Linearized Longitudinal Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 018 Learning Objectives 6 th -order -> 4 th -order -> hybrid equations Dynamic stability derivatives Long-period phugoid) mode Short-period mode Reading: Flight Dynamics 45-464, 48-486 Copyright 018 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 Fairchild-Republic A-10 Nonlinear Dynamic Equations!u = X / m gsinθ qw!w = Z / m + gcosθ + qu!x I = cosθ )u + sinθ )w!z I = sinθ )u + cosθ )w!q = M / I yy!θ = q 6 th -Order Longitudinal Equations of Motion Symmetric aircraft Motions in the vertical plane Flat earth State Vector, 6 components u Axial Velocity w Vertical Velocity x Range z = Altitude ) q Pitch Rate θ Pitch Angle x 1 x x 3 x 4 x 5 x 6 = x Lon6 Range has no dynamic effect Altitude effect is minimal air density variation) 1
4 th -Order Longitudinal Equations of Motion Nonlinear Dynamic Equations, neglecting range and altitude!u = f 1 = X / m gsinθ qw!w = f = Z / m + gcosθ + qu!q = f 3 = M / I yy!θ = f 4 = q State Vector, 4 components x 1 x x 3 x 4 = x Lon4 u w = q θ Axial Velocity, m/s Vertical Velocity, m/s Pitch Rate, rad/s Pitch Angle, rad 3 Fourth-Order Hybrid Equations of Motion 4
Transform Longitudinal Velocity Components Replace Cartesian body components of velocity by polar inertial components!u = f 1 = X / m gsinθ qw!w = f = Z / m + gcosθ + qu!q = f 3 = M / I yy!θ = f 4 = q x 1 x x 3 x 4 u Axial Velocity w = Vertical Velocity = q Pitch Rate θ Pitch Angle 5 Transform Longitudinal Velocity Components Replace Cartesian body components of velocity by polar inertial components!v = f 1 = T cos α + i! γ = f = T sin α + i ) D mgsinγ ) + L mgcosγ m mv Hawker P117 Kestral!q = f 3 = M / I yy!θ = f 4 = q x 1 x x 3 x 4 V Velocity γ = Flight Path Angle = q Pitch Rate θ Pitch Angle i = Incidence angle of the thrust vector with respect to the centerline 6 3
Hybrid Longitudinal Equations of Motion Replace pitch angle by angle of attack α = θ γ!v = f 1 = T cos α + i! γ = f = T sin α + i!q = f 3 = M / I yy!θ = f 4 = q ) D mgsinγ ) + L mgcosγ m mv x 1 x x 3 x 4 V Velocity γ = Flight Path Angle = q Pitch Rate θ Pitch Angle 7 Hybrid Longitudinal Equations of Motion Replace pitch angle by angle of attack α = θ γ!v = f 1 = T cos α + i! γ = f = T sin α + i!q = f 3 = M / I yy ) D mgsinγ ) + L mgcosγ m mv!α =!θ! γ = f 4 = q f = q 1 mv T sin α + i ) + L mgcosγ x 1 x x 3 x 4 V Velocity γ = Flight Path Angle = q Pitch Rate α Angle of Attack θ = α +γ 8 4
Why Transform Equations and State Vector? x 1 x x 3 x 4 V γ = = q α Velocity Flight Path Angle Pitch Rate Angle of Attack 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!xt) =!x N t) + Δ!xt) f[x N t),u N t),w N t),t] + F t)δxt) + G t)δut) + L t)δwt) Steady, Level Flight!x N t) 0 f[x N t),u N t),w N t),t] Δ!xt) FΔxt) + GΔut) + LΔwt) Rates of change are small 10 5
