Aircraft Flight Dynamics Robert F. Stengel Lecture12 LateralDirectional Dynamics

Linearized Lateral-Directional Equations of Motion 
 Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012" •  Spiral, Dutch roll, and roll modes" •  Stability derivatives" 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 ! 6-Component " Lateral-Directional Equations of Motion" State Vector, 6 components! Nonlinear Dynamic Equations!  v = Y / m + gsin φ cos θ − ru + pw  y I = cos θ sin ψ ( ) u + cos φ cos ψ + sin φ sin θ sin ψ ( ) v + −sin φ cos ψ + cos φ sin θ sin ψ ( ) w  p = I zz L + I xz N − I xz I yy − I xx − I zz ( ) p + I xz 2 + I zz I zz − I yy ( ) % & ' ( r { } q ( ) ÷ I xx I zz − I xz 2 ( )  r = I xz L + I xx N − I xz I yy − I xx − I zz ( ) r + I xz 2 + I xx I xx − I yy ( ) % & ' ( p { } q ( ) ÷ I xx I zz − I xz 2 ( )  φ = p + q sin φ + r cos φ ( ) tan θ  ψ = qsin φ + r cos φ ( ) sec θ x 1 x 2 x 3 x 4 x 5 x 6 ! " # # # # # # # # $ % & & & & & & & & = x LD 6 = v y p r φ ψ ! " # # # # # # # # $ % & & & & & & & & = Side Velocity Crossrange Body − Axis Roll Rate Body − Axis Yaw Rate Roll Angle about Body x Axis Yaw Angle about Inertial x Axis ! " # # # # # # # # $ % & & & & & & & & Douglas A-4! 4- Component " Lateral-Directional Equations of Motion" State Vector, 4 components! Nonlinear Dynamic Equations, neglecting crossrange and yaw angle!  v = Y / m + g sin φ cos θ − ru + pw  p = I zz L + I xz N − I xz I yy − I xx − I zz ( ) p + I xz 2 + I zz I zz − I yy ( ) $ % & ' r { } q ( ) ÷ I xx I zz − I xz 2 ( )  r = I xz L + I xx N − I xz I yy − I xx − I zz ( ) r + I xz 2 + I xx I xx − I yy ( ) $ % & ' p { } q ( ) ÷ I xx I zz − I xz 2 ( )  φ = p + qsin φ + r cos φ ( ) tan θ x 1 x 2 x 3 x 4 ! " # # # # # $ % & & & & & = x LD 4 = v p r φ ! " # # # # # $ % & & & & & = Side Velocity Body − Axis Roll Rate Body − Axis Yaw Rate Roll Angle about Body x Axis ! " # # # # # $ % & & & & & Eurofighter Typhoon! Lateral-Directional Equations of Motion Assuming Steady, Level Longitudinal Flight" Nonlinear dynamic equations, assuming steady, level, flight (longitudinal variables are constant )!  v = Y B / m + gsin φ cos θ N − ru N + pw N = Y B / m + gsin φ cos α N − ru N + pw N  p = I zz L B + I xz N B ( ) I xx I zz − I xz 2 ( )  r = I xz L B + I xx N B ( ) I xx I zz − I xz 2 ( )  φ = p + r cos φ ( ) tan θ N = p + r cos φ ( ) tan α N q N = 0 γ N = 0 θ N = α N Lockheed F-117! Lateral-Directional Force and Moments" Y B = C Y B 1 2 ρ N V N 2 S; Body − Axis Side Force L B = C l B 1 2 ρ N V N 2 Sb; Body − Axis Rolling