Time-domain Aeroelastic Analysis of Bridge using a Truncated Fourier Series of the Aerodynamic Transfer Function

Transcription

1 Time-domai Aeroelasic Aalysis of ridge usig a Trucaed Fourier Series of he Aerodyamic Trasfer Fucio Jiwook PA Graduae Sude Seoul aioal iversiy Seoul, orea jwpark7@su.ac.kr H Sug LEE Professor Seoul aioal iversiy Seoul, orea chslee@ su.ac.kr ilje JG Visiig Scholar iversiy of ore Dame ore Dame, SA ilje.jug.9@d.edu Ho-yug Associae Professor Seoul aioal iversiy Seoul, orea hokyugk@ su.ac.kr Summary A rucaed Fourier series approximaio mehod for he ime-domai roelasic aalysis of bridge is preseed. The rodyamic rasfer fucios are modified by approximaig hose usig a rucaed Fourier series o srogly eforce he causaliy codiio i he impulse respose fucio. The coefficies of rucaed Fourier series are deermied by miimizig he error bewee measured ad modified rasfer fucio. The impulse respose fucios correspodig o he modified rasfer fucios become a series of Dirac dela fucios wih he same umber of he erms used i he Fourier series, which cosiderably reduces compuaioal effor. eywords: impulse respose fucio; rasfer fucio; Fourier series; causaliy codiio; covoluio iegral; roelasic aalysis; fluer derivaive. roducio The imporace of ime-domai roelasic aalysis has bee emphasized i rece decades o cosider various olieariies of a srucural sysem ad/or o-saioary effecs of air flows, alhough he rodyamic forces are basically defied i he frequecy domai wih he fluer derivaives[]. The key issue of he ime-domai roelasic aalysis is evaluaio of rodyamic impulse respose fucios obaied by akig iverse Fourier rasform o he measured rasfer fucios defied i erms of he fluer derivaives. Oce he impulse respose fucios are kow, he rodyamic forces i he ime-domai ca be evaluaed hrough a oe-sided covoluio iegral. Here, he impulse respose fucios should saisfy he causaliy codiio which saes ha he impulse respose fucio i egaive ime-domai is zero from he physical poi of view. However, because he measured fluer derivaives are raced wihou ay cosideraio of he causaliy codiio, he measured rasfer fucios should be modified o saisfy he causaliy codiio. The raioal fucio approximaio (FA has bee widely used o modify he rodyamic rasfer fucios []. However, he FA, which is based o he ac soluio o he ideally hi secio, has a cerai limiaio o is applicabiliy o bluff secios [3]. To overcome he limiaio of he FA, Jug e al. [4] proposed he pealy fucio approach (. heir approach, he causaliy codiio is weay imposed as a pealy fucio i he miimizaio o modify rodyamic rasfer fucios usig he cubic splie ierpolaio. Alhough heir approach yields accurae ad sable resuls eve for a bluff secio, a raher complicaed FE-based formulaio is required, ad he pealy umber should be deermied ieraively. oreover, he covoluio iegrals should be evaluaed hrough umerical iegraio from o he curre ime, which requires a huge compuaioal effor for large-scale srucures. This paper preses he ac relaio bewee he real ad imagiary pars of rodyamic rasfer fucios for derivig impulse respose fucios ha saisfy he causaliy codiio usig a rucaed Fourier series. The coefficies of rucaed Fourier series are deermied by miimizig

