Periodic and quasiperiodic galloping of a wind-excited tower under external excitation

Transcription

1 Periodic nd qusiperiodic glloping of wind-excied ower under exernl exciion Mohmed Belhq, Ilhm Kirrou, Lhcen Mokni Lborory of Mechnics, Universiy Hssn II-Csblnc, Morocco Absrc The glloping of ll srucures excied by sedy nd unsedy wind my be periodic or qusiperiodic () wih mpliudes hving he sme order of mgniude. While he onse of periodic nd glloping ws sudied, heir conrol on he oher hnd hs received less enion. In his pper, we conduc nlyicl sudy on he effec of fs hrmonic exciion on he onse of periodic nd glloping in he presence of sedy nd unsedy wind. We consider he cses where he unsedy wind cives eiher exernl exciion, prmeric one or boh. A perurbion nlysis is performed o obin close expressions of soluion nd he corresponding modulion envelopes. We show h vrious loding siuions, he periodic nd glloping onse is significnly influenced by he mpliude of he fs exernl exciion. In he cse where he unsedy wind cives prmeric exciion, he glloping occurs wih higher frequency modulion compred o he cse where he unsedy wind cives exernl exciion. In he cse where exernl nd prmeric exciions re cived simulneously, fs hrmonic exciion elimines bisbiliy in he mpliude response nd gives rise o new smll modulion envelope. Keywords: Qusiperiodic glloping, wind effec, srucurl dynmics, perurbion nlysis, conrol, fs exciion. 1

2 1 Inroducion Dynmic nlysis of ll srucures under unsedy (urbulen) wind exciion hs been mjor concern in consrucing nd designing sble buildings. Indeed, wind-induced vibrions of such buildings my cuse glloping bove cerin hreshold of he wind speed [1 5]. In his conex, considerble effors hve been done o conrol he mpliude of such wind-induced vibrions; see for insnce [6] in which review of he min clsses of semi-cive conrol devices is given nd heir full-scle implemenion o civil infrsrucure pplicions is presened. The effec of unsedy wind on he periodic glloping of ll prismic srucures hs been sudied considering lumped single degree of freedom (sdof) model [4]. The muliple scles mehod (MSM) ws pplied nd he response of he sysem exmined ner primry nd secondry resonnces. The resuls shown h he unsedy wind decreses he wind speed onse (he criicl wind speed bove which glloping occurs) ner he primry resonnce nd hs no significn influence ner secondry resonnces. This sudy [4] hs been exended o nlyze he effec of self- nd exernl or/nd prmeric exciions on periodic glloping of ower ner he primry resonnce [7]. Using he MSM, he effec of unsedy wind on Hopf bifurcion ws nlyzed nd he influence of sedy wind speed on response ws repored using numericl simulion. In ll srucures subjeced o sedy wind, Hopf bifurcion my occur criicl wind speed. However, when srucure is under sedy nd unsedy wind, he induced moion of he srucure my be periodic or [7]. The periodic glloping usully occurs round he resonnce, while he one ppers wy from he resonnce resuling from he inercion beween selfnd exernl or/nd prmeric exciions [8 11]. In his siuion, he exisence of periodic nd responses my led o frequency-locking (or synchronizion) beween he frequency of he unsedy force nd he frequency of he self exciion. Such mechnism is produced by he disppernce of slow flow limi cycle hrough homoclinic or heeroclinic bifurcion [12 15]. Due o he fc h he mpliude of glloping of ower is found o be of he sme order of mgniude s h of he periodic response [7], he effec of wind speed on he onse of glloping should no be ignored nd hs o be ken ino considerion nd nlyzed crefully. For insnce, i ws shown numericlly using 3D model h wind-excied lrge sress 2

3 ower develops response rher hn periodic oscillions [16]. Recenly, he effec of wind speed on he onse of glloping hs been exmined nlyiclly [17] nd i ws shown h glloping cn occur even for smll vlues of he wind velociy wih mpliude hving he sme order of mgniude s h of periodic glloping. In his pper, we exends he previous sudies [4,7,17] by focusing principlly on he effec of exernl fs hrmonic exciion (FHE) on he periodic nd glloping onse. Specificlly, we firs conduc nlyicl remen o pproxime he response nd is modulion envelope. Then, we exmine he effec of FHE on he onse of periodic nd glloping in he presence of unsedy wind. In his cse, he soluions nd he modulion envelopes cn be obined using he procedure known s hree sge perurbion nlysis (TSPA) [18, 19] which consiss of pplying he mehod of direc priion of moion (DPM) [2 22] followed by wo sges of MSM [23]. The pper is orgnized s follows: In Secion 2, he equion of moion is given nd he DPM followed by firs MSM re pplied o derive he modulion equions of he slow dynmic ner he primry resonnce. In Secion 3 we nlyze he effec of he FHE mpliude on he periodic glloping onse in he cses where he unsedy wind cives eiher n exernl exciion, prmeric one or boh. In Secion 4 we pply second MSM on he modulion equions o pproxime he soluions nd he modulion envelopes. A creful nlysis is conduced o exmine he effec of he FHE on glloping onse in he cse of vrious loding siuions. A summry of he resuls is provided in he concluding secion. 2 Equions of moion nd slow flow A single mode pproch of he srucure moion is considered nd modelled by sdof lumped mss sysem [4, 7]. I is ssumed h he ower is under sedy nd unsedy wind flow nd suppose h FHE cn be inroduced o excie he srucure. In his cse, he dimensionless sdof equion of moion cn be wrien in he form [7] ẍ + x + [ c (1 Ū) b 1u() ] ẋ + b 2 ẋ 2 + [ b 31 Ū + b 32 Ū 2 u()] ẋ 3 = η 1 Ūu() + η 2 Ū 2 + Y cos ν (1) 3

