Aircraft Flight Dynamics Robert F. Stengel Lecture10 Linearized Equations and Modes of Motion

The Mystery Airplane: Convair XF-92A (1948) Linearized Equations 
 and Modes of Motion
 Robert Stengel, Aircraft Flight Dynamics
 MAE 331, 2012 " •  Linearization of nonlinear dynamic models" –  Nominal flight path" –  Perturbations about the nominal flight path" •  Modes of motion" –  Longitudinal" –  Lateral-directional" 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 ! !  !  !  !  Precursor to F-102, F-106, B-58, and F2Y" M = 1.05 in a dive; under-powered, pre-area rule" Landed at θ = 45°, V = 67 mph (108 km/h)" Violent pitchup during high-speed turns, alleviated by wing fences" !  Poor high-speed and good low-speed handling qualities" Nominal and Actual Trajectories " •  Nominal (or reference) trajectory and control history" Nominal and Actual Flight Paths {x N (t), u N (t), w N (t)} x : dynamic state u : control input w : disturbance input for t in [t o ,t f ] •  Actual trajectory perturbed by" € –  Small initial condition variation, Δxo(to)" –  Small control variation, Δu(t)" {x(t), u(t), w(t)} for t in [to ,t f ] = {x N (t) + Δx(t), u N (t) + Δu(t), w N (t) + Δw(t)} Both Paths Satisfy the Dynamic Equations " •  Dynamic models for the actual and the nominal problems are the same" Approximate Neighboring Trajectory as a Linear Perturbation to the Nominal Trajectory "  x N (t) = f[x N (t), u N (t), w N (t),t]   x(t) = x N (t) + Δ (t) = f[x N (t) + Δx(t), u N (t) + Δu(t), w N (t) + Δw(t),t] x  x N (t) = f[x N (t), u N (t), w N (t)], x N ( t o ) given •   x(t) = f[x(t), u(t), w(t)], x ( t o ) given •  Differences in initial condition and forcing " Δx(t o ) = x(t o ) − x N (t o ) # Δu(t) = u(t) − u N (t) % $ % Δw(t) = w(t) − w N (t) & ' % ( in *t o ,t f , + % ) •    x x(t) = x N (t) + Δ (t) perturb rate of change and the state" ≈ f[x N (t), u N (t), w N (t),t] + # x(t) = x N (t) + Δ (t) '  x % %  ( in *t o ,t f , $ + x(t) − x N (t) + Δx(t) % % ) & Linearized Equation Approximates Perturbation Dynamics " •  Solve for the nominal and perturbation trajectories separately" Nominal Equation"  x N (t) = f[x N (t), u N (t), w N (t),t], x N ( t o ) given dim(x) = n × dim(u) = m × dim(w) = s × Perturbation Equation" $ ' $ ' $ ' &∂ f ) &∂ f ) &∂f ) Δw(t)) Δ (t) ≈ & x=x (t ) Δx(t)) + & x=x (t ) Δu(t)) + & x x=x ∂ x u=uN (t ) ∂ u u=uN (t ) ∂ w u=uN (t )) N (t & ) & ) & ) % w=wNN (t ) ( % w=wNN (t ) ( % ( w=w N (t )  F(t)Δx(t)+ G(t)Δu(t)+ L(t)Δw(t), Δx (t o ) given Approximate the new trajectory as the sum of the nominal path plus a linear perturbation" ∂f ∂f ∂f Δx(t) + Δu(t) + Δw(t) ∂x ∂u ∂w Jacobian Matrices Express Solution Sensitivity to Small Perturbations " •  Sensitivity to state perturbations: stability matrix, F, is square" " ∂f ∂ f1 $ 1∂x  ∂ xn $ ∂f F(t) = =$    x=x ∂ x u=uN (t )) $ (t N ∂f ∂ fn w=w N (t ) $ n∂x  ∂ xn # % ' ' dim(F) = n × n ' ' 'x=x (t ) &u=uN (t ) N w=w N (t ) dim(Δx) = n × dim(Δu) = m × dim(w) = s ì ã Sensitivity to control and disturbance perturbations is similar, but matrices, G and L, may not be square" G(t) = dim(G) = n × m ∂f x = x N (t ) ∂ u u= uN (t ) w = w N (t ) L(t) = ∂f ∂w dim(L) = n × s x = x N (t ) u= u N (t ) w = w N (t ) Numerical Integration: " MATLAB Ordinary Differential Equation * Solvers" How Is System Response Calculated?" •  –  Numerical integration ( time domain )" •  Adams-Bashforth-Moulton Algorithm" •  Numerical Differentiation Formula" •  •  Linear and nonlinear, time-varying and timeinvariant dynamic models" Explicit Runge-Kutta Algorithm" Modified Rosenbrock Method" •  Trapezoidal Rule" •  Trapezoidal Rule w/Back Differentiation" •  Linear, time-invariant (LTI) dynamic models" –  Numerical integration ( time domain )" –  State transition ( time domain )" –  Transfer functions ( frequency domain )" * http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/ref/ode23.html.! Shampine, L F and M W Reichelt, "The MATLAB ODE Suite," SIAM Journal on Scientific Computing, Vol 18, 1997, pp 1-22.! MATLAB Simulation of Linear and Nonlinear Dynamic Systems " •  MATLAB Main Script" % Nonlinear and Linear Examples" clear" tspan = [0 10]; " xo = [0, 10];" [t1,x1 = ode23('NonLin',tspan,xo); " xo = [0, 1];" € [t2,x2] = ode23('NonLin',tspan,xo); " xo = [0, 10];" [t3,x3] = "ode23('Lin',tspan,xo);" xo = [0, 1];" [t4,x4] = ode23('Lin',tspan,xo);" " subplot(2,1,1)" plot(t1,x1(:,1),'k',t2,x2(:,1),'b',t3,x3(:,1),'r',t4,x4(:,1),'g')" ylabel('Position'), grid" subplot(2,1,2)" plot(t1,x1(:,2),'k',t2,x2(:,2),'b',t3,x3(:,2),'r',t4,x4(:,2),'g')" € xlabel('Time'), ylabel('Rate'), grid" •  Linear System" ˙ x1 (t) = x (t) ˙ x (t) = −10x1 (t) − x (t) function xdot = Lin(t,x)" % Linear Ordinary Differential Equation" % x(1) = Position" % x(2) = Rate" xdot = [x(2)" -10*x(1) - x(2)];" •  Nonlinear System" ˙ x1 (t) = x (t) ˙ x (t) = −10x1 (t) + 0.8x1 (t) − x (t) function xdot = NonLin(t,x)" % Nonlinear Ordinary Differential Equation" % x(1) = Position" % x(2) = Rate "" xdot = [x(2)" -10*x(1) + 0.8*x(1)^3 - x(2)];" Comparison of Damped Linear and Nonlinear Systems" Linear Spring"  x1 (t) = x2 (t)  x2 (t) = −10x1 (t) − x2 (t) Spring" Spring Force vs Displacement " Displacement" Rate of Change" Damper" Linear plus Stiffening Cubic Spring"  x1 (t) = x2 (t)  x2 (t) = −10x1 (t) −10x13 (t) − x2 (t) Linear plus Weakening Cubic Spring"  x1 (t) = x2 (t)  x2 (t) = −10x1 (t)+ 0.8x13 (t) − x2 (t) Linear and Stiffening Cubic Springs: Small and Large Initial Conditions " •  Linear and Weakening Cubic Springs: Small and Large Initial Conditions" Linear and nonlinear responses are indistinguishable with small initial condition" Stiffening Linear-Cubic Spring Example" !  Nonlinear, time-invariant (NTI) equation" Linear, Time-Varying (LTV) Approximation of Perturbation Dynamics  x1 (t) = f1 = x2 (t)  x2 (t) = f2 = −10x1 (t) − 10x13 (t) − x2 (t) !  Integrate equations to produce nominal path" ! x (0) $ # 1N &⇒ # x2 (0) & " N % ! f ∫# f N # " N tf $ & dt ⇒ & % ! x (t) $ # 1N & in !0,t $ " f% # x2 (t) & " N % !  Analytical evaluation of partial derivatives" ∂ f1 ∂ x1 = 0; ∂ f1 ∂ x2 = ∂ f1 ∂ f2 ∂ f2 ∂ x1 = −10 − 30x1N(t); ∂ x2 = −1 ∂ f2 ∂ u = 0; ∂ u = 0; ∂ f1 ∂ f2 ∂w = ∂w = Nominal (NTI) and Perturbation (LTV) Dynamic Equations " !  Nonlinear, time-invariant (NTI) nominal equation"  x N (t) = f[x N (t)], x N (0) given  x1N (t) = x2 N (t)  x2 N (t) = −10x1N (t) − 10x13N(t) − x2 N (t) Example" ! x (0) $ ! $ # 1N &=# & # x2 (0) & " % " N % !  