Stability Analysis of a Tethered Airfoil

Transcription

1 13 Amricn Control Confrnc ACC Wshington, DC, USA, Jun 17-19, 13 Stbilit Anlsis of Tthrd Airfoil Sigits Rimkus 1, Tuhin Ds 1 nd Rnjn Mukhrj Abstrct Th stbilit nlsis of tthrd irfoil sstm is prsntd nd conditions rquird for th xistnc of stbl quilibrium points r drivd. Two css r invstigtd: cs whr th tthr is ssumd to b stright; nd scond, mor gnrl cs, whr th tthr is ssumd to tk ctnr gomtr. For ch cs, th rlvnt qutions of motion r drivd nd simultion rsults r usd to vlidt th mthmticl modls. Spcificll, for th stright tthr cs, th nlticl conditions for stbilit r drivd nd simultd. For th ctnr cs, simultions wr prformd to invstigt how th quilibrium point movs s oprting conditions r vrid. NOMENCLATURE m k Mss of th irfoil l t,m t Lngth nd mss of tthr l,m Lngth nd mss of ch tthr lmnt Inclintion of irfoil with horiontl φ i Inclintion of i th tthr lmnt with horiontl n, n nd co-ordints of position of nth tthr lmnt, Fr strm ir vlocit componnt in dirction, Fr strm ir vlocit componnt in dirction U rl, Rltiv vlocit of wind in dirction U rl, Rltiv vlocit of wind in dirction U rl Fr strm ir vlocit rltiv to irfoil L,D Lift nd drg forcs cting on th irfoil prpndiculr nd prlll to th dirction of th vlocit of th wind rltiv to th irfoil F c Forc on irfoil from tthr ρ Fr strm ir dnsit C L,C D Co-fficints of lift nd drg A Airfoil r α Angl of ttck α L, Angl of ttck for ro lift ǫ Spn ffctivnss fctor AR Aspct Rtio b, s Wing spn nd wing r rspctivl C d Profil drg c, c Co-ordints of cntr of mss of irfoil I O Momnt of inrti for stright tthr cs Subscripts: Vlus t quilibrium 1 S. Rimkus nd T. Ds r with th Dprtmnt of Mchnicl nd Arospc Enginring, Univrsit of Cntrl Florid, Orlndo, FL 3816, USAtds@ucf.du R. Mukhrj is with th Dprtmnt of Mchnicl Enginring, Michign Stt Univrsit, Est Lnsing, MI 4884, USA INTRODUCTION Mchins usd to convrt th kintic nrg contind within wind into lctricl powr r bcoming mor nd mor commonplc. Argubl, th most populr tp of ths mchins is th wind turbin which mplos lrg rotting wings to gnrt lctricit. Howvr, fr mor sotric clss of mchins, consisting of tthrd irfoils, usd for powr gnrtion lso xist. Mchins lik Ockls Lddrmill 1 nd tthr-irfoil dsign discussd in hv bn dscribd in litrtur. A smll-scl tst of tthrd irfoil sstm ws dscribd b Cnl t l. 3 furthr vlidtd th thor with xprimntl dt. Work b Lod 4 found tht kit of comprbl si to lrg irlinr could produc mor thn 3 MW of powr in 1 m/s wind, figur similr to tht producd b lrg wind turbin. In thir rlir work 5, th uthors prsntd simpl modl of two-dimnsionl tthr-irfoil sstm mountd on bs cpbl of linr horiontl motion. B oscillting th bs in prticulr mnnr nd chnging th ngl of ttck snchronousl, usful powr ws gnrtd. Th uthors lso brifl prsntd n mpiricl mthod of dtrmining th stbilit of th tthr-irfoil sstm. B injcting smll prturbtions into th sstm in th form of stp chngs in wind vlocit nd intil conditions, th uthors wr bl to show tht th sstm tjctoris convrgd to quilibrium. In this ppr, w invstigt