Linearized Longitudinal Equations of Motion

11/13/18

Linearized Longitudinal Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018 Learning Objectives • 6th-order -> 4th-order -> hybrid equations • Dynamic stability derivatives • Long-period (phugoid) mode • Short-period mode

Reading: Flight Dynamics 452-464, 482-486

6th-Order Longitudinal Equations of Motion Fairchild-Republic A-10 Symmetric aircraft Motions in the vertical plane Flat earth Nonlinear Dynamic Equations State Vector, 6 components

u! = X / m − gsinθ − qw ⎡ Axial Velocity ⎤ ⎡ x ⎤ ⎡ u ⎤ ⎢ ⎥ ⎢ 1 ⎥ w! = Z / m + gcosθ + qu ⎢ ⎥ w ⎢ Vertical Velocity ⎥ ⎢ x2 ⎥ x! = cosθ u + sinθ w ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ I ( ) ( ) ⎢ x ⎥ Range x3 = ⎢ ⎥ ⎢ ⎥ = x ⎢ z ⎥ Lon6 z!I = (−sinθ )u + (cosθ )w ⎢ Altitude(–) ⎥ ⎢ x ⎥ ⎢ ⎥ 4 q M / I q ⎢ Pitch Rate ⎥ ⎢ x ⎥ ! = yy ⎢ ⎥ ⎢ ⎥ ⎢ 5 ⎥ ⎢ ⎥ ⎣ θ ⎦ ⎢ Pitch Angle ⎥ ⎢ x ⎥ θ! = q ⎣ ⎦ ⎣ 6 ⎦

Range has no dynamic effect Altitude effect is minimal (air density variation) 2

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4th-Order Longitudinal Equations of Motion

Nonlinear Dynamic Equations, neglecting range and altitude

u! = f1 = X / m − gsinθ − qw

w! = f2 = Z / m + gcosθ + qu

q! = f3 = M / I yy

! θ = f4 = q State Vector, 4 components ⎡ ⎤ ⎡ ⎤ x1 ⎡ u ⎤ Axial Velocity, m/s ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x2 ⎥ w ⎢ Vertical Velocity, m/s ⎥ = x ⎢ ⎥ = ⎢ ⎥ Lon4 ⎢ ⎥ x3 ⎢ q ⎥ Pitch Rate, rad/s ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x θ Pitch Angle, rad 3 ⎣⎢ 4 ⎦⎥ ⎣ ⎦ ⎣⎢ ⎦⎥

Fourth-Order Hybrid Equations of Motion

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Transform Longitudinal Velocity Components Replace Cartesian body components of velocity by polar inertial components

u! = f1 = X / m − gsinθ − qw

w! = f2 = Z / m + gcosθ + qu

q! = f3 = M / I yy

! θ = f4 = q

⎡ ⎤ ⎡ ⎤ x1 ⎡ u ⎤ Axial Velocity ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x2 ⎥ w ⎢ Vertical Velocity ⎥ = ⎢ ⎥ = ⎢ x ⎥ ⎢ q ⎥ ⎢ Pitch Rate ⎥ ⎢ 3 ⎥ ⎢ ⎥ x ⎢ θ ⎥ Pitch Angle ⎣⎢ 4 ⎦⎥ ⎣ ⎦ ⎣⎢ ⎦⎥ 5

Transform Longitudinal Velocity Components

Replace Cartesian body components of velocity by polar inertial components

V! = f = T cos α + i − D − mgsinγ m 1 ⎣⎡ ( ) ⎦⎤ Hawker P1127 Kestral

γ! = f2 = ⎣⎡T sin(α + i) + L − mgcosγ ⎦⎤ mV

q! = f3 = M / I yy ! θ = f4 = q

⎡ ⎤ ⎡ ⎤ x1 ⎡ V ⎤ Velocity ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ i = Incidence angle of the thrust vector x γ Flight Path Angle ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ with respect to the centerline ⎢ ⎥ = = ⎢ ⎥ x3 ⎢ q ⎥ Pitch Rate ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x θ Pitch Angle ⎣⎢ 4 ⎦⎥ ⎣ ⎦ ⎣⎢ ⎦⎥ 6

