Aircraft Flight Dynamics Robert F. Stengel Lecture19 Advanced LateralDirectional Dynamics

Advanced Problems of Lateral- Directional Dynamics 
 Robert Stengel, Aircraft Flight Dynamics
 MAE 331, 2012" •  Fourth-order dynamics" –  Steady-state response to control" –  Transfer functions" –  Frequency response" –  Root locus analysis of parameter variations " •  Residualization" •  Roll-spiral oscillation" 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 ! Stability-Axis Lateral-Directional Equations" Δ  r(t) Δ  β (t) Δ  p(t) Δ  φ (t) $ % & & & & & ' ( ) ) ) ) ) = N r N β N p 0 −1 Y β V N 0 g V N L r L β L p 0 0 0 1 0 $ % & & & & & & & ' ( ) ) ) ) ) ) ) Δr(t) Δ β (t) Δp(t) Δ φ (t) $ % & & & & & ' ( ) ) ) ) ) + ~ 0 N δ R 0 0 0 Y δ SF V N L δ A ~ 0 0 0 0 0 $ % & & & & & & ' ( ) ) ) ) ) ) Δ δ A Δ δ R Δ δ SF $ % & & & ' ( ) ) ) Δx 1 Δx 2 Δx 3 Δx 4 " # $ $ $ $ $ % & ' ' ' ' ' = Δr Δ β Δp Δ φ " # $ $ $ $ $ % & ' ' ' ' ' = Yaw Rate Perturbation Sideslip Angle Perturbation Roll Rate Perturbation Roll Angle Perturbation " # $ $ $ $ $ % & ' ' ' ' ' Δu 1 Δu 2 " # $ $ % & ' ' = Δ δ A Δ δ R " # $ % & ' = Aileron Perturbation Rudder Perturbation " # $ $ % & ' ' •  With idealized aileron and rudder effects (i.e., N δ A = L δ R = 0) " Lateral-Directional Characteristic Equation" Δ LD (s) = s − λ S ( ) s − λ R ( ) s 2 + 2 ζω n s + ω n 2 ( ) DR •  Typically factors into real spiral and roll roots and an oscillatory pair of Dutch roll roots " Δ LD (s) = s 4 + L p + N r + Y β V N # $ % & ' ( s 3 + N β − L r N p + L p Y β V N + N r Y β V N + L p # $ % & ' ( * + , - . / s 2 + Y β V N L r N p − L p N r ( ) + L β N p − g V N ( ) * + , - . / s + g V N L β N r − L r N β ( ) = s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 = 0 Business Jet Example of Lateral-Directional Characteristic Equation" Δ LD (s) = s − 0.00883 ( ) s + 1.2 ( ) s 2 + 2 0.08 ( ) 1.39 ( ) s + 1.39 2 # $ % & Slightly unstable Spiral! Stable Roll! Lightly damped Dutch roll! Dutch roll! Spiral! Dutch roll! Roll! Steady-State Response Δx S = −F −1 G Δu S Equilibrium Response of" 2 nd -Order Dutch Roll Model" Δr SS Δ β SS # $ % % & ' ( ( = − Y β V N −N β 1 N r # $ % % % & ' ( ( ( Y β V N N r + N β * + , - . / N δ R 0 # $ % % & ' ( ( Δ δ R SS Δr S = − Y β V N N δ R % & ' ( ) * Y β V N N r + N β % & ' ( ) * Δ δ R S Δ β S = − N δ R Y β V N N r + N β % & ' ( ) * Δ δ R S •  Equilibrium response to constant rudder" •  Steady yaw rate and sideslip angle are not zero" •  What is the corresponding ground track of the aircraft [y(t) vs. x(t)]?" Equilibrium Response of Roll-Spiral Model" Δp S = − L δ A L p Δ δ A S Δ φ (t) S = − L δ A L p Δ δ A S dt 0 t ∫ Δp SS Δ φ SS # $ % % & ' ( ( = − L p 0 1 0 # $ % % & ' ( ( −1 L δ A 0 # $ % % & ' ( ( Δ δ A SS L p 0 1 0 ! " # # $ % & & −1 = 0 0 −1 L p ! " # # $ % & & 0 but" •  Steady roll rate proportional to aileron" •  Roll angle, integral of roll rate, continually increases" •  Equilibrium state with constant aileron" Δp S = −L p −1 L δ A Δ δ A S