4 where he do denoes differeniion wih respec o he non-dimensionl ime. Equion (1) conins, in ddiion o he elsic, viscous nd ineril liner erms, qudric nd cubic componens in he velociy genered by he erodynmic forces. The sedy componen of he wind velociy is represened by Ū nd he urbulen wind flow is pproximed by periodic force, u(), which is ssumed o include he wo firs hrmonics, u() = u 1 sin Ω + u 2 sin 2Ω, where u 1, u 2 nd Ω re, respecively, he mpliudes nd he fundmenl frequency of he response. We shll nlyze he cse of exernl exciion (u 1, u 2 = ), he cse of prmeric one (u 1 =, u 2 ), nd he cse where exernl nd prmeric exciions re presen simulneously. The coefficiens of Eq. (1) nd numericl vlues of prmeers re given in Appendix I nd Y, ν re he mpliude nd he frequency of he fs exciion, respecively. Noice h he cse of wo owers linked by nonliner viscous device submied o urbulen wind flow hs been considered in [24]. The inroducion of FHE s possible conrol sregy ws moived by previous experimenl work done for vibring esing purpose of full size ower [25]. The mechnicl vibrion excier sysem used in such n experimen consiss of pir of couner-roing eccenric weighs so rrnged h reciliner sinusoidlly vrying horizonl ineri force is genered. The wo weighs roe bou common vericl shf, nd re driven in opposie direcions by chindrive sysem. This vibrion excier is plced on he op of he srucure nd debis hrmonic force ble o excie he srucure. Here we ssume h he genered frequency of he vibrion excier is relively higher hn he nurl frequency of he firs mode of he ower such h he oher lower modes of he ower cnno be cived. Equion (1) includes slow dynmic due o he sedy nd unsedy wind nd fs dynmic induced by he FHE. To exmine he influence of he FHE on periodic nd glloping, we firs perform he mehod of DPM which consiss in inroducing wo differen ime scles, fs ime T = ν nd slow ime T 1 =, nd spliing up x() ino slow pr z(t 1 ) nd fs pr ϕ(t, T 1 ) s x() = z(t 1 ) + µϕ(t, T 1 ) (2) where z describes he slow min moions ime-scle of oscillions, µϕ snds for n overly 4

5 of he fs moions nd µ indices h µϕ is smll compred o z. Since ν is considered s lrge prmeer, we choose µ ν 1 for convenience. The fs pr µϕ nd is derivives re ssumed o be 2π periodic funcions of fs ime T wih zero men vlue wih respec o his ime, so h < x() >= z(t 1 ) where <> 1 2π () dt 2π defines ime-verging operor over one period of he fs exciion wih he slow ime T 1 fixed. Averging procedure gives he following equion governing he slow dynmic of moion z+z+ [ c (1 Ū) b 1u()+H( b 31 Ū +b 32 Ū 2 u())] ż+b 2 ż 2 + [ b 31 Ū +b 32 Ū 2 u()] ż 3 = G+η 1 Ūu()+η 2 Ū 2 (3) where H = 3Y 2 nd G = b 2Y 2. Deils on he verging procedure nd he derivion of he 2ν 2 2ν 2 slow dynmic (3) re given in Appendix II. Noe h he cse wihou fs exciion (Y = or H = G = ) ws considered in [7]. To obin he modulion equions of he slow dynmic (3) ner primry resonnce, he MSM ws performed by inroducing bookkeeping prmeer ε, scling s z = ε 1 2 z, b 1 = εb 1, b 2 = ε 1 2 b 2, η 1 = ε 3 2 η 1, η 2 = ε 3 2 η 2 nd ssuming h Ū = 1 + εv (V snds for he men wind velociy) nd he resonnce condiion Ω = 1 + εσ where σ is deuning prmeer [7]. Scling lso H = εh, wo-scle expnsion of he soluion is sough in he form = z (, 1 ) + εz 1 (, 1 ) + O(ε 2 ) (4) where i = ε i (i =, 1). In erms of he vribles i, he ime derivives become d d = d + εd 1 + O(ε 2 ) nd d2 d 2 = d 2 + 2εd d 1 + O(ε 2 ), where d j i = j j i. Subsiuing Eq. (4) ino Eq. (3), equing coefficiens of he sme power of ϵ, we obin he wo firs orders d 2 z + z = G (5) d 2 z 1 + z 1 = 2d d 1 z + (c V + b 1 u( ) H(b 31 + b 32 u( )))(d z ) b 2 (d z ) 2 (b 31 + b 32 u( ))(d z ) 3 + η 1 u( ) + η 2 (6) A soluion o he firs order of sysem (5) is given by z = A( 1 ) exp(i ) + Ā( 1) exp( i ) G (7) 5

