Wind Tunnel Study the Turbulence Effects on Aerodynamics of Suspended Truss Bridge

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Dominic McCoy
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1 Wnd Tnnl Sdy Trblnc Effcs on rodynamcs of Sspndd Trss rdg oang Trong Lam, ros asc and os Yamada,, Dp of vl Engnrng, Yooama Naonal Unvrsy, Yooama , Japan STRT Ts papr prsns rsls from a sa spac sysm dnfcaon racon flr drvavs ndr dffrnc rbln flows T am of sdy s o clarfy ffcs of oncomng rblnc on flr of sspndd long span brdgs dc by sng scon modl wnd nnl s Svral wnd nnl ss on a rssd dc scon av bn carrd o w dffrn oncomng rbln proprs nvolv rdcd rblnc nnss and rbln scals T analyss nclds nvsgaon on: ffc of modal dynamc rsponss on oncomng flows N, ransn and bffng rsponss daa from wnd nnl s av bn analyss by sysm dnfcaon cnq n racng flr drvavs FDs and dffcls nvolvd n s mod ar dscssd T m doman analyss socasc sysm dnfcaon DT-SSI s proposd o rac smlanosly all FDs from ons and wo dgr of frdom sysms Fnally, rsls ndr dffrn condons was dscssd and concldd INTRODUTION T wnd n amosprc bondary layr s always rblnc ny sdy of wnd-ndcd vbraon problms ms confron s ss r by macng rblnc caracrscs complly or by acnowldgng ncrany n conclsons as a rsl of mprfc smlaons No many rsarcs av focsd clarly on ffcs of rblnc on arolasc forcs Scanlan & Ln 978 ar ponrs wo sd a rssd dc scon modl n concldd a flr drvavs as an nsgnfcan dffrnc bwn smoo and rbln flows owvr, son condcd s on a modl of Goldn Ga rdg dc scon and rsls ar dffrn from a by Scanlan & Ln 978 aan & arm 007 Sdy on ffcs of rbln flows on flr drvavs, fr vbraon cnq of sconal modl sd and applcaon sysm dnfcaons cnq o rac FDs s wdly Varos sysm dnfcaons PD Sdn, Profssor

2 cnq was dvlopd by many aors: Endd alman Flr lgorm Yamada al 99, odfd Ibram Tm Doman Sara al 994, Unfyng las-sqars mod G al 000, Irav las-sqars mod owdrry and Sarar 00 In s sysms bffng forc and r rspons ar consdrd as rnal nos, so s cass mor dffcls a g wnd vlocy and spcally, appars rblnc arol and Rg 006 sd SI combnd sysm dnfcaon mod s basd on Sarar ITD o rac smlanosly all FDs from a DOF rcanglar scon modl T conclson s a dnfcaon of flr drvavs n rbln flow rsld sasfacory n sp of dffcls nconrd d o procss casd by locally ndcd nos owng o sgnar of rblnc T man rason s a, SI s drmnsc sysm dnfcaon and ffcs of rblnc ar rgardd as a mor nosy-np sgnal o sysm mas mor problms n dnfcaon procss Nas, acdonal and Jaobsn 0 ar mployd o rac FDs from ambn vbraon daa from fll-scal monorng sng mor labora socasc dnfcaon cnq Jaobsn al 995, and sdy also llsrad vably of sysm dnfcaon cnqs for racng valabl rsl from fll-scal daa oonyapnyo al 00 appld daadrvn socasc sbspac dnfcaon cnq SSI-DT n sor o rac FDs of brdg dc from wnd nnl s ndr bo smoo and rbln flows T conclson of s papr s a SSI-DT can b sd o sma FDs from bffng rsponss w rlabl rsls and an advanag of socasc sysm s a consdrs bffng forc and rspons l np nsad of nos Trfor, rao of sgnal o nos s no affcd by wnd spd and flr drvavs a g wnd spds ar radly avalabl rgaard & ndrsn 997 compard r sa spacs sysms: socasc sbspac dnfcaon SSI, socasc ralzaon smaor mar bloc anl and prdcon rror mod PE T rsls sow a SSI gvs good rsls n rms of smad modal paramrs and mod saps; s sn o gv poor smas of dampng raos and mod saps compard w or wo cnqs; and SSI s appromaly n ms fasr an PE In addon, r s a sorcomng nvolvd n fr vbraon cnq; a g wnd vlocs racon FDs canno b oband accraly bcas arodynamc dampng of vrcal mod s oo g and vrcal fr-vbraon daa s no long nog for analyss From s consdraons wc brng da for applyng socasc sysm dnfcaon SSI o sma FDs from gs rsponss of brdg rssd dc scon In s papr concnras on nvsgaon of rbln ffcs on FDs of brdg dc by mployng a scon modl wnd nnl s n a rbln flow T op only sysm dnfcaon DT-SSI s proposd o rac FDs from bffng rspons Tss ar also carrd o w fr vbraon mod n ordr o compar w proposd mod EPIREENTL SETUP ND TURULENT GENERTION wnd nnl s was condcd n a closd-crc wnd nnl a Yooama Naonal Unvrsy T worng cross-scon s 8m wd and 8m g T

