航空器的稳定与控制（英文版）：lecture_7

16.333: Lecture # 7 Approximate Longitudinal Dynamics Models ? A couple more stability derivatives ? Given mode shapes found identify simpler models that capture the main re- sponses Fall 2004 16.333 6–1 More Stability Derivatives ? Recall from 6–2 that the derivative stability derivative terms Z w˙ and M w˙ ended up on the LHS as modi?cations to the normal mass and inertia terms – These are the apparent mass e?ects – some of the surrounding displaced air is “entrained” and moves with the aircraft – Acceleration derivatives quantify this e?ect – Signi?cant for blimps, less so for aircraft. ? Main e?ect: rate of change of the normal velocity w˙ causes a transient in the downwash ? from the wing that creates a change in the angle of attack of the tail some time later – Downwash Lag e?ect ? If aircraft ?ying at U 0 , will take approximately Δt = l t /U 0 to reach the tail. – Instantaneous downwash at the tail ?(t) is due to the wing α at time t ? Δt. ?? ?(t) = ?α α(t ? Δt) – Taylor series expansion α(t ? Δt) ≈ α(t) ? α˙Δt – Note that Δ?(t) = ?Δα t . Change in the tail AOA can be com- puted as d? d? l t Δ?(t) = ? α˙Δt = α˙ = ?Δα t dα ? dα U 0 ? ? ? ? Fall 2004 16.333 6–2 ? For the tail, we have that the lift increment due to the change in downwash is d? l t ΔC L t = C L α t Δα t = C L α t α˙ dαU 0 The change in lift force is then 1 ΔL t = ρ(U 0 2 ) t S t ΔC L t 2 In terms of the Z-force coe?cient ? ΔL t S t S t d? l t ΔC Z = 1 = ?η ΔC L t = ?η C L α t α˙? ρU 0 2 S S S dαU 0 2 ? c/(2U 0 ) to nondimensionalize time, so the appropriate stabil-We use ˉ ity coe?cient form is (note use C z to be general, but we are looking at ΔC z from before): ?C Z 2U 0 ?C Z = =C Z α˙ αˉ 0 ? ( ˙c/2U 0 ) cˉ ?α˙ 0 2U 0 S t l t d? = ?η cˉ S U 0 C L α t dα d? = ?2ηV H C L α t dα ? The pitching moment due to the lift increment is ΔM cg = ?l t ΔL t 1 ρ(U 2 0 ) t S t ΔC L t 1 → ΔC M cg = ?l t 2 ρU 0 2 Scˉ 2 d? l t = ?ηV H ΔC L t = ?ηV H C L α t α˙ dαU 0 ? ? ? ? ? Fall 2004 16.333 6–3 So that ?C M 2U 0 ?C M = =C M α˙ αˉ 0 ? ( ˙c/2U 0 ) cˉ ?α˙ 0 d? l t 2U 0 = ?ηV H C L α t dαU 0 cˉ d? l t = ?2ηV H C L α t dα cˉ l t C Z α˙ ≡ cˉ ? Similarly, pitching motion of the aircraft changes the AOA of the tail. Nose pitch up at rate q, increases apparent downwards velocity of tail by ql t , changing the AOA by ql t Δα t = U 0 which changes the lift at the tail (and the moment about the cg). ? Following same analysis as above: Lift increment ΔL t = C L α t ql t U 0 1 2 ρ(U 2 0 ) t S t ΔC Z = ? ΔL t 1 2 ρ(U 2 0 )S = ?η S t S C L α t ql t U 0 ? ? ? ? C Z q ≡ ?C Z ?(qˉc/2U 0 ) 0 = 2U 0 ˉc ?C Z ?q 0 = ?