th stbilit problm of th tthr-irfoil sstm. W considr two css, stright tthr cs, nd ctnr tthr cs. Th stright tthr cs is good pproximtion of th tthr shp for smll tthr lngths nd high wind spd oprting conditions whr rodnmic forcs cting on th irfoil rsult in lrg tnsion forcs in th tthr. Th ctnr cs is mor gnrl nd is vlid for lssr wind spds. W first provid n ovrviw outlining vrious ssumptions nd gnrl rodnmic qutions which will llow for th drivtion of th qutions of motion for th stright tthr nd ctnr gomtr. Nxt, w driv th qutions of motion nd linri thm bout th quilibrium point. Thn, w provid th rsults of svrl simultions. Finll, w mk concluding rmrks. SYSTEM DESCRIPTION AND MODEL DEVELOPMENT Assumptions W ssum tthr-irfoil sstm s shown in Fig.1. For modling th sstm, w mk th following ssumptions: A1. Th tthr-irfoil sstm movs ntirl within th pln /$ AACC 561

2 A. Th irfoil is squr, flt plt, nd its instntnous ngl of ttck α is sufficintl smll tht th th foil is not in stlld condition. A3. Th tthr is ngligibl thin, inxtnsibl, nd is not subjct to rodnmic lods. A4. Th vlocit of th oncoming wind hs tim-std mgnitud nd dirction. A5. A control sstm is usd to mintin constnt inclintion of th irfoil, i.. φk =. A6. Th tthr is connctd to sttionr bs O. α c, c whr ǫ is obtind from xprimntl dt, nd with tpicl vlus in th rng.8 ǫ 1. Th drg cofficints is modld s C D = C d α+ C L α 3 πǫar STABILITY ANALYSIS Loction of Sttic Equilibrium Point At sttic quilibrium, th forcs on th tthr r shown in Fig.3. From Figs. nd 3, nd using subscript to rprsnt F,, θl F, g Kit, mss m k F,1 O Fig. 1. Tthr, mss m t, lngth l t Schmtic of tthr-irfoil sstm Forcs on th Airfoil: Lift nd Drg Formultion A fr bod digrm of th irfoil is shown in Fig.. For th sk of concisnss, th nt momnt on th irfoil du to xtrnl forcs is not shown sinc it ssumd tht = through n ctiv control, ssumption A5. Th lift nd drg α U rl Fig.. L F c γ m k g forcs cting on th irfoil r whr D Fr-bod Digrm of Airfoil L = 1 ρc LA U rl, D = 1 ρc LA U rl 1 U rl = U rl, +U rl,, tn = U rl, /U rl, Th clcultion of C L nd C D r dscribd in dtil in 5, which r bsd on thin irfoil thor discussd in 6, 7. Th lift cofficint is modld s C L α = dc L dα α α L,, dc L dα = π 1+, AR = b s ǫar, Fig. 3. F,1 θo m g t quilibrium condition, w hv Tthr with Forcs t Equilibrium D cos L sin = F, 4 D sin +L sin m k g = F,. 5 whr, is th sttic quilibrium position of th cntr of mss of th irfoil. From forc blnc t quilibrium configurtion of th tthr, w obtin F,1 = F,, F,1 = m t g F, 6 At quilibrium, th tthr tks th shp of ctnr, with qution q = cosh +h 7 whr h, q nd r constnt prmtrs of th ctnr. Th lngth of th tthr is constnt nd from Eq.7 l t = 1+ d = sinh q d d sinh q. 8 W lso not in Fig.3, tht th tnsion t n point of th ctnr is dirctd tngntil to th ctnr. Hnc, d d, = tnθ d d, = tnθ l = sinh q F =,1 F,1 = sinh q F =, 9 F, Noting tht L nd D cn b clcultd for std oprting condition nd noting tht tn =, /,, th bov qutions cn b solvd to clcult th sttic quilibrium position,. 56

