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Maneuvering at High Angles and Angular Rates Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018 Learning Objectives • High angle of attack and angular rates • Asymmetric flight • Nonlinear aerodynamics • Inertial coupling • Spins and tumbling
Flight Dynamics 681-785 Airplane Stability and Control Chapter 8
Copyright 2018 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE331.html 1 http://www.princeton.edu/~stengel/FlightDynamics.html
Tactical Airplane Maneuverability
• Maneuverability parameters – Stability – Roll rate and acceleration – Normal load factor – Thrust/weight ratio – Pitch rate – Transient response – Control forces • Dogfights – Preferable to launch missiles at long range – Dogfight is a backup tactic – Preferable to have an unfair advantage • Air-combat sequence – Detection – Closing – Attack – Maneuvers, e.g., • Scissors • High yo-yo – Disengagement
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Coupling of Longitudinal and Lateral-Directional Motions
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Longitudinal Motions can Couple to Lateral-Directional Motions • Linearized equations have limited application to high-angle/high-rate maneuvers – Steady, non-zero sideslip angle (Sec. 7.1, FD) – Steady turn (Sec. 7.1, FD) – Steady roll rate
" F FLon % F = $ Lon Lat−Dir ' $ FLat−Dir F ' # Lon Lat−Dir &
Lon Lat−Dir FLat−Dir , FLon ≠ 0 4
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Stability Boundaries Arising
From Asymmetric Flight Northrop F-5E
NASA CR-2788 5
Stability Boundaries with Nominal Sideslip, βo, and Roll Rate, po
NASA CR-2788 6
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Pitch-Yaw Coupling Due To Steady Roll Rate, po • Combine 2nd-order short period and Dutch roll modes – Body axes
– Constant roll rate = po, rad/s State vector Control input vector " Δw % Normal velocity, m / s # ΔδE & Elevator, deg or rad " Δx % $ ' Lon $ Δq ' Pitch rate, rad / s % ( Δx(t) = $ ' = Δu(t) = % Δδ A ( Ailerons, deg or rad $ ΔxLD ' $ Δv ' Side velocity, m / s # & $ ' $% Δδ R '( Rudder, deg or rad # Δr & Yaw rate, rad / s 4th-order dynamic model
" Δx % " F FLon % " Δx % " G % $ Lon ' $ Lon LD ' $ Lon ' $ Lon ' u = LD + Δ $ Δx LD ' $ FLon FLD ' $ ΔxLD ' $ GLD ' # & # & # & # & 7
Time Response to Elevator Step Input
• When po = 0 /s – Elevator input produces longitudinal response but no lateral-directional response
• At po = 60 /s – Short-period (faster) mode dominates longitudinal response – Dutch-roll (slower) mode dominates lateral directional response • At po = 120 /s – Both modes are evident in both responses – Fast mode is even faster – Slow mode is even slower 8
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Pitch-Yaw Coupling Due To Steady Roll Rate, po • 4th-order stability matrix – Body axes
– Neglible vo, uo ~ VN – Negligible coupling aerodynamic effects • Constant roll rate is only source of coupling
Short Period Yaw-to-Pitch Coupling ⎡ ⎤ ⎡ − p 0 ⎤ ⎢ ⎡ ⎤ o ⎥ Zw uo ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ I − I ⎥ ⎥ M M ( zz xx ) ⎢ ⎢ w q ⎥ ⎢ 0 po ⎥ ⎥ " Δw % Lon ⎣ ⎦ I $ ' ⎡ ⎤ ⎢ yy ⎥ Δq FLon FLD ⎣⎢ ⎦⎥ Δx(t) = $ ' ⎢ ⎥ = ⎢ ⎥ $ Δv ' LD ⎢ ⎥ $ ' ⎢ FLon FLD ⎥ ⎡ p 0 ⎤ # Δr & ⎣ ⎦ ⎢ o ⎡ ⎤ ⎥ ⎢ ⎥ Yv –uo ⎢ ⎢ I − I ⎥ ⎢ ⎥ ⎥ 0 ( xx yy ) p N N ⎢ ⎢ o ⎥ ⎣⎢ v r ⎦⎥ ⎥ ⎢ ⎢ Izz ⎥ ⎥ ⎣ ⎣ ⎦ ⎦ Pitch-to-Yaw Coupling Dutch Roll 9
Pitch-Yaw Coupling Due To Steady Roll Rate, po Characteristic Polynomial # %# % Δrolling (s) = {$(s − Zw )(s − M q ) −uoM w &$(s −Yv )(s − Nr )+uoNv &} ' + 2 ) (Izz − I xx ) (I xx − I yy ) (I xx − I yy ) (Izz − I xx )) + po ((s − M q )(s − Nr ) −(s − Zw )(s −Yv ) −uoM w −uoNv , *) I yy Izz Izz I yy -)
4 (Izz − I xx ) (I xx − I yy ) − po I yy Izz
2 4 • Coupling effect is proportional to po and po
• Effect on roots is independent of the sign of po 2 2 • Cannot use Evans s root-locus rules with k = po , as k also appears 2 • Can compute effect of po on roots using MATLAB’s eig
I − I I − I " $ 2 " $ 4 ( zz xx ) ( xx yy ) Δrolling (s) = #ΔSP (s)ΔDR (s)%+ po # fcn(s, M q , Nr ,Zw ,Yv , I xx , I yy , Izz ,uo, M w , Nv )%− po I yy Izz
Thunderbird F-16 Barrel Roll http://www.youtube.com/watch?v=ovSOStlncbU 10
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Effect of Steady Roll Rate, po, on Pitching and Yawing Roots