Nominal Equations of Motion in Equilibrium Trimmed Condition) x N t) = 0 = f[x N t),u N t),w N t),t] x T N = # γ N 0 α $ N T % = constant T, D, L, and M contain state, control, and disturbance effects! = 0 = f 1 = T cos α N + i! γ N = 0 = f = T sin α N + i ) D mgsinγ N ) + L mgcosγ N m m!q N = 0 = f 3 = M I yy!α N = 0 = f 4 = 0) 1 m ) + L mgcosγ N T sin α N + i 11 Flight Conditions for Steady, Level Flight Nonlinear longitudinal model V = f 1 = 1 m $ % T cos α + i) D mgsinγ ' γ = f = 1 mv $ % T sin α + i) + L mgcosγ ' q = f 3 = M / I yy α = f 4 = θ γ = q 1 mv $ % T sin α + i ) + L mg cosγ ' Nonlinear longitudinal model in equilibrium 0 = f 1 = 1 m $ % T cos α + i) D mgsinγ ' 0 = f = 1 mv $ % T sin α + i ) + L mg cosγ ' 0 = f 3 = M / I yy 0 = f 4 = θ γ = q 1 mv $ % T sin α + i ) + L mg cosγ ' 1 6
Numerical Solution for Level Flight Trimmed Condition Specify desired altitude and airspeed, h N and Guess starting values for the trim parameters, δt 0, δe 0, and θ 0 Calculate starting values of f 1, f, and f 3 f 1 =!V = 1 T δt,δ E,θ,h,V m )cos α + i f =! γ = 1 mv T δt,δ E,θ,h,V )sin α + i N f 3 =!q = M δt,δ E,θ,h,V ) / I yy ) D δt,δ E,θ,h,V ) ) + L δt,δ E,θ,h,V ) mg f 1, f, and f 3 = 0 in equilibrium, but not for arbitrary δt 0, δe 0, and θ 0 Define a scalar, positive-definite trim error cost function, e.g., J δt,δe,θ) = a f 1 )+ b f )+ c f 3 ) 13 Minimize the Cost Function with Respect to the Trim Parameters Error cost is bowl-shaped J δt,δe,θ) = a f 1 )+ b f )+ c f 3 ) Cost is minimized at bottom of bowl, i.e., when $ % J δt J δe J θ ' ) = 0 Search to find the minimum value of J 14 7
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 Δ!x Lon = Δ!V Δ! γ Δ!q Δ!α = F Lon ΔV Δγ Δq Δα ΔδT + G Lon Δδ E " +" 16 8
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 perturbations on phugoid motion F Lon = Effects of phugoid perturbations on short-period motion F Ph small F Ph SP F Ph small F SP Effects of short-period perturbations on phugoid motion F SP Ph F SP Effects of short-period perturbations on shortperiod motion F Ph 0 0 F SP 18 9
Sensitivity Matrices for Longitudinal LTI Model Δ x Lon t) = F Lon Δx Lon t) + G Lon Δu Lon t) + L Lon Δw Lon t) F Lon = F 11 F 1 F 13 F 14 F 1 F F 3 F 4 F 31 F 3 F 33 F 34 F 41 F 4 F 43 F 44 G Lon = G 11 G 1 G 13 G 1 G G 3 G 31 G 3 G 33 G 41 G 4 G 43 L Lon = L 11 L 1 L 1 L L 31 L 3 L 41 L 4 19 Dimensional Stability and Control Derivatives 0 10
Dimensional Stability-Derivative Notation Redefine force and moment symbols as acceleration symbols Dimensional stability derivatives portray acceleration sensitivities to state perturbations Drag mass m) D!V, m/s Lift mass L V! γ, m/s L V! γ, rad/s Moment M!q, rad/s moment of inertia I yy ) 1 Dimensional Stability-Derivative Notation ) ρ N F 11 = D V = D! 1 V m C T V cosα N C DV Thrust and drag effects are combined and represented by one symbol F 4 = L α = L α! 1 m Thrust and lift effects are combined and represented by one symbol S + C TN cosα N C DN )ρ N S C TN cosα N + C Lα ) ρ N S F 34 = M α = M α! 1 ρ mα I yy Sc 11
Longitudinal Stability Matrix Effects of phugoid perturbations on phugoid motion Effects of short-period perturbations on phugoid motion! F Lon = # # " F Ph SP F Ph F SP Ph F SP! # # $ # = # # % # # # " D V gcosγ N D q D α L V g sinγ N L q L α M V 0 M q M α L V g sinγ N * 1 L - q, + / L α. $ % Effects of phugoid perturbations on shortperiod motion Effects of short-period perturbations on short-period motion 3 Comparison of nd - and 4 th - Order Model Response 4 1