nvsgad profl s rssd dc scon Fg I was fabrcad by wood w a scal :80

scon-modl ar 6mm, and 65mm, rspcvly T n mass 8095 g/m and momn of

lnar sprng w sffnss T monng poson of sprng was adjsd for lasc cnr

moons n a dsrd dgr of frdom wl prformng ss T ss av bn carrd o n

flows sd n s sdy wr gnrad w bplan woodn grd T rbln proprs ar

3 nvsgad profl s rssd dc scon Fg I was fabrcad by wood w a scal :80 o rprsn cross-scon of long-span sspndd brdg T wd and dp of scon-modl ar 6mm, and 65mm, rspcvly T n mass 8095 g/m and momn of nra 08 gm/m T scon s aacd o a rgd fram and a ac cornr sppord by a lnar sprng w sffnss T monng poson of sprng was adjsd for lasc cnr and gravy cnr of cross scon concdd T cross-scon was also o rsran moons n a dsrd dgr of frdom wl prformng ss T ss av bn carrd o n bo smoo and dffrn rblnc flows Fg Trssd dc sconal modl T rbln flows sd n s sdy wr gnrad w bplan woodn grd T rbln proprs ar conrolld by cangng dsancs o modl T flow condons and rblnc proprs ar followng:

4 a along wnd rblnc b vrcal wnd rblnc Fg Probably dnsy fncon of longdnal and vrcal vlocy flcaon Probably dnsy fncon T probably dnsy fncon of flcad wnd spd s csomarly w Gass dsrbon Fg sows bo longdnal and vrcal wnd spd flcaon farly good agrmn w Gass dsrbon Rdcd rblnc nnsy asd on macng powr spcrm of rbln flow n wnd nnl and fllscal spcral dnsy fncon, asc and Yamada 0 nrodcd rdcd rblnc nnsy wc can b wrn as sown blow: I I r L / D / wr I rblnc nnsy for along-wnd rblnc componn s dfnd as: I s sandard dvaon of rblnc componn and U s man wnd vlocy; D s g of cross-scon modl; L s ngral lng scal for rblnc componn n longdnal drcon In s sdy ngral scal s dfnd as: L U n n pa s frqncy a wc crv rdcd spcrm racs a mamm T lng scal calclad from Eq s rasonabl compard w aocorrlaon mod, bcas aocorrlaon of rblnc componn n svral cass s js asympoc o abscssa d o ara ndr aocorrlaon crv and abscssa q larg owvr, was pond n E Sm and Scanlan 996 a smaon of pa L basd on masrd val of U and n pa can b n rror svral fold, owng o snsvy of L o assmpon concrnng spcral sap bwn n = 0 and n = n pa Fg dpcs rsls of r rblnc paramrs I, I r, and L corrspondng o dffrn grd-o-modl dsancs, cas dsanc arond 48m, cas arond 4m and cas arond m w vary man wnd vlocs s n Fg sown a avrag paramrs cas : I = 67%, I r = 697% and L = 6cm; cas : I = 9%, I r =09% and L = 904cm; cas : I = 56%, I r = 0% and L = 679cm T rdcd rblnc nnss ncras proporonal w rblnc nnsy b nvrs s r for ngral lng scal Tabl Trbln flow paramrs Paramr as as as I %

L cm 6 904 679 I r % 697 09 0 Trblnc powr spcral dnsy T rblnc nnsy and ngral lng scal do no flly dscrb a proprs of rbln oncomng flows, bcas ans o Y Naamra and S Ozono 987 sdd on blff-body arodynamc

vlocy componn vrss non-dmnsonal powr spcral dnsy fncon and macng bwn powr spcrm of smlad daa from wnd nnl and mprcal amosprc rblnc von arman and Erocod ompard w von arman spcrm, masrd daa conccdd wll