taken alone" Equilibrium Response of 4 th -Order Model" •  Equilibrium state with constant aileron, rudder, and side-force panel deflection" Δr S Δ β S Δp S Δ φ S $ % & & & & & ' ( ) ) ) ) ) = − N r N β N p 0 −1 Y β V N 0 g V N L r L β L p 0 0 0 1 0 $ % & & & & & & & ' ( ) ) ) ) ) ) ) −1 ~ 0 N δ R 0 0 0 Y δ SF V N L δ A ~ 0 0 0 0 0 $ % & & & & & & ' ( ) ) ) ) ) ) Δ δ A S Δ δ R S Δ δ SF S $ % & & & ' ( ) ) ) Equilibrium Response of the 4 th -Order Lateral-Directional Model" Δy S = H x Δx S = −H x F −1 GΔu S •  With H x = Identity matrix " •  Observations" –  Steady-state roll rate is zero" –  Aileron and rudder commands produce steady-state yaw rate, sideslip angle, and roll angle" –  Side force command produces steady-state roll angle but has no effect on steady-state yaw rate or sideslip angle " Δr S Δ β S Δp S Δ φ S $ % & & & & & ' ( ) ) ) ) ) = g V N L δ A N β − g V N L β N δ R 0 g V N L δ A N r g V N L r N δ R 0 0 0 0 N β + N r Y β V N , - . / 0 1 L δ A − L β + L r Y β V N , - . / 0 1 N δ R L r N β − L β N r ( ) Y δ SF V N $ % & & & & & & & & & ' ( ) ) ) ) ) ) ) ) ) g V N L β N r − L r N β ( ) Δ δ A S Δ δ R S Δ δ SF S $ % & & & ' ( ) ) ) Stability and Transient Response 4 th -Order Initial-Condition Responses of Business Jet" •  Initial roll angle and rate have little effect on yaw rate and sideslip angle responses" •  Initial yaw rate and sideslip angle have large effect on roll rate and roll angle responses" Initial ! yaw rate! Initial ! sideslip angle! Initial ! roll rate! Initial ! roll angle! Effects of Variation in Primary Stability Derivatives N β Effect on 4 th -Order Roots! •  Group Δ(s) terms multiplied by N β to form numerator" •  Denominator formed from remaining terms of Δ(s)" Δ LD (s) = d(s) + N β n(s) = 0 kn(s) d(s) = −1 = N β s − z 1 ( ) s − z 2 ( ) s − λ 1 ( ) s − λ 2 ( ) s 2 + 2 ζω n s + ω n 2 ( ) N ! > 0" N ! < 0" •  Positive N Β " –  Increases Dutch roll natural frequency " –  Damping ratio decreases but remains stable" –  Spiral mode drawn toward origin" –  Roll mode unchanged" •  Negative N β destabilizes Dutch roll mode" Root Locus Gain = Directional Stability! Roll! Spiral! Dutch Roll! Dutch Roll! Zero! Zero! N r Effect on 4 th -Order Roots" Δ LD (s) = d(s) + N r n(s) = 0 kn(s) d(s) = −1 = N r s − z 1 ( ) s 2 + 2 µν n s + ν n 2 ( ) s − λ 1 ( ) s − λ 2 ( ) s 2 + 2 ζω n s + ω n 2 ( ) •  Negative N r " –  Increases Dutch roll damping " –  Draws spiral and roll modes together drawn toward origin" •  Positive N r destabilizes Dutch roll mode" N r < 0" N r > 0" Root Locus Gain = Yaw Damping! Roll! Spiral! Zero! Dutch Roll! Dutch Roll! Zero! Zero! L p Effect on 4 th -Order Roots" Δ LD (s) = d(s) + L p n(s) = 0 kn(s) d(s) = −1 = L p s s 2 + 2 µν n s + ν n 2 ( ) s − λ 1 ( ) s − λ 2 ( ) s 2 + 2 ζω n s + ω n 2 ( ) L p < 0" L p > 0" •  Negative L p " –  Decreases roll mode time constant" –  Draws spiral and roll modes together drawn toward origin" •  Positive L p destabilizes roll mode" •  L p has negligible