6 where i is he imginry uni nd A is n unknown complex mpliude. Equion (6) cn be solved for he complex mpliude A by inroducing is polr form s A = 1 2 eiϕ. Subsiuing he expression of A ino Eq. (6) nd elimining he seculr erms, he modulion equions of he mpliude nd he phse ϕ cn be exrced s ȧ = [S 1 S 3 sin(2ϕ)] + [ S 2 + 2S 4 sin(2ϕ)] 3 β cos(ϕ) ϕ = [σ S 3 cos(2ϕ)] + [S 4 cos(2ϕ)] 3 + β sin(ϕ) (8) where S 1 = 1(c 2 V Hb 31 ), S 2 = 3b 8 31, S 3 = 1(b 4 1 Hb 32 )u 2, S 4 = b 32 u 8 2 nd β = η 1u 1. I is 2 ineresing o observe from he ler expressions h he FHE influences he dynmic of he ower hrough he prmeer H (= 3Y 2 2ν 2 ) inroduced ino he coefficiens S 1 nd especilly S 3 which is reled o he prmeric exciion u 2. 3 Periodic glloping In his secion, we repor on he effec of he mpliude of he FHE on he glloping of ower. To be consisen wih previous resuls, he numericl vlues used here re picked from [7]. Periodic soluions of Eq. (3) corresponding o equilibri of he slow flow (8) re given by seing ȧ = ϕ = in (8). We obin rivil soluion = nd nonrivil one 4(c V Hb 31 ) = (9) 3b 31 which corresponds o he periodic glloping mpliude of he ower. Figure 1 shows he glloping mpliude versus he wind velociy V in he bsence of he unsedy wind (u 1 =, u 2 = ) nd for differen vlues of he FHE mpliude Y. Herefer, he frequency of he FHE will be fixed, ν = 8. This vlue of he frequency is chosen such h oher modes of he ower re no excied. Figure 1 shows h incresing he mpliude of he FHE rerds subsnilly he glloping onse. Insed, in he bsence of FHE incresing he mpliude of he unsedy wind componen decreses rpidly he glloping onse [4, 7]. 6

7 Figure 1: Effec of he mpliude Y on glloping onse in he bsence of urbulen wind (u 1 =, u 2 = ). Solid line: sble; dshed line: unsble. In he cse of urbulen wind wih exernl exciion (u 1, u 2 = ), he mpliude-response equion obined from he slow flow (8) reds S S 1 S (S σ 2 ) 2 β 2 = (1) In Fig. 2 we show he effec of he mpliude Y on he glloping mpliude, s given by Eq. (1), for given vlues of he exciion u 1. The solid line corresponds o he sble brnch, while he dshed line corresponds o he unsble one. The resuls of numericl simulions (circles) re obined using he fourh-order Runge-Ku mehod. One observes in Fig. 2 h he FHE increses he criicl wind speed by shifing he glloping mpliude owrd higher wind velociies resuling in decrese of he corresponding mpliude for given wind velociy V. Figure 2b shows he effec of he mpliude Y on he glloping versus σ indicing effecively h he glloping mpliude decreses wih incresing Y. The resuls of numericl simulions (circles) re ploed for comprison wih he nlyicl finding (solid line). 7

8 Figure 2: Effec of Y on glloping mpliude; () u 1 =.1, σ =, (b) u 1 =.33, V =.117. Solid line: sble; dshed line: unsble; circle: numericl simulion. In he cse of urbulen wind wih prmeric exciion (u 1 =, u 2 ), he mpliuderesponse equion is given by ( S 1 + S 2 3 ) 2 ( S 3 + 2S 4 3 ) 2 + ( σ) 2 ( S 3 + S 4 3 ) 2 = 1 (11) Figure 3 shows, for given vlue of he exciion u 2, he effec of he mpliude Y on he glloping mpliude versus he wind velociy V, s given by (11). A solid line corresponds o sble brnch nd doed line corresponds o unsble one. I cn be seen from his figure h in he bsence of he fs exciion (Y =) he prmeric componen shifs he sble brnch righ. As resul, he vlue of he wind velociy V which glloping occurs is decresed [4, 7]. Insed, rerding effec of he glloping onse is chieved for incresing vlues of he FHE mpliude Y. Figure 3b shows he mpliude of he response versus σ for differen vlues of Y nd for fixed V nd u 2 indicing decrese of he glloping mpliude s Y is incresed. 8