2 • Factor Δrolling(s) for various values of po 2 • po = root locus gain, k • Faster mode gets faster • Slower mode gets slower and may become unstable
No Damping With Typical Damping
Short Period
Dutch Roll
Dutch Roll
Short Period
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Steady Roll Rate, po, Effect Expressed by Root Locus or Parameter Plot Parameter plot: variation of real and imaginary parts of roots vs. roll rate, Divergent root at high roll rate po
Dutch roll: solid Dutch roll: solid Short period: dashed Short period: dashed Bifurcation
Instability
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Steady-State Response as Well as Stability is
Affected by High Roll Angle of Atack Response to Elevator Rate vs. Roll Rate
f (v,w, p,q,r,Δδ A,ΔδE,Δδ R)SS = 0
Sideslip Response to Elevator vs. Roll Rate
• Effects of steady roll rate on nonlinear equilibrium control response – Pitch-yaw coupling – p jump or p acceleration
• Multiple equilibria for same control Aileron Angle settings vs. Roll Rate – Up to 9 possible roll rates for one aileron setting
– Sensitivity to elevator setting Rhoads, Schuler, • Flight Dynamics, 7.3 1957 13
The Butterfly Catastrophe*
f1 (v,w, p,q,r,Δδ A,ΔδE,Δδ R)SS = 0
pSS = f2 (v,w,q,r,Δδ A,ΔδE,Δδ R)SS • Surface of equilibrium solutions for roll rate • Possibility of an unrecoverable spin
Roll Rate Equilibrium Surface The Butterfly Catastrophe
after Mehra, Carroll, 1979 * René Thom, 1974 14
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Tumbling and Spins
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Tumbling, Spins, and Recovery • Strong nonlinear effects • Aircraft-specific control strategy for recovery
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Wind Tunnel Spin Testing • Sidney B. Gates, RAE: "The Spinning of Aeroplanes" (with L.W. Bryant, 1926), neutral and maneuver points, stick force per g • Continued research on stalls and spins at NASA, USAF, and in many other countries NASA Langley Spin Tunnel
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NASA Langley Spin Tunnel Testing
http://www.youtube.com/watch?v=u7FCqLpTgkk http://www.youtube.com/watch?v=tQwMCmI55Q0 http://www.youtube.com/watch?v=VUKTBUY1RII 18
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Tails with Negative Dihedral (Anhedral) • Horizontal tail below wing's wake • May have adverse effect on spin characteristics • F-4 model test
Yaw coefficient, Anti-spin moment Cn
Pro-spin moment
Spin rate, pb/2V 19
Yawing Moment at High Angle of Attack
• Dynamic as well as static effects,e.g., hysteresis • Random asymmetric yawing moments (left or right) – generated by slender nose at zero sideslip angle – may exceed rudder control power F-111
Vortex-induced side force on nose
F-4
F-104
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Controlling Yawing Moment at High Angle of Attack
McDonnell-Douglas F/A-18 • Sucking, blowing, or movable strakes to control nose vortices • X-29, F/A-18 HARV • Vortex bursting effect on tail
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Control Effectiveness at High Angle of Attack and Deflection Angle
• Assumption of Newtonian flow
Elevator Effect Aileron Effect
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Control at High Aerodynamic Angles
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Supermaneuverability
• Means of forcing opponent to overshoot • Pugachev s Cobra maneuver, first done in Sukhoi Su-27 • Beneficial effect of thrust-vector control (X-31) • Mongoose maneuver (X-31) • Essentially low-speed maneuvers, not where you want to be in air combat (i.e., high energy-state)
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Thrust Vector Control
Pitch and Yaw Control (X-31) Pitch Control (F-22)
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Next Time: Aeroelasticity and Fuel Slosh Flight Dynamics 418-419, 549-569, 665-678 Airplane Stability and Control Chapter 19
Learning Objectives • Aerodynamic effects of bending and torsion • Modifications to aerodynamic coefficients • Dynamic coupling • Fuel shift and sloshing dynamics
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Supplemental Material
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Stall-Spin Studies of General Aviation Aircraft
http://www.youtube.com/watch?v=
TmWB6oyJ9lE 28
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