Advanced Problems of Lateral-Directional Dynamics

Home , Directional stability, Dutch roll

12/4/18

Advanced Problems of Lateral- Directional Dynamics Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018

Learning Objectives

• 4th-order dynamics – Steady-state response to control – Transfer functions – Frequency response – Root locus analysis of parameter variations • Residualization

Flight Dynamics 595-627

Stability-Axis Lateral-Directional Equations

With idealized aileron and rudder effects (i.e., NδA = LδR = 0)

⎡ ⎤ Nr Nβ N p 0 ⎡ Δr!(t) ⎤ ⎢ ⎥⎡ Δr(t) ⎤ ⎡ ~ 0 N ⎤ ⎢ ⎥ Y ⎢ ⎥ δ R ! ⎢ β g ⎥ ⎢ ⎥ ⎢ Δβ(t) ⎥ −1 0 ⎢ Δβ(t) ⎥ 0 0 ⎡ Δδ A ⎤ = ⎢ V V ⎥ + ⎢ ⎥ ⎢ ⎥ ⎢ N N ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ Δp!(t) Δp(t) Lδ A ~ 0 ⎣ Δδ R ⎦ ⎢ ⎥ ⎢ L L L 0 ⎥⎢ ⎥ ⎢ ⎥ ⎢ Δφ!(t) ⎥ r β p Δφ(t) 0 0 ⎣ ⎦ ⎢ ⎥⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎢ 0 0 1 0 ⎥ ⎣ ⎦

" % " % Δx1 " Δr % Yaw Rate Perturbation $ ' $ ' $ ' " % " % $ Δx2 ' Δβ $ Sideslip Angle Perturbation ' Δu1 " Δδ A % Aileron Perturbation = $ ' = $ ' = = $ ' $ ' $ p ' $ ' $ ' Δx3 Δ Roll Rate Perturbation Δu Δδ R Rudder Perturbation $ ' $ ' $ ' #$ 2 &' # & #$ &' $ Δx ' Δφ $ Roll Angle Perturbation ' # 4 & #$ &' # &

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Lateral-Directional Characteristic Equation

4 ⎛ Yβ ⎞ 3 Δ LD (s) = s + ⎜ Lp + Nr + ⎟ s ⎝ VN ⎠ ⎡ Y Y ⎤ N L N L β N ⎛ β L ⎞ s2 + ⎢ β − r p + p V + r ⎜ V + p ⎟ ⎥ ⎣ N ⎝ N ⎠ ⎦

⎡Yβ g ⎤ + (Lr N p − Lp Nr ) + Lβ N p − s ⎣⎢ VN ( VN )⎦⎥ g + Lβ Nr − Lr Nβ VN ( ) 4 3 2 = s + a3s + a2s + a1s + a0 = 0 Typically factors into real spiral and roll roots and an oscillatory pair of Dutch roll roots

Δ (s) = s − λ s − λ s2 + 2ζω s + ω 2 LD ( S )( R )( n n )DR 3

Business Jet Example of Lateral-Directional Characteristic Equation

Dutch roll

Roll Spiral

Dutch roll

2 2 Δ LD (s) = (s − 0.00883)(s + 1.2)$#s + 2(0.08)(1.39)s + 1.39 &% Slightly Stable Lightly damped 4 unstable Spiral Roll Dutch roll

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4th-Order Initial-Condition Responses of Business Jet

Initial yaw rate

Initial sideslip angle

Initial roll rate

Initial roll angle

• Initial roll angle and rate have little effect on yaw rate and sideslip angle responses • Roll angle is slightly divergent (spiral mode) • Initial yaw rate and sideslip angle have large effect on roll rate and roll angle responses 5

Approximate Roll and Spiral Modes

• Roll rate is damped by Lp • Roll angle is a pure integral of roll rate

# & # & Δp # Lp 0 & Δp # L & % ( = % ( % ( + % δ A ( Δδ A % Δφ ( % 1 0 ( % Δφ ( % 0 ( $ ' $ ' $ ' $ ' Characteristic polynomial has real roots (s) s s L Δ RS = ( − p )