effect on spiral mode" •  Normally negative ; however, can become positive at high angle of attack" Root Locus Gain = Roll Damping! Roll! Spiral!Zero! Dutch Roll & Zero! Dutch Roll & Zero! Coupling Stability Derivatives and Their Effects Dihedral Effect: Roll Acceleration Sensitivity to Sideslip Angle, L β ! L β ≈ C l β ρ V 2 2I xx $ % & ' ( ) Sb C l β ≈ C l β ( ) Wing + C l β ( ) Wing− Fuselage + C l β ( ) Vertical Tail •  Wing, wing-fuselage interference, and vertical tail are principal contributors" Typically < 0 for stability! Dihedral Effect: Roll Acceleration Sensitivity to Sideslip Angle, L β ! L β ≈ C l β ρ V 2 2I xx $ % & ' ( ) Sb •  Dihedral and sweep effect" C l β ( ) Wing = 1 + 2 λ 6 1 + λ ( ) ΓC L α wing + C L tan Λ 1 − M 2 cos 2 Λ ' ( ) * + , •  Tapered, trapezoidal, swept wing" Wing and Tail Location Effects on L β ! •  High/low wing effect" C l β ( ) Wing− Fuselage = 1.2 AR z Wing h + w ( ) b 2 C l β ( ) Vertical Tail ≈ z vt b C Y β ( ) Vertical Tail •  Vertical tail effect" L β Effect on 4 th -Order Roots! •  Negative L β " –  Stabilizes spiral and roll modes but " –  Destabilizes Dutch roll mode" •  Positive L β does the opposite" Root Locus Gain = Dihedral Effect! Δ LD (s) = d(s) + L β g V N − N p ( ) n(s) = 0 n(s) d(s) = −1 = L β g V N − N p ( ) s − z 1 ( ) s − λ S ( ) s − λ R ( ) s 2 + 2 ζω n DR s + ω n DR 2 ( ) L !!! < 0" L ! > 0" Bizjet Example! Δ LD (s) = s − 0.00883 ( ) s + 1.2 ( ) s 2 + 2 0.08 ( ) 1.39 ( ) s + 1.39 2 # $ % & Roll! Spiral!Zero! Dutch Roll! Dutch Roll! Stabilizing Lateral- Directional Motions" •  Provide sufficient L β (–) to stabilize the spiral mode" •  Provide sufficient N r (–) to damp the Dutch roll mode" How can L β and N r be adjusted  artificially  , i.e., by closed-loop control?! Original Root Locus! Increased |N r |! Solar Impulse! Fourth-Order Frequency Response Yaw Rate and Sideslip Angle Frequency Responses of Business Jet" 2 nd -Order Response to Rudder" •  Yawing response to aileron is not negligible" •  Yaw rate response is poorly characterized by the 2 nd -order model below the Dutch roll natural frequency " •  Sideslip angle response is adequately characterized by the 2 nd -order model" 4 th -Order Response to Aileron and Rudder" Δr j ω ( ) Δ δ A j ω ( ) Δ β j ω ( ) Δ δ A j ω ( ) Δr j ω ( ) Δ δ R j ω ( ) Δ β j ω ( ) Δ δ R j ω ( ) Δr j ω ( ) Δ δ R j ω ( ) Δ β j ω ( ) Δ δ R j ω ( ) Roll Rate and Roll Angle Frequency Responses of Business Jet" 2 nd -Order Response to Aileron" •  Roll response to rudder is not negligible" •  Roll rate response is marginally well characterized by the 2 nd -order model" •  Roll angle response is poorly characterized at low frequency by the 2 nd - order model" Δp j ω ( ) Δ δ R j ω ( ) Δ φ j ω ( ) Δ δ R j ω ( ) Δp j ω ( ) Δ δ A j ω ( ) Δ φ j ω ( ) Δ δ A j ω ( ) Δp j ω ( ) Δ δ A j ω ( ) Δ φ j ω ( ) Δ δ A j ω ( ) 4 th -Order Response to Aileron and Rudder" Frequency and Step Responses to Aileron Input" •  Roll rate response is relatively benign" •  