Unsteady Aerodynamic Forces: Experiments, Simulations, and Models Steve Brunton & Clancy Rowley FAA/JUP Quarterly Meeting April 6, Wednesday, March 8,
Motivation Applications of Unsteady Models Conventional UAVs (performance/robustness) Micro air vehicles (MAVs) Flow control, flight dynamic control Autopilots / Flight simulators Gust disturbance mitigation Predator (General Atomics) Need for State-Space Models Need models suitable for control Combining with flight models FLYIT Simulators, Inc. Daedalus Dakota Wednesday, March 8,
Flight Dynamic Control coupled model reference trajectory, wind disturbances flight dynamics deviation from desired path, or state thrust, elevator, aileron, blowing/suction aerodynamics position, aerodynamic state controller estimator Wednesday, March 8,
Stall velocity and size Smaller, lower stall velocity RQ- Predator (7 m/s stall) Daedalus Dakota (8m/s stall) Puma AE ( m/s stall) S Wing surface area V stall = ρ (C L max S) W W L C L V Aircraft weight Lift force Lift coefficient Velocity of aircraft Wednesday, March 8,
Lift vs. Angle of Attack.6.4. Average Lift pre Shedding Average Lift post Shedding Min/Max of Limit Cycle Lift Coefficient, C L.8.6.4...4 3 4 5 6 7 8 9 Angle of Attack, (deg) Need model that captures lift due to moving airfoil! Wednesday, March 8,
Lift vs. Angle of Attack.6.4. Average Lift pre Shedding Average Lift post Shedding Min/Max of Limit Cycle Sinusoidal (f=.,a=3) Lift Coefficient, C L.8.6.4...4 3 4 5 6 7 8 9 Angle of Attack, (deg) Need model that captures lift due to moving airfoil! Wednesday, March 8,
Lift vs. Angle of Attack.6.4. Average Lift pre Shedding Average Lift post Shedding Min/Max of Limit Cycle Sinusoidal (f=.,a=3) Canonical (a=,a=) Lift Coefficient, C L.8.6.4...4 3 4 5 6 7 8 9 Angle of Attack, (deg) Need model that captures lift due to moving airfoil! Wednesday, March 8,
Added-Mass D Model Problem Transient Periodic Vortex Shedding Lift Drag Re = 3 α = 3 Wednesday, March 8,
Added-Mass D Model Problem Transient Periodic Vortex Shedding Lift Drag Re = 3 α = 3 Wednesday, March 8,
Reduced Order Indicial Response C L (t) =C S L(t)α() + t!"#$%&$'(#)*+,+#))()+-#$$ C L α C S L(t τ) α(τ)dτ fast dynamics d dt x A r x α = α + α α Reduced-order model B r α input C L = x C r C Lα C L α α + C L α α α α C L α s + C L quasi-steady and added-mass C Lα s G(s) Model Summary Linearized about α = Based on experiment, simulation or theory.#$'+)*/#-%$ Recovers stability derivatives C Lα,C L α,c L α associated with quasi-steady and added-mass Brunton and Rowley, in preparation. ODE model ideal for control design Wednesday, March 8,
Lift vs. Angle of Attack.6.4. Average Lift pre Shedding Average Lift post Shedding Min/Max of Limit Cycle Lift Coefficient, C L.8.6.4.!"#$%&$'(#)*+,+#))()+-#$$ C L α. C L α s α + C L Models linearized at α =.4 3 4 5 6 7 8 9 Angle of Attack, (deg) C Lα s G(s).#$'+)*/#-%$ Wednesday, March 8,
Bode Plot - Pitch (QC) Frequency response input is α ( α is angle of attack) 6 4 Quarter-Chord Pitching output is lift coefficient C L Pitching at quarter chord Magnitude (db) Reduced order model with ERA r=3 accurately reproduces Indicial Response 4 Indicial Response and ROM agree better with DNS than Theodorsen s model. Asymptotes are correct for Indicial Response because it is based on experiment Model for pitch/plunge dynamics [ERA, r=3 (MIMO)] works as well, for the same order model Phase (deg) 5 5 Frequency (rad U/c) Indicial Response ROM, r=3 Wagner/Theodorsen DNS ROM, r=3 (MIMO) Brunton and Rowley, in preparation. Wednesday, March 8,
Lift vs. Angle of Attack.6.4. Average Lift pre Shedding Average Lift post Shedding Min/Max of Limit Cycle Lift Coefficient, C L.8.6.4.!"#$%&$'(#)*+,+#))()+-#$$ C L α. C L α s α + C L Models linearized at α =.4 3 4 5 6 7 8 9 Angle of Attack, (deg) C Lα s G(s).#$'+)*/#-%$ Wednesday, March 8,
Magnitude (db) 6 4 Bode Plot of Model (-) vs Data (x) Frequency Response Linearized at various! ERA,!= DNS,!= ERA,!= DNS,!= ERA,!= DNS,!= 4 5 Phase 5 Brunton and Rowley, AIAA ASM Wednesday, March 8, Frequency Direct numerical simulation confirms that local linearized models are accurate for small amplitude sinusoidal maneuvers
(Indicial) Step Response u A u u T Time u k u(t) y(t) y k PLANT Previously, models are based on aerodynamic step response Idea: Have pilot fly aircraft around for 5- minutes, back out the Markov parameters, and construct ERA model. Wednesday, March 8,
