16.333: Lecture # 7. Approximate Longitudinal Dynamics Models

Transcription

1 16.333: Lecture # 7 Approxiate Longitudinal Dynaics Models A couple ore stability derivatives Given ode shapes found identify sipler odels that capture the ain responses

2 Fall More Stability Derivatives Recall fro 6 2 that the derivative stability derivative ters Z ẇ and M ẇ ended up on the LHS as odifications to the noral ass and inertia ters These are the apparent ass effects soe of the surrounding displaced air is entrained and oves with the aircraft Acceleration derivatives quantify this effect Significant for blips, less so for aircraft. Main effect: rate of change of the noral velocity ẇ causes a transient in the downwash ɛ fro the wing that creates a change in the angle of attack of the tail soe tie later Downwash Lag effect If aircraft flying at U, will take approxiately Δt = l t /U to reach the tail. Instantaneous downwash at the tail ɛ(t) is due to the wing α at tie t Δt. ɛ ɛ(t) = α α(t Δt) Taylor series expansion α(t Δt) α(t) α Δt Note that Δɛ(t) = Δα t. Change in the tail AOA can be coputed as dɛ dɛ l t Δɛ(t) = α Δt = α = Δα t dα dα U

3 Fall For the tail, we have that the lift increent due to the change in downwash is dɛ l t ΔC Lt = C Lαt Δα t = C Lαt α dα U The change in lift force is then 1 ΔL t = ρ(u 2 ) t S t ΔC Lt 2 In ters of the Z force coefficient ΔL t S t S t dɛ l t ΔC Z = 1 = η ΔC Lt = η C Lαt α ρu 2 S S S dα U 2 We use c/(2u ) to nondiensionalize tie, so the appropriate stability coefficient for is (note use C z to be general, but we are looking at ΔC z fro before): ( ) ( ) C Z 2U C Z C Zα = = ( α c/2u ) c α 2U S t l t dɛ = η C Lαt c S U dα dɛ = 2ηV H C Lαt dα The pitching oent due to the lift increent is ΔM cg = l t ΔL t 1 ρ(u 2 2 ) t S t ΔC Lt ΔC Mcg = l t 1ρU 2 Sc 2 dɛ l t = ηv H ΔC Lt = ηv H C Lαt α dα U

4 Fall So that ( ) ( ) C M 2U C M C Mα = = ( α c/2u ) c α dɛ l t 2U = ηv H C Lαt dα U c dɛ l t = 2ηV H C Lαt dα c l t C c Zα Siilarly, pitching otion 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 which changes the lift at the tail (and the oent about the cg). Following sae analysis as above: Lift increent ΔL t = C Lαt ql t U 1 2 ρ(u 2 ) t S t C Zq ΔC Z = ΔL t )S = η S t S C ql t L αt U 1 2 ρ(u 2 ( ) ( ) C Z = 2U C Z (q c/2u ) c q = η 2U l t S t c U S C L αt = 2ηV H C Lαt Can also show that l t C Mq = C Zq c

5 Fall Approxiate Aircraft Dynaic Models It is often good to develop sipler odels of the full set of aircraft dynaics. Provides insights on the role of the aerodynaic paraeters on the frequency and daping of the two odes. Useful for the control design work as well Basic approach is to recognize that the odes have very separate sets of states that participate in the response. Short Period priarily θ and w in the sae phase. The u and q response is very sall. Phugoid priarily θ and u, and θ lags by about 9. The w and q response is very sall. Full equations fro before: X u u ẇ q θ Z u w [M u+z uγ] = Z X w Z w Z ẇ [M w+z wγ] Z q+u Z ẇ [M q+(z q+u )Γ] 1 g cos Θ [ u ] [ ΔX ] c g sin Θ Z w ΔZ c w q + ΔM c θ g sin Θ Γ

6 Fall For the Short Period approxiation, 1. Since u in this ode, then u and can eliinate the X force equation. Z w Z q +U g sin Θ ẇ Z Z w ΔZ c ẇ Z ẇ w [M q +(Z q +U q = [M w +Z w Γ] )Γ] + g sin Θ Γ q ΔM c θ 1 θ 2. Typically find that Z ẇ and Z q U. Check for 747: Z ẇ = 199 = Z q = U = M ẇ M ẇ Γ = Zẇ Γ Z w U [ ] [ g sin Θ ] ΔZ c ẇ w M M w +Z ẇ w M q +(U ) M ẇ q = g sin Θ M + ΔM c ẇ q θ 1 θ 3. Set Θ = and reove θ fro the odel (it can be derived fro q) With these approxiations, the longitudinal dynaics reduce to ẋ sp = A sp x sp + B sp δ e where δ e is the elevator input, and [ ] [ ] w Z w / U x sp =, A q sp = Iyy 1 (M w + M ẇ Z w /) Iyy 1 (M q + M ẇ U ) [ ] Z δe / B sp = Iyy 1 (M δe + M ẇ Z δe /)

7 Fall Characteristic equation for this syste: s 2 + 2ζ sp ω sp s + ωsp 2 =, where the full approxiation gives: ( ) Z w M q M ẇ 2ζ sp ω sp = + + U ω 2 Z w M q U M w sp = Given approxiate agnitude of the derivatives for a typical aircraft, can develop a coarse approxiate: 2ζ sp ω sp M q ζ sp M q 1 2 U M w ω 2 U M w sp ω sp U M w Nuerical values for 747 Frequency Daping rad/sec Full odel Full Approxiate Coarse Approxiate Both approxiations give the frequency well, but full approxiation gives a uch better daping estiate Approxiations showed that short period ode frequency is deterined by M w easure of the aerodynaic stiffness in pitch. Sign of M w negative if cg sufficient far forward changes sign (ode goes unstable) when cg at the stick fixed neutral point. Follows fro discussion of C Mα (see 2 11)