3 O Fig. 4. θ 8U m g t F c L m g k D c, c Stright Tthr-irfoil with Extrnl Forcs Stbilit Anlsis with Stright Tthr Assumption Th stright tthr ssumption cn b vlid t high wind spds whn th tthr tnsion is high. W ppl this ssumption to driv simpl nlticl conditions for stbilit. Th ssumption lso rducs th tthr-irfoil to 1DOF sstm. For stright tthr, c = l t cosθ nd c = l t sinθ, nd th qution of motion in θ cn b writtn s I O θ = lt Lcosθ l t Dsinθ 1 m k + m t 1 gl t cosθ, I O = 3 m t +m k lt 11 Th formultions of th lift forc L nd drg forc D r L = 1 ρc LA U +l t θsinθ cosθ+l t θ D = 1 ρc DA U + l t θsinθ cosθ+l t θ 1 rspctivl, whr is U, ż, l t θcosθ = rctn = rctn, ẏ, +l t θsinθ 13 In ordr to invstigt th stbilit of th sttic quilibrium, w linri Eq.11 bout th quilibrium ngl θ I O δ = fθ,+ f δ + f θ θ δ 14 θ, θ, whr δ = θ θ, nd fθ, θ is th right-hnd sid of Eq.11. Crring out th drivtivs nd grouping lik trms ilds th linrid qution of motion I O δ +c δ +kδ = 15 whr 1 c = ρal t C D 1+sin θ 1 ρal t C L cosθ sinθ 16 1 k = ρal tc D U cosθ m k + m t gl t sinθ 1 ρal tc L U sinθ 17 For th bov scond ordr linr sstm, th ncssr nd sufficint conditions for stbl quilibrium t θ = θ r I O >, c >, nd k >. Th inrti I O is lws positiv, so th inqulit conditions for th cofficints c nd k dtrmin stbilit. Evluting th inqulit condition for c ilds C D > cosθ sinθ C L 1+sin θ Similrl, vluting th inqulit condition for k ilds C D cosθ C L sinθ > m k +m t g ρau Stbilit Anlsis with Ctnr Tthr 18 sinθ 19 In this sction, w will rlx th stright tthr ssumption nd instd considr th tthr to rtin th ctnr shp for smll prturbtions bout th quilibrium. This mounts to using sttics bsd modl for th ctnr whil studing th dnmic bhvior of th tthr-irfoil sstm clos to quilibrium. Now w hv DOF sstm sinc c nd c cn chng indpndntl s long s th tthr lngth l t rmins unchngd. Th qutions of motion r O Fig. 5. 8U m g t L F c m g k D c, c Ctnr Tthr-Airfoil with Extrnl Forcs m k ÿ c = Dcos Lsin F c, m k c = Dsin +Lcos F c, m k g 1 W linri th qutions of motion bout th quilibrium point, m k δÿ = Dcos ẏ δẏ + Dcos ż δż Lsin ẏ δẏ Lsin ż δż F c, δ F c, nd m k δ = Dsin ẏ δẏ + Dsin ż δż + Lcos ẏ δẏ + Lcos ż δż F c, δ F c, 3 whr δ = nd = nd rprsnts clcultion t th quilibrium. Crring out th drivtivs for th rodnmic forcs in Eq. using Eq.13, nd 563

4 noting in Fig. tht α + = π ilds m k δÿ = 1 U ρa, +U C L, cos C D +, cos dc L sin δẏ + 1 U ρa, +U C L C D, cos +, cos dc L +sin δż F c, δ F c, 4 Prforming th sm stps on Eq.3 ilds m k δ = 1 U ρa, +U C L +C D, cos +, sin dc L +cos δẏ 1 U ρa, + C L, cos +C D, sin dc L +cos δż F c, δ F c, 5 whr, in Eqs.4 nd 5 dc L π = 1+, = AR dcd dα + C L πar dc L dα 6 r obtind from Eqs. nd 3. To solv th sptil drivtivs of F c, nd F c,, w procd s follows. W first not from Eq.7 tht q cosh = h, q = cosh +h 7 Upon diffrntition nd rrrnging Eqs.7 nd 8, A d dq δ = B 8 dh whr A R 3 3 nd B R 3. Th ntris for A r q q 11 = qsinh +h, 1 = sinh q q q 1 = sinh cosh q = sinh 31 = l t q q cosh q q cosh q q 3 = cosh cosh 13 =, 3 = 1, 33 = 9 nd th ntris for