Random Input Maneuver α α α C L (t) Idea: Have pilot fly aircraft around for 5- minutes, back out the Markov parameters, and construct ERA model. Wednesday, March 8,
Wind Tunnel Setup Test section NACA 6 Airfoil (4.6 cm chord) Push rods and sting Servo tubes Wednesday, March 8,
Experimental Information Andrew Fejer Unsteady Flow Wind Tunnel (.6m x.6m x 3.5m test section) NACA 6 Airfoil Chord Length:.46 m Free Stream Velocity: 4. m/s. Convection time =.6 seconds Reynolds Number: 65, Pitch point x/c =. (% chord) Velocity measurement: Pitot tube, Validyne DP-3 pressure transducer Force measurement: ATI Nano5 force transducer Pushrod position measurement: linear potentiometer Pushrod actuation: Copley servo tubes Acknowledgments: Professor David Williams Seth Buntain and Vien Quatch Wednesday, March 8,
Phase Averaged Data.5.5.5.5 5 5 5 3 35 4 Convective Time.5.5 94 95 96 97 98 99..8 Step Up, Step Down, 5 degrees Phase averaged over cycles.6.4...4 Wednesday, March 8, 3 4 5 6 7 8 Convective Time
Wing Maneuver 8 6 4 Commanded Angle Measured Angle Angle (degrees) 4 6 8 3 4 5 6 7 8 9 Convective Times (s=tu/c) Wednesday, March 8,
What are we modeling? Angle (degrees) 3 4 5 6 7 8 9 Angle (degrees) 3 4 5 6 7 8 9 3 Measured Force ROM, r=3 3 Measured Force ROM, r=3 3 4 5 6 7 8 9 Model using command acceleration 3 4 5 6 7 8 9 Model using measured acceleration Wednesday, March 8,
What are we modeling? Angle (degrees) 3 4 5 6 7 8 9 Angle (degrees) 3 4 5 6 7 8 9 3 Measured Force ROM, r=3 3 Measured Force ROM, r=3 3 4 5 6 7 8 9 Model using command acceleration 3 4 5 6 7 8 9 Model using measured acceleration our model α pos α cmnd α pot C Simulink Actuator Aerodynamics L Wednesday, March 8,
Four Test Maneuvers Angle (degrees) 3 4 5 6 7 8 9 Angle (degrees) Maneuver Maneuver 3 4 5 6 7 8 9 3 Measured Force ROM, r=3 3 Measured Force ROM, r=3 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Angle (degrees) 3 4 5 6 7 8 9 Maneuver 3 Maneuver 4 Angle (degrees) 3 4 5 6 7 8 9 3 Measured Force ROM, r=3 3 Measured Force ROM, r=3 3 4 5 6 7 8 9 Wednesday, March 8, 3 4 5 6 7 8 9
Bode Plots for AoA= 6 Model using measured acceleration 4 Magnitude (db) 4 6 3 Phase (degrees) 5 5 maneuver maneuver maneuver 3 maneuver 4 3 Frequency (rad/s c/u) Idea: lets combine all maneuvers into one large system ID maneuver! Wednesday, March 8,
Bode Plot for AoA= Angle (degrees) 5 5 5 3 35 4 45 3 Measured Force ROM, r=3 Magnitude (db) Phase (deg) Bode Diagram 5 5 5 3 35 4 45 8 6 4 4 45 45 9 35 8 Wednesday, March 8, Resonant peak Added-mass bump
C Lα C L α{ { Model using ALL data Angle (degrees) 5 5 5 3 35 4 45 3 Measured Force ROM, r=3 5 5 5 3 35 4 45 Theory Experimental H i from OKID H i αc Lα H i αc Lα C L α. C L α Impulse response in α Markov parameter.5..5 Wednesday, March 8, 5 5
3Hz Mechanical Oscillation. Step Up, Step Down, 5 degrees.8.6.4...4 3 4 5 6 7 8 Convective Time Wednesday, March 8,
Models agree with data Angle (degrees) 3 4 5 6 7 8 9 Angle (degrees) 3 4 5 6 7 8 9 3 Experiment Model Model Model 3 Model 4 3 Experiment Model Model Model 3 Model 4 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Angle (degrees) 3 4 5 6 7 8 9 Angle (degrees) 3 4 5 6 7 8 9 3 Experiment 3 Model Model Model 3 Model 4 3 Experiment 4 Model Model Model 3 Model 4 3 4 5 6 7 8 9 Wednesday, March 8, 3 4 5 6 7 8 9
Model for Plunging Magnitude (db) Phase (deg) 6 4 8 35 9 45 Bode Diagram Frequency (rad/sec) Vertical Position (inches) 3 3 4 5 6 7 8 9 Measured Force ROM, r=3 3 4 5 6 7 8 9 Wednesday, March 8,
Conclusions Reduced order model based on indicial response at non-zero angle of attack - Based on eigensystem realization algorithm (ERA) - Models appear to capture dynamics up to Hopf bifurcation Observer/Kalman Filter Identification with more realistic input/output data - Efficient computation of reduced-order models - Ideal for simulation or experimental data Confirmation with experimental data - Tested modeling procedure in Dave Williams wind tunnel experiment - Flexible procedure works with various geometry, Reynolds number Wagner, 95. Theodorsen, 935. Leishman, 6. OL, Altman, Eldredge, Garmann, and Lian, Wednesday, March 8, Brunton and Rowley, AIAA ASM 9- Juang and Pappa, 985. Ma, Ahuja, Rowley,. Juang, Phan, Horta, Longman, 99.