Neutral stability λS = 0 λ = L Generally < 0 R p 6

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Approximate Dutch Roll Mode

# N N & # & r β NδR # Δr & % (# Δr & % ( % ( % ( = * Y - Y % (+% YδR (ΔδR % Δβ ( % , r −1/ β (% Δβ ( $ ' $ ' % V ( % +VN . VN ( $ N ' $ '

2 ⎛ Yβ ⎞ ⎡ Yr Yβ ⎤ • Characteristic Δ DR (s) = s − ⎜ Nr + ⎟ s + Nβ 1− + Nr ⎝ VN ⎠ ⎢ ( VN ) VN ⎥ polynomial, ⎣ ⎦ Y Y natural ω = N 1− r + N β nDR β V r V frequency, and ( N ) N

damping ratio ⎛ Yβ ⎞ Yr Yβ ζ DR = −⎜ Nr + ⎟ 2 Nβ 1− + Nr ⎝ VN ⎠ ( VN ) VN

Y • With negligible N N β ωnDR = β + r side-force VN sensitivity to % Y ( Y N β 2 N N β yaw rate, Yr ζ DR = −' r + * β + r 7 & VN ) VN

Effects of Variation in Primary Stability Derivatives

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th Nβ Effect on 4 - Order Roots

• Group Δ(s) terms multiplied by Nβ to form numerator • Denominator formed from remaining terms of Δ(s)

Nβ > 0 Root Locus Gain = Directional Stability Dutch Roll

Zero Δ LD (s) = d(s) + Nβ n(s) = 0 Zero Roll Spiral n(s) Nβ s − z1 s − z2 ( )( ) Dutch Roll k = −1 = 2 2 d(s) (s − λ1 )(s − λ2 )(s + 2ζω ns +ω n ) Nβ < 0

• Positive Nβ – Increases Dutch roll natural frequency – Damping ratio decreases but remains stable – Spiral mode drawn toward origin – Roll mode unchanged • Negative N destabilizes Dutch roll mode β 9

th Nr Effect on 4 - Order Roots

Dutch Roll Root Locus Gain = Yaw Damping Nr < 0 Nr > 0

Zero Δ LD (s) = d(s) + Nrn(s) = 0 Zero 2 2 n(s) Nr (s − z1 )(s + 2µν ns +ν n ) Roll Spiral k = −1 = 2 2 Zero d(s) (s − λ1 )(s − λ2 )(s + 2ζω ns +ω n )

Dutch Roll • Negative Nr – Increases Dutch roll damping – Draws spiral and roll modes together

• Positive Nr destabilizes Dutch roll mode

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th Lp Effect on 4 - Order Roots

Root Locus Gain = Roll Damping Dutch Roll & Zero

Δ LD (s) = d(s) + Lpn(s) = 0 2 2 n(s) Lps(s + 2µν ns +ν n ) Lp < 0 Lp > 0 k = −1 = 2 2 d(s) (s − λ1 )(s − λ2 ) s + 2ζω ns +ω n ( ) Roll Zero Spiral

• Negative Lp – Roll mode time constant – Draws spiral mode toward origin Dutch Roll & Zero • Positive Lp destabilizes roll mode • Lp: negligible effect on Dutch roll mode

• Lp can become positive at high angle of attack 11

Coupling Stability Derivatives

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th Dihedral (Lβ) Effect on 4 -Order Roots

Root Locus Gain = Dihedral Effect Bizjet Example g Δ LD (s) = d(s) + Lβ − N p n(s) = 0 ( VN ) Dutch Roll g L − N s − z n(s) β V p ( 1 ) Lβ< 0 Lβ > 0 k = −1 = ( N ) d(s) s − λ s − λ s2 + 2ζω s +ω 2 ( S )( R )( nDR nDR ) Roll Zero Spiral

Dutch Roll Δ LD (s) = 2 2 (s − 0.00883)(s + 1.2)$#s + 2(0.08)(1.39)s + 1.39 &%

• Negative Lβ – Stabilizes spiral and roll modes but ... – Destabilizes Dutch roll mode

• Positive Lβ does the opposite 13

Stabilizing Lateral- Solar Impulse Directional Motions

• Provide sufficient Lβ (–) to stabilize the spiral mode • Provide sufficient Nr (–) to damp the Dutch roll mode