Moment N B = C n B 1 2 ρ N V N 2 Sb; Body − Axis Yawing Moment Linearized Equations of Motion Body-Axis Perturbation Equations of Motion" Δ  v(t) Δ  p(t) Δ  r(t) Δ  φ (t) # $ % % % % % & ' ( ( ( ( ( = ∂ f 1 ∂ v ∂ f 1 ∂ p ∂ f 1 ∂ r ∂ f 1 ∂φ ∂ f 2 ∂ v ∂ f 2 ∂ p ∂ f 2 ∂ r ∂ f 2 ∂φ ∂ f 3 ∂ v ∂ f 3 ∂ p ∂ f 3 ∂ r ∂ f 3 ∂φ ∂ f 4 ∂ v ∂ f 4 ∂ p ∂ f 4 ∂ r ∂ f 4 ∂φ # $ % % % % % % % % % % % & ' ( ( ( ( ( ( ( ( ( ( ( Δv(t) Δp(t) Δr(t) Δ φ (t) # $ % % % % % & ' ( ( ( ( ( + Control [ ] + Disturbance [ ] Body-Axis Perturbation Variables" Δu 1 Δu 2 " # $ $ % & ' ' = Δ δ A Δ δ R " # $ % & ' = Aileron Perturbation Rudder Perturbation " # $ $ % & ' ' Δw 1 Δw 2 " # $ $ % & ' ' = Δ δ A Δ δ R " # $ % & ' = Side Wind Perturbation Vortical Wind Perturbation " # $ $ % & ' ' Δv Δp Δr Δ φ # $ % % % % % & ' ( ( ( ( ( = Side Velocity Perturbation Body − Axis Roll Rate Perturbation Body − Axis Yaw Rate Perturbation Roll Angle about Body x Axis Perturbation # $ % % % % % & ' ( ( ( ( ( Linearized Lateral-Directional Response to Yaw Rate Initial Condition" ~Roll-mode response of roll angle! ~Spiral-mode response of crossrange! ~Spiral-mode response of yaw angle! ~Dutch-roll- mode response of side velocity! ~Dutch-roll-mode response of roll and yaw rates! Dimensional Stability-and-Control Derivatives" ∂ f 1 ∂ v ∂ f 1 ∂ p ∂ f 1 ∂ r ∂ f 1 ∂φ ∂ f 2 ∂ v ∂ f 2 ∂ p ∂ f 2 ∂ r ∂ f 2 ∂φ ∂ f 3 ∂ v ∂ f 3 ∂ p ∂ f 3 ∂ r ∂ f 3 ∂φ ∂ f 4 ∂ v ∂ f 4 ∂ p ∂ f 4 ∂ r ∂ f 4 ∂φ # $ % % % % % & ' ( ( ( ( ( Stability Matrix! = Y v Y p + w N ( ) Y r −u N ( ) g cos θ N L v L p L r 0 N v N p N r 0 0 1 tan θ N 0 # $ % % % % % % & ' ( ( ( ( ( ( Dimensional Stability-and- Control Derivatives" ∂ f 1 ∂δ A ∂ f 1 ∂δ R ∂ f 2 ∂δ A ∂ f 2 ∂δ R ∂ f 3 ∂δ A ∂ f 3 ∂δ R ∂ f 4 ∂δ A ∂ f 4 ∂δ R # $ % % % % % & ' ( ( ( ( ( = Y δ A Y δ R L δ A L δ R N δ A N δ R 0 0 # $ % % % % % & ' ( ( ( ( ( ∂ f 1 ∂ v wind ∂ f 1 ∂ p wind ∂ f 2 ∂ v wind ∂ f 2 ∂ p wind ∂ f 3 ∂ v wind ∂ f 3 ∂ p wind ∂ f 4 ∂ v wind ∂ f 4 ∂ p wind " # $ $ $ $ $ % & ' ' ' ' ' = Y v Y p L v L p N v N p 0 0 " # $ $ $ $ $ % & ' ' ' ' ' Control Effect Matrix! Disturbance Effect Matrix! Stability Axes Stability Axes" •  Alternative set of body axes" •  Nominal x axis is offset from the body centerline by the nominal angle of attack, α N " Transformation from Original Body Axes to Stability Axes" H B S = cos α N 0 sin α N 0 1 0 −sin α N 0 cos α N # $ % % % & ' ( ( ( Δu Δv Δw " # $ $ $ % & ' ' ' S = H B S Δu Δv Δw " # $ $ $ % & ' ' ' B Δp Δq Δr " # $ $ $ % & ' ' ' S = H B S Δp Δq Δr " # $ $ $ % & ' ' ' B •  Side