8 Fall For the Phugoid approxiation, start again with: X u X w u g cos Θ u ΔX c Z g sin Θ u Z w Z q +U ΔZ c ẇ Z ẇ Z ẇ Z ẇ Z ẇ w = q [M q +(Z q +U )Γ] + [M u +Z u Γ] [M w +Z w Γ] g sin Θ Γ q ΔM c θ 1 θ 1. Changes to w and q are very sall copared to u, so we can Set ẇ and q Set Θ = X u X w u g u ΔX c Zu Z w Z q +U Z ΔZ c ẇ Z ẇ Z w = ẇ [M q +(Z q +U )Γ] + [M u +Z u Γ] [M w +Z w Γ] q ΔM c θ 1 θ 2. Use what is left of the Z equation to show that with these approxiations (elevator inputs) Zw Z q +U Z [ ] Z u δ e Z ẇ Z ẇ Z Z w ẇ ẇ = u δ q e [M u +Z u Γ] [M w +Z w Γ] [M q +(Z q +U )Γ] [M δ e +Z δe Γ ] Iyy 3. Use (Z ẇ so Γ M ẇ ) and (Z q U ) so that: [ ] [ ] [ Z w ] U w M M w + Z ẇ w [M q + U M ẇ ] q [ ] [ ] [ Z u ] [ Z δe = ] M M u + Z ẇ u M ẇ δ u M δe + Z e δe

9 Fall Solve to show that U M u Z u M q [ ] w = Z w M q U M w q Z u M w Z w M u Z w M q U M w u + U M δe Z δe M q Z w M q U M w Z δe M w Z w M δe Z w M q U M w δ e 5. Substitute into the reduced equations to get full approxiation: ( ) [ ] Xu X U M u Z u M q + w [ ] Z w M q U M w g u u = ( ) θ θ + Z u M w Z w M u Z w M q U M w ( ) X δ e X w U M δ e Z δe M q + Z w M q U M w Z δ e M w Z w M δ e Z w M q U M w δ e 6. Still a bit coplicated. Typically get that M u Z w M w Z u (1.4:4) M w U M q Z w (1:.13) M u X w /M w X u sall 7. With these approxiations, the longitudinal dynaics reduce to the coarse approxiation where δ e is the elevator input. ẋ ph = A ph x ph + B ph δ e

10 Fall And [ ] u x ph = A ph = θ X u g Z u U ( [ ] ) X X δe w Mw M δe B ph = ( [ ] ) Z Z + w δe Mw M δe U 8. Which gives 2ζ ph ω ph = X u / ω 2 gz u ph = U Nuerical values for 747 Frequency Daping rad/sec Full odel Full Approxiate Coarse Approxiate

11 Fall Further insights: recall that ( ) ( ) ( ) ( ) U Z U L = (C Lu + 2C L ) QS u QS u M 2 = C 1 M 2 L 2C L 2C L so ( ) ( ) Z ρu o S 2g Z u = ( 2C L ) = u 2 U Then gz g ω 2 u ph = = U g = 2 U U 2 which is exactly what Lanchester s approxiation gave Ω 2 g U Note that ( ) ( ) X ρu o S X u = ( 2C D ) = ρu o SC D u 2 and so 2g = ρu 2 SC L o X u X u U ζ ph = = 2ωph 2 2g ( ) 1 ρu 2 SC D = o 2 ρu ( 2 o SC ) L 1 C D = 2 C L so the daping ratio of the approxiate phugoid ode is inversely proportional to the lift to drag ratio.

12 Fall Transfer function fro elevator to flight variables G ude G γde arg G ude arg G γde Freq Coparison fro elevator (Phugoid Model) B747 at M=.8. Blue Full odel, Black Full approxiate odel, Magenta Coarse approxiate odel

13 Fall Transfer function fro elevator to flight variables G αde G γde arg G ude arg G γde Freq Coparison fro elevator (Short Period Model) B747 at M=.8. Blue Full odel, Magenta Approxiate odel

14 Fall Suary Approxiate longitudinal odels are fairly accurate Indicate that the aircraft responses are ainly deterined by these stability derivatives: Property Daping of the short period Frequency of the short period Daping of the Phugoid Frequency of the Phugoid Stability derivative M q M w X u Z u

15 Fall Given a change in α, expect changes in u as well. These will both ipact the lift and drag of the aircraft, requiring that we re tri throttle setting to aintain whatever aspects of the flight condition ight 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 aintain L = W, want ΔL =, so u = ( ) Giving ΔD = L α D u + D α Δα L u 2C L CDα = C Lα D α = QSC Dα πear L α = QSC Lα L α L u Δα QS D u = (2C D ) (4 16) U QS L u = (2C L ) (4 17) U ( ( ) ) C Lα 2C D ΔD = QS + CDα Δα 2CL /U ( U ) QS 2CL 2 = C D + C Lα Δα CL πear (T + ΔT ) (D + ΔD) ΔD tan Δγ = = ( L + Δ ) L L C D 2C L C Lα Δα = C L πear CL For 747 (Reid 165 and Nelson 416), AR = 7.14, so πear 18, C L =.654 C D =.43, C Lα = 5.5, for a Δα =.185rad (6 7) Δγ =.6rad. This is the opposite sign to the linear siulation results, but they are both very sall nubers.