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Hybrid Longitudinal Equations of Motion • Replace pitch angle by angle of attack α = θ − γ ! V = f1 = ⎣⎡T cos(α + i) − D − mgsinγ ⎦⎤ m

γ! = f2 = ⎣⎡T sin(α + i) + L − mgcosγ ⎦⎤ mV

q! = f3 = M / I yy ! θ = f4 = q

⎡ ⎤ ⎡ ⎤ x1 ⎡ V ⎤ Velocity ⎢ ⎥ ⎢ ⎥ x ⎢ ⎥ Flight Path Angle ⎢ 2 ⎥ ⎢ γ ⎥ ⎢ ⎥ ⎢ ⎥ = = ⎢ ⎥ x3 ⎢ q ⎥ Pitch Rate ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x θ Pitch Angle 7 ⎣⎢ 4 ⎦⎥ ⎣ ⎦ ⎣⎢ ⎦⎥

Hybrid Longitudinal Equations of Motion • Replace pitch angle by angle of attack α = θ − γ

! V = f1 = ⎣⎡T cos(α + i) − D − mgsinγ ⎦⎤ m

γ! = f2 = ⎣⎡T sin(α + i) + L − mgcosγ ⎦⎤ mV

q! = f3 = M / I yy

! 1 α! = θ −γ! = f4 = q − f2 = q − ⎡T sin(α + i) + L − mgcosγ ⎤ mV ⎣ ⎦

⎡ ⎤ ⎡ ⎤ x1 ⎡ V ⎤ Velocity ⎢ ⎥ ⎢ ⎥ x ⎢ ⎥ Flight Path Angle ⎢ 2 ⎥ ⎢ γ ⎥ ⎢ ⎥ ⎢ ⎥ = = ⎢ ⎥ x3 ⎢ q ⎥ Pitch Rate ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x α Angle of Attack ⎣⎢ 4 ⎦⎥ ⎣ ⎦ ⎣⎢ ⎦⎥ θ = α +γ 8

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Why Transform Equations and State Vector?

⎡ ⎤ ⎡ ⎤ x1 ⎡ V ⎤ Velocity ⎢ ⎥ ⎢ ⎥ x ⎢ ⎥ Flight Path Angle ⎢ 2 ⎥ ⎢ γ ⎥ ⎢ ⎥ ⎢ ⎥ = = ⎢ ⎥ x3 ⎢ q ⎥ Pitch Rate ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x α Angle of Attack ⎣⎢ 4 ⎦⎥ ⎣ ⎦ ⎣⎢ ⎦⎥

Velocity and flight path angle typically have slow variations

Pitch rate and angle of attack typically have quicker variations

Coupling typically small 9

Small Perturbations from Steady Path

x!(t) = x! N (t) + Δx!(t)

≈ f[xN (t),uN (t),wN (t),t] + F(t)Δx(t) + G(t)Δu(t) + L(t)Δw(t)

Steady, Level Flight

x! N (t) ≡ 0 ≈ f[xN (t),uN (t),wN (t),t] Δx!(t) ≈ FΔx(t) + GΔu(t) + LΔw(t) Rates of change are “small” 10

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Nominal Equations of Motion in Equilibrium (Trimmed Condition) x (t) 0 f[x (t),u (t),w (t),t]  N = = N N N

T xT = # V γ 0 α % = constant N $ N N N & T, D, L, and M contain state, control, and disturbance effects

! VN = 0 = f1 = ⎣⎡T cos(α N + i) − D − mgsinγ N ⎦⎤ m

γ!N = 0 = f2 = ⎣⎡T sin(α N + i) + L − mgcosγ N ⎦⎤ mVN

q!N = 0 = f3 = M I yy

1 α! N = 0 = f4 = (0) − ⎣⎡T sin(α N + i) + L − mgcosγ N ⎦⎤ mVN 11

Flight Conditions for Steady, Level Flight Nonlinear longitudinal model

 1 V = f1 = $T cos(α + i) − D − mgsinγ & m % ' 1 γ = f2 = $T sin(α + i) + L − mgcosγ & mV % '

q = f3 = M / I yy  1 α = f4 = θ − γ = q − %$T sin(α + i) + L − mgcosγ '& mV Nonlinear longitudinal model in equilibrium

