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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 Copyright 2018 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE331.html http://www.princeton.edu/~stengel/FlightDynamics.html 1 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 1 11/13/18 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 4 2 11/13/18 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 3 11/13/18 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 4 11/13/18 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 5 11/13/18 Nominal Equations of Motion in Equilibrium (Trimmed Condition) x N (t) = 0 = f[xN (t),uN (t),wN (t),t] 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 6 11/13/18 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 14 7 11/13/18 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)&% 15 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 ⎦ 16 8 11/13/18 Linearized Equations of Motion Phugoid (Long-Period) Motion Short-Period Motion 17 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 9 11/13/18 Sensitivity Matrices for Longitudinal LTI Model Δx (t) = F Δx (t) + G Δu (t) + L Δw (t) Lon Lon Lon Lon Lon Lon 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 ⎦⎥ 19 Dimensional Stability and Control Derivatives 20 10 11/13/18 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 11 11/13/18 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 24 12 11/13/18 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 25 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 ⎣ ⎦ ⎣ ⎦ 13 11/13/18 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)] 28 14 11/13/18 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)] 29 Approximate Phugoid Response to a 10% Thrust Increase What is the steady-state response? 30 15 11/13/18 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.
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