Original Root Locus Increased |Nr|

How can Lβ and Nr be adjusted artificially, i.e., by closed-loop control? 14

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Oscillatory Roll-Spiral Mode

Δ = s − λ s − λ or s2 + 2ζω s +ω 2 RSres ( S )( R ) ( n n )RS The characteristic equation factors into real or complex roots Real roots are roll mode and spiral mode when

Lβ Nr < Lr Nβ

Complex roots produce roll-spiral oscillation or lateral phugoid mode when

Lβ Nr > Lr Nβ and ⎡ g ⎤ N p ⎢(Lβ + LrYβ /VN ) 2 (Lβ Nr − Lr Nβ )⎥ < 1 ⎣ VN ⎦ 15

Roll-Spiral Oscillation of the M2-F2 Lifting Body Test Vehicle

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4th-Order Frequency Response

Yaw Rate and Sideslip Angle Frequency Responses of Business Jet 4th-Order Response to 2nd-Order Response Aileron and Rudder to Rudder

Δr( jω ) Δr( jω ) Δr( jω ) Δδ A( jω ) Δδ R( jω ) Δδ R( jω )

Δβ ( jω ) Δβ ( jω ) Δβ ( jω ) Δδ R( jω ) Δδ A( jω ) Δδ R( jω )

Yawing response to aileron is not negligible Yaw rate response is poorly characterized by the 2nd-order model below the Dutch roll natural frequency Sideslip angle response is adequately characterized by the 2nd-order model 18

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Roll Rate and Roll Angle Frequency Responses of Business Jet 4th-Order Response to 2nd-Order Response Aileron and Rudder to Aileron

p j Δp( jω ) Δ ( ω ) Δp( jω ) R j Δδ A( jω ) Δδ ( ω ) Δδ A( jω )

Δφ( jω ) Δφ( jω ) Δφ( jω ) Δδ A( jω ) Δδ R( jω ) Δδ A( jω )

Roll response to rudder is not negligible Roll rate response is marginally well characterized by the 2nd-order model Roll angle response is poorly characterized at low frequency by the 2nd-order model 19

Frequency and Step Responses to Aileron Input

Δr( jω ) Δβ ( jω ) Δp( jω ) Δφ( jω ) Δδ A( jω ) Δδ A( jω ) Δδ A( jω ) Δδ A( jω )

Δv(t)

Δy(t)

Δr(t) Δψ (t)

Δp(t)

Δφ(t)

Yaw/sideslip sensitivity in the vicinity Roll rate response is relatively benign of the Dutch roll natural frequency Ratio of roll angle to sideslip response is important to the pilot 20

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Frequency and Step Responses to Rudder Input Δr( jω ) Δβ ( jω ) Δp( jω ) Δφ( jω ) Δδ R( jω ) Δδ R( jω ) Δδ R( jω ) Δδ R( jω )

Δv(t) Δy(t)

Δr(t) Δφ(t) Δp(t) Δψ (t)

Yaw response variability near and below Lightly damped yaw/sideslip response the Dutch roll natural frequency would be hard to control precisely Significant roll rate response near the Dutch roll natural frequency 21

Reduction of Model Order by Residualization

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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

" fast % " Δx fast % Ffast Fslow " Δx fast % " G fast % $ ' = $ ' $ ' + $ ' Δu $ slow ' $ Δx slow ' Ffast Fslow $ Δxslow ' $ Gslow ' # & # & # & # & Express as 2 separate equations

f Δx f = Ff Δx f + Fs Δxs + G f Δu Δx = FsΔx + F Δx + G Δu s f f s s s 23

Approximation for Fast-Mode Response

Assume that fast mode reaches steady state very quickly compared to slow-mode response

f Δx! f ≈ 0 ≈ Ff Δx f + Fs Δxs + G f Δu s Δx! s = Ff Δx f + FsΔxs + GsΔu

Steady-state solution for Δxfast f 0 ≈ Ff Δx f + Fs Δxs + G f Δu f Ff Δx f = −Fs Δxs − G f Δu

−1 f Δx f = −Ff (Fs Δxs +G f Δu) 24

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Adjust Slow-Mode Equation for Fast-Mode Steady State