1 0 = f1 = $T cos(α + i) − D − mgsinγ & m % ' 1 0 = f2 = $T sin(α + i) + L − mgcosγ & mV % '

0 = f3 = M / I yy 1  12 0 = f4 = θ − γ = q − %$T sin(α + i) + L − mgcosγ '& mV

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Numerical Solution for Level Flight Trimmed Condition

• Specify desired altitude and airspeed, hN and VN • Guess starting values for the trim parameters, δT0, δE0, and θ0

• Calculate starting values of f1, f2, and f3

! 1 f1 = V = ⎡T (δT,δ E,θ,h,V )cos(α + i) − D(δT,δ E,θ,h,V )⎤ m ⎣ ⎦ 1 f2 = γ! = ⎣⎡T (δT,δ E,θ,h,V )sin(α + i) + L(δT,δ E,θ,h,V ) − mg⎦⎤ mVN

f3 = q! = M (δT,δ E,θ,h,V ) / I yy

• f1, f2, and f3 = 0 in equilibrium, but not for arbitrary δT0, δE0, and θ0 • Define a scalar, positive-definite trim error cost function, e.g.,

2 2 2 J δT,δE,θ = a f1 +b f2 + c f3 ( ) ( ) ( ) ( ) 13

Minimize the Cost Function with Respect to the Trim Parameters

Error cost is bowl-shaped

2 2 2 J (δT,δE,θ) = a( f1 )+b( f2 )+ c( f3 ) Cost is minimized at bottom of bowl, i.e., when

$ ∂ J ∂ J ∂ J ' & ) = 0 % ∂δT ∂δE ∂θ ( Search to find the minimum value of J

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Example of Search for Trimmed Condition (Fig. 3.6-9, Flight Dynamics)

In MATLAB, use fminsearch or fsolve to find trim settings

(δT*,δE*,θ *) = fminsearch$#J,(δT,δE,θ)&%

Small Perturbations from Steady Path Approximated by Linear Equations

Linearized Equations of Motion

⎡ ΔV! ⎤ ⎡ ΔV ⎤ ⎢ ⎥ ⎢ ⎥ ⎡ ΔδT ⎤ Δγ! Δγ ⎢ ⎥ Δx! = ⎢ ⎥ = F ⎢ ⎥ + G Δδ E +" Lon ⎢ Δq! ⎥ Lon ⎢ Δq ⎥ Lon ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢ " ⎦⎥ Δα! Δα ⎣⎢ ⎦⎥ ⎣ ⎦

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Linearized Equations of Motion

Phugoid (Long-Period) Motion

Short-Period Motion

Approximate Decoupling of Fast and Slow Modes of Motion Hybrid linearized equations allow the two modes to be examined separately

Effects of phugoid Effects of short-period perturbations on perturbations on phugoid motion phugoid motion ⎡ Ph ⎤ FPh FSP FLon = ⎢ ⎥ ⎢ FSP F ⎥ ⎣ Ph SP ⎦ Effects of phugoid Effects of short-period perturbations on perturbations on short- short-period motion period motion

⎡ F small ⎤ ⎡ F 0 ⎤ = ⎢ Ph ⎥ ≈ ⎢ Ph ⎥ small F 0 F ⎣⎢ SP ⎦⎥ ⎣⎢ SP ⎦⎥ 18

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Sensitivity Matrices for Longitudinal LTI Model x (t) F x (t) G u (t) L w (t) Δ Lon = LonΔ Lon + LonΔ Lon + LonΔ Lon

⎡ F F F F ⎤ ⎢ 11 12 13 14 ⎥ ⎢ F21 F22 F23 F24 ⎥ F = Lon ⎢ F F F F ⎥ ⎢ 31 32 33 34 ⎥ F F F F ⎣⎢ 41 42 43 44 ⎦⎥

⎡ G G G ⎤ ⎡ L L ⎤ ⎢ 11 12 13 ⎥ ⎢ 11 12 ⎥ ⎢ G21 G22 G23 ⎥ ⎢ L21 L22 ⎥ G = L = Lon ⎢ G G G ⎥ Lon ⎢ L L ⎥ ⎢ 31 32 33 ⎥ ⎢ 31 32 ⎥ G G G L L ⎣⎢ 41 42 43 ⎦⎥ ⎣⎢ 41 42 ⎦⎥