5 L cm I r % Trblnc powr spcral dnsy T rblnc nnsy and ngral lng scal do no flly dscrb a proprs of rbln oncomng flows, bcas ans o Y Naamra and S Ozono 987 sdd on blff-body arodynamc sown a small-scal rblnc affcs flow flds and arodynamc parmrs mor an largr on Trfor powr spcral dsrbon of rblnc scals was also qanfd for s rsarc Fg 4a sows non-dmnsonal frqncy dsrbon of rbln along-wnd vlocy componn vrss non-dmnsonal powr spcral dnsy fncon and macng bwn powr spcrm of smlad daa from wnd nnl and mprcal amosprc rblnc von arman and Erocod ompard w von arman spcrm, masrd daa conccdd wll w n g frqncy and was a ll gr n low frqncy Trbln nrgy s gnrad n largr dds low frqncy For mos srcr, s low-frqncy flcaons gv no sgnfcan rspons conrbon In Fg 4b, r spcra oband a dffrn rdcd rbln nnsy rbln nnsy ar sown Vals of spcral dnsy fncon ncras as rdcd rblnc nnsy s ncrasd Non-dmnson PSD-nSn/ Non-dmnsonal frqncy, f L =nl /U a b Fg 4 Powr spcral dnsy fncon for longdnal rblnc componn PSD-Sn Frqncy ODEL DYNI RESPONSES Tss av bn condcd ndr bo smoo and rblnc flows T am of s sng s o qanfy ffc of oncomng rblnc flows on dynamc rsponss of scon modl Fg 5 llsras vbraon ampld of wo dgr of frdom avng and orsonal mod vrss rdcd wnd vlocs ndr smoo and dffrn rblnc flows bbrvaon of RS s roo-man-sqar; a s mamm ampld of vbraon; Smoo s smoo flow condon In smoo flow, vrcal vbraon s lmd wn rdcd wnd spd s from 0 o 9 and n consdrably ncrass b vrcal flr dos no occr n s s Fg 5a On

or and, orsonal dsplacmn s vry small ll sddn ncrmn and flr occr a rdcd wnd vlocy abo 55 Fg 5b a avng dsplacmn b Torsonal dsplacmn Fg 5 odl dsplacmn w vary rdcd rblnc nnsy Wn modl s mmrsd n rbln

occr In cas of I r = 697% and I r = 09%, orsonal vbraon gradally ncrass w rdcd wnd vlocy and dvrgnc occrs a Vrd = 7 and Vrd = 75, rspcvly, lar compard w smoo flow On or and, I r = 0% dvrgnc dos no

6 or and, orsonal dsplacmn s vry small ll sddn ncrmn and flr occr a rdcd wnd vlocy abo 55 Fg 5b a avng dsplacmn b Torsonal dsplacmn Fg 5 odl dsplacmn w vary rdcd rblnc nnsy Wn modl s mmrsd n rbln flows, vrcal and orsonal moons vbra n sp of small rdcd wnd vlocs T vrcal rspons ncrass proporonally w V rd and wn rdcd rbln nnsy ncrass, ampld of vbraon lgly ncras In rbln flows vrcal flr also dos no occr In cas of I r = 697% and I r = 09%, orsonal vbraon gradally ncrass w rdcd wnd vlocy and dvrgnc occrs a Vrd = 7 and Vrd = 75, rspcvly, lar compard w smoo flow On or and, I r = 0% dvrgnc dos no occr In gnral, ffcs of rbln flows on scon modl ndc largr vbraon an smoo flow b vbraon proporonally ncras w man vlocs and wo sddn ncras occrrng l smoo flow T moons ndr rbln flows ar nown as bffng rspons wc can ffc on srvc sa dsgn of brdg sc as fag problm 4 IDENTIFITION OF FLUTTER DERIVTIVES 4 Socasc sa-spac modls onsdrng a DOFs scon modl of brdg dc n rbln flow Flcang wnd load a acs on a dc can b prssd by a combnaon of a sl-cd forc and a bffng forc: m[ ] L I[ ] s L s b wr m and I ar mass and mass momn of nral pr n lng rspcvly; and a a ar crclar frqncs of avng and pcng mod n sll ar; and a ar crcal dampng rao; L s and s ar arodynamc slf-cd forc and momn, rspcvly, gvn by: b