9 Figure 3: Effec of Y on he glloping mpliude; () u 2 =.1, σ =, (b) u 2 =.1, V =.167. Solid line: sble; dshed line: unsble; circle: numericl simulion. In he cse where he urbulen wind cives boh exernl nd prmeric exciions (u 1, u 2 ), Figs. 4,b show, respecively, he periodic mpliude in he bsence nd presence of he FHE mpliude Y. The comprison beween he nlyicl predicions (solid lines) nd he numericl simulions (circles) shows good greemen. These figures indice h incresing Y elimines he bisbiliy indiced by loop in he mpliude response which corresponds o he coexisence of wo differen mpliudes of (periodic) oscillion () Y =.35.3 (b) Y = σ x Figure 4: Effec of Y on he periodic glloping; V =.11, u 1 =.1, u 2 =.1. Solid line: sble; dshed line: unsble; circle: numericl simulion. σ x 1 3 9

10 4 Qusi-periodic glloping To pproxime periodic soluions of he slow flow (8) corresponding o responses of he slow dynmic (3) we perform he hird sep of he TSPA. To his end one rnsforms he slow flow from he polr form (8) o he following Cresin sysem using he vrible chnge u = cos ϕ nd v = sin ϕ du d = (σ + S 3)v β + η{s 1 u (S 2 u + S 4 v)(u 2 + v 2 ) 2S 4 u 2 v} dv d = (σ S 3)u + η{s 1 v (S 2 v + S 4 u)(u 2 + v 2 ) 2S 4 uv 2 } where η is new bookkeeping prmeer inroduced in dmping nd nonlineriy such h he unperurbed sysem of Eq. (12) hs bsic soluion (Eq. (14)). Following [26], [27], [18] by using he MSM, periodic soluion of he slow flow (12) cn be sough in he following form u() = u (T 1, T 2 ) + ηu 1 (T 1, T 2 ) + O(η 2 ) (12) v() = v (T 1, T 2 ) + ηv 1 (T 1, T 2 ) + O(η 2 ) (13) where T 1 = nd T 2 = η. Inroducing D i = T i yields d d = D 1 + ηd 2 + O(η 2 ), subsiuing (13) ino (12) nd collecing erms, we ge differen order of η D 2 1u + λ 2 u = αv = D 1 u + β (14) D 2 1u 1 + λ 2 u 1 = α[ D 2 v + S 1 v (S 2 v + S 4 u )(u 2 + v 2 ) 2S 4 u v 2 ] D 1 D 2 u + S 1 D 1 u D 1 [(S 2 u + S 4 v )(u 2 + v 2 ) + 2S 4 u 2 v ] αv 1 = D 1 u 1 + D 2 u S 1 u + (S 2 u + S 4 v )(u 2 + v 2 ) + 2S 4 u 2 v (15) where α = σ + S 3 nd λ = σ 2 S 2 3 (16) defines he frequency of he periodic soluion of he slow flow (12) corresponding o he frequency of he modulion. I should be noed h his frequency λ depends on he prmeric 1

11 exciion u 2 vi he coefficien S 3 given in (8) indicing h he frequency of he modulion is influenced by he prmeric exciion. The soluion of he firs-order sysem (14) is given by u (T 1, T 2 ) = R(T 2 ) cos(λt 1 + θ(t 2 )) v (T 1, T 2 ) = λ α R(T 2) cos(λt 1 + θ(t 2 )) + β α (17) Subsiuing (17) ino (15) nd removing seculr erms gives he following uonomous slow slow flow sysem on R nd θ dr d = ( S 1 2 β2 α S ) (1 2 2 R R dθ d = ( 3β 2 2αλ S 4 3λβ2 2α S S 2 + λ2 ) ( 3λ 3 R 2α S ) 2 2 R 3 8α 3 S 4 3α 8λ S 4 ) R 3 Thus, he pproxime periodic soluion of he slow flow (12) is given by u() = R cos(ϕ) v() = λ α R cos(ϕ) + β α where he mpliude R obined by seing dr = is given by d 2α R = 2 S 1 4β 2 S 2 S 2 (α 2 + λ 2 ) (18) (19) nd ϕ is given in he Appendix by Eq. (29). Using (18), he moduled mpliude of he oscillions is pproximed by () = [ 1 2 R2 + λ2 2α 2 R2 + β2 α 2 ] [2λβ α 2 R sin(ϕ) ( 1 2 R2 λ2 2α 2 R2) cos(2ϕ) ] (2) nd he envelope is delimied by min nd mx given by { [1 min = min 2 R2 + λ2 2α 2 R2 + β2 ] 2λβ ± α 2 α R ± ( R2 λ2 2α R2)} (21) 2 { [1 mx = mx 2 R2 + λ2 2α 2 R2 + β2 ] 2λβ ± α 2 α R ± ( R2 λ2 2α R2)} (22) 2 11