Dimensional Stability and Control Derivatives

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Dimensional Stability-Derivative Notation § Redefine force and moment symbols as acceleration symbols § Dimensional stability derivatives portray acceleration sensitivities to state perturbations Drag ⇒ D ∝V!, m/s2 mass (m) Lift ⇒ L ∝Vγ!, m/s2 ⇒ L V ∝γ!, rad/s2 mass

Moment 2 ⇒ M ∝ q!, rad/s moment of inertia (I yy ) 21

Dimensional Stability-Derivative Notation

∂D 1 ⎡ ρ V 2 ⎤ F = = −D ! C cosα − C N N S + C cosα − C ρ V S 11 V ⎢( TV N DV ) ( TN N DN ) N N ⎥ ∂V m ⎣ 2 ⎦

Thrust and drag effects are combined and represented by one symbol

2 ∂L Lα 1 ⎡ ρNVN ⎤ F24 = VN = ! CT cosα N + CL S V ⎢( N α ) ⎥ ∂α N mVN ⎣ 2 ⎦

Thrust and lift effects are combined and represented by one symbol

∂M 1 ⎡ ρ V 2 ⎤ F = = M ! C N N Sc 34 α ⎢ mα ⎥ ∂α I yy ⎣ 2 ⎦ 22

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Longitudinal Stability Matrix

Effects of phugoid Effects of short-period perturbations on perturbations on phugoid phugoid motion motion

! −D −gcosγ −D −D $ # V N q α & # L g L L & V sin q α ! Ph $ # γ N & F F VN V VN VN F = # Ph SP & = # N & Lon SP # F F & # MV 0 M q Mα & " Ph SP % # & g * L - # LV q Lα & − − sinγ N ,1− / − # VN V VN VN & " N + . %

Effects of phugoid Effects of short-period perturbations on short- perturbations on period motion short-period motion 23

Comparison of 2nd- and 4th- Order Model Response

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4th-Order Initial-Condition Responses of Business Jet at Two Time Scales Plotted over different periods of time 4 initial conditions [V(0), γ(0), q(0), α(0)]

0 - 100 sec 0 - 6 sec • Reveals Phugoid Mode • Reveals Short-Period Mode

2nd-Order Models of Longitudinal Motion Assume off-diagonal blocks of (4 x 4) stability matrix are negligible

⎡ F ~ 0 ⎤ F = ⎢ Ph ⎥ Lon ~ 0 F ⎣⎢ SP ⎦⎥ Approximate Phugoid Equation

Δx! Ph = ⎡ ⎤ −DV −gcosγ N ⎡ ΔV! ⎤ ⎢ ⎥⎡ ΔV ⎤ ⎢ ⎥ ≈ ⎢ LV g ⎥⎢ ⎥ ⎢ Δγ! ⎥ sinγ N ⎢ Δγ ⎥ ⎣ ⎦ ⎢ VN V ⎥⎣ ⎦ ⎣ N ⎦ ⎡ T ⎤ ⎡ −D ⎤ ⎢ δT ⎥ ⎢ V ⎥ + ΔδT + ΔV ⎢ LδT ⎥ ⎢ LV ⎥ wind ⎢ VN ⎥ ⎢ VN ⎥ 26 ⎣ ⎦ ⎣ ⎦

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2nd-Order Models of Longitudinal Motion

⎡ F ~ 0 ⎤ F = ⎢ Ph ⎥ Lon ~ 0 F ⎣⎢ SP ⎦⎥ Approximate Short-Period Equation

Δx! SP = ⎡ ⎤ M q Mα ⎡ Δq! ⎤ ⎢ ⎥⎡ Δq ⎤ ≈ ⎢ ⎥ ⎢ ⎛ Lq ⎞ L ⎥⎢ ⎥ ⎢ Δα! ⎥ ⎜1− ⎟ − α ⎢ Δα ⎥ ⎣ ⎦ ⎢ ⎝ VN ⎠ VN ⎥⎣ ⎦ ⎣ ⎦ ⎡ M ⎤ ⎡ M ⎤ ⎢ δ E ⎥ ⎢ α ⎥ + Δδ E + Δα ⎢ −Lδ E ⎥ ⎢ −Lα ⎥ wind ⎢ VN ⎥ ⎢ VN ⎥ 27 ⎣ ⎦ ⎣ ⎦