η 2U 0 ˉc l t U 0 S t S C L α t = ?2ηV H C L α t Can also show that ? l t C M q = C Z q cˉ Fall 2004 16.333 6–4 Approximate Aircraft Dynamic Models ? It is often good to develop simpler models of the full set of aircraft dynamics. – Provides insights on the role of the aerodynamic parameters on the frequency and damping of the two modes. – Useful for the control design work as well ? Basic approach is to recognize that the modes have very separate sets of states that participate in the response. – Short Period – primarily θ and w in the same phase. The u and q response is very small. – Phugoid – primarily θ and u, and θ lags by about 90 ? . The w and q response is very small. ? Full equations from before: ? ? ? X u u˙ m ? w˙ Z u w [M u +Z u Γ] ? = ? m?Z ˙ q˙ θ ˙ I yy 0 X w m Z w m?Z ˙w [M w +Z w Γ] I yy 0 0 Z q +mU 0 m?Z ˙w [M q +(Z q +mU 0 )Γ] I yy 1 ?g cos Θ 0 ? ? ? ? ? u ΔX c ?mg sin Θ 0 m?Z ˙ w ΔZ c ? mg sin Θ 0 Γ w ? q + ΔM c I yy 0 θ 0 ? ? ? ? ? ? ? ? ? ? Fall 2004 16.333 6–5 ? For the Short Period approximation, 1. Since u ≈ 0 in this mode, then u˙ ≈ 0 and can eliminate the X-force equation. ?? ? ? ???? Z q +mU 0 ?mg sin Θ 0 m?Z ˙ Z w ΔZ c w˙ w m?Z w˙ m?Z w˙ [ M q +(Z q +mU 0 )Γ ] ? ? ? ? ? ? w ? ? = ? ? + ? [M w +Z w Γ] ΔM c ? mg sin Θ 0 Γ I yy q˙ q I yy I yy θ ˙ θ 0 0 1 0 2. Typically ?nd that Z w˙ ? m and Z q ? mU 0 . Check for 747: – Z w˙ = 1909 ? m = 2.8866 × 10 5 – Z q = 4.5 × 10 5 ? mU 0 = 6.8 × 10 7 M w˙ M w˙ Γ = m ? Z w˙ ? Γ ≈ m ?? ? ? ???? Z w U 0 ?g sin Θ 0 ΔZ c w˙ w m ? ? ? ? ? ? M w +Z w M w˙ M q +(mU 0 ) M w˙ m ? ? = ? ? + ? ΔM c ? mg sin Θ 0 M ˙w I yy q˙ m q I yy I yy m θ ˙ θ 0 0 1 0 3. Set Θ 0 = 0 and remove θ from the model (it can be derived from q) ? With these approximations, the longitudinal dynamics reduce to x˙ sp = A sp x sp + B sp δ e where δ e is the elevator input, and w Z w /m U 0 x sp = q , A sp = I ?1 (M w + M w˙ Z w /m) I ?1 (M q + M w˙ U 0 ) yy yy ? ? Z δ e /m B sp = I ?1 (M δ e + M w˙ Z δ e /m) yy ? ? ? ? Fall 2004 16.333 6–6 ? Characteristic equation for this system: s 2 + 2ζ sp ω sp s + ω 2 = 0, sp where the full approximation gives: Z w M q M w˙ 2ζ sp ω sp = ? + + U 0 m I yy I yy ω 2 Z w M q U 0 M w = sp mI yy ? I yy ? Given approximate magnitude of the derivatives for a typical aircraft, can develop a coarse approximate: 2ζ sp ω sp ≈? M q I yy ? ? ζ sp ≈? M q 2 ?1 U 0 M w I yy ω 2 U 0 M w ? → sp ≈? ω sp ≈ ?U 0 M w I yy I yy Numerical values for 747 ? Frequency Damping rad/sec Full model 0.962 0.387 Full Approximate 0.963 0.385 Coarse Approximate 0.906 0.187 Both approximations give the frequency well, but full approximation gives a much better damping estimate ? Approximations showed that short period mode frequency is deter- mined by M w – measure of the aerodynamic sti?ness in pitch. – Sign of M w negative if cg su?cient far forward – changes sign (mode goes unstable) when