12 Using (21) nd (22), he criicl deuning prmeer σ c given by he condiions 2β σ c = S 3 ± 2 S 2 S 1 (23) σ c = ±S 3 defines he inervl [ σ c, σ c ] wihin which he glloping mpliude is periodic. Ouside his inervl he glloping mpliude is. One observes h in he cse of exernl exciion (u 2 = ), S 3 = nd in he cse of prmeric one (u 1 = ), β =. These wo cses will be highlighed below. 4.1 Cse of urbulen wind wih exernl exciion Nex, we explore he modulion envelope nd he influence of he HFE on he glloping onse. When he ower is under exernl exciion (u 1, u 2 = ), Figs. 5,b show he modulion envelope, s given by Eqs. (21), (22), nd he periodic mpliude response, s given by Eq. (1), for given vlues of V nd u 1 nd for wo differen vlues of he HFE mpliude Y. The comprison beween he nlyicl predicions (solid lines) nd he numericl simulions obined by using Runge-Ku mehod (circles) revels h he nlyicl pproch predic well he envelope of he modulion () Y =.3 (b) Y = σ x Figure 5: Periodic nd glloping versus σ for V =.117 nd u 1 =.33. Solid lines: sble; dshed lines: unsble; circle: numericl simulion. σ x 1 3 The effec of he mpliude Y on he envelope is shown in Fig. 6. One cn observe 12

13 h s Y is incresed, he men mpliude of he response reduces subsnilly, while he modulion envelope moves wy from he resonnce region o dispper, s shown in Fig (c) Y= Y=.1.1 Y=.12.5 Y= σ Figure 6: glloping envelope versus σ for he prmeer vlues of Fig. 5. x 1 3 Figure 8 presens exmples of ime hisories of he slow dynmic obined by performing numericl simulion of Eq. (3) for some vlues of Y picked from Figs. 5, 6 nd 7. In he bsence of he HFE (Y = ), Figs. 8,b show, respecively, he periodic response for σ = nd he one for σ = Figures 8c,d show, for Y =.12 nd Y =.14, respecively, response wih smll mpliude nd sligh modulion (Fig.8c) nd periodic response wih smll mpliude (Fig.8d), which is coheren wih he nlyicl predicions shown in Figs. 6 nd 7b () Y = (b) Y = σ x Figure 7: Periodic nd glloping versus σ for he prmeer vlues of Fig. 5. σ x 1 3 Becuse he mpliude of he modulions is found o be of he sme order of mgniude 13

14 s h of he periodic response (Fig. 8), i is required h he glloping onse should be nlyzed crefully. Bsed on Eq. (23), glloping my be expeced minly in cerin region wy from he resonnce, s depiced in Figs. 5 nd 7. Figure 9 shows he envelope versus he wind velociy V for given vlue of u 2 nd in he bsence of he HFE (Y = ). For very smll vlues of V, smll periodic response indiced by he horizonl line is observed, mening h he ower lwys performs smll periodic oscillion due o he exernl exciion. As V is incresed slighly, smll modulion of he periodic response ppers ( he locion where he horizonl line mees he envelope) giving rise o glloping onse. Incresing V furher, he envelope delimied by mx nd min increses, s shown by ime hisories of he slow dynmic inse Fig. 9 obined by numericl simulion of Eq. (3)..3 () Y = σ =.3 (b) Y = σ = x x (c) Y=.12 σ =.5.3 (d) Y =.14 σ = x x 1 4 Figure 8: Exmples of ime hisories of he slow dynmics for he prmeer vlues of Fig. 5. Vlues of Y nd σ re picked from Figs. 5, 6 nd 7b. 14

15 Figure 9b shows he glloping mpliude in he presence of FHE. I cn be seen h he FHE rerds significnly he onse of glloping, keeping he ower oscilling periodiclly wih smll mpliude in lrge inervl of he wind velociy indiced by he (hick) horizonl line. The boxes inse he figure show ime hisories before nd fer he glloping onse ().2 Y=.35.3 (b).2 Y= V=.5 6 x 1 4 V=.8.2 mx.2 min x V V=.16 x V= V=.16 x x 1 4 mx min V Figure 9: The envelope versus V for he prmeer vlues of Fig. 5 wih σ =.5. In Fig. 1 we show in he prmeer plne σ c versus V (Fig. 1) nd σ c versus Y (Fig. 1b) he curves delimiing he regions of periodic (hched) nd glloping (unhched), s given by Eq. (23). One observes h in he bsence of he FHE (Y = ) he domin of periodic glloping decreses wih incresing V (Fig. 1) nd increses wih Y (Fig. 1b), which is coheren wih Fig

16 1.5 x () Y= 1.5 x (b) V=.117 σ c.5 Periodic glloping glloping σ c.5 glloping Periodic glloping.5 glloping.5 glloping V Y Figure 1: Periodic nd glloping domins for u 1 = Cse of urbulen wind wih prmeric exciion In he cse of prmeric exciion (u 1 =, u 2 ), Figs. 11,b show he periodic mpliude nd he envelope of he oscillions for wo differen vlues of he mpliude Y. The nlyicl predicions (solid lines) nd he numericl simulions obined by using Runge-Ku mehod (circles) re ploed for vlidion. The effec of he mpliude Y on he envelope is shown in Fig. 11c. One observes h he envelope nd is men mpliude decrese subsnilly wih incresing Y, while he frequency inervl of he response remins consn. Exmples of ime hisories of he slow dynmic re shown in Fig. 12 for some vlues of Y picked from Fig. 11. Figures 12,b show for σ = nd σ =.1, respecively, he periodic nd he responses in he bsence of he HFE. The responses re shown in Figs. 12c,d for wo vlues of Y confirming he decresing in he men mpliude nd in he envelope, s indiced in Fig. 11c. 16

