion was ca ied ou and a e he ins alla ion, an in-si u alida ion was pe o med by compa ing agains con en ional piezoelec ic accele ome e s. 3. PEERED ANALYSIS OF ONE TEST The unce ain ies associa ed o he modal iden i ica ion o one es , co esponding o he eco d measu ed on 01/05/2013 a 19:16 (60-minu e es ), a e analyzed he ein. The dynamic beha io o he s uc u e is mainly go e ned by he e ical esponse (p e ious ime-his o y analyses ha e shown ha 44 la e al and longi udinal accele a ions a e small and negligible, espec i ely, as compa ed o e ical ones). Tha is he eason why his analysis concen a es on he e ical ib a ion. The p ocess ollowed is, i s ly, he signal p ocessing unde aken be o e he modal iden i ica ion is p esen ed. A e wa ds, using SSI-co echnique is applied o di e en 20-minu e ime windows o he 60-minu e es . 3.1. Da a p ocessing The e ical esponse o he s uc u e o 1-hou es is analyzed (wi h an ini ial sampling equency o 200 Hz). The signal is il e ed by a low-pass Bu e wo h il e o o de 4 wi h a cu o equency o 5 Hz. A decima ion ac o o 16 is applied ob aining a Nyquis equency o 6.25 Hz. As an example, Figu e 3a and b show he il e ed and decima ed esponse in ime and equency domain, espec i ely, co esponding o he ou cen al channels o ups eam side. Up o en peaks below 4 Hz can be obse ed in Figu e 3b. (a) Time domain. (b) F equency domain. Figu e 3. P ocessed aw da a o 4 cen al channels o ups eam side. 3.2. Ope a ional Modal Analysis using he same SSI echnique The modal pa ame e s ha e been acked o e ime o cap u e he dynamic beha io o he s uc u e as bes as possible. As a ule o humb, he minimum du a ion o he measu emen should be a leas 45 1000 cycles o he lowes na u al equency is expec ed o be iden i ied [19]. This du a ion is usually ecommended as he minimum one o noise-con amina ed signals and close ib a ion modes. The e o e, his du a ion, which co esponds app oxima ely o 20 minu es, has been adop ed he e o ca y ou he modal acking. Th ee consecu i e windows o 20-minu e da a a e conside ed o iden i y he modal pa ame e s using only SSI-co . These esul s a e compa ed wi h hose ob ained using he 60-minu e eco d. Table 1 shows he es ima es ob ained om he ou da a blocks. A MAC co ela ion be ween he modal shapes es ima ed by h ee echniques has been ca ied ou . Modes ha exhibi a MAC alue g ea e han 0.95 o all o he c oss alues a e highligh ed in bold. Modes wi h na u al equencies o e 4 Hz (0.8 imes he cu o equency o he il e ) a e no included. Modes ha exhibi a MAC alue g ea e han 0.95 o all o he c oss alues a e highligh ed in bold. The a e age empe a u e o each ime window was 16.42, 13.84 and 11.56 oC, espec i ely, and 13.94 oC o he 60-minu e eco d. Table 1. Na u al equencies and damping a ios iden i ied by SSI-co and di e en ime blocks. Mode 60 min. 1s 20 min. 2nd 20 min. 3 d 20 min. (Hz) 𝜻𝜻 (%) (Hz) 𝜻𝜻 (%) (Hz) 𝜻𝜻 (%) (Hz) 𝜻𝜻 (%) - 0.8671 0.3352 0.8663 0.2526 - - 0.8660 1.0824 - - - 1.0251 0.3793 1.0243 0.1433 1.0281 1.8147 1 1.0548 0.9622 1.0536 0.3755 1.0518 0.3513 1.0535 0.4883 2 1.4077 0.3792 1.4073 0.3523 1.4059 0.5874 1.4093 0.2594 3 1.5408 0.6762 1.5408 0.5343 1.5395 0.7945 1.5431 0.7658 4 1.7869 0.4483 1.7884 0.5036 1.7882 0.6227 1.7833 0.4014 5 1.8571 0.6381 1.8567 0.6340 1.8574 0.6775 1.8569 