velocity (Δv) and pitch rate (Δq) are unchanged by the transformation " Stability-Axis State " •  Rotate body axes to stability axes" Δv(t) Δp(t) Δr(t) Δ φ (t) # $ % % % % % & ' ( ( ( ( ( Body−Axis ⇒  α N ⇒ Δv(t) Δp(t) Δr(t) Δ φ (t) # $ % % % % % & ' ( ( ( ( ( Stability−Axis Stability-Axis State" Δv(t) Δp(t) Δr(t) Δ φ (t) # $ % % % % % & ' ( ( ( ( ( Stability−Axis ⇒ Δ β ≈ Δv V N ⇒ Δ β (t) Δp(t) Δr(t) Δ φ (t) # $ % % % % % & ' ( ( ( ( ( Stability−Axis •  Replace side velocity by sideslip angle" Stability-Axis State" Δ β (t) Δp(t) Δr(t) Δ φ (t) $ % & & & & & ' ( ) ) ) ) ) Stability−Axis ⇒ Δr(t) Δ β (t) Δp(t) Δ φ (t) $ % & & & & & ' ( ) ) ) ) ) Stability−Axis = Stability − Axis Yaw Rate Perturbation Sideslip Angle Perturbation Stability − Axis Roll Rate Perturbation Stability − Axis Roll Angle Perturbation $ % & & & & & ' ( ) ) ) ) ) •  Revise state order" Stability-Axis Lateral-Directional Equations" Δ  r(t) Δ  β (t) Δ  p(t) Δ  φ (t) $ % & & & & & ' ( ) ) ) ) ) S = N r N β N p 0 Y r V N −1 + , - . / 0 Y β V N Y p V N g cos γ N V N L r L β L p 0 tan γ N 0 1 0 $ % & & & & & & & ' ( ) ) ) ) ) ) ) S Δr(t) Δ β (t) Δp(t) Δ φ (t) $ % & & & & & ' ( ) ) ) ) ) S + N δ A N δ R Y δ A V N Y δ R V N L δ A L δ R 0 0 $ % & & & & & & ' ( ) ) ) ) ) ) S Δ δ A(t) Δ δ R(t) $ % & & ' ( ) ) + N β N p Y β V N Y p V N L β L p 0 0 $ % & & & & & & & ' ( ) ) ) ) ) ) ) S Δ β wind Δp wind $ % & & ' ( ) ) Why Modify the Equations?" •  Dutch-roll mode is primarily described by stability-axis yaw rate and sideslip angle" •  Roll and spiral mode are primarily described by stability-axis roll rate and roll angle" •  Linearized equations allow the three modes to be studied" Stable Spiral! Unstable Spiral! Roll! Dutch Roll, top! Dutch Roll, front! Why Modify the Equations?" F LD = F DR F RS DR F DR RS F RS ! " # # $ % & & = F DR small small F RS ! " # # $ % & & ≈ F DR 0 0 F RS ! " # # $ % & & Effects of Dutch roll perturbations on Dutch roll motion" Effects of Dutch roll perturbations on roll-spiral motion" Effects of roll-spiral perturbations on Dutch roll motion" Effects of roll-spiral perturbations on roll-spiral motion" but are the off-diagonal blocks really small?! Dassault Rafale! Stability, Control, and Disturbance Matrices" F LD = F DR F RS DR F DR RS F RS ! " # # $ % & & = N r N β N p 0 Y r V N −1 ) * + , - . Y β V N Y p V N g cos γ N V N L r L β L p 0 tan γ N 0 1 0 ! " # # # # # # # $ % & & & & & & & G LD = N δ A N δ R Y δ A V N Y δ R V N L δ A L δ R 0 0 " # $ $ $ $ $ $ % & ' ' ' ' ' ' L LD = N β N p Y β V N Y