Perturbations approximated by linear, time-varying (LTV) equation" Δ (t) = F(t)Δx(t), Δx(0) given x " Δ (t) % " x1 $ '=$ $ Δ2 (t) ' $ − (10 + 30x1N(t)) −1 # x & # Example" %" Δx (t) % '$ ' '$ Δx2 (t) ' & &# " Δx (0) % " % $ '=$ ' $ Δx2 (0) ' # & # & Comparison of Approximate and Exact Solutions" Initial Conditions x2 N (0) = Δx2 (0) = x2 N (t)+ Δx2 (t) = 10 x2 (t) = 10 x N (t) Δx(t) x N (t)+ Δx(t) x(t)  x N (t) Δ (t) x  x x N (t) + Δ (t)  x(t) Suppose Nominal Initial Condition is Zero " •  Nominal solution remains at equilibrium"  x N (t) = f[x N (t)], x N (0) = 0, x N (t) = in [ 0, ∞ ] •  Perturbation equation is linear and time-invariant (LTI)" " Δ (t) % " %" Δx1 (t) % x1 '$ $ '=$ ' x $ Δ2 (t) ' $ "−10 − 30 ( )% −1 '$ Δx2 (t) ' # & # & # & &# Separation of the Equations of Motion into Longitudinal and LateralDirectional Sets Grumman F9F" Rigid-Body Equations of Motion (Scalar Notation) " Reorder the State Vector " State Vector" •  •  Rate of change of Translational Velocity " Rate of change of Translational Position "  u = X / m − g sin θ + rv − qw  v = Y / m + g sin φ cosθ − ru + pw  w = Z / m + g cos φ cosθ + qu − pv  x I = ( cosθ cosψ ) u + ( − cos φ sin ψ + sin φ sin θ cosψ ) v + ( sin φ sin ψ + cos φ sin θ cosψ ) w  yI = ( cosθ sin ψ ) u + ( cos φ cosψ + sin φ sin θ sin ψ ) v + ( − sin φ cosψ + cos φ sin θ sin ψ ) w  zI = ( − sin θ ) u + ( sin φ cosθ ) v + ( cos φ cosθ ) w •  Rate of change of Angular Velocity (Ixy = Iyz = 0) " ( { ( ( ( ) }) ( ) ( ) }) ( )  # q = " M − ( I xx − I zz ) pr − I xz p − r $ ÷ I yy % ( { (  r = I xz L + I xx N − I xz I yy − I xx − I zz •  ) ) )r + "I # 2  p = I zz L + I xz N − I xz I yy − I xx − I zz p + " I xz + I zz I zz − I yy $ r q ÷ I xx I zz − I xz # % Rate of change of Angular Position" xz + I xx I xx − I yy $ p q ÷ I xx I zz − I xz %  φ = p + ( q sin φ + r cos φ ) tan θ  θ = q cos φ − r sin φ  ψ = ( q sin φ + r cos φ ) sec θ ! # # # # # # # # # # # # # # # # # # " x1 $ & ! x2 & # & # x3 & # x4 & # & # x5 & # x6 & # &=# x7 & # x8 & # & # x9 & # & # x10 & # x11 & # & # " x12 % & u v w x y z p q r φ θ ψ $ & & & & & & & & & & & & & & & & % ! # # # # # # # # # # # # # # # # # # " x1 $ & ! x2 & # & # x3 & # x4 & # & # x5 & # x6 & # &=# x7 & # x8 & # & # x9 & # & # x10 & # x11 & # & # " x12 & % u v w x y z p q r φ θ ψ $ & & & & & & & & & & & & & & & & % First six elements of the state are longitudinal variables " " Second six elements of the state are lateraldirectional variables " € •  Dynamics of position, velocity, angle, and angular rate in the vertical plane "   x I = ( cosθ cosψ ) u + ( − cos φ sin ψ + sin φ sin θ cosψ ) v + ( sin φ sin ψ + cos φ sin θ cosψ ) w = x3 = f3 ( )  & q = % M − ( I xx − I zz ) pr − I xz p − r ' ÷ I yy (  θ = q cos φ − r sin φ  x Lon (t) = f[x Lon (t), u Lon (t), w Lon (t)] ! # # # # # # $ # &=# & # % # # # # # # # " u w x z q θ v y p r φ ψ $ & & & & & & & & & & & & & & & & % •  Dynamics of position, velocity, angle, and angular rate out of the vertical plane "  = x1 = f1  = x = f2  zI = ( − sin θ ) u + ( sin φ cosθ ) v + ( cos φ cosθ ) w x1 $ & x2 & & x3 & x4 & & x5 & ! x x6 & Lon & =# x7 & # x Lat−Dir " x8 & & x9 & & x10 & x11 & & x12 & %new Lateral-Directional Equations of Motion" Longitudinal Equations of Motion"  u = X / m − g sin θ + rv − qw  w = Z / m + g cos φ cosθ + qu − pv ! # # # # # # # # # # # # # # # # # # "  = x4 = f4  = x = f5  = x = f6  = x7 = f7  v = Y / m + g sin φ cosθ − ru + pw  yI = ( cosθ sin ψ ) u + ( cos φ cosψ + sin φ sin θ sin ψ ) v + ( − sin φ cosψ + cos φ sin θ sin ψ ) w  = x = f8 (  r = (I L + I  = x = f9 { ( N − {I ( I ) )r + %I & ( (I ) }) ( ) ' p} q ) ÷ ( I ( 2  p = I zz L + I xz N − I xz I yy − I xx − I zz p + % I xz + I zz I zz − I yy ' r q ÷ I xx I zz − I xz & ( xz xx xz yy − I xx − I zz  φ = p + ( q sin φ + r cos φ ) tan θ xz + I xx xx − I yy I −I xx zz xz ) )  = x10 = f10  = x11 = f11  ψ = ( q sin φ + r cos φ ) sec θ  = x12 = f12  x LD (t) = f[x LD (t), u LD (t), w LD (t)] Sensitivity to Control Inputs " Sensitivity to Small Motions " •  (12 x 12) stability matrix for the entire system " " ∂f ∂ f1 $ ∂ x1 ∂ x2 $ ∂ f2 $ ∂ f2 ∂ x1 ∂ x2 F(t) = $ $ $ ∂ f12 $ ∂ f12 ∂ x1 ∂ x2 $ # •  % "   ∂u ∂u ∂ x12 ' $ ∂u ∂w ' $ ' $ ∂w   ∂w ∂ f2 ∂u ∂w ∂ x12 ' = $ ' $ ' $   ∂ψ ' $ ∂ψ ∂ f12 ∂u ∂w ∂ x12 ' # & ∂ f1  ∂ u ∂ψ  ∂ w ∂ψ  ∂ψ ∂ψ % ' ' ' ' ' ' ' & •  " $ $ $ u(t) = $ $ $ $ $ # Four (6 x 6) blocks distinguish longitudinal and lateral-directional effects " Effects of longitudinal perturbations on longitudinal motion" " FLon F = $ Lat−Dir $ FLon # Effects of longitudinal perturbations on lateral-directional motion" •  Effects of lateral-directional perturbations on longitudinal motion" Control input vector and perturbation " δ E(t) δT (t) δ F(t) δ A(t) δ R(t) δ SF(t) % ' ' ' ' ' ' ' ' & Throttle, % Flaps, deg or rad Ailerons, deg or rad Rudder, deg or rad Side Force Panels, deg or rad Δδ E(t) ΔδT (t) Δδ F(t) Δδ A(t) Δδ R(t) Δδ SF(t) & ( ( ( ( ( ( ( ( ' Four (6 x 3) blocks distinguish longitudinal and lateral-directional control effects " Effects of longitudinal controls on longitudinal motion" % Lon FLat−Dir ' FLat−Dir ' & # % % % Δu(t) = % % % % % $ Elevator, deg or rad " G Lon G = $ Lat−Dir $ G Lon # Effects of lateral-directional perturbations on lateral-directional motion" Effects of longitudinal controls on lateral-directional motion" Effects of lateral-directional controls on longitudinal motion" % Lon G Lat−Dir ' G Lat−Dir ' & Effects of lateral-directional controls on lateral-directional motion" Restrict the Nominal Flight Path to the Vertical Plane " •  Decoupling Approximation for Small Perturbations from Steady, Level Flight •  Nominal longitudinal equations reduce to" x Lat − DirN =  u N = X / m − g sin θ N − qN wN  wN = Z / m + g cosθ N + qN u N  x I N = ( cosθ N ) u N + ( sin θ N ) wN  zI N = ( − sin θ N ) u N + ( cosθ N ) wN  qN = M I yy  θ N = qN •  BUT, Lateraldirectional perturbations need not be zero" Nominal State Vector"  x Lat − DirN = Nominal lateraldirectional motions are zero" Δ Lat − DirN ≠ x Δx Lat − DirN ≠ ! # # # # # # # # # # # # # # # # # # " x1 $ & x2 & & x3 & x4 & & x5 & x6 & ! x Lon & =# x7 & # x Lat−Dir " x8 & & x9 & & x10 & x11 & & x12 % &N $ & & %N ! # # # # # # # =# # # # # # # # # " uN $ & wN & & xN & zN & & qN & θN & & & & & & & & & % Restrict the Nominal Flight Path to Steady, Level Flight " •  Specify nominal airspeed (VN) and altitude (hN = –zN) " •  Small Perturbation Effects are Uncoupled in Steady, Symmetric, Level Flight " Calculate conditions for trimmed (equilibrium) flight" –  See Flight Dynamics and FLIGHT program for a solution method " = X / m − g sin θ N − qN wN = Z / m + g cosθ N + qN u N " $ $ $ $ $ $ $ # VN = ( cosθ N ) u N + ( sin θ N ) wN = ( − sin θ N ) u N + ( cosθ N ) wN 0= M I yy = qN •  Assume the airplane is symmetric and its nominal path is steady, level flight" –  Small longitudinal and lateral-directional perturbations are approximately uncoupled from each other" –  (12 x 12) system is " Trimmed State Vector is constant " " uTrim % $ ' $ wTrim ' $ ' V (t − t ) =$ N ' $ zN ' $ ' $ θ 'Trim $ θ & Trim # u w x z q % ' ' ' ' ' ' ' ' & •  block diagonal" •  constant, i.e., linear, time-invariant (LTI)" •  decoupled into two separate (6 x 6) systems " " F Lon F =$ $ # 0 FLat−Dir % ' ' & Dynamic Equation" " L Lon L =$ $ # % ' ' & L Lat−Dir Δ Lat − Dir (t) = FLat − Dir Δx Lat − Dir (t) + G Lat − Dir Δu Lat − Dir (t) + L Lat − Dir Δw Lat − Dir (t) x State Vector" Δx Lon G Lat−Dir % ' ' & Dynamic Equation" Δ Lon (t) = FLon Δx Lon (t) + G Lon Δu Lon (t) + L Lon Δw Lon (t) x Δx1 % " Δu % ' ' $ Δx2 ' $ Δw ' ' Δx3 ' = $ Δx ' $ Δz ' Δx4 ' ' $ ' $ Δq ' Δx5 ' $ Δθ ' & # Δx6 ' & Lon (6 x 6) LTI Lateral-Directional Perturbation Model " (6 x 6) LTI Longitudinal" Perturbation Model " " $ $ $ =$ $ $ $ $ # " G Lon G =$ $ # State Vector" Control " Vector" Δu Lon # Δδ T & = % Δδ E ( ( % % Δδ F ( ' $ Disturbance" Vector" Δw Lon " Δuwind $ = $ Δwwind $ Δq wind # % ' ' ' & Δx Lat − Dir # % % % =% % % % % $ Δx1 & # Δv ( % Δx2 ( % Δy ( % Δp Δx3 ( =% Δx4 ( % Δr ( % Δφ Δx5 ( % Δψ ( Δx6 $ ' Lat − Dir % & ( ( ( ( ( ( ( ( ' Control " Vector" $ Δδ A ' Δu Lat − Dir = & Δδ R ) ) & ) & % Δδ SF ( Disturbance" Vector" " Δvwind $ Δw Lon = $ Δpwind $ Δr wind # % ' ' ' & Fourier Transform of a Scalar Variable" •  Transformation from time domain to frequency domain ! ∞ Frequency Domain Description of LTI System Dynamics F [ Δx(t)] = Δx( jω ) = ∫ Δx(t)e − jω t dt, ω = frequency, rad / s −∞ jω : Imaginary operator, rad/s Δx(t) : real variable Δx( jω ) : complex variable = a(ω )+ jb(ω ) = A(ω )e jϕ (ω ) A : amplitude ϕ : phase angle Fourier Transform of a Scalar Variable" Laplace Transform of a Scalar Variable" •  Δx(t) Laplace transformation from time domain to frequency domain ! ∞ L [ Δx(t)] = Δx(s) = ∫ Δx(t)e− st dt s = σ + jω = Laplace (complex) operator, rad/s Δx( jω ) = a(ω ) + jb(ω ) Δx(t) : real variable Δx(s) : complex variable = a(s)+ jb(s) = A(s)e jϕ (s ) Laplace Transformation is a Linear Operation" •  Sum of Laplace transforms! L [ Δx1 (t)+ Δx2 (t)] = L [ Δx1 (t)] + L [ Δx2 (t)] = Δx1 (s)+ Δx2 (s) Laplace Transforms of Vectors and Matrices" •  Laplace transform of a vector variable! " Δx1 (s) % ' $ L [ Δx(t)] = Δx(s) = $ Δx2 (s) ' $ ' & # •  Laplace transform of a matrix variable! •  Multiplication by a constant! ! a11 (s) a12 (s) # L [ A(t)] = A(s) = # a21 (s) a22 (s) # " L [ aΔx(t)] = aL [ Δx(t)] = aΔx(s) $ & & & % •  Laplace transform