2 he error bewee measured ad modified rasfer fucio. The impulse respose fucios correspodig o he modified rasfer fucios become a series of Dirac dela fucios, which cosiderably reduces compuaioal effor. The validiy ad effeciveess of he proposed mehod are demosraed hrough umerical amples for he bluff H-ype secio ad large-scale bridge, d Jido cable sayed bridge, by performig ime-domai roelasic aalysis.. Causaliy equireme i Aerodyamic Forces The rodyamic forces iduced by moio of a objec i a saioary wid flow are pressed by he covoluio iegrals i he ime-domai []. L τ = ρ ( ( ( ( τ dτ hα τ α τ dτ τ = ρ ( ( ( ( αh τ dτ αα τ α τ dτ where L ad = he rodyamic lif force ad mome, respecively; h ad α = he verical ad roaioal displaceme, respecively; ρ = air desiy; = mea cross wid velociy; ad = widh of he secio. The real fucio, for k, l = h, α is he -compoe of he impulse respose fucio represeig he rodyamic force i he k direcio a ime iduced by he ui impulse moio of a objec i he l direcio a =. The oe-sided covoluio iegrals i Eq. ( are valid if ad oly if every compoe of he impulse respose fucio vaishes for he egaive ime domai [4], ha is, for <, which is he causaliy codiio. The impulse respose fucio is defied as he iverse Fourier rasform of he rasfer fucio of he rodyamic forces i he frequecy domai: i = i e ω ( ( dω ( π = he -compoe of he rodyamic rasfer fucio; i = he imagiary ui; ad Here, superscrip ad idicae he imagiary ad real par of a compl variable, respecively. The rodyamic rasfer fucio i Eq. ( is pressed i erms of fluer derivaives ideified i wid uel ess [5]: i i where = ω/ = he o-dimesioal reduced frequecy where ω = he agular frequecy of * * oscillaio; ad H m ad A m ( m =,, 3, 4 = he fluer derivaives. As he impulse respose fucio is a real fucio, (ω ad (ω are a eve ad odd fucio, respecively. The impulse respose fucio for ad he causaliy codiio for < becomes as follows: * * * = i H H 4 ihα hα = i H * * * α h αh = i A A4 iαα αα = i A π ( = ( ( ω cosω ( ωsi ω dω for = ( ω cosω, H, A ( ωsi ω dω for < * 3 * 3 ( (4 (5 The causaliy codiio i Eq. (5 implies ha a cerai relaioship iss bewee he real ad imagiary par of he rodyamic rasfer fucio. Sice, however, such a relaio is geerally o cosidered i he ideificaio of fluer derivaives, he measured rasfer fucios should be modified so as o saisfy he causaliy codiio. (3

3 3. Eforceme of he Causaliy Codiio usig a Trucaed Fourier Series The rodyamic rasfer fucio is geerally pressed i erms of he reduced frequecy, ad is defied up o he imum reduced frequecy,, adoped i acual wid-uel ess. Therefore, each compoe of he modified rasfer fucio is pressed as a rucaed Fourier series wih he period of. Sice he real ad imagiary par of he rodyamic rasfer fucio are a eve ad odd fucio, respecively, he Fourier cosie series ad he Fourier sie series are separaely adoped for he idividual par as follows: where = he compoe of he modified rasfer fucio; a ad = ukow coefficies of he Fourier series; ad = he umber of erms i he Fourier series. The liear erm is iroduced i he imagiary par of Eq. (6 o preve oscillaios of he Fourier sie series caused by a discoiuiy bewee he Fourier sie series ad he measured rasfer fucio a =. Subsiuig Eq. (6 io Eq. (4 yields he pressio for he modified impulse respose fucio. = π = = a ( ( cosω π [ δ( ( cos( b δ ( si ω dω δ( π = ( si( ] d a b δ( π = where = he -compoe of he modified impulse respose fucio. The firs hree erms i he las equaio of Eq. (7 vaish for <, while he las erm does o uless b = a. Therefore, he causaliy codiio for he modified rasfer fucio i Eq. (6 is defied as: b = a for =,, (8 Eforceme of Eq. (8 o Eq. (6 ad Eq. (7 leads o he fial pressios for he modified rasfer fucio ad he correspodig impulse respose fucio ha acly saisfy he causaliy requireme. = a = b π a cos = π a si = = aδ( b δ π aδ( = The ukow coefficies i Eq. (9 are easily deermied by miimizig he errors bewee he measured ad modified rasfer fucios. The miimizaio process proposed by Jug e al. [4] is give as: i Π a ( = a ( = b ( a a cos = π b si = = L π L a b w ( w ( ( ( ( a d a d T where a ; = a prescribed weighig facor. Each erm i he objec = (a b a a w fucio is ormalized wih respec o is ow L-orm deoed as o level he magiude of L b (6 (7 (9 ( (

