Flight Dynamics & Control Dynamics Near Equilibria

Transcription

1 Flight Dynmic & Contol Dynmic Ne Equilibi Hy G. Kwtny Deptment of Mechnicl Engineeing & Mechnic Dexel Univeity

2 Outline Pogm Stight & Level Flight the pimy tedytte Petubtion Eqution fo Stight & Level Flight Longitudinl & Ltel Dynmic Coodinted Tun Stedy Sidelip

3 Pogm Detemine tim condition Define nd compute tedy-tte flight condition Stight nd level flight (cuie, climb, decend) Coodinted tun Stedy idelip Detemine petubtion model Smll petubtion fom tim condition Exmine dynmicl behvio

4 Reduction in Dimenion coodinte: x, y, z, φθψ,, velocitie: uvwpq,,,,, 12 tte Suppoe 1) the eth i flt nd 2) we ignoe vition of i denity Note: Thee popetie do not equie lineity ( x y z ) the dynmic e invint w..t. loction,, the dynmic e invint w..t. (inetil) heding Conequently, we could dop x, y, z, ψ nd tudy n 8 tte ytem Thee omitted vible cn lwy be included by dding the ppopite kinemtic eqution we often include z nd/o ψ

5 Reduced Eqution m u v qw inθ X + T m v = m pw u + mg coθinφ + Y m w qu pv coθcoφ Z ( I y I z ) q I xz pq 2 2 ( I x I y ) pq I xzq Ix Ixz p + L I y q = I z I x q + I xz p + M + Ixz Iz + N φ 1 inφtnθ coφtnθ p θ coφ inφ q = ψ inφecθ coφecθ T T

6 Equilibi The eqution e ognized in fit ode tndd fom, involving the tte vecto x, input vecto u, nd output vecto y. = = h( xu) E x x f xu, tte eqution y, output eqution A et of vlue ( x, u, y ) i clled n equilibium (o tim) point if it tifie: ( x u) (, ) = f, y = h x u We e inteeted in motion tht emin cloe to the equilibium point.

7 The Pimy Stedy Stte: Stight & Level flight The tight nd level flight condition equie equilibium flight long line pth with contnt flight pth ngle, contnt velocity V, zeo idelip nd wing level. Thu, we impoe the following condition fo tight nd level flight: Equilibium: ( α β ) u =, v =, w = V =, =, =, p =, q =, =, φ =, θ =, ψ = ω = Output: peed: V = V flight pth ngle: γ : = θ α = γ idelip: β = oll: φ = γ

8 Stight & Level Flight Popoition : An equilibium point tifying the tight nd level flight condition exit if nd only if thee exit α, δe, T which tify the equ (, α, δe ) i ( α γ ) T Z( V α δe ) M ( V, α, δe,) + TT = tion: X V mg n + + =,,, + mg co α + γ =, In thi ce the equilibium vlue of the tte nd contol e: longitudinl vible; tte: V = V, α = α, q =, θ = γ + α contol: δe = δe, T = T ltel vible; tte: β =, p =, =, φ =, contol: δ =, δ =

9 Lineiztion = + δ = + δ = + δ E( x ) δx = f x + δx( t), u + δu( t) Define: x t x x t, u t u u t, y t y y t The eqution become: Now, contuct Tylo eie fo f, h (, ) (, ) (, ) y + δy t = h x + δx t u + δu t (, ) (, ) f x u f x u f x + δx u + δu = f x u + δx + δu + hot x u h( x, u) h( x, u) h( x + δxu, + δu) = h( x, u) + δx + δu + hot x u Notice tht f x, u = nd h x, u = y, o f ( x, u ) f ( x, u ) δ = δ + δ x u h( x, u) h( x, u) δy = δx+ δu x u E x x x u Eδx = Aδx+ Bδu δy = Cδx+ Dδu

10 Petubtion fo Stight & Level Flight ~ longitudinl eqution mcoα mv inα mv inα V minα mv coα mv coα d α = Iy dt q 1 θ 2 V QSCX ( αδ, e) QSCx (, e) mg co V α αδ θ 2 V QSCZ ( αδ, e) QSCz α ( αδ, e) V coα α 2 2 V QS CM ( αδ, e ) QS Cm (, e ) V QS Cmq (, e ) q α αδ αδ 1 θ 1 QSCx δ ( αδ, ) e e QSCz δ ( αδ, ) e e T + T QSCm δ ( αδ, ) δ e e e

