12/3/18 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 2 1 12/3/18 Coupling of Longitudinal and Lateral-Directional Motions 3 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 2 12/3/18 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 3 12/3/18 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 ' Δu Δx $ FLD F ' Δx G #$ LD &' # Lon LD & #$ LD &' #$ LD &' 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 4 12/3/18 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 Evanss 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 5 12/3/18 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 11 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 12 6 12/3/18 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 7 12/3/18 Tumbling and Spins 15 Tumbling, Spins, and Recovery • Strong nonlinear effects • Aircraft-specific control strategy for recovery 16 8 12/3/18 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 17 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 9 12/3/18 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 20 10 12/3/18 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 21 Control Effectiveness at High Angle of Attack and Deflection Angle • Assumption of Newtonian flow Elevator Effect Aileron Effect 22 11 12/3/18 Control at High Aerodynamic Angles 23 Supermaneuverability • Means of forcing opponent to overshoot • Pugachevs 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) 24 12 12/3/18 Thrust Vector Control Pitch and Yaw Control (X-31) Pitch Control (F-22) 25 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 26 13 12/3/18 Supplemental Material 27 Stall-Spin Studies of General Aviation Aircraft http://www.youtube.com/watch?v= TmWB6oyJ9lE 28 14.