Advanced Problems of Lateral-Directional Dynamics

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 & #$ &' # &

1 12/4/18

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

2 12/4/18

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

3 12/4/18

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

4 12/4/18

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

5 12/4/18

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

6 12/4/18

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

7 12/4/18

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

8 12/4/18

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

9 12/4/18

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

10 12/4/18

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

11 12/4/18

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

12 12/4/18

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 ⎥ ⎣ ⎦⎣ ⎦

13 12/4/18

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

14 12/4/18

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

15 12/4/18

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 )

16 12/4/18

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 )

17 12/4/18

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

18 12/4/18

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