cg at the stick ?xed neutral point. Follows from discussion of C M α (see 2–11) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Fall 2004 16.333 6–7 ? For the Phugoid approximation, start again with: ?? ? ? X u X w 0 ?g cos Θ 0 m m ?mg sin Θ 0 ΔX c u˙ u ? ? ? ? ? ? ? ? ? ? Z q +mU 0 ? ? ? ? Z u Z w ? ? ? ? ? ? ? ? ? ? ? ? ΔZ c ΔM c w˙ q˙ w q m?Z w˙ m?Z w˙ m?Z w˙ m?Z w˙ [ M q +(Z q +mU 0 )Γ ] + = [M u +Z u Γ] [M w +Z w Γ] ? mg sin Θ 0 Γ I yy I yy I yy I yy θ ˙ θ 0 0 0 1 0 1. Changes to w and q are very small compared to u, so we can – Set w˙ ≈ 0 and q˙ ≈ 0 – Set Θ 0 = 0 X u X w 0 ?g m m ? ? ? ? ???? ΔX c u˙ u ? ? ? ? ? ? ? ? ? ? Z q +mU 0Z u Z w ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 0 ΔZ c ΔM c 0 0 w q m?Z w˙ m?Z w˙ m?Z w˙ [ M q +(Z q +mU 0 )Γ ] + = [M u +Z u Γ] [M w +Z w Γ] I yy I yy I yy θ ˙ θ 0 0 0 1 0 2. Use what is left of the Z-equation to show that with these ap- proximations (elevator inputs) ?? ?? ?? Z δ eZ u Z q +mU 0Z w m?Z w˙ m?Z w˙ m?Z w˙ m?Z w˙ ? ? ? ? ? ? ? ? w ? ? u? δ e = ? [M w +Z w Γ] [ M q +(Z q +mU 0 )Γ ] [ M δ e +Z δ e Γ ] [M u +Z u Γ]q I yy I yy I yy I yy 3. Use (Z w˙ ? m so Γ ≈ M w˙ ) and (Z q ? mU 0 ) so that: m Z w mU 0 w M w + Z w M w˙ [M q + U 0 M w˙ ] q m ???? ? ? ? ? ? ? ? ? Z u Z δ e = M u + Z u M w˙ u ? M w˙ ? m M δ e + Z δ e m δ e ? ? ? ? ? ? ? ? ? ? ? Fall 2004 16.333 6–8 4. Solve to show that ? ? ? ? w q = ? ? ? mU 0 M u ? Z u M q Z w M q ? mU 0 M w Z u M w ? Z w M u ? ? ? u + ? ? ? mU 0 M δ e ? Z δ e M q Z w M q ? mU 0 M w Z δ e M w ? Z w M δ e ? ? ? δ e Z w M q ? mU 0 M w Z w M q ? mU 0 M w 5. Substitute into the reduced equations to get full approximation: ?? mU M Z M? 0 u u q Z w M q ?mU 0 M w ? X u X w + ?g u˙ θ ˙ ? ? m m ? ? u = θ Z u M w ?Z w M u 0 Z w M q ?mU 0 M w ? ? X δ e X w mU 0 M δ e ?Z δ e M q + m m Z w M q ?mU 0 M w ? ? ? ? δ e + Z δ e M w ?Z w M δ e Z w M q ?mU 0 M w 6. Still a bit complicated. Typically get that (1.4:4) – |M u Z w |?|M w Z u | (1:0.13) – |M w U 0 m|?|M q Z w | – M u X w /M w |? X u small | 7. With these approximations, the longitudinal dynamics reduce to the coarse approximation x˙ ph = A ph x ph + B ph δ e where δ e is the elevator input. ? ? ? ? ? ? ? ? ? ? Fall 2004 16.333 6–9 And ?? u x ph = A ph = θ ? ? ? X u ?g m ?Z u 0 mU 0 ? ? ? ?? B ph = ? ? ? ? ? ? M δ e M w X δ e ? X w m Z w +?Z δ e M w M δ e ? ? ? ? ? ? mU 0 8. Which gives 2ζ ph ω ph = ?X u /m ω 2 gZ u = ph ? mU 0 Numerical values for 747 Frequency Damping rad/sec Full model 0.0673 0.0489 Full Approximate 0.0670 0.0419 Coarse Approximate 0.0611 0.0561 ? ? ? ? ? ? ? ? ? ? Fall 2004 16.333 6–10 ? Further insights: recall that ? ?? ? ? ?? ? U 0 ?Z U 0 ?L = + 2C L 0 ) QS ?u 0 ? QS ?u 0 ≡?(C L u M 2 = ? 