B r q b 11 =, b 1 =, b 1 = sinh, b = 1 q b 31 = cosh, b 3 = 3 From Eqs.6 nd 9 w hv df,1 = df,, df,1 = df, 31 Now, for smll chng in th ctnr, th corrsponding chng in slop nd tnsion forcs t, cn b rltd s d d = d, F,1 +df,1, 3 F,1 +df,1 which upon simplifiction ilds F,1 +df,1 = F,1 1 q F,1 +df,1 F,1 cosh = F,1 F,1 µma 1 B q 1 d dq dh δ = F,1 F,1 pδ q 33 Eqution 33 cn b xprssd s F,1 F,1 T df, df, = F,1 p q Tking similr pproch for, ilds T F, df, u ν = F F, df,, v whr, ν = 1 cosh q nd u nd v r dfind s q, N = T δ T δ , 36 νna 1 Bδ T = uδ +v 37 Combining Eqs.34 nd 35 w gt F c, F c, T = F, F, T = Q 1 U T 38 whr Q = F1 F 1, U = F F pf 1 qf1 vf u νf 39 Whn ssmbld s mtrix qution, Eqs.4 nd 5 tk th form mk δÿ m k δ δẏ +C δż δ +K. = 4 564

5 Th mtrix C consists soll of trms from rodnmic forcs, nd K consists of trms rltd to th tnsion in th tthr. For stbilit, th ignvlus of th mtrix I M 1 K M 1 C 41 must hv ngtiv rl prts. Solving for th ignvlus of Eq.41 nlticll is tdious nd hnc w rsort to numricl computtions to gnrt root-locus, shown in th nxt sction. SIMULATIONS AND OBSERVATIONS Bfor prsnting th simultion rsults, th spcific formul forc L n dvlopd for this modl r givn. W ssum squr shpd wing spn with on of th cornrs fcing th hd wind. For squr of sid, from Eq. AR = / =. Th prmtr ǫ is tkn s.8 nd α L, s -.35rd bsd on xprimntl dt from vrious NACA irfoils prsntd in 6, 7, 5. This ilds C L =.793α Th profil drg C d ws obtind from 8, bsd on th rng of Rnolds numbrs xpctd for wind spds vring from 1 m/s to 3 m/s. Th rsulting drg cofficint is C D =.1943α C L 43 Stbilit Conditions: Stright Tthr Assumption Tbl I shows th prmtrs usd in simulting th irfoil s oprtion. Our prliminr stimts of th totl TABLE I SIMULATION PARAMETERS Prmtr Vlu l t 1 m m t.5 kg m k kg A m I O.5 kg m ρ 1.3 kg m 3 U,i 15 m s 1 U,f m s 1 n+1 1 tthr drg, ssuming it to b vrticl clindr, is 5N vn with consrvtiv stimt of th tthr dimtr ssuming tthr md of Kvlr. Hnc, w ssrt tht th ngligibl tthr drg ssumption is plusibl, 8. Using th sm numricl s in 5, simultion is run for 3 sconds with th stp wind chng from U,i to U,f occurring t sconds. Th rsults of th simultion r shown in Fig. 6. As shown in Fig. 6, th vrg tthr ngl φ i rngs from bout 1.34 rdins bfor th stp chng to bout 1.45 rdins ftr th stp chng. Th tight bnds of φ i vlus indict high lvl of tnsion within th tthr. Rfrring to Eq.18 nd Fig. 6, thr r two vlus of θ to considr, nml θ = tn /.9 = rds, nd θ = tn 1 99./1.5 = rds. Th clcultd rtio nd th simultd rtio of C D to C L r both shown in Fig.7. Th clcultd rtio is found using Eq.18 nd th vlus for θ mntiond bov. Th CD/CL b CD cosθ- - CL sinθ- Fig RHS of Eq.18 RHS of Eq tim s Vrifing Stbilit Critri from Stright Tthr Assumption vlu of θ is ssumd to chng instntnousl with th stp wind chng. Th simultd rtio is found using th simultd vlus of C D nd C L t ch tim stp. As shown in Fig.7, for th first sconds of simultion, th first stbilit critrion dfind b Eq.18 is violtd, t th quilibrium is stbl. Howvr, onc th wind spd is incrsd, th stbilit