0.4995 - - - 1.9260 0.3870 - - - - 6 2.3020 0.4799 2.3022 0.4444 2.3020 0.5353 2.3014 0.4788 - 2.6819 0.3206 - - 2.6802 0.2709 2.6836 0.2546 - - - - - - - 2.7306 0.4594 - - - - - 2.7848 0.3825 - - - 2.8051 0.5159 2.8097 0.7520 - - 2.8041 0.4918 - - - - - 2.8147 0.1976 - - - 2.8682 0.3238 - - - - 2.8631 0.3248 - - - 2.9171 0.2887 - - - - 7 3.3674 0.4485 3.3680 0.4194 3.3659 0.4449 3.3709 0.4587 8 3.5375 0.8723 3.5356 0.6417 3.5361 0.9920 3.5343 0.8376 9 3.9487 0.5106 3.9443 0.4427 3.9458 0.5759 3.9552 0.3504 Table 2 p esen s a summa y o he selec ed modes. The able shows he mean alues o he 20-minu e es s and he maximum e o s compa ed wi h he esul s ob ained wi h he 60-minu e es . In his case, he maximum e o s ob ained a e 0.2855% and 61.09% o equencies and damping a ios espec i ely. The e o s in na u al equency es ima es using 20 minu es as compa ed wi h 60 a e negligible. 46 Table 2. Summa y o iden i ied mode equencies and s a is ical compa ison o he 20-minu e agains 1-hou ime blocks: mean equency, mean damping and co esponding e o s. Mode F equency Damping 𝒇𝒇� (Hz) E o (%) 𝜻𝜻 �(%) E o (%) 1 1.0530 0.2849 0.4050 61.09 2 1.4075 0.2416 0.3997 32.80 3 1.5411 0.2336 0.6982 26.02 4 1.7866 0.2855 0.5092 22.13 5 1.8570 0.0377 0.6037 17.80 6 2.3019 0.0348 0.4862 9.090 7 3.3683 0.1484 0.4410 3.930 8 3.5353 0.0905 0.8238 35.05 9 3.9484 0.2761 0.4563 22.55 Figu e 4 shows an example o iden i ied modal shapes co esponding o he lowes six modes ob ained wi h SSI-co o he 60-minu e es . (a) = 0.87 Hz (b) = 1.05 Hz (c) = 1.41 Hz (d) = 1.55 Hz (e) = 1.79 Hz ( ) = 1.86 Hz Figu e 4. Fi s six modes shapes ob ained wi h SSI-co . 47 4. CONTINUOUS DYNAMIC ANALYSIS The p ocedu e ca ied ou o ack he ib a ion modes is depic ed in Figu e 5. The esul s ob ained om 1-yea o con inuous dynamic moni o ing a e desc ibed om now on. A me hod o ack he e olu ion o he main ib a ion modes o e ime is desc ibed. Then, he in luence o he en i onmen al/ope a ional ac o s on he modal es ima es is s udied. 4.1. T acking o modal p ope ies Fo a single es , no special ea men on SSI i sel o sepa a e closely-spaced equencies is conside ed. Howe e , in a dynamic con inuous moni o ing amewo k, he ib a ions modes mus be ca e ully acked since hei na u al equencies can be simila and e en can change hei o de . The e o e, a p ocedu e o link modal pa ame e s iden i ied om each da a se ha a e associa ed wi h he same ib a ion mode is needed. A c i e ion based on selec ing a mode wi hin a ixed equency ange could lead o w ong esul s due o he a ia ion o equency anges be ween di e en es s. As a consequence, i p oposes a acking me hod in o de o ollow he e olu ion o he main ib a ion modes o e ime. F om he acqui ed da a, an au oma ed OMA has been implemen ed using SSI-co . The p ocedu e ca ied ou is depic ed in Figu e 5. The p ocess can be di ided in o he ollowing s eps: Figu e 5. T acking p ocedu e o modal pa ame e s. 48 •Take he las N es s, N being he numbe o es s. N should be ep esen a i e o he a iabili y o he modal es ima es. •Fo each es , he OMA is ca ied ou . Thus, a numbe o modes o each es is iden i ied and deno ed as 𝑟𝑟𝑖𝑖 (wi h i = 1 . . .N). •The objec i e is o ind modal es ima es ha co espond o he same pe sis en mode o e ime. Each modal shape es ima e is compa ed wi