Comparison of Bizjet 4th- and 2nd-Order Model Responses

Phugoid Time Scale, ~100 s

4th Order, 4 initial conditions [V(0), γ(0), q(0), α(0)]

2nd Order, 2 initial conditions [V(0), γ(0)]

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Comparison of Bizjet 4th- and 2nd-Order Model Responses

Short-Period Time Scale, ~10 s

4th Order, 4 initial conditions [V(0), γ(0), q(0), α(0)]

2nd Order, 2 initial conditions [q(0), α(0)]

Approximate Phugoid Response to a 10% Thrust Increase

What is the steady-state response? 30

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Approximate Short-Period Response to a 0.1-Rad Pitch Control Step Input

Pitch Rate, rad/s Angle of Attack, rad

What is the steady-state response? 31

Normal Load Factor Response to a 0.1-Rad Pitch Control Step Input

Grumman X-29 • Normal load factor at the center of mass

VN VN ⎛ Lα Lδ E ⎞ Δnz = (Δα! − Δq) = Δα + Δδ E g g ⎜ V V ⎟ ⎝ N N ⎠ • Pilot focuses on normal load factor during rapid maneuvering

Normal Load Factor, gs at c.m. Normal Load Factor, gs at c.m. Aft Pitch Control (Elevator) Forward Pitch Control (Canard)

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Curtiss Autocar, 1917 Waterman Arrowbile, 1935

Stout Skycar, 1931 ConsolidatedVultee 111, 1940s

ConvAIRCAR 116 Taylor AirCar, 1950s (w/Crosley auto), 1940s

Hallock Road Wing , 1957

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Mitzar SkyMaster Pinto, 1970s Lotus Elise Aerocar, concept, 2002

Pinto separated from the airframe. Two killed. Proposed, not built.

Haynes Skyblazer, concept, 2004 Aeromobil, 2014

Proposed, not built.

Aircraft entered a spin, ballistic 35 parachute was deployed. No fatality.

Terrafugia Transition Terrafugia TF-X, concept

[2019, $400K (est)] Date? Cost? … or, now, for the same price Jaguar F Type Tecnam Astore PLUS

plus $120K in savings account 36

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Next Time: Lateral-Directional Dynamics Reading: Flight Dynamics 574-591

Learning Objectives

• 6th-order -> 4th-order -> hybrid equations • Dynamic stability derivatives • Dutch roll mode • Roll and spiral modes

Supplemental Material

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Elements of the Stability Matrix Stability derivatives portray acceleration sensitivities to state perturbations ∂ f ∂ f ∂ f ∂ f 1 ≡ −D ; 1 = −gcosγ ; 1 ≡ −D ; 1 ≡ −D ∂V V ∂γ N ∂q q ∂α α

f f g f f ∂ 2 LV ∂ 2 ∂ 2 Lq ∂ 2 Lα ≡ ; = sinγ N ; ≡ ; ≡ VN VN VN ∂V ∂γ VN ∂q ∂α

∂ f ∂ f ∂ f ∂ f 3 ≡ M ; 3 = 0; 3 ≡ M ; 3 ≡ M ∂V V ∂γ ∂q q ∂α α

f f g f f ∂ 4 LV ∂ 4 ∂ 4 Lq ∂ 4 Lα ≡ − ; = − sinγ N ; ≡ 1− ; ≡ − VN VN VN ∂V ∂γ VN ∂q ∂α

Velocity Dynamics First row of nonlinear equation 1 V! = f = T cosα − D − mgsinγ 1 m [ ] 1 ⎡ ρV 2 ρV 2 ⎤ = C cosα S − C S − mgsinγ m ⎢ T 2 D 2 ⎥ ⎣ ⎦ First row of linearized equation ! ΔV(t) = [F11ΔV(t) + F12Δγ (t) + F13Δq(t) + F14 Δα(t)]