1?M 2 C L 0 ?2C L 0 ≈?2C L 0 so ? ? ? ? ?Z ρU o S 2mg Z u ≡ = (?2C L 0 ) = ? ?u 0 2 U 0 Then ? mg 2 ω ph = ?gZ u = mU 0 2 mU 0 g = √ 2 U 0 which is exactly what Lanchester’s approximation gave Ω ≈ √ 2 g U 0 Note that ?X ρU o S X u ≡ = (?2C D 0 ) = ?ρU o SC D 0 ?u 0 2 and 2mg = ρU 2 SC L 0o so X u X u U 0 ζ ph = ? 2mω ph = ? 2 √ 2mg 1 ρU 2 SC D 0o = √ 2 ρU 2 SC L 0o ? 1 C D 0 = √ 2 C L 0 so the damping ratio of the approximate phugoid mode is inversely proportional to the lift to drag ratio. Fall 2004 16.333 6–11 10 ?2 10 ?1 10 0 10 0 10 1 10 2 10 3 10 4 |G u d e | Freq (rad/sec) Transfer function from elevator to flight variables 10 ?2 10 ?1 10 0 10 ?1 10 0 10 1 10 2 10 3 10 4 |G g d e | Freq (rad/sec) 10 ?2 10 ?1 10 0 ?250?200?150?100 ?50 0 50 arg G u d e Freq (rad/sec) 10 ?2 10 ?1 10 0 ?350?300?250?200?150?100 ?50 0 arg G g d e Freq (rad/sec) Freq Comparison from elevator (Phugoid Model) – B747 at M=0.8. Blue– Full model, Black– Full approximate model, Magenta– Coarse approximate model Fall 2004 16.333 6–12 10 ?2 10 ?1 10 0 10 ?1 10 0 10 1 10 2 |G a d e | Freq (rad/sec) Transfer function from elevator to flight variables 10 ?2 10 ?1 10 0 10 ?1 10 0 10 1 10 2 |G g d e | Freq (rad/sec) 10 ?2 10 ?1 10 0 0 50 100150200250300 arg G u d e Freq (rad/sec) 10 ?2 10 ?1 10 0 ?350?300?250?200?150?100 ?50 0 arg G g d e Freq (rad/sec) Freq Comparison from elevator (Short Period Model) – B747 at M=0.8. Blue– Full model, Magenta– Approx- imate model Fall 2004 16.333 6–13 Summary ? Approximate longitudinal models are fairly accurate ? Indicate that the aircraft responses are mainly determined by these stability derivatives: Property Stability derivative Damping of the short period M q Frequency of the short period M w Damping of the Phugoid X u Frequency of the Phugoid Z u ? ? ? ? ? ? Fall 2004 16.333 6–14 ? Given a change in α, expect changes in u as well. These will both impact the lift and drag of the aircraft, requiring that we re-trim throttle setting to maintain whatever aspects of the ?ight condition might have changed (other than the ones we wanted to change). We have: ? ? ? ?? ? ΔL L u L α u = ΔD D u D α Δα But to maintain L = W , want ΔL = 0, so u = L u Δα ? ? ? L α Giving ΔD = ? L α D u + D α Δα L u 2C L 0 C D α = C L α → D α = QSC D α πeAR → L α = QSC L α QS D u = (2C D 0 ) (4 ? 16) U 0 QS L u = (2C L 0 ) (4 ? 17) U 0 C L α 2C D 0 ΔD = QS + C D α Δα? 2C L 0 /U 0 U 0 QS 2C 2 L 0 = C L 0 ?C D 0 + πeAR C L α Δα (T 0 + ΔT ) ? (D 0 + ΔD) ?ΔD tan Δγ = = ? L 0 + Δ ? L L 0 C D 0 2C L 0 C L α Δα = C L 0 ? πeAR C L 0 For 747 (Reid 165 and Nelson 416), AR = 7.14, so πeAR ≈ 18, C L 0 = 0.654 C D 0 = 0.043, C L α = 5.5, for a Δα = ?0.0185rad (6–7) Δγ = ?0.0006rad. This is the opposite sign to the linear simulation results, but they are both very small numbers.