of a time-derivative! L [ Δ (t)] = sΔx(s) − Δx(0) x Laplace Transform of a Dynamic System" Laplace Transform of a Dynamic System" •  Rearrange Laplace transform of dynamic equation! •  System equation! Δ (t) = F Δx(t) + G Δu(t) + LΔw(t) x •  F to left, I.C to right! dim(Δx) = (n × 1) dim(Δu) = (m × 1) dim(Δw) = (s × 1) •  Laplace transform of system equation! sΔx(s) − Δx(0) = F Δx(s) + GΔ u(s) + LΔw(s) sΔx(s) − FΔ x(s) = Δx(0)+ GΔ u(s)+ LΔw(s) •  Combine terms! [ sI − F] Δx(s) = Δx(0)+ GΔ u(s)+ LΔw(s) •  Multiply both sides by inverse of (sI – F)! Δx(s) = [ sI − F] −1 [Δx(0)+ G Δu(s)+ LΔw(s)] Matrix Inverse Examples" Matrix Inverse" Forward" Inverse" y = Ax; x = A −1y −1 [A] Adj( A ) Adj( A ) = = A det A T = A = a; A = a dim(A) = (2 × 2) −1 dim(x) = dim(y) = (n × 1) dim(A) = (n × n) (n × n) (1 × 1) C ; C = matrix of cofactors det A dim(A) = (1 × 1) T Cofactors are signed minors of A" ijth minor of A is the determinant of A with the ith row and jth column removed" ! a11 a12 # A = # a21 a22 # a a32 " 31 a13 a23 a33 ! (a a − a a ) − (a a − a a ) (a a − a a ) $ 22 33 23 32 21 33 23 31 21 32 22 31 & # # − ( a12 a33 − a13a32 ) ( a11a33 − a13a31 ) − ( a11a32 − a12 a31 ) & & # $ # ( a12 a23 − a13a22 ) − ( a11a23 − a13a21 ) ( a11a22 − a12 a21 ) & & % " −1 &; A = a11a22 a33 + a12 a23a31 + a13a21a32 − a13a22 a31 − a12 a21a33 − a11a23a32 & % dim(A) = (3 ì 3) ã Numerator is a square matrix" •  Denominator is a scalar" ! a11 a12 A=# # a21 a22 " T ! a22 −a21 $ ! a22 −a12 $ & & # # −a21 a11 & $ # −a12 a11 & % =# % " " −1 &; A = a11a22 − a12 a21 a11a22 − a12 a21 & % ! (a a − a a ) − (a a − a a ) (a a − a a ) $ 22 33 23 32 12 33 13 32 12 23 13 22 & # # − ( a21a33 − a23a31 ) ( a11a33 − a13a31 ) − ( a11a23 − a13a21 ) & & # # ( a21a32 − a22 a31 ) − ( a11a32 − a12 a31 ) ( a11a22 − a12 a21 ) & % = " a11a22 a33 + a12 a23a31 + a13a21a32 − a13a22 a31 − a12 a21a33 − a11a23a32 Characteristic Polynomial of a LTI Dynamic System" −1 Δx(s) = [ sI − F ] [ Δx(0) + G Δu(s) + LΔw(s)] •  Inverse of characteristic matrix! Modes of Motion [ sI − F] −1 = Adj ( sI − F ) (n x n) sI − F •  Characteristic polynomial of the system " –  is a scalar" –  defines the system’s modes of motion! sI − F = det ( sI − F ) ≡ Δ(s) = s n + an −1s n −1 + + a1s + a0 Rigid-Body Motion of a Linear, Time-Invariant Aircraft Model " Eigenvalues (or Roots) of a Dynamic System" •  •  •  •  •  Characteristic equation of the system ! Δ(s) = s n + an −1s n −1 + + a1s + a0 = = ( s − λ1 ) ( s − λ2 ) ( ) ( s − λn ) = •  12th-order system of LTI equations" 12 eigenvalues of the stability matrix, F" 12 roots of the characteristic equation" Characteristic equation of the system ! Δ(s) = s12 + a11s11 + + a1s + a0 = where λi are the eigenvalues of F or the roots of the characteristic polynomial! = ( s − λ1 ) ( s − λ2 ) ( ) ( s − λ12 ) = •  Eigenvalues are real or complex numbers that can be plotted in the s plane! •  Real root! Up to 12 modes of motion! λi = σ i In steady, level flight, longitudinal and lateral-directional LTI models are uncoupled! •  Complex roots occur in conjugate pairs! s Plane! λi = σ i + jω i λ * = σ i − jω i i Δ(s) = $( s − λ1 )( s − λ6 )&long $( s − λ1 )( s − λ6 )&lat−dir = % ' % ' Positive real part represents instability" Complex Conjugate Roots Form a Single Oscillatory Mode of Motion " Longitudinal Modes of Motion in Steady, Level Flight " Phugoid Roots" Δ Lon (t) = FLon Δx Lon (t) + G Lon Δu Lon (t) + L Lon Δw Lon (t) x •  roots of the longitudinal characteristic equation! = %s − (σ phugoid & Δ Lon (s) = ( s − λ1 ) ( s − λ2 ) ( ) ( s − λ6 ) = ( )( )( )( = s − λrange s − λheight s − λ phugoid s − λ Real" Real" Complex" )( )( s − λ Complex" Δ Lon (s) = ( s − λran ) s − λhgt s + 2ζ Pω nP s + ω Range" Height" * phugoid phugoid phugoid phugoid * phugoid short period )( s − λ Complex" * short period ) Complex" Phugoid" )( s nP P − jω phugoid )' ( nP nP ω n : Natural frequency, rad/s ζ : Damping ratio, Short Period Roots" •  modes of motion (typical)! ( ) (s − λ ) + jω )'%s − (σ (& = ( s + 2ζ ω s + ω ) (s − λ + 2ζ SPω nSP s + ω Short Period" nSP )=0 (s − λ short period = %s − (σ short period + jω short period & ) (s − λ )'%s − (σ (& * short period ) short period = ( s + 2ζ SPω nSP s + ω nSP ) − jω short period )' ( Longitudinal Modes of Motion " Lateral-Directional Modes of Motion in Steady, Level Flight " •  Eigenvalues determine the damping and natural frequencies of the linear system s modes of motion" •  Longitudinal characteristic equation has eigenvalues" –  eigenvalues normally appear as complex pairs" –  Range and height modes usually inconsequential" λran : range mode ≈ λhgt : height mode ≈ (ζ (ζ SP P , ω nP ) : phugoid mode , ω nSP ) : short - period mode Δ Lat − Dir (t) = FLat − Dir Δx Lat − Dir (t) + G Lat − Dir Δu Lat − Dir (t) + L Lat − Dir Δw Lat − Dir (t) x •  Roots of the lateral-directional characteristic equation! Δ LD (s) = ( s − λ1 ) ( s − λ2 ) ( ) ( s − λ6 ) = ( )( )( ( ) )( = s − λcrossrange s − λheading s − λspiral ( s − λroll ) s − λ Dutch roll s − λ * roll Dutch ) •  modes of motion (typical)! ( ) Δ LD (s) = ( s − λcr ) ( s − λhead ) ( s − λS ) ( s − λ R ) s + 2ζ DRω nDR s + ω nDR = Crossrange" Lateral-Directional Modes of Motion " •  Lateral-directional characteristic equation has eigenvalues" –  eigenvalues normally appear as a complex pair" –  Crossrange and heading modes usually inconsequential" λcr : crossrange mode ≈ λhead : heading mode ≈ λS : spiral mode λ R : roll mode (ζ DR , ω nDR ) : Dutch roll mode Heading" Spiral" Roll" Dutch Roll" Next Time: Longitudinal Dynamics Reading Flight Dynamics, 452-464, 482-486 Virtual Textbook, Part 11 Sensitivity to Small Control and Disturbance Perturbations " F(t) = Supplemental Material ∂f ∂ u x = x N (t )) u= u N (t w = w N (t ) How Do We Calculate the Partial Derivatives? " ∂f ∂x x = x N (t ) u= u N (t ) w = w N (t ) G(t) = ∂f ∂ u x = x N (t )) u= u N (t L(t) = x = x N (t ) u= u N (t ) w = w N (t ) ; G(t) = ∂f ∂w w = w N (t ) x = x N (t ) u= u N (t ) w = w N (t ) •  Numerically" –  First differences in f(x,u,w)" •  Analytically" –  Symbolic evaluation of analytical models of F, G, and L" " ∂ f1 ∂ f1 $ ∂ u1 ∂ u2 $ ∂ f2 $ ∂ f2 ∂ u1 ∂ u2 =$ $ $ ∂ fn $ ∂ fn ∂ u1 ∂ u2 $ # ∂ f1 ∂ f2 ∂ um ∂ um ∂ fn ∂ um ∂f ∂ u x = x N (t )) u= u N (t ; L(t) = ∂f ∂w w = w N (t ) •  Control-effect matrix" G(t) = F(t) = ∂f ∂x x = x N (t ) u= u N (t ) w = w N (t ) •  Disturbance-effect matrix" % ' ' ' ' ' ' ' ' x = x N (t ) & u= uN (t ) L(t) = ∂f ∂w x = x N (t ) u= u N (t ) w = w N (t ) w = w N (t ) " ∂ f1 ∂ f1 $ ∂ w1 ∂ w2 $ ∂ f2 $ ∂ f2 ∂ w1 ∂ w2 =$ $ $ ∂ fn $ ∂ fn ∂ w1 ∂ w2 $ # % ∂ ws ' ' ∂ f2 ' ∂ ws ' ' ' ∂ fn ' ∂ ws ' x = x N (t ) & u= uN (t ) ∂ f1 w = w N (t ) Numerical Estimation of the Jacobian Matrices " # # ( x + Δx ) % % % % x2 % f1 % %  % % % xn $ % % % % # ( x + Δx ) % % % % x2 f2 % %  % F(t) ≈ % % % xn $ % % % % % # ( x + Δx ) % % % % x2 % fn %  % % % % xn $ % % $ & # ( x − Δx ) ( % ( % x2 − f1 % (  ( % ( % xn ' $ 2Δx1 & ( ( ( ( ( ' # x1 % % ( x + Δx2 ) f1 %  % % xn $ & # x1 ( % ( % ( x + Δx2 ) − f1 % (  ( % ( % xn ' $ 2Δx2 & ( ( ( ( ( ' & # ( x − Δx ) ( % ( % x2 − f2 % (  ( % ( % xn ' $ 2Δx1 & ( ( ( ( ( ' # x1 % % ( x + Δx2 ) f2 %  % % xn $ & # x1 ( % ( % ( x + Δx2 ) − f2 % (  ( % ( % xn ' $ 2Δx2 & ( ( ( ( ( '   & # ( x − Δx ) ( % ( % x2 ( − fn %  ( % ( % xn ' $ 2Δx1  & ( ( ( ( ( '   # x1 % x2 % f1 %  % % ( xn + Δxn ) $ & ( ( ( ( ( '    & # x1 ( % x2 ( % − f1 % (  ( % ( % ( xn + Δxn ) ' $ 2Δxn # x1 % x2 % fn %  % % ( x + Δxn ) $ n  & # x1 ( % x2 ( % ( − fn %  ( % ( % ( x + Δxn ) ' $ n 2Δxn & ( ( ( ( ( ' & ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (x=x N (t) 'u=uN (t) w=w N (t) Sensitivity to Disturbance Inputs" •  •  Rectangular (Euler) Integration" Disturbance input vector and perturbation " ! # # # w(t) = # # # # # " •  Integration Algorithms" uw (t) $ & ww (t) & qw (t) & & vw (t) & & pw (t) & rw (t) & % " $ $ $ Δw(t) = $ $ $ $ $ # Axial wind, m / s Normal wind, m / s Pitching wind shear, deg / s or rad / s Lateral wind, m / s Rolling wind shear, deg / s or rad / s Yawing wind shear, deg / s or rad / s Δuw (t) % ' Δww (t) ' Δqw (t) ' ' Δvw (t) ' ' Δpw (t) ' Δrw (t) ' & Four (6 x 3) blocks distinguish longitudinal and lateral-directional effects " Effects of longitudinal disturbances on longitudinal motion" " L Lon L = $ Lat−Dir $ L Lon # Effects of longitudinal disturbances on lateral-directional motion" x(t k ) = x(t k−1 ) + δx(t k−1,t k ) ≈ x(t k−1 ) + f [ x(t k−1 ),u(t k−1 ),w(t k−1 )] δt , δt = t k − t k−1 •  Trapezoidal (modified Euler) Integration (~MATLAB s ode23)" € x(t k ) ≈ x(t k−1 ) + where Effects of lateral-directional disturbances on longitudinal motion" LLon Lat−Dir L Lat−Dir % ' ' & Effects of lateral-directional disturbances on lateral-directional motion" [δx1 + δx ] δx1 = f [ x(t k−1 ),u(t k−1 ),w(t k−1 )] δt δx = f [ x(t k−1 ) + δx1,u(t k ),w(t k )] δt •  See MATLAB manual for descriptions of ode45 and ode15s" € ... Longitudinal Modes of Motion " Lateral-Directional Modes of Motion in Steady, Level Flight " •  Eigenvalues determine the damping and natural frequencies of the linear system s modes of motion" • ... (n × n) (1 × 1) C ; C = matrix of cofactors det A dim(A) = (1 × 1) T Cofactors are signed minors of A" ijth minor of A is the determinant of A with the ith row and jth column removed" ! a11 a12... polynomial of the system " –  is a scalar" –  defines the system’s modes of motion! sI − F = det ( sI − F ) ≡ Δ(s) = s n + an −1s n −1 + + a1s + a0 Rigid-Body Motion of a Linear, Time-Invariant Aircraft