Substitute quasi-steady Δxfast in differential equation for Δxslow

x Fs ⎡F −1 F f x G u ⎤ F x G u Δ! s = − f ⎣ f ( s Δ s + f Δ )⎦ + sΔ s + sΔ s −1 f s −1−1

= ⎣⎡Fs − Ff Ff Fs ⎦⎤Δxs + ⎣⎡Gs − Ff Ff G f ⎦⎤Δu

Residualized differential equation for Δxslow x F' x G' u Δ s = s Δ s + s Δ where

s −1 f F's = ⎣⎡Fs − Ff Ff Fs ⎦⎤ s −1 G's = ⎣⎡Gs − Ff Ff G f ⎦⎤ 25

Model of the Residualized Roll-Spiral Mode Yawing motion is assumed to be instantaneous compared to rolling motions Residualized roll/spiral equation

⎡ ⎡ ⎛ Y ⎞ ⎤ ⎡ ⎤ ⎤ ⎢ β g ⎥ ⎢ N p Lr + Lβ ⎥ ⎢ Lr Nβ − Lβ Nr ⎥ ⎢ ⎝⎜ V ⎠⎟ V ( ) ⎥ ⎡ Δp! ⎤ ⎢L − N ⎥ ⎢ N ⎥ ⎡ Δp ⎤ ⎢ ⎥ " ⎢ ⎢ p ⎛ Y ⎞ ⎥ ⎢ ⎛ Y ⎞ ⎥ ⎥⎢ ⎥ +# Δφ! ⎢ N + N β N + N β ⎥ Δφ ⎣⎢ ⎦⎥ ⎢ ⎜ β r ⎟ ⎥ ⎢ ⎜ β r ⎟ ⎥ ⎣⎢ ⎦⎥ ⎢ ⎣ ⎝ VN ⎠ ⎦ ⎣ ⎝ VN ⎠ ⎦ ⎥ ⎢ ⎥ ⎣ 1 0 ⎦ ⎡ f f ⎤⎡ Δp ⎤ = ⎢ 11 12 ⎥⎢ ⎥ +# ⎢ 1 0 ⎥⎢ Δφ ⎥ ⎣ ⎦⎣ ⎦

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Roots of the 2nd-Order Residualized Roll-Spiral Mode

⎡ 1 0 ⎤ ⎡ f f ⎤ sI − F' = s − 11 12 = Δ RS ⎢ ⎥ ⎢ ⎥ RSres ⎣ 0 1 ⎦ ⎣⎢ 1 0 ⎦⎥

2 ⎡ ⎛ Lβ + LrYβ /VN ⎞ ⎤ g ⎛ Lβ Nr − Lr Nβ ⎞ = s − ⎢Lp − N p ⎜ ⎟ ⎥s + ⎜ ⎟ N + NrY /VN VN N + NrY /VN ⎣⎢ ⎝ β β ⎠ ⎦⎥ ⎝ β β ⎠ = s − λ s − λ or s2 + 2ζω s +ω 2 = 0 ( S )( R ) ( n n )RS For the business jet model s2 1.0894s 0.0108 0 Δ RSres = + − =

= (s − 0.0098)(s +1.1) = (s − λS )(s − λR ) Slightly unstable spiral mode Similar to 4th-order roll-spiral results

2 2 Δ LD (s) = (s − 0.00883)(s + 1.2)#s + 2(0.08)(1.39)s + 1.39 % $ & 27

Next: Flying Qualities

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Supplemental Material

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 Unstable -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

• 2nd-order-model eigenvalues are close to those of the 4th-order model • Eigenvalue magnitudes of Dutch roll and roll roots are similar 30

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Roll Acceleration Due to Yaw Rate, Lr

⎛ ρV 2 ⎞ L ≈ C N Sb r lr ⎜ ⎟ ⎝ 2I xx ⎠ ⎛ b ⎞ ⎛ ρV 2 ⎞ ⎛ ρV ⎞ = C N Sb = C N Sb2 lrˆ ⎜ ⎟ ⎜ ⎟ lrˆ ⎜ ⎟ ⎝ 2VN ⎠ ⎝ 2I xx ⎠ ⎝ 4I xx ⎠