+[G11Δδ E(t) + G12ΔδT (t) + G13Δδ F(t)] + L ΔV + L Δα [ 11 wind 12 wind ]

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Sensitivity of Velocity Dynamics to State Perturbations

% 2 (  ρV V = '(CT cosα −CD ) S − mgsinγ* m 2 & ) Coefficients in first row of F

1 ⎡ ρ V 2 ⎤ F = C cosα − C N N S + C cosα − C ρ V S 11 ⎢( TV N DV ) ( TN N DN ) N N ⎥ m ⎣ 2 ⎦ ∂C F12 = −gcosγ N C ≡ T TV ∂V 2 1 V ∂CD − ⎡ ρN N ⎤ C ≡ F = C S DV 13 ⎢ Dq ⎥ m ⎣ 2 ⎦ ∂V ∂C C ≡ D Dq ∂q −1 ⎡ ρ V 2 ⎤ F = C sinα + C N N S ∂C 14 ⎢( TN N Dα ) ⎥ D m 2 CD ≡ ⎣ ⎦ α ∂α 41

Sensitivity of Velocity Dynamics to Control and Disturbance Perturbations Coefficients in first rows of G and L

−1 ⎡ ρ V 2 ⎤ G = C N N S 11 ⎢ Dδ E ⎥ m ⎣ 2 ⎦ ∂ f1 L11 = − 1 ⎡ ρ V 2 ⎤ ∂V G = C cosα N N S 12 ⎢ TδT N ⎥ m ⎣ 2 ⎦ ∂ f1 L12 = − 1 ⎡ V 2 ⎤ ∂α − ρN N ∂C G = C S C ≡ T 13 D TδT m ⎢ δ F 2 ⎥ ∂δT ⎣ ⎦ ∂C C ≡ D DδE ∂δE

∂CD CD ≡ δF ∂δF 42

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Flight Path Angle Dynamics Second row of nonlinear equation 1 γ! = f = T sinα + L − mgcosγ 2 mV [ ] 1 ⎡ ρV 2 ρV 2 ⎤ = ⎢CT sinα S + CL S − mgcosγ ⎥ mV ⎣ 2 2 ⎦ Second row of linearized equation

Δγ!(t) = [F21ΔV(t) + F22Δγ (t) + F23Δq(t) + F24 Δα(t)]

+[G21Δδ E(t) + G22ΔδT (t) + G23Δδ F(t)] + L ΔV + L Δα [ 21 wind 22 wind ]

Sensitivity of Flight Path Angle Dynamics to State Perturbations % ρV 2 ( γ = '(CT sinα +CL ) S − mgcosγ* mV 2 & ) Coefficients in second row of F 1 ⎡ ρ V 2 ⎤ F = C sinα + C N N S + C sinα + C ρ V S 21 ⎢( TV N LV ) ( TN N LN ) N N ⎥ mVN ⎣ 2 ⎦ 1 ⎡ ρ V 2 ⎤ − C sinα + C N N S − mgcosγ 2 ⎢( TN N LN ) N ⎥ mVN ⎣ 2 ⎦ ∂C C T TV ≡ F22 = gsinγ N VN ∂V ∂C C L LV ≡ 2 ∂V 1 ⎡ ρNVN ⎤ ∂C F23 = CL S C L ⎢ q ⎥ Lq ≡ mVN ⎣ 2 ⎦ ∂q ∂C C ≡ L 1 ⎡ ρ V 2 ⎤ Lα ∂α F = C cosα + C N N S 24 ⎢( TN N Lα ) ⎥ 44 mVN ⎣ 2 ⎦

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Pitch Rate Dynamics Third row of nonlinear equation

2 M Cm (ρV 2)Sc q = f = = 3 I I yy yy Third row of linearized equation

Δq!(t) = [F31ΔV(t) + F32Δγ (t) + F33Δq(t) + F34 Δα(t)]

+[G31Δδ E(t) + G32ΔδT (t) + G33Δδ F(t)] + L ΔV + L Δα [ 31 wind 32 wind ]