p V N L β L p 0 0 " # $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' Δx 1 Δx 2 Δx 3 Δx 4 " # $ $ $ $ $ % & ' ' ' ' ' = Δr Δ β Δp Δ φ " # $ $ $ $ $ % & ' ' ' ' ' Δu 1 Δu 2 " # $ $ % & ' ' = Δ δ A Δ δ R " # $ % & ' Δw 1 Δw 2 " # $ $ % & ' ' = Δ δ A Δ δ R " # $ % & ' Lateral-Directional Stability Derivatives 2 nd -Order Approximate Modes of Lateral- Directional Motion 2 nd -Order Approximations in System Matrices" F LD = F DR 0 0 F RS ! " # # $ % & & = N r N β 0 0 Y r V N −1 ) * + , - . Y β V N 0 0 0 0 L p 0 0 0 1 0 ! " # # # # # # # $ % & & & & & & & G LD = N δ A 0 Y δ A V N 0 0 L δ R 0 0 " # $ $ $ $ $ $ % & ' ' ' ' ' ' L LD = N β 0 Y β V N 0 0 L p 0 0 " # $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' Second-Order Models of Lateral-Directional Motion" •  Approximate Spiral-Roll Equation" •  Approximate Dutch Roll Equation" Δ  x DR = Δ  r Δ  β # $ % % & ' ( ( ≈ N r N β Y r V N −1 + , - . / 0 Y β V N # $ % % % % & ' ( ( ( ( Δr Δ β # $ % % & ' ( ( + N δ R Y δ R V N # $ % % % & ' ( ( ( Δ δ R + N β Y β V N # $ % % % % & ' ( ( ( ( Δ β wind Δ  x RS = Δ  p Δ  φ # $ % % & ' ( ( ≈ L p 0 1 0 # $ % % & ' ( ( Δp Δ φ # $ % % & ' ( ( + L δ A 0 # $ % % & ' ( ( Δ δ A + L p 0 # $ % % & ' ( ( Δp wind Approximate Roll and Spiral Modes" Δ  p Δ  φ # $ % % & ' ( ( = L p 0 1 0 # $ % % & ' ( ( Δp Δ φ # $ % % & ' ( ( + L δ A 0 # $ % % & ' ( ( Δ δ A Δ RS (s) = s s − L p ( ) λ S = 0 λ R = L p •  Characteristic polynomial has real roots" •  Roll rate is damped by L p " •  Roll angle is a pure integral of roll rate" Δp t ( ) Δ φ t ( ) •  Initial condition response" Neutral stability! Generally < 0! Roll Damping Due to Roll Rate, L p ! L p ≈ C l p ρ V N 2 2I xx # $ % & ' ( Sb = C l ˆ p b 2V N # $ % & ' ( ρ V N 2 2I xx # $ % & ' ( Sb = C l ˆ p ρ V N 4 I xx # $ % & ' ( Sb 2 C l ˆ p ( ) Wing = ∂ ΔC l ( ) Wing ∂ ˆ p = − C L α 12 1 + 3 λ 1 + λ & ' ( ) * + •  Wing with taper" •  Thin triangular wing" C l ˆ p ( ) Wing = − π AR 32 C l ˆ p ≈ C l ˆ p ( ) Vertical Tail + C l ˆ p ( ) Horizontal Tail + C l ˆ p ( ) Wing •  Vertical tail, horizontal tail, and wing are principal contributors" < 0 for stability! NACA-TR-1098, 1952! NACA-TR-1052, 1951 ! Roll Damping Due to Roll Rate, L p ! •  Tapered vertical tail" •  Tapered horizontal tail" ˆ p = pb 2V N C l ˆ p ( ) ht = ∂ ΔC l ( ) ht ∂ ˆ p = − C L α ht 12 S ht S % & ' ( ) * 1+ 3 λ 1+ λ % & ' ( ) * •  pb/2V N describes helix angle for a steady roll" C l ˆ p ( ) vt = ∂ ΔC l ( ) vt ∂ ˆ p = − C Y β vt 12 S vt S % & ' ( ) * 1+ 3 λ 1+ λ % & ' ( ) * Approximate Dutch Roll