7 U U U U U U L s s /U and =,,,4 ar flr drvavs L b and b ar bffng forcs n vrcal and orsonal drcons y sbsng abov qaon Eq 4 no Eq and movng arodynamc dampng and sffnss rms o lf and sd, Eq can b rwrn as follow: } { } ]{ [ } ]{ [ } ]{ [ f q q q 5 T s gnralzd bffng rspons; {f} ={L b b} T s bffng forc; {f} s facorzd no mar and np vcor ; [] = mass mar; [ ] = gross dampng mar ncldng pyscal dampng of srcr and arodynamc dampng; [ ] = gross sffnss mar T scond-ordr of dffrnal qaon, Eq 5 s gnralzd n -DOF can b ransformd no a frs-ordr sa qaon Eq q q I q q c m n nn n n n n 6 T combnaon of sa qaon and obsrvaon qaon flly dscrb np and op bavors of srcral sysm and ar as sc namd saspac sysm D y c c 7 wr c dsgnad sa mar s a n-by-n n=n ; s sa vcor; s np mar; c s op and D s drc ransmsson mar a connos m In modal analyss, somms np s nnown and masrmns ar mosly sampld a dscr-m On or and, s mpossbl o masr all DOFs and wn masrmns always av dsrbanc ffcs For all s rasons, connos drmnsc sysm wll b convrd o sabl form: dscr-m socasc sa-spac modl as follow: v y w 8 wr T q {q } ar dscr-m sa vcor conanng dscr

8 q sampl dsplacmn q and vlocy ; w s procss nos d o dsrbancs and modllng naccracs; v s masrmn nos d o snsor naccracy Followng assmpon w and v s zro man and w covaranc mar: wp E v p Q S T T wq vq pq wr nd p and q ar m- pq s roncr dla s corrlaon Ew p w T q and Ev p v T q ar qal zro n cas of dffrn m-nsan Frr socasc modl s assmd a { }, {w } and {v } ar mal ndpndn: E w T =0 and E v T =0 ccordng o Prs & Roc 999 provn a op covaranc R=E[y + y T ] for any arbrary mcan b consdrd as mpls rspons Eq 0 of drmnsc lnar mnvaranc sysm,, G; wr G= E[ + y T ] s n sa-op covaranc mar T S R R G 0 T classfcaon of SSI basd on y sp of s mods; by followng Prs & Roc 999, y ar covaranc-drvn socasc sbspac dnfcaon OV-SSI and daa-drvn socasc sbspac dnfcaon DT-SSI In s sdy DT-SSI sd o rac FDs 4 DT-SSI DT-SSI wors drcly w m-srs of prmnal daa, wo nd o convr o-p daa o corrlaon, covaranc or spcra T man sp of DT-SSI s a projcon of row spac of fr ops no row of pas ops T orogonal projcon P s dfnd as: f p f p p T p P Y / Y Y Y Y Y Y Wr mar Y f and Y p ar ndr alf par and ppr par alf of a boc anl mar, dfnd as: Y Y Y : j : j : j Y Y p f p l l wr ar nmbr of bloc rows, j ar nmbr of daa pons, l ar nmbr of op snsors T man orm of socasc sbspac dnfcaon sas a projcon P can b facorzd as prodc of obsrvably mar O and alman flr sa sqnc Prs & Roc 999: 9

9 j O P T obsrvably mar O and alman flr sqnc ar oband by applyng SVD o projcon mar: T V U S P 4 ombnng Eq and Eq 4 gvs: P O U S O /, 5 If sfd pas and fr ops of anal mar anor projc s oband: / f O Y Y P p 6 O - s oband from O afr dlng las l rows and sfd sa sqnc can b compd n Eq 6 as: P O 7 From Eq 5 and Eq 7, alman sa sqncs, ar calclad sng only op daa T sysm marcs can now b rcovrd from ovr drmnd s of lnar qaons, oband by ndng Eq 8: v w Y / 8 wr Y / s a anl mar w only on bloc row Snc alman sa sqnc T w T v T ar ncorrlad w, s of qaon can b solvd for, n a las-sqars: / Y 9 4 Idnfcaon of flr drvavs T modal paramrs of sysm can b oband by solvng gnval problm sa mar Eq 9: ; 0

mod sap mar Wn compl modal paramrs nown, gross dampng and gross sffnss n Eq 5 s drmnd by followng: L 0 0 ; ; wr 0 and 0 mcancal dampng and sffnss mar of sysm ndr no-wnd condon Ts, flr drvavs of wo