Root Locus Gain = Roll Due to Yaw Rate Lr < 0 Lr > 0

Δ LD (s) = d(s) + Lr N pn(s) = 0 kn(s) Lr N p (s − z1 )(s − z2 ) = −1 = 2 2 d(s) (s − λ1 )(s − λ2 )(s + 2ζω ns +ω n )

Yaw Acceleration Due to Roll Rate, Np

# ρV 2 & N ≈ C N Sb p np % ( $ 2Izz ' # b & # ρV 2 & # ρV & = C N Sb = C N Sb2 npˆ % ( % ( npˆ % ( $ 2VN ' $ 2Izz ' $ 4I xx '

Np > 0 Np < 0 Root Locus Gain = Yaw due to Roll Rate

Δ LD (s) = d(s) + N pn(s) = 0

kn(s) N ps(s − z1 ) = −1 = 2 2 d(s) (s − λ1 )(s − λ2 )(s + 2ζω ns +ω n )

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Steady-State Response

−1 ΔxS = −F GΔuS

Equilibrium Response of 2nd-Order Dutch Roll Model Equilibrium response to constant rudder

# & Yβ % −Nβ ( % VN ( # & % 1 N (# & ΔrSS $ r ' N % ( = − % δR (ΔδR *Y - SS $% ΔβSS '( β $% 0 '( , Nr + Nβ / +VN .

%Yβ ( ' NδR * &VN ) Δr = − ΔδR S %Y ( S Steady yaw rate and sideslip β N N ' r + β * angle are not zero &VN ) N Δβ = − δR ΔδR S %Y ( S β N N ' r + β * 34 &VN )

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Equilibrium Response of 2nd- Order Roll-Spiral Model

Equilibrium state with constant aileron

−1 # & ΔpSS # Lp 0 & # L & % ( = − % ( % δ A ( Δδ A Δφ SS $% SS '( $% 1 0 '( $% 0 '(

but ! 0 0 $ −1 # & ! $ −1 Lp Lp 0 "# %& # & = 0 "# 1 0 %&

taken −1 alone ΔpS = −Lp LδAΔδAS

Lδ A ΔpS = − Δδ AS • Steady roll rate Lp proportional to aileron t L • Roll angle, integral of roll Δφ(t) = − δ A Δδ A dt S ∫ S rate, continually increases 0 Lp 35

Equilibrium Response of 4th-Order Model Equilibrium state with constant aileron and rudder deflection

−1 ⎡ N N N 0 ⎤ ⎡ ⎤ r β p ΔrS ⎢ ⎥ ⎡ ~ 0 N ⎤ ⎢ ⎥ Y δ R ⎢ β g ⎥ ⎢ ⎥⎡ ⎤ ⎢ ΔβS ⎥ −1 0 0 0 Δδ AS = − ⎢ V V ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ N N ⎥ ΔpS ⎢ Lδ A ~ 0 ⎥⎢ Δδ RS ⎥ ⎢ ⎥ ⎢ L L L 0 ⎥ ⎢ ⎥⎣ ⎦ Δφ r β p 0 0 ⎣⎢ S ⎦⎥ ⎢ ⎥ ⎣⎢ ⎦⎥ ⎣⎢ 0 0 1 0 ⎦⎥

⎡ g g ⎤ ⎢ Lδ A Nβ − Lβ Nδ R ⎥ V V ⎢ N N ⎥ ⎢ g g ⎥ ⎢ Lδ A Nr Lr Nδ R ⎥ V V ⎢ N N ⎥ ⎢ 0 0 ⎥ ⎢ ⎥ ⎡ ⎤ Y Y ΔrS ⎢ ⎛ β ⎞ ⎛ β ⎞ ⎥ ⎢ ⎥ Nβ + Nr Lδ A − Lβ + Lr Nδ R ⎢ ⎝⎜ V ⎠⎟ ⎝⎜ V ⎠⎟ ⎥ ⎡ ⎤ ⎢ ΔβS ⎥ N N Δδ AS = ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ g ΔpS ⎢ Δδ RS ⎥ ⎢ ⎥ (Lβ Nr − Lr Nβ ) ⎣ ⎦ Δφ VN ⎣⎢ S ⎦⎥ 36

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NTSB Simulation of American Flight 587 Flight simulation derived from digital flight data recorder (DFDR) tape