Mode" Δ  r Δ  β # $ % % & ' ( ( = N r N β Y r V N −1 * + , - . / Y β V N # $ % % % % & ' ( ( ( ( Δr Δ β # $ % % & ' ( ( + N δ R Y δ R V N # $ % % % & ' ( ( ( Δ δ R Δ DR (s) = s 2 − N r + Y β V N $ % & ' ( ) s + N β 1− Y r V N ( ) + N r Y β V N * + , - . / ω n DR = N β 1− Y r V N ( ) + N r Y β V N ζ DR = − N r + Y β V N $ % & ' ( ) 2 N β 1− Y r V N ( ) + N r Y β V N ω n DR = N β + N r Y β V N ζ DR = − N r + Y β V N % & ' ( ) * 2 N β + N r Y β V N •  With negligible side-force sensitivity to yaw rate, Y r " •  Characteristic polynomial, natural frequency, and damping ratio" Initial Condition Response of Approximate Dutch Roll Mode" Δr t ( ) Δ β t ( ) Side Force due to Sideslip Angle " Y ≈ ∂C Y ∂ β qS • β = C Y β qS • β C Y β ≈ C Y β ( ) Fuselage + C Y β ( ) Vertical Tail + C Y β ( ) Wing C Y β ( ) Vertical Tail ≈ ∂C Y ∂ β $ % & ' ( ) vt η vt S VerticalTail S C Y β ( ) Fuselage ≈ −2 S Base S ; S B = π d Base 2 4 C Y β ( ) Wing ≈ −C D Parasite, Wing − kΓ 2 η vt = Vertical tail efficiency k = π AR 1+ 1+ AR 2 Γ = Wing dihedral angle, rad •  Fuselage, vertical tail, and wing are main contributors" Yawing Moment due to Sideslip Angle! N ≈ ∂ C n ∂β ρ V 2 2 % & ' ( ) * Sb • β = C n β ρ V 2 2 % & ' ( ) * Sb • β !  Side force contributions times respective moment arms" –  Non-dimensional stability derivative" C n β ≈ C n β ( ) Vertical Tail + C n β ( ) Fuselage + C n β ( ) Wing + C n β ( ) Propeller C n β ( ) Vertical Tail ≈ −C Y β vt η vt S vt l vt Sb  −C Y β vt η vt V VT Vertical tail contribution" V VT = S vt l vt Sb = Vertical Tail Volume Ratio η vt = η elas 1+ ∂σ ∂β ( ) V vt 2 V N 2 % & ' ( ) * Yawing Moment due to Sideslip Angle! l vt  Vertical tail length (+) = distance from center of mass to tail center of pressure = x cm − x cp vt [x is positive forward; both are negative numbers] C n β ( ) Fuselage = −2K Volume Fuselage Sb K = 1− d max Length fuselage " # $ % & ' 1.3 Fuselage contribution" C n β ( ) Wing = 0.75C L N Γ + fcn Λ, AR, λ ( ) C L N 2 Wing (differential lift and induced drag) contribution" Yawing Moment due to Sideslip Angle! Yaw Damping Due to Yaw Rate, N r ! •  Dimensional stability derivative" N r ≈ C n r ρ V N 2 2I zz # $ % & ' ( Sb = C n ˆ r b 2V N # $ % & ' ( ρ V N 2 2I zz # $ % & ' ( Sb = C n ˆ r ρ V N 4 I zz # $ % & ' ( Sb 2 < 0 for stability! •  High wing- sweep angle can lead to N r > 0" Martin Marietta X-24B! Yaw Damping Due to Yaw Rate, N r ! C n ˆ r ≈ C n ˆ r ( ) Vertical Tail + C n ˆ r ( ) Wing C n ˆ r ( ) Wing = k 0 C L 2 + k 1 C D Parasite, Wing k 0 and k 1 