4 each erm. Wih his ormalizaio, a equal weighig of w = / for all ca be adoped. Sice he modified impulse respose fucios become a series of Dirac-dela fucios, he rodyamic forces i Eq. ( are evaluaed wihou umerical iegraio. L = ρ a hα = ρ a ( a b αα hα ( a b b αh αα α b α h a = ahα = αh π h aαh aαα = π = π Eq. ( coais oly pas displacemes, while he covoluio iegral wih he impulse respose fucios obaied by he requires he complee ime hisories of he displacemes i a ime-marchig algorihm. The proposed mehod grealy improves compuaioal efficiecy i he evaluaio of he rodyamic forces, especially, for a large-scale srucure. 4. Applicaios ad Verificaio For he verificaio of he proposed mehod, ime-domai rodyamic aalyses are performed for he H-ype bluff secio ad a real bridge, d Jido cable sayed bridge. The fluer derivaives of he bluff H-ype secio are raced by im ad ig [6] a he oudary Layer Wid Tuel Laboraory of he iversiy of Weser Oario i Oario, Caada, ad he fluer derivaives of d Jido cable sayed bridge are raced a he wid uel laboraory of Seoul aioal iversiy i Seoul, orea. 4. H-ype secio Jug e al. [4] demosraed he limiaio of he FA for his secio. appears ha a compariso of he proposed mehod wih he FA would be meaigless, ad hus he resuls obaied by he proposed mehod are compared wih hose by he proposed by Jug e al. [4]. For roelasic aalyses i he ime domai, a elasically suppored sysem is cosidered. The equaios of moio per ui legh are defied as follows: m h h c h h k = L L m α c α k α = α α α where m, c j j ad k j are he mass, dampig ad siffess i he direcio of j = h, α, respecively. L ad are he eral ciaio forces i he h ad α direcio, respecively. Fig. shows he modified rasfer fucios for he lif force evaluaed usig he proposed mehod ad, respecively, alog wih he measured oes. The proposed mehod ad he yield almos he same resuls, eve hough some differeces are observed i he compoes. Sice he proposed mehod yields a closer soluio o he measured rasfer fucio, i is believed ha he proposed mehod represes acual physical pheomea beer ha he. To esure he covergece of he proposed mehod, he umber of series erms is varied as, 5 ad. As show i Fig., he modified rasfer fucios wih 5 erms are closely coverge o hose wih erms. The accuracy of he proposed mehod is amied for he secio moued o sprigs ad subjeced o harmoic ciaios. The applied ciaio forces are as follows: L L ω = si (4 where L ; ; ad = /m = m/m ω = 8π rad/s. Fig. shows he verical displaceme of π ( (3

5 he secio. The rasie resposes ad paricular resposes are show i Figs. (a ad (b, respecively. o oiceable differece is foud amog he resposes by he proposed mehod, he ad he paricular soluio of Eq. (3. The resuls of he rasfer fucios for he mome ad he roaioal displacemes by he proposed mehod ad he are also almos he same bu o preseed, here. magiay par of compoe (a (b =ω/ =ω/ 4 5 easured rasfer fucio Proposed mehod (= Proposed mehod (=5 Proposed mehod (= easured rasfer fucio - Proposed mehod (= - Proposed mehod (=5 Proposed mehod (= (c =ω/ (d =ω/ 4 5 Fig.. Trasfer fucios of he H-ype secio for he lif force: (a imagiary par of he compoe; (b imagiary eal par of compoe - easured rasfer fucio Proposed mehod (= Proposed mehod (=5 Proposed mehod (= par of he hα compoe; (c real par of he compoe; ad (d real par of he hα compoe magiary par of hα compoe eal par of hα compoe - easured rasfer fucio Proposed mehod (= Proposed mehod (=5 Proposed mehod (= Verical displaceme (cm 5 Proposed mehod (= (a Time (s (b Time (s Fig.. Forced vibraio resposes a a wid velociy of 6.m/s for he H-ype secio: (a verical displaceme for =~ s; (b verical displaceme for =8~ s 4. d Jido Cable Sayed ridge d Jido Cable Sayed ridge has mai spa legh of 344 m ad is locaed bewee Jido ad Ham, orea. The modal aalysis is used for a real bridge wih muli-degree of freedom ulike a secio wih wo-degree of freedom[7]. o oly self-ied forces bu also buffeig forces are cosidered i he ime-domai aalysis of his real bridge. The velociy flucuaios are geeraed by AA (Auo-regressio movig-average echique hrough vo-arma specrum wihou cosiderig he admiace fucio. The modes which have a high level of coribuio i he Verical displaceme (cm 5 Paricular sol. Proposed mehod (=