11 Petubtion fo Stight & Level Flight ~ ltel eqution mv β Ix Ixy d p = Ixy Iz dt 1 φ b b 2V β 2V b b ( α) 2V ( α) 2V ( α) b b ( α) ( α) ( α) QSCyβ α QS Cy α QS Cy α mu mg coθ β QS Cl QS Clp QS Cl p β QS Cn QS 2V Cnp QS 2V Cn β 1 tnθ φ Cy δ ( α) C yδ α Clδ C lδ C ( α) C ( α) α α δ + QS δ nδ nδ

12 Longitudinl Dynmic Ne tight nd level flight the genel eqution of motion my be divided into two et which e lmot decoupled. Longitudinl Dynmic : o olling v =, φ =, ψ =, p =, =. Longitudinl Vible : Motion in body x-z plne, without ywing uwq,,, θ, x, y the x- z plne i plne of ymmety Uncoupled longitudinl motion exit povided oto gyocopic effect e bent the flt eth ppoximtion i vlid m u qw inθ X Foce eqution: m mg m w = qu + + coθ Z Moment eqution: Iq = M y x coθ inθ u Kinemtic: =, θ q z = inθ coθ v

13 Ltel Dynmic Ltel Dynmic : ( w, u u, θ = θ, q=, α α ) Ltel Vible : motion in body x-y plne, no pitching vp,,, ψφ,, y motion e mll-tjectoy emin cloe to tight & level eodynmic co-coupling tem e negligible Uncoupled ltel motion exit povided oto gyocopic effect e bent the flt eth ppoximtion i vlid Foce eqution: co in Moment eqution: Kinemtic: mv = mu + mg θ φ+ Y I x I xz p L = Ixz I z N φ 1 coφtnθ = ψ p coφecθ y = coθ inφu + inφinθ inψ + coφcoψ v

14 Bnked Coodinted Tun The bnked tun i defined by the following condition: 1) equilibium - V, αβ, uvw,,, pq,, e contnt v qw inθ X + T = m pw u + mg coθinφ + Y qu pv coθcoφ Z I y I z q + I xz pq L q+ I p + M + TT I N x I y pq + I xzq = Iz Ix ( 2 2 ) xz

15 Bnked Coodinted Tun~2 2) bnked tun condition: the (inetil) ngul velocity i veticl nd contnt p inθ ω = q = coθinφ ω ω co θcoφ 3) coodinted tun condition - um of gvity nd inetil foce lie in plne of ymmety ( x z plne) mpw mu + mg coθinφ = pv co βinα V co β coα + g coθinφ = 4) climb condition: V = V, γ = γ

16 Bnked Coodinted Tun ~ 3 Thee e 12 eqution in 12 unknown V, αβ,, pq,,, θφ,, T, δe, δ, δ The fct tht the velocity i contnt in body fme, with contnt ngul velocity bout z inuue tht gound tck i cicul. The coodinted tun condition inue tht pilot nd penge will not expeience ny ide foce. A pilot chieve coodintion by uing the udde in conjunction with n intument clled tun coodinto which meue the diffeence between the inetil nd gvity foce cting long the y xi. The udde lo induce moment which countect the dvee yw moment eulting fom inceed (deceed) dg on the outide (inide) wing poduced by ileon poition nd which cn be ignificnt duing the olling phe of the tun.

17 Bnked Coodinted Tun ~ 4 ( pqvα θ) 5 eqution explicitly yield 5 unknown,,,, o in tem of the othe, o thee cn be eliminted, leving: pv co βinα V co βcoα + g coθinφ = v qw inθ X + T m + mg + Y = qu pv coθcoφ Z ( I y I z ) q + I xz pq 2 2 ( I x I y ) pq + I xzq L I z I x q + I xz p + M TT + = N with pq,, known function of ω

18 Bnked Coodinted Tun ~ 5 Conide the poibility of olution in which ω i mll ( R lge). Popoition : Thee exit olution coeponding to ω = if nd only if thee exit α, δe, T tifying ( α δe ) mg ( α + γ ) + T = M ( V α δe ) X V,, in, Z,,, + T = T ( V α δe ) mg ( α γ ),, + co + =, Thi i the ce of tight nd level flight in which the equilibium vlue of the tte nd contol e: In thi ce the equilibium vlue of the tte nd contol e: longitudinl vible; tte: V = V, α = α, q =, θ = γ + α contol: δ = δ, T = T ltel vible; e e tte: β =, p =, =, φ =, contol: δ =, δ =