10 mod sap mar Wn compl modal paramrs nown, gross dampng and gross sffnss n Eq 5 s drmnd by followng: L 0 0 ; ; wr 0 and 0 mcancal dampng and sffnss mar of sysm ndr no-wnd condon Ts, flr drvavs of wo DOF can b dfnd as:,,,, I m I m I m I m 5 FLUTTER DERIVTIVES ND OPRISON T dcay and bffng rsponss ar acqrd a a samplng frqncy 00z and s sampls ar zros sng bfor o oprang w alab Fg 6 a Fr dcay rspons V=5m/s b ffng rspons V=5m/s Fg 6 Rspons of brdg dc scon-modl - orsonal 5 Erac of flr drvavs from bffng rspons mm dg mm dg

11 g wnd vlocy, arodynamc dampng of avng mod s oo g and vrcal Fg 7 FDs of brdg scon-modl by SDOF s and copld s by fr dcay and bffng rsponss cas Ir=09% Fg 8 FDs of brdg scon-modl by SDOF s and copld s by fr dcay and bffng rsponss cas Ir=09%

12 fr rspons s oo sor; rfor, racon of FDs canno b accomplsd g accracy T brdg dc scon-modl wll vbra ndr caon of rbln flows vn a small wnd vlocy, s rasonabl o rac FDs from bffng rspons Fg 7&8 sow flr drvavs of brdg dc by DT- SSI mod from bo fr dcay and bffng rsponss of DOF and DOF sysms ndr rblnc flows Ir=09% Ovrall, mos FDs ar n good agrmn w bo fr dcay and bffng rspons of DOF and DOF sysms, cp FDs rlad o vrcal frqncy and vrcal dampng and 4 T dffrnc may b pland by sor daa o rcod ndr fr dcay s T copld arodynamc drvavs and racd from bffng rsponss ar mor scarng an from fr dcay, spcally a g rdcd wnd spd small wnd vlocy, ampld of bffng vbraon s small and g dampng orsonal mod s no flly cd, so FDs s lss accra 5 T ffcs of rblnc on flr drvavs Fg 0 & llsra flr drvavs of avng and orsonal mod ndr smoo and rbln flows w dffrnc rdcd rblnc nnsy In s drvavs, orsonal dampng rm ; plays an mporan rol on orsnal flr sably, snc s posv/ngav val corrsponds o arodynamc nsably/sably of orsnal flr On or and, copld rm, and, and arodynamcally n-copld drvav av sgnfcan rol on avngorsonal of DOF copld flr nsably asmoo 00 s sown n Fg 9 &, ndr smoo flow, posv val a g rdcd wnd spd Vrd>55 and oal orsonal dampng ar almos conrold by, coplng a Dampng rao of avng mod b Dampng rao of orsonal mod Fg 9 Dampng rao of brdg scon-modl ndr smoo and rblnc flows

13 Fg 0 FDs of brdg scon-modl ndr smoo and rbln flows by fr dcay rspons sc as and dsablz orsonal flr T sgnfcan ffcs of rblnc flows on flr drvavs ar also llsrad basd on rm and ardynamc orsonal dampng owvr, ffcs of dffrn rbln nnsy on s arodynamc rms ar farly mods Wn r s an ncras n rdcd rblnc nnsy, crcal rdcd vlocy of orsonal flr s also ncrasd possy val corspodng rblnc: Vrd=68-7 w Ir=697%; Vrd=75 w Ir=09%; s sabl compard w dynamc rsponss of brdg dc Fg 5 In gnral, flr drvavs as a sgnfcan dffrnc bwn smoo and rblnc flows, rsac of wnd-ndcd vbraon problms ms b consdrd on s ss