6 direcio of verical ad roaioal displaceme for he secio are used for he modal aalysis. Fig. 3 shows he measured rasfer fucio ad modified rasfer fucio for he lif force obaied by ad proposed mehod for =, 5, ad. As i he previous ample, he propose mehod ad yields almos he same resuls. Fig. 4 shows he verical displaceme a he middle of deck spa. The resposes by proposed mehod are also almos he same wih hose by. To check he applicabiliy of he proposed mehod for a real bridge, he compuaio imes for he ad proposed mehod durig covoluio iegraio are compared. The compuaio ime is 5.7 sec for ad 7.9 sec for he proposed mehod. This differece of he compuaio ime ca be cosidered as a meaigless differece. However, if he olieariies of srucural sysem are cosidered, ime ierval is decreased ad he umber of cosidered mode for modal aalysis is icreased, he differece of compuaio ime bewee he ad proposed mehod ca be huge. magiary par of compoe easured rasfer fucio Pealy mehod - Proposed mehod (= Proposed mehod (=5 Proposed mehod (= - (a =ω/ easured rasfer fucio Pealy mehod Proposed mehod (= Proposed mehod (=5 Proposed mehod (= eal par of compoe - magiary par of hα compoe - easured rasfer fucio Pealy mehod Proposed mehod (= Proposed mehod (=5 Proposed mehod (= - (b =ω/ 4 easured rasfer fucio Pealy mehod Proposed mehod (= Proposed mehod (=5 Proposed mehod (= eal par of hα compoe (c =ω/ (d =ω/ Fig. 3. Trasfer fucios of he secio of d Jido bridge for he lif force: (a imagiary par of he compoe; (b imagiary par of he hα compoe; (c real par of he compoe; ad (d real par of he hα compoe Verical displaceme (cm. Proposed mehod (= Time (s Fig. 4. esposes a he middle of deck spa of d Jido bridge for a wid velociy of 3. m/s

7 5. Coclusios The causaliy codiio, required o perform oe-sided covoluio iegrals for a ime-domai roelasic aalysis, is srogly eforced by pressig each par of he rodyamic rasfer fucio wih a rucaed Fourier series. The coefficies of he Fourier series are deermied hrough he miimizaio of errors bewee he measured rasfer fucio ad he Fourier series. The covoluio iegrals coai oly a few erms o he curre ad pas displacemes, ad he compuaioal efficiecy is grealy improved compared wih he. The applicabiliy ad validiy of he proposed mehod are demosraed hrough he amples for he H-ype secio ad d Jido cable sayed bridge. From he resuls of he amples, cosequely, he proposed mehod ca be applied o he bluff secio ad large-scale bridge, efficiely. 6. Ackowledgeme This research was suppored by he gra (9CCT-A from he iisry of Lad, Traspor ad ariime Affairs of orea goverme hrough he Core esearch siue a Seoul aioal iversiy for Core Egieerig Techology Developme of Super Log Spa ridge &D Ceer. 7. efereces [] CHE, X., ad AEE, A., Aeroleasic Aalysis of ridges: Effecs of Turbulece ad Aerodyamic olieariies. ASCE Joural of Egieerig echaics, Vol. 9, o. 8, 3, pp [] CHE X., ATSOTO., ad AEE A., Aerodyamic Couplig Effecs o Fluer ad uffeig of ridges. ASCE Joural of Egieerig echaics, Vol. 6, o.,, pp [3] CAACOGLA, L., ad JOES,.P., Time Domai vs. Frequecy Domai Characerizaio of Aeroelasic Forces for ridge Deck Secios. Joural of Wid Egieerig ad dusrial Aerodyamics, Vol. 9, o. 3, 3, pp [4] JG,.,, H.., ad LEE, H. S., Evaluaio of mpulse espose Fucios for Covoluio egrals of Aerodyamic Forces by Opimizaio wih a Pealy Fucio. ASCE Joural of Egieerig echaics, Vol. 38, o. 5,, pp [5] SCALA,. H., ad TOO, J. J., Airfoil ad ridge Deck Fluer Derivaives. ASCE Joural of Egieerig echaics Divisio, Vol. 97, o. 6, 97, pp [6] J.D., ad G J.P.C., The Developme of Wid Tuel Tes Techique for a Aeroelasic uffeig Aalysis of Log-spa ridges, LWTL-SS9-7-DAFT repor submied o orea Wid Egieerig esearch Ceer, he oudary Layer Wid Tuel Laboraory of he iversiy of Weser Oario. [7] ATSCH, H., JOES,. P., ad SCALA,. H., ulimode Coupled Fluer ad uffeig Aalysis of he Akashi-aikyo bridge. Joural of Srucural Egieerig, Vol. 5, o., 999, pp. 6-7