4 th -Order Initial-Condition Responses of Business Jet at Two Time Scales Plotted over different periods of time 4 initial conditions [V0), γ0), q0), α0)] 0-100 sec Reveals Phugoid Mode 0-6 sec Reveals Short-Period Mode 5 nd -Order Models of Longitudinal Motion Assume off-diagonal blocks of 4 x 4) stability matrix are negligible Approximate Phugoid Equation Δ!V Δ! γ + F Lon = D V T δt L δt L V F Ph ~ 0 ~ 0 F SP Δ!x Ph = ΔδT + gcosγ N g sinγ N D V L V ΔV Δγ ΔV wind 6 13
nd -Order Models of Longitudinal Motion F Ph ~ 0 F Lon = ~ 0 F SP Approximate Short-Period Equation Δ!x SP = Δ!q Δ!α + M δ E L δ E M q 1 L q Δδ E + M α L α M α L α Δq Δα Δα wind 7 Comparison of Bizjet 4 th - and nd -Order Model Responses Phugoid Time Scale, ~100 s 4 th Order, 4 initial conditions [V0), γ0), q0), α0)] nd Order, initial conditions [V0), γ0)] 8 14
Comparison of Bizjet 4 th - and nd -Order Model Responses Short-Period Time Scale, ~10 s 4 th Order, 4 initial conditions [V0), γ0), q0), α0)] nd Order, initial conditions [q0), α0)] 9 Approximate Phugoid Response to a 10% Thrust Increase What is the steady-state response? 30 15
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 Normal load factor at the center of mass Grumman X-9 Δn z = g Δ!α Δq ) = g L α Δα + L δ E Δδ E Pilot focuses on normal load factor during rapid maneuvering Normal Load Factor, g s at c.m. Aft Pitch Control Elevator) Normal Load Factor, g s at c.m. Forward Pitch Control Canard) 3 16
Curtiss Autocar, 1917 Waterman Arrowbile, 1935 Stout Skycar, 1931 ConsolidatedVultee 111, 1940s 33 ConvAIRCAR 116 w/crosley auto), 1940s Taylor AirCar, 1950s Hallock Road Wing, 1957 34 17
Mitzar SkyMaster Pinto, 1970s Lotus Elise Aerocar, concept, 00 Pinto separated from the airframe. Two killed. Haynes Skyblazer, concept, 004 Proposed, not built. Aeromobil, 014 Proposed, not built. Aircraft entered a spin, ballistic parachute was deployed. No fatality. 35 Terrafugia Transition Terrafugia TF-X, concept [019, $400K est)] Date? Cost? or, now, for the same price Jaguar F Type Tecnam Astore PLUS plus $10K in savings account 36 18
Next Time: Lateral-Directional Dynamics Reading: Flight Dynamics 574-591 Learning Objectives 6 th -order -> 4 th -order -> hybrid equations Dynamic stability derivatives Dutch roll mode Roll and spiral modes 37 Supplemental Material 38 19
Elements of the Stability Matrix Stability derivatives portray acceleration sensitivities to state perturbations f 1 V D V ; f 1 γ = gcosγ N ; f 1 q D q ; f 1 α D α f V L V ; f γ = g sinγ N ; f q L q ; f α L α f 3 V M V ; f 3 γ = 0; f 3 q M q ; f 3 α M α f 4 V L V ; f 4 γ = g sinγ N ; f 4 q 1 L q ; f 4 α L α 39 Velocity Dynamics First row of nonlinear equation!v = f 1 = 1 [ T cosα D mgsinγ ] m = 1 m C T cosα ρv S C ρv D S mgsinγ First row of linearized equation [ ] [ ] + [ L 11 ΔV wind + L 1 Δα wind ] Δ!Vt) = F 11 ΔVt) + F 1 Δγ t) + F 13 Δqt) + F 14 Δαt) + G 11 Δδ Et) + G 1 ΔδT t) + G 13 Δδ Ft) 40 0