critrion is stisfid. Th scond stbilit critrion coms from Eq.19 nd is rprsntd grphicll in Fig.7b. In this cs, th scond stbilit critrion is stisfid t ll tims. Th bov rsults show th vlidit of th stright tthr ssumption t highr wind spds. Stbilit Conditions: Ctnr As mntiond rlir, th ctnr modl rprsnts mor ccurt gomtric configurtion of th tthr undr std stt oprting conditions. Fig. 8 grphicll illus- Fig. 8. Equilibrium Tthr Angls for Vring U nd m k trts th quilibrium ngl θ for rng of wind spdsu nd irfoil mss m k. Th ngl θ rprsnts th ngl md b th tthr with th horiontl t th bs,. Th plot confirms th xpctd rsult tht s th mss incrss nd/or th wind spd rducs, th quilibrium point trnsitions 565

6 φi nd φ rd b c tim s α d g φ i φi r d/s tim s 5 F T N tim s d 15 f 1.9, c m 1.5, 99. c, c m 6 4 c c 1 3 tim s Fig c m Rspons to Stp Chng in Wind Spd 1 3 tim s from on with positiv ltitud to on with ngtiv ltitud. Rcll tht solving for th ignvlus of Eq.41 will provid insight to th stbilit of givn quilibrium point. W nxt gnrt th root locus plot whr, in Fig.9 th wind spd U is hld constnt nd th irfoil mss m k is vrid ovr rng. In Fig.9b, th irfoil mss m k ws chosn to b kg, Imλ i Imλ i 4 - λ 1 λ λ 3 λ λ 1 λ λ 3 λ 4 b R λ i Fig. 9. Root Locus from Ctnr Tthr Modl Vring m k nd b Vring U nd th wind spd U ws vrid. All of th ignvlus shown in Fig. 9 hv ngtiv rl prts, thus indicting tht th sttic quilibrium points r stbl. CONCLUSION A stbilit nlsis for tthr-irfoil hs bn prsntd, nd conditions for th xistnc of stbl quilibri hv bn invstigtd. For stright tthr, n ssumption tht would b vlid for high wind spds, nlticl conditions for stbilit wr drivd nd confirmd through simultions. Sinc th tthr lngth nd wight cn b substntil, th tthr modl ws rfind to tht of ctnr. A sttics bsd modl of th ctnr ws ugmntd to th dnmic modl of th irfoil, to prform mor rlistic stud of stbilit. For th lttr cs, stbilit of quilibrium points for vring wind spd nd vring irfoil mss wr vrifid numricll. Futur work will includ nling stbilit whn th tthr-irfoil sstm is usd for hrnssing wind nrg. Also, linrition pproch, which givs ncssr nd sufficint conditions for stbilit, cnnot b usd to dtct priodic orbits, such s crosswind flight. Futur work will invstigt th nonlinr bhvior of th sstm. REFERENCES 1 W.J. Ockls. Lddrmill, novl concpt to xploit th nrg in th irspc. Aircrft Dsign, , 1. P. Willms, B. Lnsdorp, nd W. Ockls. Optiml cross-wind towing nd powr gnrtion with tthrd kits. AIAA Journl of Guidnc, Control nd Dnmics, 31, 8. 3 M. Cnl, L. Fgino, nd M. Milns. High ltitud wind nrg gnrtion using controlld powr kits. IEEE Trnsctions on Control Sstms Tchnolog, 18 :79 93, 1. 4 M.L. Lod. Crosswind kit powr. Journl of Enrg, 4 3:16 111, T. Ds, R. Mukhrj,, R. Sridhr, nd A. Hllum. Two dimnsionl modling nd simultion of tthrd irfoil sstm for hrnssing wind nrg. ASME Dnmic Sstms nd Control Confrnc, J. D. Andrson Jr. Fundmntls of Arodnmics. McGrw-Hill, 1. 7 J. D. Andrson Jr. Introduction to Flight. McGrw-Hill, F.M. Whit. Fluid Mchnics. McGrw-Hill,. 566