Aircraft Flight Dynamics Robert F. Stengel Lecture20 Flying Qualities Criteria

Flying Qualities Criteria
 Robert Stengel, Aircraft Flight Dynamics
 MAE 331, 2012 " Copyright 2012 by Robert Stengel. All rights reserved. For educational use only.! http://www.princeton.edu/~stengel/MAE331.html ! http://www.princeton.edu/~stengel/FlightDynamics.html ! •  MIL-F-8785C criteria! •  CAP, C*, and other longitudinal criteria" •  ϕ/β, ω ϕ /ω  , and other lateral- directional criteria" •  Pilot-vehicle interactions" •  Flight control system design" Design for Satisfactory Flying Qualities" •  Satisfy procurement requirement (e.g., Mil Standard)" •  Satisfy test pilots (e.g., Cooper-Harper ratings)" •  Avoid pilot-induced oscillations (PIO)" •  Minimize time-delay effects" •  Time- and frequency-domain criteria" MIL-F-8785C Identifies Satisfactory, Acceptable, and Unacceptable Response Characteristics" Damping Ratio" Step Response" Frequency Response" Short-period angle-of-attack response to elevator input! Longitudinal Criteria Long-Period Flying Qualities Criteria (MIL-F-8785C)! •  Static speed stability" –  No tendency for aperiodic divergence" •  Phugoid oscillation -> 2 real roots, 1 that is unstable" –  Stable control stick position and force gradients" •  e.g., Increasing pull position and force with decreasing speed" A.  Non-terminal flight requiring rapid maneuvering" B.  Non-terminal flight requiring gradual maneuvering" C.  Terminal flight" 1.  Clearly adequate for the mission" 2.  Adequate to accomplish the mission, with some increase in workload" 3.  Aircraft can be controlled safely, but workload is excessive" Level of Performance! Flight Phase! Long-Period Flying Qualities Criteria (MIL-F-8785C)! •  Flight path stability [Phase C]" 1.  (Δγ/ΔV) SS < 0.06 deg/kt" 2.  (Δγ/ΔV) SS < 0.15 deg/kt" 3.  (Δγ/ΔV) SS < 0.24 deg/kt" ΔV SS = aΔ δ E SS + 0 ( ) Δ δ T SS + bΔ δ F SS Δ γ SS = cΔ δ E SS + dΔ δ T SS + eΔ δ F SS •  Lecture 19" Δ γ SS ΔV SS = c a (with appropriate scaling) •  From 4 th -order model" Long-Period Flying Qualities Criteria (MIL-F-8785C)! •  Phugoid stability" 1. Damping ratio ≥ 0.04" 2. Damping ratio ≥ 0" 3. Time to double, T 2 ≥ 55 sec" € T 2 Ph = −0.693/ ζ Ph ω n Ph Time to Double! Short Period Criteria" •  Important parameters" –  Short-period natural frequency" –  Damping ratio" –  Lift slope" –  Step response" •  Over-/under-shoot" •  Rise time" •  Settling time" •  Pure time delay" –  Pitch angle response" –  Normal load factor response" –  Flight path angle response (landing)" Space Shuttle Pitch-Response Criterion" Short-Period Approximation Transfer Functions" •  Elevator to pitch rate" Δq(s) Δ δ E(s) = k q s − z q ( ) s 2 + 2 ζ SP ω n SP s + ω n SP 2 ≡ k q s + 1 T θ 2 ( ) * + , - s 2 + 2 ζ SP ω n SP s + ω n SP 2 •  Pure gain or phase change in feedback control cannot produce instability" Bode Plot! Nichols Chart! Root Locus! Short-Period Approximation Transfer Functions" •  Elevator to pitch angle" •  Integral of prior