Ratio of roll angle to sideslip response is important to the pilot" •  Yaw/sideslip sensitivity in the vicinity of the Dutch roll natural frequency" Δr j ω ( ) Δ δ A j ω ( ) Δ β j ω ( ) Δ δ A j ω ( ) Δp j ω ( ) Δ δ A j ω ( ) Δ φ j ω ( ) Δ δ A j ω ( ) Δv t ( ) Δy t ( ) Δr t ( ) Δp t ( ) Δ ψ t ( ) Δ φ t ( ) Frequency and Step Responses to Rudder Input" •  Lightly damped yaw/sideslip response would be hard to control precisely" •  Yaw response variability near and below the Dutch roll natural frequency" •  Significant roll rate response near the Dutch roll natural frequency" Δr j ω ( ) Δ δ R j ω ( ) Δ β j ω ( ) Δ δ R j ω ( ) Δp j ω ( ) Δ δ R j ω ( ) Δ φ j ω ( ) Δ δ R j ω ( ) Δv t ( ) Δy t ( ) Δr t ( ) Δp t ( ) Δ ψ t ( ) Δ φ t ( ) Order Reduction by Residualization Approximate Low- Order Response" •  Dynamic model order can be reduced when" –  One mode is stable and well-damped, and it and is faster than the other" –  The two modes are coupled" Δ  x fast Δ  x slow " # $ $ % & ' ' = F fast F slow fast F fast slow F slow " # $ $ % & ' ' Δx fast Δx slow " # $ $ % & ' ' + G fast G slow " # $ $ % & ' ' Δu Δ  x f = F f Δx f + F s f Δx s + G f Δu Δ  x s = F f s Δx f + F s Δx s + G s Δu or! Residualization Provides an Approximation for Low-Order Dynamics" •  Assume that fast mode reaches steady state on a time scale that is short compared to the slow mode" Δ  x f ≈ 0 ≈ F f Δx f + F s f Δx s + G f Δu Δ  x s = F f s Δx f + F s Δx s + G s Δu •  Algebraic solution for Δx fast " 0 ≈ F f Δx f + F s f Δx s + G f Δu F f Δx f = −F s f Δx s − G f Δu Δx f = −F f −1 F s f Δx s + G f Δu ( ) •  Substitute quasi-steady Δx fast in differential equation for Δx slow " Δ  x s = −F f s F f −1 F s f Δx s + G f Δu ( ) # $ % & + F s Δx s + G s Δu = F s − F f s F f −1 F s f # $ % & Δx s + G s − F f s F f −1−1 G f # $ % & Δu •  Residualized equation for Δx slow " Δ  x s = F ' s Δx s + G ' s Δu F ' s = F s − F f s F f −1 F s f " # $ % G ' s = G s − F f s F f −1 G f " # $ % where! Residualization" Residualized Roll-Spiral Mode" •  Assume that the Dutch roll mode is stable and faster than the roll mode" •  Calculate effect of the quasi-steady Dutch roll on the roll and spiral modes" Δ  x DR Δ  x RS " # $ $ % & ' ' ≈ 0 Δ  x RS " # $ $ % & ' ' = F DR F RS DR F DR RS F RS " # $ $ % & ' ' Δx DR Δx RS " # $ $ % & ' ' + G DR G RS " # $ $ % & ' ' Δ δ A Δ δ R " # $ % & ' " # $ $ % & ' ' Residualized Roll-Spiral Mode" •  Assume that the Dutch roll mode is stable and faster than the roll mode" •  Calculate effect of the quasi-steady Dutch roll on the roll and spiral modes" Δx DR = −F DR −1 F RS DR Δx RS + G DR Δ δ A Δ δ R $ % & ' ( ) * + , - , . / , 0 , Δ  x RS = F RS Δx RS − F DR RS F DR −1 F RS DR Δx RS + G DR Δ δ A Δ δ R $ % & ' ( ) * + , - , . / , 0 , + G RS Δ δ A Δ δ R $ % & ' ( ) = F' RS Δx RS + G ' RS Δ δ A Δ δ R $ % & ' ( ) Model of the Residualized Roll-Spiral Mode" •  2 nd -order approximation for roll and spiral modes" Δ  p Δ  φ # $ % % & ' ( ( = L p 0 1 0 # $ % % & ' ( ( Δp Δ φ # $ % % & ' ( ( − L r L β 0 0 # $ % % & ' ( ( N r N β −1 Y β V N # $ % % % % & ' ( ( ( ( −1 N p 0 0 g V N # $ % % % % & ' ( ( ( ( Δp Δ φ # $ % % & ' ( ( + N δ A N δ R 0 Y δ R V N # $ % % % & ' ( ( ( Δ δ A Δ δ R # $ % & ' ( , - . . / . . 