Sensitivity of Pitch Rate Dynamics to State Perturbations

q = C ρV 2 2 Sc I m ( ) yy Coefficients in third row of F

1 ⎡ ρ V 2 ⎤ F = C N N Sc + C ρ V Sc 31 ⎢ mV mN N N ⎥ I yy ⎣ 2 ⎦

F32 = 0

2 C 1 ⎡ ρ V ⎤ ∂ m N N Cm ≡ F33 = Cm Sc V V I ⎢ q 2 ⎥ ∂ yy ⎣ ⎦ ∂C C ≡ m mq ∂q 2 1 ⎡ ρNVN ⎤ ∂Cm F34 = Cm Sc C ≡ ⎢ α ⎥ mα I yy 2 ∂α ⎣ ⎦ 46

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Angle of Attack Dynamics Fourth row of nonlinear equation

 1 α = f4 = θ − γ = q − [T sinα + L − mgcosγ ] mV Fourth row of linearized equation

Δα!(t) = [F41ΔV(t) + F42Δγ (t) + F43Δq(t) + F44 Δα(t)]

+[G41Δδ E(t) + G42ΔδT (t) + G43Δδ F(t)] + L ΔV + L Δα [ 41 wind 42 wind ]

Sensitivity of Angle of Attack Dynamics to State Perturbations  α =θ −γ = q −γ Coefficients in fourth row of F

F41 = −F21 F43 = 1− F23

F42 = −F22 F44 = −F24

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Alternative Approach: Numerical Calculation of the Sensitivity Matrices (“1st Differences”)

⎡ (V + ΔV ) ⎤ ⎡ (V − ΔV ) ⎤ ⎡ V ⎤ ⎡ V ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ γ ⎥ ⎢ γ ⎥ ⎢ (γ + Δγ ) ⎥ ⎢ (γ − Δγ ) ⎥ f1 − f1 f − f ⎢ q ⎥ ⎢ q ⎥ 1 ⎢ q ⎥ 1 ⎢ q ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ α ⎥ ⎢ α ⎥ x=xN (t ) ⎢ α ⎥ ⎢ α ⎥ x=xN (t ) ⎣ ⎦ ⎣ ⎦ u=uN (t ) ⎣ ⎦ ⎣ ⎦ u=uN (t ) ∂ f w w (t ) ∂ f w w (t ) 1 (t) ≈ = N ; 1 (t) ≈ = N ∂V 2ΔV ∂γ 2Δγ

⎡ (V + ΔV ) ⎤ ⎡ (V − ΔV ) ⎤ ⎡ V ⎤ ⎡ V ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ γ ⎥ ⎢ γ ⎥ ⎢ (γ + Δγ ) ⎥ ⎢ (γ − Δγ ) ⎥ f2 − f2 f − f ⎢ q ⎥ ⎢ q ⎥ 2 ⎢ q ⎥ 2 ⎢ q ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ α ⎥ ⎢ α ⎥ x=xN (t ) ⎢ α ⎥ ⎢ α ⎥ x=xN (t ) ⎣ ⎦ ⎣ ⎦ u=uN (t ) ⎣ ⎦ ⎣ ⎦ u=uN (t ) ∂ f w w (t ) ∂ f w w (t ) 2 (t) ≈ = N ; 2 (t) ≈ = N ∂V 2ΔV ∂γ 2Δγ Remaining elements of F(t), G(t), and L(t) calculated accordingly 49

Control and Disturbance Sensitivities in Flight Path Angle, Pitch Rate, and Angle-of-Attack Dynamics

2 2 ∂ f 1 $ ρV ' ∂ f3 1 $ ρVN ' 2 = C N S = C Sc f f LδE & mδE ) ∂ 4 ∂ 2 ∂δE mV & 2 ) ∂δE I 2 = − N % ( yy % ( ∂δE ∂δE 2 2 ∂ f2 1 $ ρVN ' f 1 $ V ' ∂ f4 ∂ f2 = C sinα S ∂ 3 ρ N = − & TδT N ) = C Sc ∂δT mV 2 & mδT ) ∂δT ∂δT N % ( ∂δT I yy % 2 ( 2 ∂ f4 ∂ f2 ∂ f2 1 $ ρVN ' 2 = − = C S ∂ f3 1 $ ρVN ' ∂δF ∂δF & LδF ) = C Sc ∂δF mVN % 2 ( & mδF ) ∂δF I yy % 2 (