14 Fg FDs of brdg scon-modl ndr smoo and rbln flows by fr dcay rspons 6 ONLUSIONS Ts sdy nvsgad ffcs of rblnc on flr drvavs of rssd brdg dc scon by sng wnd-nnl s and o-p only sa spac socasc sysm dnfcaon cnq dnfd flr drvavs from vary DOF and caon onclsons from s sdy ar smmarzd r: DT-SSI mods sow a good rsl vn ndr rblnc flows bcas an advanag of os mods s consdrd bffng forc and rspons as nps nsad of noss n dnfcaon of flr drvavs from bffng rsponss s plasbl, advanag of s cnq s asr o oban bffng rspons, and sconmodl wll b oscllad ndr wnd flow, morovr, n rblnc flows Ts s lss m consmng an fr dcay s Espcally a g wnd vlocy vrcal fr dcay daa s oo sor, s cass lss accracy Trblnc flows sgnfcanly affc on dynamc rsponss and flr drvavs of rssd brdg dc scon, spcfcally, rblnc ndcs largr dsplacmn b ncas crcal dvrgnc vlocy ar also llsrad basd on bffng rspons and ardynamc orsonal dampng rm Usng proposd framwor, vary of arodynamc fars wr addrssd, wc lpd s br ndrsand ffc of dffrn flows on arodynamcs of rssd brdg dc scon REFERENES ar Prs and Gdo D Roc 999, Rfrnc-basd socasc sbspac dnfcaon for op-only modal analyss, cancal Sysms and Sgnal Procssng, 6, arol, G, onr, S, annn, and Rg, 009, Toward an mprovmn n dnfcaon of brdg dc flr drvavs, J of Eng c SE, Vol 5, arol, G, Rg, 006, Flr mcansm for rcanglar prsm n smoo and rbln flow, J Wnd Eng Ind rodyn, Vol 94, 75-9

15 oonyapnyo, V, Janspasar, T 00, Daa-drvn socasc sbspac dnfcaon of flr drvavs of brdg dcs, J Wnd Eng Ind rodyn, Vol 98, owdry, G, Sarar, PP 00, nw cnq for dnfcaon of gn flr drvavs sng r-dgrs-of-frdom scon modl, Engnrng Srcrs, Vol 5, las Dyrby and Svnd O ansn 997, Wnd Loads on Srcrs, Jon Wly & Sons, Nw Yor aan, FL 000, T ffcs of rblnc on arodynamcs of long-span brdgs, PD Dssraon; Unvrsy of Nor Dam, US aan, FL, arm, 007, T ffcs of rblnc on arodynamcs of oscllang Prsms, IWE IRNS, 85-8 Yamada,, yaa, T and Icawa, 99, asrmn of arodynamc coffcns by sysm dnfcaon mod, J Wnd Eng Ind rodyn, Vol 4, 55-6 asc, and Yamada, 0, Sdy on rblnc paral smlaon for wndnnl sng of brdg dc, Proc of IWE, msrdam, Nrlands rgaar, P and ndrsn, P 997, Sa spac dnfcaon of cvl ngnrng srcrs from op masrmn, Proc of 5 Inrnaonal odal nalyss onfrnc, Orlando, Florda, US asmoo,, Tamwa, Y and Sjo, R 00, Frqncy caracrscs n varos flr nsabls of brdg grdrs, J Wnd Eng Ind rodyn, Vol 89, Nas, N, acdonal, JG and Jaobsn, J 0, Idnfcaon of flr drvavs from fll-scal ambn vbraon masrmns of lfon sspnson brdg, J Wnd and Src, Vol 4, -8 Naamra, Y and Ozono, S 987, T ffcs of rblnc on a sparad and raacng flow, J Fld c, Vol 78, Scanlan, R and Ln, W 978, Effcs of rblnc on brdg flr drvavs, J Eng c Dv, Vol04, 79-7 Scanlan, R 997, mpld and rblnc ffcs on brdg flr drvavs, J Src Eng SE, Vol, -6 Sm, E and Scanlan, R 996, Wnd Effcs on Srcrs, rd don, Jon Wly & Sons, Nw Yor Sarar, PP, Jons, NP, Scanlan, R 99, Sysm dnfcaon for smaon of flr drvavs, J Wnd Eng Ind rodyn, Vol 4, 4-54 Prs, and Roc, GD 999, Rfrnc basd socasc sbspac dnfcaon for op-only modal analyss, c Sysms and Sgnal Procssng, 6, Jang, JN, and Pappa, RS 985, n gnsysm ralzaon algorm ER for modal paramr dnfcaon and modl rdcon, J Gd onr Dyn, 99-8 Jaobsn, J, jor-ansn, E 995, Drmnaon of arodynamc drvavs by a sysm dnfcaon mod, J Wnd Eng Ind rodyn, Vol 57, 95-05

"Science Stays True Here" Journal of Mathematics and Statistical Science, Volume 2016, Science Signpost Publishing

Science Stays True Here Journal of Mathematics and Statistical Science, Volume 2016, Science Signpost Publishing "Scnc Says r Hr" Jornal of Mahmacs and Sascal Scnc Volm 6 343-356 Scnc Sgnpos Pblshng Mhod for a Solon o Som Class of Qas-Sac Problms n Lnar Vscolascy hory as Appld o Problms of Lnar orson of a Prsmac