h all he emainde es ima es wi hin he N es s and a coun e inc eases each ime ha wo mode shapes ma ch. The coun e indica es he epea abili y o a ib a ion mode. This is done o all he modal shape es ima es. To ca y ou his s ep, he MAC alue is used. Tha is, a numbe o �∑𝑟𝑟𝑖𝑖 𝑁𝑁 𝑖𝑖=1 � es ima es a e compa ed using he MAC alue. Th ee ole ances ha e been de ined o his pu pose: o ol1: he i s one is a MAC alue ha allows o g oup es ima es co esponding o he same ib a ion mode. The epea abili y is he numbe o es ima es o a g oup. o ol2: he second one is included in o de o ejec g oups ha a e no epea ed su icien ly o be conside ed as signi ican modes. This ole ance is he lowe limi o success a io, ha is, g oups wi h low epea ibili y a e au oma ically ejec ed. o ol3: he hi d one is a MAC alue inally included o de ec g oups o es ima es ha a e ac ually es ima ions o he same mode. Then, i mo e han one g oup co espond o he es ima ion o he same mode, he one wi h highe epea abili y is selec ed. •F om abo e p ocedu e, he mos signi ican modes o he dynamic esponse o he s uc u e a e de ec ed o he N es s. •Va ia ions o hese selec ed modes a e s a is ically s udied and co ec ed modal pa ame e s a e de i ed. The abo e acking me hod is applied o consecu i e ime-his o y eco ds e e y 20 minu es using SSI- co . A numbe o N = 21643 es s ha e been aken o he whole yea 2013. I compu a ional bu dens appea , he p ocedu e explained abo e mus be modi ied sligh ly. In ha case, he es s should be di ided in o sub-blocks (N = 500) o sol e he p oblem. Tha done, you jus ha e o join each sub-block wi h he nex sub-block using he ol1. The ole ances chosen o ca y ou he acking we e: ol1 ≥ 0.95, ol2 ≥ 40% and ol3 ≤ 0.80. Up o nine ib a ion modes below 4 Hz ha e been acked. Table 3 shows he ollowing s a is ics o he es ima ion: mean, s anda d de ia ion, absolu e pe cen age a ia ion and hei epea abili y ( he success a io is included be ween b acke s). No e ha he ou h mode, wi h a equency a ound 1.79 Hz (and wi h a damping a io o only 0.42%) is p one o be exci ed by pedes ian walking. Table 3. Summa y o iden i ied na u al equencies and damping a ios o one yea moni o ing and hei s a is ics: mean equency, mean damping, s anda d de ia ion and co esponding a ia ion. Mode F equency Damping Repea abili y 𝒇𝒇� (Hz) S d 𝝊𝝊 (%) 𝜻𝜻 �(%) S d 𝝊𝝊 (%) 1 1.0482 0.0152 14.23 0.3665 0.1710 147.89 9667 (44.7%) 2 1.4145 0.0107 35.26 0.3381 0.1513 110.74 10619 (49.1%) 3 1.5440 0.0181 27.63 0.6498 0.2357 133.62 9886 (45.7%) 4 1.7937 0.0291 20.27 0.4192 0.1502 221.88 13817 (63.8%) 5 1.8594 0.0168 6.87 0.5718 0.1605 234.74 9936 (45.9%) 6 2.3117 0.0425 15.01 0.3753 0.1474 128.54 8746 (40.4%) 7 3.3821 0.0549 42.95 0.3868 0.1191 103.96 12210 (56.4%) 8 3.5512 0.0524 51.87 0.7226 0.1884 157.48 9237 (42.7%) 9 3.9619 0.0624 8.95 0.3853 0.1185 230.82 10183 (57.8%) 49 he p ojec a e: ISOLUX CORSÁN, FCC CO, GEOCISA, CSIC and UPM. The au ho s also acknowledge he inancial suppo p o ided by Resea ch P ojec DPI2013-47441-P. REFERENCES [1] A ci, O. (2014). Modal Pa ame e Va ia ions due o Jois Bo om Cho d Ex ension Ins alla ions on Labo a o y Foo b idges. Jou nal o Pe o mance o Cons uc ed Facili ies 04014140. DOI: 10.1061/(ASCE)CF.1943-5509.0000635. 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