∂ f2 ∂ f2 ∂ f3 ∂ f3 ∂ f ∂ f = − = − 4 = 2 ∂Vwind ∂V ∂Vwind ∂V ∂Vwind ∂V

∂ f2 ∂ f2 ∂ f ∂ f ∂ f3 ∂ f2 = − 3 = − 3 = ∂α ∂α ∂α ∂α wind ∂α wind ∂α wind 50

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Velocity-Dependent Derivative Definitions Air compressibility effects are a principal source of velocity dependence

a = Speed of Sound ∂CD ∂CD ∂CD CD ≡ = = a M ∂M ∂ (V / a) ∂V M = Mach number = V a

∂C #1 & C ≡ D = % (C DV ∂V $a ' DM C > 0 C < 0 DM DM ∂C #1 & C ≡ L = % (C C 0 LV V a LM DM ≈ ∂ $ ' ∂C #1 & C ≡ m = % (C mV ∂V $a ' mM 51

Wing Lift and Moment Coefficient Sensitivity to Pitch Rate Straight-wing incompressible flow estimate (Etkin) C = −2C h − 0.75 Lqˆwing Lαwing ( cm ) 2 C = −2C h − 0.5 mqˆwing Lαwing ( cm ) Straight-wing supersonic flow estimate (Etkin)

C = −2C h − 0.5 Lqˆwing Lαwing ( cm )

2 2 C 2C h 0.5 m ˆ = − − L ( cm − ) qwing 3 M 2 −1 αwing Triangular-wing estimate (Bryson, Nielsen)

2π CL = − CL qˆwing 3 αwing π Cm = − qˆwing 3AR 52

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Control- and Disturbance- Effect Matrices • Control-effect derivatives portray acceleration sensitivities to control input perturbations

# −D T −D & % δ E δT δ F ( % Lδ E / VN LδT / VN Lδ F / VN ( G = Lon % M M M ( % δ E δT δ F ( % −L / V −L / V −L / V ( $ δ E N δT N δ F N ' • Disturbance-effect derivatives portray acceleration sensitivities to disturbance input perturbations # −D −D & % Vwind αwind ( L / V L / V % Vwind N αwind N ( L = % ( Lon M M % Vwind αwind ( % ( −LV / VN −Lα / VN $% wind wind '( 53

Primary Longitudinal Stability Derivatives

# 2 & −1 ρVN DV  % CT −CD S + CT −CD ρVN S( m ( V V ) 2 ( N N ) $ '

L 1 " ρV 2 % 1 " ρV 2 % V  C N S + C ρV S − C N S − mg $ LV LN N ' 2 $ LN ' VN mV 2 mV 2 N # & N # &

1 " ρV 2 % 1 # ρV 2 & M = C N Sc M = C N Sc q $ mq ' α % mα ( I yy # 2 & I yy $ 2 '

L 1 # ρV 2 & α  C + C N S % TN Lα ( VN mV ( ) 2 N $ '

Small angle assumptions 54

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Primary Phugoid Control Derivatives

−1 ⎡ ρV 2 ⎤ D ! C N S δT ⎢ TδT ⎥ m ⎣ 2 ⎦ L 1 ⎡ ρV 2 ⎤ δ F ! C N S ⎢ Lδ F ⎥ VN mV 2 N ⎣ ⎦

Primary Short-Period Control Derivatives

⎛ ρ V 2 ⎞ M = C N N Sc δ E mδ E ⎜ ⎟ ⎝ 2I yy ⎠ L ⎛ ρ V 2 ⎞ δ E = C N N S V Lδ E ⎝⎜ 2m ⎠⎟

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Flight Motions

Dornier Do-128D

Dornier Do-128 Short-Period Demonstration http://www.youtube.com/watch?v=3hdLXE0rc9Q

Dornier Do-128 Phugoid Demonstration http://www.youtube.com/watch?v=jzxtpQ30nLg&feature=related 57