example" Δ θ (s) Δ δ E(s) = k q s − z q ( ) s s 2 + 2 ζ SP ω n SP s + ω n SP 2 ( ) •  Pure gain or phase change in feedback control cannot produce instability" Bode Plot! Nichols Chart! Root Locus! Normal Load Factor" •  Therefore, with negligible L δ E (aft tail/canard effect)" Δn z = V N g Δ  α − Δq ( ) = − V N g L α V N Δ α + L δ E V N Δ δ E % & ' ( ) * ∂ Δn z (s) ∂ Δ δ E(s) = 1 g L α ∂ Δ α (s) ∂ Δ δ E(s) + L δ E % & ' ( ) * ≈ L α g % & ' ( ) * ∂ Δ α (s) ∂ Δ δ E(s) positive down! positive up! Δ α (s) Δ δ E(s) ≈ k α s 2 + 2 ζ SP ω n SP s + ω n SP 2 •  Elevator to angle of attack (L δ E = 0)" http://www.youtube.com/watch?v=xFemVFgsJAw! Control Anticipation Parameter, CAP" •  Inner ear senses angular acceleration about 3 axes" € Δ ˙ q (0) = M δ E − M α V N + L α L δ E & ' ( ) * + Δ δ E SS Δn SS = V N g Δq SS = − V N g & ' ( ) * + M δ E L α V N − M α L δ E V N & ' ( ) * + M q L α V N + M α & ' ( ) * + Δ δ E SS CAP = Δ  q(0) Δn SS = − M δ E − M α V N + L α L δ E % & ' ( ) * M q L α V N + M α ( ) L α M δ E − L δ E M α ( ) g •  Inner ear cue should aid pilot in anticipating commanded normal acceleration" Initial Angular Acceleration! Desired Normal Load Factor! Control Anticipation Factor! MIL-F-8785C Short-Period Flying Qualities Criterion" CAP = − M q L α V N + M α ( ) L α g ≈ ω n SP 2 n z / α € ω n SP vs. n z α 1.  Clearly adequate for the mission" 2.  Adequate to accomplish the mission, with some increase in workload" 3.  Aircraft can be controlled safely, but workload is excessive" Level of Performance! with L δ E = 0! •  CAP = constant along Level Boundaries" CAP! Control Anticipation Parameter vs. Short-Period Damping Ratio " (MIL-F-8785C, Category A)" CAP = − M q L α V N + M α ( ) L α g ≈ ω n SP 2 n z / α C* Criterion" !  Below V crossover , Δq is pilots primary control objective" !  Above V crossover , Δn pilot is the primary control objective" C* = Δn pilot + V crossover g Δq = l pilot Δ  q +Δn cm ( ) + V crossover g Δq = l pilot Δ  q + V N g Δq −Δ  α ( ) $ % & ' ( ) + V crossover g Δq Fighter Aircraft: V crossover ≈ 125 m / s •  Hypothesis" –  C* blends normal load factor at pilots location and pitch rate" –  Step response of C* should lie within acceptable envelope" Gibson Dropback Criterion for Pitch Angle Control " •  Step response of pitch rate should have overshoot for satisfactory pitch and flight path angle response ! Δq(s) Δ δ E(s ) = k q s + 1 T θ 2 $ % & & ' ( ) ) s 2 + 2 ζ SP ω n SP s + ω n SP 2 = k q s + ω n SP ζ SP $ % & ' ( ) s 2 + 2 ζ SP ω n SP s + ω n SP 2 z q  − 1 T θ 2 = − ω n SP ζ SP % & ' ( ) * •  Criterion is satisfied when! Gibson, 1997! Lateral-Directional Criteria Lateral-Directional Flying Qualities Parameters" • Lateral Control Divergence Parameter, LCDP! •  ϕ/β Effect" •  ω ϕ /ω  Effect! Lateral Control Divergence Parameter (LCDP)" •  Aileron deflection produces yawing as well as rolling moment" –  Favorable yaw aids the turn command" –  Adverse yaw opposes it" •  Equilibrium response to constant aileron