19 Bnked Coodinted Tun ~ 6 Popoition : eqution nd α δ T Suppoe, e, contitute egul olution of the longitudinl equilibium ( α) Cy δ ( α) ( α) ( α) ( α) ( α) ( α) ( α) C yβ det Clβ Clδ Clδ Cnβ Cnβ Cnβ Then thee exit olution to the coodinted bnk tun eqution, with βδ,, δ mll nd αδ α δ, e, T ne, e, T. To fit ode tem, the olution i ωv tnφ = coγ g Cyβ ( α) Cy δ ( α) β Cyp ( α) Cy ( α) Clβ ( α) Clδ ( α) C lδ α δ = Clp α Cl α Cnβ ( α) Cnβ ( α) Cnβ ( α) δ Cnp ( α) Cn ( α) ωb coφ 2V Cmα Cm δ e α Cmq ωc inφ C Clq 2 ( ec 1 l C l ) C α δ δ = + e e V φ lw

20 Cowind The tedy idelip i flight condition tht my be ued duing lnding ppoche in the peence of cowind. It i n equilibium condition in the ene tht ll cceletion nd ngul te e zeo. In ddition, the icft tck line gound pth coeponding to y =, i.e., ligned with the x xi. Flight pth ngle nd peed e lo pecified. cowind β = ψ = Advee wind condition wee involved in 33% of 76 lnding ccident between 1984 nd The key iue with the cbbed ppoch i the pottouchdown, on the gound dynmic. Some combintion i genelly ued.

21 Cowind ~ 2 Conide n inetil (flt eth fixed) fme with coodinte on the ufce nd z down. The intent i to lnd long the x -xi (o we equie y = ). Aume the cowind velocity i v. Thu, we hve the following condition: Regulted output: y = y = v + V co βcoαcoθinψ + Vin β inφinθinψ + coφcoψ V = V + V co βinα coφinθinψ inφcoψ = γ : = θ α = γ Equilibium condition: V =, α =, β =, q =, θ =, p =, =, φ =, ψ = x y

22 Cowind ~ 3 Popoition : (cbbing olution) A olution to the gound tcking poblem exit with β =, φ = if nd only if thee exit α, δe nd T which tify (, α, δe ) in ( α γ ), (, α, δe ) co ( α γ ) M ( V, α, δe,) + TT = X V mg + + T = Z V + mg + = nd V v coγ 1 :,

23 Cowind ~ 4 Popoition : (tedy idelip) A olution to the gound tcking poblem exit with ψ = if nd only if thee exit α, δe nd T which tify: (, α, δ ) in( ), (,, ) co( e α γ α δe α γ ) M ( V, α, δe,) + TT = X V mg + + T = Z V + mg + = nd φ, β, δ, δ tifying mg coθinφ + Cyβ ( α) β + Cy δ ( α) δ + Cy δ α δ = QS ( α) β ( α) δ C α β + C α δ + C α δ = lβ lδ lδ C + C + C α δ = nβ nδ nδ v V + coφin β =,

24 Cowind Exmple A hypotheticl ubonic tnpot (dpted fom Etkin) h the following dt C = -.168, C =.67, C =, yβ yδ yδ C = -.47, C =.3, C = -.4, lβ lδ lδ C =.3625, C = -.16, C = -.5 nβ nδ nδ Suppoe the cowind i v obtined (in degee) =.15 V. Then the following eult e β = , φ= , δ = , δ = Slip to the left, oll to the left, hd ight udde, left ileon (into wind)

25 Comment fom Aibu Pilot I do not find cowind to be nymoe chllenging in thi iplne thn ny othe. You hve to undetnd tht you cnnot "lip" thi iplne becue you e commnding ROLL RATE with the Side Stick Contolle, not BANK ANGLE. Hee i uggetion: Allow the iplne to do n Auto Lnd in cowind when it i convenient, nd VFR. You will be hocked t the timing of when the iplne leve the CRAB nd pplie udde to lign the noe pllel with the unwy. You think it jut in't going to do it, nd t the vey lt econd, it lide it in pefectly. I would gue in the lt 2 feet o le. My technique i jut the me ny iplne I've eve flown. C-15 to B-767. Cb it down to the fle, pply enough udde to tighten the noe, dop the up-wind wing to pevent dift, lnd on the up-wind min fit Ye Aibu Cptin.