17 Figure 11: Ampliude versus σ for V =.167 nd u 2 =.1. Figure 13 shows, in he bsence of he HFE, he glloping mpliude versus he wind velociy V for given vlue of he exciion u 2. I cn be seen h s V is incresed from zero, he glloping ppers direcly from he res posiion wih smll modulion nd increses wih he wind velociy. The smll boxes inse he figure show ime hisories of he slow dynmic for wo differen vlues of V. For Y =.12 nd given vlue of V, Fig. 13b depics he glloping mpliude nd ime hisory inse he figure showing h significn rerding of he glloping onse cn be chieved by incresing he mpliude Y, s clerly shown in Fig

18 .4.3 () Y = σ =.4.3 (b) Y = σ = x x (c) Y =.12 σ = (d) Y =.15 σ = x x 1 4 Figure 12: Exmples of ime hisories of he slow dynmics for he prmeer vlues of Fig. 11 wih σ =.5. Vlues of prmeers Y nd σ re picked from Fig. 11. I is worh noicing h he frequency modulion cused by he prmeric exciion (see ime hisories inse Fig. 13) is higher hn he frequency modulion produced by he exernl exciion (see inse Fig. 9). This resul is consisen wih he expression of he frequency λ of he modulion given by (16) which depends effecively (o he leding order) on he mpliude of he prmeric exciion u 2. 18

19 .35.3 () Y= (b) Y= V=.8 6 x V=.16 x min mx V=.16 6 x min mx V V Figure 13: The envelope versus V for he prmeer vlues of Fig. 11 wih σ = (c).25.2 Y=.15.1 Y=.15 Y= V Figure 14: Effec of Y on he envelope for he prmeer vlues of Fig. 11 wih σ =.1. In Fig. 15 we show he curves delimiing he regions of periodic (hched) nd glloping (unhched), s given by (23). One observes h he domin of periodic glloping remins consn wih incresing V (Fig. 15) nd lmos consn s Y increses (Fig. 15b), which is lso coheren wih Fig. 11c. 19

20 1.5 x x 1 3 () Y= (b) V= glloping.5 glloping σ c Periodic glloping σ c Periodic glloping.5 glloping.5 glloping V Y Figure 15: Periodic nd glloping domins for u 2 =.1. Figure 16 illusres he effec of he mpliude of he FHE on he glloping domin in he prmeer plne u 2 versus σ c. The plos show h he glloping domin decreses rpidly wih u 2 nd slighly wih Y Y= Y=.2 periodic glloping u glloping glloping σ c x 1 3 Figure 16: Periodic nd glloping domins in he prmeer plne u 2 versus σ c for V =.167. Figure 17 shows he effec of he unsedy wind componens u 2 on he frequency λ of he modulion, s given by Eq. (16). Inspecion of his figure indices h in he cse of prmeric exciion (u 2 ) he frequency λ decreses rpidly wih u 2. The decresing of λ is 2

21 more pronounced s he mpliude Y is incresed. Noice h in he cse of exernl exciion (u 1 ), he frequency λ remins unchnged (o he leding order) s shown in he figure by he horizonl line. x frequency λ u 2 u 1 Y=.2 Y= u 1, u 2 Figure 17: Vriion of he modulion frequency of he glloping u 1 nd u 2 for V =.167 nd σ =.1. To suppor he prediced resuls shown in Fig. 17, exmples of ime hisories of he slow dynmic re given in Fig. 18 in he bsence of he FHE for vlues of he unsedy componens picked from Fig. 17. Figures 18,b show he response in he presence of exernl exciion (u 1 ), while Figs. 18c,d illusre he response in he presence of prmeric exciion (u 2 ). I cn be clerly seen he significn influence of he prmeric exciion on he frequency of he modulion. 21

22 .5.4 () u 1 =.2 u 2 =.5.4 (b) u 1 =.25 u 2 = x x (c) u 1 = u 2 = (d) u 1 = u 2 = x x 1 4 Figure 18: Exmples of ime hisories of he slow dynmics for Y =, V =.167 nd σ = Cse of urbulen wind wih exernl nd prmeric exciions In he cse where urbulen wind cives boh exernl nd prmeric exciions (u 1, u 2 ), Figs. 19,b show he periodic mpliude nd he modulion envelope in he bsence nd presence of he FHE, respecively. As menioned in Secion 3 hese figures indice h incresing he mpliude of he FHE Y elimines he loop in he mpliude response nd gives rise o smll new modulion envelope emning from he originl envelope. The effec of he mpliude Y on he originl envelope is illusred in Fig. 19c showing h he men mpliude s well s he envelope of he response decrese while he envelopes move wy from he resonnce genering he new smll modulion domin delimied by he wo hick lines. Figure 2 shows exmples of ime hisories of he slow dynmic for some vlues of Y 22