input " Δ φ S Δ δ A S = N β + N r Y β V N % & ' ( ) * L δ A − L β + L r Y β V N % & ' ( ) * N δ A g V N L β N r − L r N β ( ) •  Large-enough N δ A effect can reverse the sign of the response" –  Can occur at high angle of attack " –  Can cause departure from controlled flight" •  Lateral Control Divergence Parameter provides simplified criterion" LCDP ≡ C n β − C n δ A C l δ A C l β N β ( ) L δ A − L β ( ) N δ A L δ A = N β − N δ A L δ A L β ω Φ / ω d Effect! • Aileron-to-roll-angle transfer function " Δ φ (s) Δ δ A(s) = k φ s 2 + 2 ζ φ ω φ s + ω φ 2 ( ) s − λ S ( ) s − λ R ( ) s 2 + 2 ζ DR ω n DR s + ω n DR 2 ( )   ω ϕ is the natural frequency of the complex zeros"   ω d = ω nDR is the natural frequency of the Dutch roll mode" • Conditional instability may occur with closed- loop control of roll angle, even with a perfect pilot" ω ϕ /ω  Effect" •  As feedback gain increases, Dutch roll roots go to numerator zeros " •  If zeros are over poles, conditional instability results" Δ φ (s) Δ δ A(s) = k φ s 2 + 2 ζ φ ω φ s + ω φ 2 ( ) s − λ S ( ) s − λ R ( ) s 2 + 2 ζ DR ω n DR s + ω n DR 2 ( ) ϕ/β Effect" •  ϕ/β measures the degree of rolling response in the Dutch roll mode" –  Large ϕ/β : Dutch roll is primarily a rolling motion" –  Small ϕ/β : Dutch roll is primarily a yawing motion" • Eigenvectors, e i , indicate the degree of participation of the state component in the i th mode of motion" det sI − F ( ) = s − λ 1 ( ) s − λ 2 ( ) s − λ n ( ) λ i I − F ( ) e i = 0 Eigenvectors! • Eigenvectors, e i , are solutions to the equation" λ i I − F ( ) e i = 0, i = 1, n or λ i e i = Fe i , i = 1, n • For each eigenvalue, the corresponding eigenvector can be found (within an arbitrary constant) from" Adj λ i I − F ( ) = a 1 e i a 2 e i … a n e i ( ) , i = 1, n MATLAB V,D ( ) = eig F ( ) V: Modal Matrix (i.e., Matrix of Eigenvectors) D: Diagonal Matrix of Corresponding Eigenvalues ϕ/β Effect" •  With λ i chosen as a complex root of the Dutch roll mode, the corresponding eigenvector is" e DR+ = e r e β e p e φ # $ % % % % % & ' ( ( ( ( ( DR+ = σ + j ω ( ) r σ + j ω ( ) β σ + j ω ( ) p σ + j ω ( ) φ # $ % % % % % % & ' ( ( ( ( ( ( DR+ = AR e j φ ( ) r AR e j φ ( ) β AR e j φ ( ) p AR e j φ ( ) φ # $ % % % % % % % & ' ( ( ( ( ( ( ( DR+ •  ϕ/β is the magnitude of the ratio of the ϕ and β eigenvectors" φ β = AR ( ) φ AR ( ) β = V N g # $ % & ' ( ζ DR ω n DR + Y β V N + L β L r # $ % & ' ( 2 + ω n DR 1− ζ DR 2 ( ) , - . . / 0 1 1 1 2 ϕ/β Effect for the Business Jet Example! e DR+ = e r e β e p e φ # $ % % % % % % % & ' ( ( ( ( ( ( ( DR+ = 0.525 0.416 0.603 0.433 # $ % % % % & ' ( ( ( ( DR+ φ β = 1.04 Roll/Sideslip Angle ratio in the Dutch roll mode! Early Lateral-Directional Flying Qualities Criteria ! T 1 2 = 0.693 / ζ ω n v = V N β O  Hara, via Etkin! Ashkenas, via Etkin! Time to Half! Criteria for Lateral-Directional Modes (MIL-F-8785C)" Maximum Roll- Mode Time Constant" Minimum Spiral-Mode Time to Double" Minimum Dutch Roll Natural Frequency and Damping (MIL-F-8785C)" Pilot-Vehicle Interactions Pilot-Induced Roll Oscillation" Δ φ (s) Δ δ A(s) pilot in loop = K p / T p s + 1 / T p $ % & ' ( ) k φ s 2 + 2 ζ φ ω φ s + ω φ 2 ( ) s − λ S ( ) s − λ R ( ) s 2 + 2 ζ DR ω n DR s + ω n DR 2 ( ) . / 0 0 1 2 3 3 Aileron-to-Roll Angle Root Locus ! Pilot-Aircraft Nichols Chart ! Pilot Transfer Function ! Aircraft Transfer Function ! YF-16! YF-17 Landing Approach Simulation " •  Original design" –  Low short-period natural frequency" –  Overdamped short period" –  Rapid roll-off of phase angle" –  PIO tendency, CHR = 10" Elevator-to-pitch angle Nichols chart (gain vs. phase angle)" 80° Phase Margin! Gibson, 1997! •  Revised DFCS design" –  Higher short-period natural frequency" –  Lower short-period damping" –  Reduced time delay in DFCS" –  CHR = 2" Input frequency, rad/s" 13 dB Gain Margin! •  Alternative pilot transfer function: gain plus pure time delay " H j ω ( ) pilot = K P e − j ωτ •  Gain = constant" •  Phase angle linear in frequency" •  As input frequency increases, ϕ(ω) eventually > –180° ! But Stability Margins Were Large … How Could CHR = 10? " K P e − j ωτ = K P  K P e − j ωτ ( ) = − j ωτ H s ( ) pilot = Δu s ( ) Δ ε s ( ) = K P e − τ s Inverse Problem of Lateral Control! •  Given a flight path, what is the control history that generates it? " –  Necessary piloting actions " –  Control-law design" •  Aileron-rudder interconnect (ARI) simplifies pilot input" Grumman F-14 Tomcat! Yaw Angle" Roll Angle" Lateral-Stick Command" Angle of attack ( α ) = 10 deg; ARI off" α = 30 deg; ARI off" α = 30 deg; ARI on" Stengel, Broussard, 1978! Flight Control System Design Control System Design Methods! •  Linear-quadratic (LQ) regulator" •  Pole placement" •  Parametric optimization" •  Nonlinear inverse dynamics" •  Neural networks" •  Noisy, incomplete measurements" –  State observer" –  Kalman filter (optimal estimator)" •  Assume Gaussian errors" •  Combine with LQ regulator" •  LQG regulator" •  Control at all points in flight envelope" –  Robustness" –  Gain scheduling" –  Adaptive control" Proportional Stability Augmentation with Command Input! Δu t ( ) = C F Δy C t ( ) − C B Δx t ( ) Section 4.7, Flight Dynamics" dim Δu t ( ) " # $ % = m × 1; dim Δx t ( ) " # $ % = n ×1 dim Δy C t ( ) " # $ % = r × 1, r ≤ m dim C F [ ] = m × r; dim C B [ ] = m × n •  Full state feedback" •  Command = desired output" –  r (≤ m) components" –  Cannot have more independent command inputs, Δy C (t), than independent control inputs, Δu(t)" Proportional Stability Augmentation with Command Input! Δ  x t ( ) = FΔx t ( ) + GΔu t ( ) Δy t ( ) = H x Δx t ( ) ; H u  0 Δu t ( ) = C F Δy C t ( ) − C B Δx t ( ) Δ  x t ( ) = FΔx t ( ) + G C F Δy C t ( ) − C B Δx t ( ) # $ % & = F − GC B Δx t ( ) # $ % & Δx t ( ) + GC F Δy C t ( ) = F CL Δx t ( ) + G CL Δy C t ( ) •  Satisfy flying qualities criteria by adjusting gains of the closed-loop command/stability augmentation system" Section 4.7, Flight Dynamics" Dynamics and Control! Substitute