0 1 . . 2 . . + L δ A L δ R 0 0 # $ % % & ' ( ( Δ δ A Δ δ R # $ % & ' ( Δ  p Δ  φ # $ % % & ' ( ( = L p − N p L r Y β V N + L β + , - . / 0 N β + N r Y β V N + , - . / 0 # $ % % % % & ' ( ( ( ( g V N L r N β − L β N r ( ) N β + N r Y β V N + , - . / 0 # $ % % % % & ' ( ( ( ( 1 0 # $ % % % % % % & ' ( ( ( ( ( ( Δp Δ φ # $ % % & ' ( ( + = f 11 f 12 1 0 # $ % % & ' ( ( Δp Δ φ # $ % % & ' ( ( + Roots of the Residualized Roll-Spiral Mode" sI − F ' RS = s 1 0 0 1 " # $ % & ' − f 11 f 12 1 0 " # $ $ % & ' ' = Δ RS res = s 2 − L p − N p L β + L r Y β / V N N β + N r Y β / V N * + , - . / " # $ $ % & ' ' s + g V N L β N r − L r N β N β + N r Y β / V N * + , - . / = s − λ S ( ) s − λ R ( ) or s 2 + 2 ζω n s + ω n 2 ( ) RS = 0 •  For the business jet model" Δ RS res = s 2 + 1.0894s − 0.0108 = 0 = s − 0.0098 ( ) s + 1.1 ( ) = s − λ S ( ) s − λ R ( ) •  Slightly unstable spiral mode" •  Similar to 4 th -order roll-spiral results" Δ LD (s) = s − 0.00883 ( ) s + 1.2 ( ) s 2 + 2 0.08 ( ) 1.39 ( ) s + 1.39 2 # $ % & Oscillatory Roll-Spiral Mode" Δ RS res = s − λ S ( ) s − λ R ( ) or s 2 + 2 ζω n s + ω n 2 ( ) RS •  The characteristic equation factors into real or complex roots" –  Real roots are roll mode and spiral mode when" L β N r > L r N β and N p L β + L r Y β / V N ( ) 2 g V N L β N r − L r N β ( ) # $ % & ' ( <1 L β N r < L r N β –  Complex roots produce roll-spiral oscillation or lateral phugoid mode when ! Roll-Spiral Oscillation of a Lifting Reentry Vehicle" Next Time: Flying Qualities Criteria  Reading Aircraft Stability and Control, Ch. 21 Virtual Textbook, Part 20 Supplemental Material Equilibrium Response of 4 th -Order Model" •  Equilibrium state with constant aileron and spiral wind perturbations" Δr SS Δ β SS Δp SS Δ φ SS $ % & & & & & ' ( ) ) ) ) ) = a b 0 c d 0 0 0 0 e f g $ % & & & & ' ( ) ) ) ) Δ δ A SS Δ δ R SS Δ δ SF SS $ % & & & ' ( ) ) ) •  Observations" –  Aileron command" –  Rudder command" –  Side-force panel command" –  Steady-state roll rate is zero" –  Steady-state roll angle is bounded ! Effects of Variation in Secondary Stability Derivatives . Advanced Problems of Lateral- Directional Dynamics 
 Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012" •  Fourth-order dynamics& quot; –  Steady-state. only.! http://www.princeton.edu/~stengel/MAE331.html ! http://www.princeton.edu/~stengel/FlightDynamics.html ! Stability-Axis Lateral-Directional Equations" Δ  r(t) Δ  β (t) Δ  p(t) Δ  φ (t) $ % & & & & & ' ( ) ) ) ) ) = N r N β N p 0 −1 Y β V N 0 g V N L r L β L p 0 0. Perturbation " # $ $ % & ' ' •  With idealized aileron and rudder effects (i.e., N δ A = L δ R = 0) " Lateral-Directional Characteristic Equation" Δ LD (s) = s − λ S ( ) s − λ R ( ) s 2 + 2 ζω n s