23 picked from Fig.19. Figures 2,b show for σ = nd σ =.1, respecively, he periodic nd he responses in he bsence of he HFE (Y = ). The response corresponding o he new born smll modulion envelopes re shown in Figs. 2c,d for given Y nd for wo vlues of σ, confirming he birh of such new modulion envelope () Y =.35.3 (b) Y = σ x σ x (c).25 Y=.2.15 Y= Y= σ Figure 19: Ampliude versus σ for V =.11, u 1 =.1 nd u 2 =.1. Thick lines: new modulion envelope. x 1 3 Figure 21 shows, in he bsence of he HFE, he (originl nd new) modulion envelopes versus he wind velociy V for given vlues of exciions u 1, u 2. One noices h s V is incresed (Fig. 21b), he originl glloping onse increses wih he wind velociy while he smll envelope persiss prior o he originl glloping envelope s shown in he lef box inse Fig. 21b. Figure 22 clerly confirms he significn rerding of he lrge originl 23

24 glloping onse by incresing he mpliude Y. One noice h he presence of he new smll modulion envelope cuses he ower o undergo smll qusi-periodic vibrions even in he presence of smll wind..4.3 () Y = σ =.4.3 (b) Y = σ = x x (c) Y =.12 σ = (d) Y =.12 σ = x x 1 4 Figure 2: Exmples of ime hisories of he slow dynmics for he prmeer vlues of Fig. 19. Vlues of prmeers Y nd σ re picked from Fig

25 () V=.2 x Y=.35.3 (b) V= V= x 1 4 Y= x min mx V=.1 x V.2.5 mx min V Figure 21: New nd originl envelopes versus V for he prmeer vlues of Fig. 19 for σ = (c) Y=.15 Y= V Figure 22: Effec of Y on envelopes for he prmeer vlues of Fig. 19 wih σ =.1. 5 Conclusion The effec of HFE on he periodic nd glloping of ower subjeced o sedy nd unsedy wind ws sudied nlyiclly ner he primry resonnce. A lumped mss sdof model ws con- 25

26 sidered nd enion ws focused on he cse where he urbulen wind cives eiher exernl exciion, prmeric one or boh. The TSPA is performed o obin explici relionships of he responses nd he modulion envelopes. In he cse of sedy wind, he FHE mpliude cuses he Hopf bifurcion locion o shif owrd higher wind velociy, hereby rerding he periodic glloping onse. In he cse of urbulen wind wih exernl exciion nd in he bsence of he FHE, he ower my experience smll periodic response even for very smll vlues of wind velociy, nd s he wind velociy increses slighly, smll modulion of he mpliude of he periodic response ppers cusing glloping. In he presence of FHE, he onse of glloping cn be rerded keeping he ower oscilling periodiclly in lrge inervl of he wind velociy. In he cse where urbulen wind cives prmeric exciion nd in he bsence of he FHE, he envelope of he glloping wih smll modulion ppers direcly from he res posiion which increses wih he wind velociy. As he HFE is inroduced, he glloping onse is significnly delyed. Moreover, in he cse of prmeric exciion, he ower develops glloping wih higher frequency modulion, compred o he cse for which he urbulen wind cives exernl exciion. In he cse where boh exernl nd prmeric exciions re cived he FHE elimines bisbiliy phenomenon which is responsible for insbiliy nd jumps in he mpliude response of he ower. One cn conclude from his work h he effec of wind speed on he onse of glloping should no be negleced. Indeed, glloping is more likely o occur in lrge inervl of frequencies nd hen is onse hs o be ken ino considerion in he design process of ll buildings o enhnce sbiliy performnce. The use of FHE my be viewed s possible lernive conrol sregy ble o rerding he onse of periodic nd glloping of owers. Such conrol cn be exploied especilly when oher sregies, such s mss uned dmpers nd uned liquid dmpers, which re ofen need lrge plces o be inslled, cnno be implemened for prevening or rerding lrge srucurl vibrion. 26