are functions of aspect ratio and sweep angle" •  Wing contribution" •  Vertical tail contribution" Δ C n ( ) Vertical Tail = − C n β ( ) Vertical Tail rl vt V N ( ) = − C n β ( ) Vertical Tail l vt b $ % & ' ( ) b V N $ % & ' ( ) r ˆ r = rb 2V N C n ˆ r ( ) vt = ∂ Δ C n ( ) Vertical Tail ∂ rb 2V N ( ) = ∂ Δ C n ( ) Vertical Tail ∂ ˆ r = −2 C n β ( ) Vertical Tail l vt b % & ' ( ) * NACA-TR-1098, 1952! NACA-TR-1052, 1951 ! Comparison of Fourth- and Second-Order Dynamic Models •  2 nd -order-model eigenvalues are close to those of the 4 th -order model" •  Eigenvalue magnitudes of Dutch roll and roll roots are similar" Bizjet Fourth- and Second-Order Models and Eigenvalues " Fourth-Order Model F = G = Eigenvalue Damping Freq. (rad/s) -0.1079 1.9011 0.0566 0 0 -1.1196 0.00883 -1 -0.1567 0 0.0958 0 0 -1.2 0.2501 -2.408 -1.1616 0 2.3106 0 -1.16e-01 + 1.39e+00j 8.32E-02 1.39E+00 0 0 1 0 0 0 -1.16e-01 - 1.39e+00j 8.32E-02 1.39E+00 Dutch Roll Approximation F = G = Eigenvalue Damping Freq. (rad/s) -0.1079 1.9011 -1.1196 -1.32e-01 + 1.38e+00j 9.55E-02 1.38E+00 -1 -0.1567 0 -1.32e-01 - 1.38e+00j 9.55E-02 1.38E+00 Roll-Spiral Approximation F = G = Eigenvalue Damping Freq. (rad/s) -1.1616 0 2.3106 0 1 0 0 -1.16 Unstable! Comparison of Second- and Fourth-Order Initial-Condition Responses of Business Jet" Fourth-Order Response! Second-Order Response! •  Speed and damping of responses is adequately portrayed by 2 nd -order models" •  Roll-spiral modes have little effect on yaw rate and sideslip angle responses" •  Dutch roll mode has large effect on roll rate and roll angle responses" Primary Lateral-Directional Control Derivatives" L δ A = C l δ A ρ V N 2 2I xx # $ % & ' ( Sb N δ R = C n δ R ρ V N 2 2I zz # $ % & ' ( Sb [...]...Next Time: Analysis of Time Response Reading Flight Dynamics, 298-314, 338-342 Virtual Textbook, Part 13 Body-Axis Perturbation Equations of Motion " •  Rolling and yawing motions" # % % % % % $ #  Δv(t) & % Yv ( Δ (t) ( % Lv p =% Δ r(t) ( % N v . Linearized Lateral-Directional Equations of Motion 
 Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012" •  Spiral, Dutch roll, and roll. only.! http://www.princeton.edu/~stengel/MAE331.html ! http://www.princeton.edu/~stengel/FlightDynamics.html ! 6-Component " Lateral-Directional Equations of Motion" State Vector, 6 components! Nonlinear. Axis ! " # # # # # $ % & & & & & Eurofighter Typhoon! Lateral-Directional Equations of Motion Assuming Steady, Level Longitudinal Flight& quot; Nonlinear dynamic equations, assuming steady,