Control in Dynamic Equation! •  Eigenvalues" •  Root loci" •  Transfer functions" •  Bode plots" •  Nichols charts" •  " Next Time: Maneuvering and Aeroelasticity  Reading Flight Dynamics, 681-785 Virtual Textbook, Part 21 Supplemental Material Large Aircraft Flying Qualities" •  High wing loading, W/S" •  Distance from pilot to rotational center" •  Slosh susceptibility of large tanks" •  High wing span -> short relative tail length" –  Higher trim drag" –  Increased yaw due to roll, need for rudder coordination" –  Reduced rudder effect" •  Altitude response during approach" –  Increased non-minimum-phase delay in response to elevator" –  Potential improvement from canard" •  Longitudinal dynamics" –  Phugoid/short-period resonance" •  Rolling response (e.g., time to bank)" •  Reduced static stability" •  Off-axis passenger comfort in BWB turns" [...]... t ) (% 0 ( % Δξ ( t ) '$ & # G ( + % CL ( % I ' $ & ( Δy C ( t ) ( ' •  Satisfy flying qualities criteria by adjusting gains of the closed-loop command/stability augmentation system" New modes of motion in augmented system" Section 4.7, Flight Dynamics" Proportional-Filter Stability Augmentation with Command Input ! Flight Testing Videos ! http://www.youtube.com/watch?v=GXdJxjvQZW4! http://www.youtube.com/watch?v=t6DdlPoPOE4!... ( t ) = C F Δy C ( t ) − C I ∫ # Δy ( t ) − Δy C ( t ) % dt − C B Δx ( t ) $ & { = [ F − GC B ] Δx ( t ) + G C F Δy C ( t ) + C I ∫ # Δy C ( t ) − H x Δx ( t ) % dt $ & Section 4.7, Flight Dynamics" Section 4.7, Flight Dynamics" } Proportional-Integral 
 Command and Stability Augmentation! Proportional-Integral
 Command and Stability Augmentation ! Δy ( t ) = H x Δx ( t ); H u  0 •  Define integral.. .Criteria for Oscillations and Excursions (MIL-F-8785C) " Criteria for Oscillations and Excursions Proportional-Integral Command and Stability Augmentation ! Proportional-Integral Command and Stability Augmentation ! (MIL-F-8785C) ! •  Full state feedback" •  Command = desired output" –  r (≤ m) components" •  Integral compensation eliminates long-term (bias) errors" Dynamics and Control!... http://www.youtube.com/watch?v=GXdJxjvQZW4! http://www.youtube.com/watch?v=t6DdlPoPOE4! http://www.youtube.com/watch?v=j85jlc1Zfk4! Δu ( t ) = + ∫ #C F Δy C ( t ) − C B Δx ( t ) − −C I Δu ( t ) % dt $ & Section 4.7, Flight Dynamics" Eigenvalues" Root loci" Transfer functions" Bode plots" Nichols charts" " . Flying Qualities Criteria Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012 " Copyright 2012 by Robert Stengel. All. workload" 3.  Aircraft can be controlled safely, but workload is excessive" Level of Performance! Flight Phase! Long-Period Flying Qualities Criteria (MIL-F-8785C)! •  Flight path stability. only.! http://www.princeton.edu/~stengel/MAE331.html ! http://www.princeton.edu/~stengel/FlightDynamics.html ! •  MIL-F-8785C criteria! •  CAP, C*, and other longitudinal criteria& quot; •  ϕ/β, ω ϕ /ω  , and other lateral- directional criteria& quot; •  Pilot-vehicle