27 Finlly, i should be noed h he innovive ide of using FHE for rerding glloping onse of ll building comes essenilly from previous experimenl work on vibring esing of full size ower [25]. The mechnicl vibrion excier sysem used in such n experimen ws designed so h sinusoidlly vrying horizonl exciion cn be genered nd pplied o he ower from he op. Appendix I The expressions of he coefficiens of Eq. (1) re 3EI ω = π hl m, c = ρa 1bhlŪc 2π 3EIm, b 1 = c, b 2 = 4ρA 2bl 3πm, b 31 = 3πρA 3bl 3EI 8hŪc m 3 b 32 = b 31, η 1 = 4ρA bh 2 lū 2 c 3π 3 EI, η 2 = η 1, U() = Ū + u(), 2 where l is he heigh of he ower, b is he cross-secion wide, EI he ol siffness of he single sory, m is he mss longiudinl densiy, h is he iner sory heigh, nd ρ is he ir mss densiy. A i, i =,...3 re he erodynmic coefficiens for he squred cross-secion. The dimensionl criicl velociy is given by Ū c = 4πξ 3EIm ρba 1 hl (24) Here ξ is he modl dmping rio, depending on boh he exernl nd inernl dmpings ξ = ηh2 24EI ω + c 2mω (25) The following numericl vlues picked from [7] re used for convenience: he heigh of he ower is l = 36m, he cross-secion is b = 16m wide, he ol siffness of he single sory is EI = Nm 2, he mss longiudinl densiy is m = 4737kg/m, he dmping rio is ζ =.5 percen (corresponding o η = Ns, c = Ns/m 2 in Eq. (25)). The iner-sory heigh is ssumed h = 4m. The erodynmic coefficiens A i, i =,..., 3 re ken from [4] for he squred cross-secion: A =.297, A 1 =.9298, A 2 =.24, A 3 = The ir mss densiy is ρ = 1.25kg/m 3. The (dimensionl) nurl frequency of he rod is ω = 5.89rd/s. The (dimensionl) criicl wind velociy ssumes he vlue U c = 3m/s. 27

28 Appendix II Inroducing D j i (2) ino Eq. (1) gives j j T i yields d = νd d d + D 1, 2 d 2 = ν 2 D 2 + 2νD D 1 + D 2 1 nd subsiuing Eq. µ 1 D 2 ϕ + D 2 1z + 2D D 1 ϕ + µd 2 1ϕ + ( c (1 Ū) b 1u() )( D 1 z + D ϕ + µd 1 ϕ ) + z +µϕ + b 2 ( (D1 z) 2 + 2D 1 z(d ϕ + µd 1 ϕ) + (D ϕ) 2 + 2µD ϕd 1 ϕ + (µd 1 ϕ) 2) + ( b 31 Ū + b 32 Ū 2 u())( (D 1 z) 3 + 3(D 1 z) 2 (D ϕ + µd 1 ϕ) + 3(D 1 z)(d ϕ + µd 1 ϕ) 2 +(D ϕ + µd 1 ϕ) 3) = η 1 Ūu() + η 2 Ū 2 + Y cos(ν) (26) Averging (26) leds o D 2 1z + ( c (1 Ū) b 1u() ) D 1 z + z + b 2 ( (D1 z) 2 + < (D ϕ) 2 > + < (2µD ϕd 1 ϕ) > + < (µd 1 ϕ) 2 > ) + ( b 31 Ū + b 32 Ū 2 u())( (D 1 z) 3 + 3D 1 z(< (D ϕ) 2 > + < (2µD ϕd 1 ϕ) > + < (µd 1 ϕ) 2 >) ) = η 1 Ūu() + η 2 Ū 2 (27) Subrcing (27) from (26) yields µ 1 D 2 ϕ + 2D D 1 ϕ + µd 2 1ϕ + ( c (1 Ū) b 1u() )( D ϕ + µd 1 ϕ ) + µϕ + b 2 ( 2D1 z(d ϕ +µd 1 ϕ) + (D ϕ) 2 < (D ϕ) 2 > +2µD ϕd 1 ϕ + (µd 1 ϕ) 2 < (µd 1 ϕ) 2 > ) + ( b 31 Ū + b 32 Ū 2 u())( 3(D 1 z) 2 (D + µd 1 ϕ) + 3D 1 z(d ϕ) 2 3D 1 z < (D ϕ) 2 > +6D 1 zµ(d ϕd 1 ϕ) +3D 1 z(µd 1 ϕ) 2 3D 1 z < (µd 1 ϕ) 2 > +(D ϕ) 3 + 3µ(D ϕ) 2 D 1 ϕ +3D ϕ(µd 1 ϕ) 2 + µd 1 ϕ ) = Y cos(t ) (28) Using he ineril pproximion [2], i.e. ll erms in he lef-hnd side of Eq. (28), excep he firs, re ignored, one obins ϕ = µy cos(t ) (29) Insering ϕ from Eq. (29) ino Eq. (27), using h < cos 2 T >= 1/2, nd keeping only erms of orders hree in z, give he equion governing he slow dynmic of he moion (3). 28

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31 [24] D. Zulli, A. Luongo, Bifurcion nd sbiliy of wo-ower sysem under wind-induced prmeric, exernl nd self-exciion, J. Sound Vib. 331, (212) [25] Keighley, W.O., Housner, G.W., Hudson, D.E.: Vibrion ess of he Encino dm inke ower, Cliforni Insiue of Technology, Repor No. 2163, Psden, Cliforni (1961) [26] Belhq, M., Houssni, M.: Qusi-periodic oscillions, chos nd suppression of chos in nonliner oscillor driven by prmeric nd exernl exciions. Nonliner Dyn. 18, 1-24 (1999) [27] Rnd, R.H., Guennoun, K., Belhq, M.: 2:2:1 Resonnce in he qusi-periodic Mhieu equion. Nonliner Dyn. 31, (23) 31