Sensitivity of Velocity Dynamics to State Perturbations % V = ' C T cosα C D ) ρ N F 11 = 1 m C T V cosα N C DV ) ρv Coefficients in first row of F S mgsinγ * m ) S + C TN cosα N C DN )ρ N S F 14 = 1 m F 1 = gcosγ N F 13 = 1 m C D q ρ N S C T N sinα N + C Dα ) ρ N S C TV C DV C Dq C T V C D V C D q C Dα C D α 41 Sensitivity of Velocity Dynamics to Control and Disturbance Perturbations Coefficients in first rows of G and L G 11 = 1 m C D δ E ρ N S G 1 = 1 m C ρ T δt cosα N N S G 13 = 1 m C D δ F ρ N S C TδT C DδE C T δt C D δe L 11 = f 1 V L 1 = f 1 α C DδF C D δf 4 1
Flight Path Angle Dynamics Second row of nonlinear equation! γ = f = 1 mv [ T sinα + L mgcosγ ] = 1 mv C T sinα ρv S + C ρv L S mgcosγ Second row of linearized equation [ ] [ ] + [ L 1 ΔV wind + L Δα wind ] Δ γ!t) = F 1 ΔVt) + F Δγ t) + F 3 Δqt) + F 4 Δαt) + G 1 Δδ Et) + G ΔδT t) + G 3 Δδ Ft) 43 Sensitivity of Flight Path Angle Dynamics to State Perturbations % γ = ' C T sinα + C L ) ρv S mgcosγ * mv ) Coefficients in second row of F F 1 = 1 C TV sinα N + C LV m ) ρ N S + C TN sinα N + C LN )ρ N S 1 C TN sinα N + C LN m ) ρ N S mgcosγ N F = gsinγ N C TV C T V F 3 = 1 ρ Lq S mv N F 4 = 1 C TN cosα N + C Lα m ) ρ V N N S C LV C Lq C L V C L q C Lα C L α 44
Pitch Rate Dynamics Third row of nonlinear equation q = f 3 = M = C m ρv )Sc I yy I yy Third row of linearized equation [ ] [ ] + [ L 31 ΔV wind + L 3 Δα wind ] Δ!qt) = F 31 ΔVt) + F 3 Δγ t) + F 33 Δqt) + F 34 Δαt) + G 31 Δδ Et) + G 3 ΔδT t) + G 33 Δδ Ft) 45 Sensitivity of Pitch Rate Dynamics to State Perturbations q = C m ρv )Sc I yy Coefficients in third row of F F 31 = 1 ρ mv I yy F 3 = 0 Sc + C mn ρ N Sc F 33 = 1 ρ mq I yy F 34 = 1 ρ mα I yy Sc Sc C mv C mq C mα C m V C m q C m α 46 3
Angle of Attack Dynamics Fourth row of nonlinear equation α = f 4 = θ γ = q 1 mv [ T sinα + L mgcosγ ] Fourth row of linearized equation [ ] [ ] [ ] Δ!αt) = F 41 ΔVt) + F 4 Δγ t) + F 43 Δqt) + F 44 Δαt) + G 41 Δδ Et) + G 4 ΔδT t) + G 43 Δδ Ft) + L 41 ΔV wind + L 4 Δα wind 47 Sensitivity of Angle of Attack Dynamics to State Perturbations α = θ γ = q γ Coefficients in fourth row of F F 41 = F 1 F 43 = 1 F 3 F 4 = F F 44 = F 4 48 4
Alternative Approach: Numerical Calculation of the Sensitivity Matrices 1 st Differences ) f 1 V t) f 1 V + ΔV ) γ q α f 1 ΔV V ΔV ) γ q α x=x N t ) u=u N t ) w=w N t ) ; f 1 γ t ) f 1 V γ + Δγ ) q α f 1 Δγ V γ Δγ ) q α x=x N t ) u=u N t ) w=w N t ) f V t) f V + ΔV ) γ q α f ΔV V ΔV ) γ q α x=x N t ) u=u N t ) w=w N t ) ; f t γ ) Remaining elements of Ft), Gt), and Lt) calculated accordingly f V γ + Δγ ) q α f Δγ V γ Δγ ) q α x=x N t ) u=u N t ) w=w N t ) 49 Control and Disturbance Sensitivities in Flight Path Angle, Pitch Rate, and Angle-of-Attack Dynamics f δe = 1 $ ρv LδE mv N % S ' ) f δt = 1 $ ρv C TδT sinα N N mv N % S ' ) f δf = 1 $ ρv LδF mv N % S ' ) f 3 δe = 1 $ ρv mδe I yy % Sc ' ) f 3 δt = 1 $ ρv mδt I yy % Sc ' ) f 3 δf = 1 $ ρv mδf I yy % Sc ' ) f 4 δe = f δe f 4 δt = f δt f 4 δf = f δf f = f V wind V f α wind = f α f 3 = f 3 V wind V f 3 α wind = f 3 α f 4 = f V wind V f 3 α wind = f α 50 5
Velocity-Dependent Derivative Definitions Air compressibility effects are a principal source of velocity dependence C DM C D M = C D V / a) = a C D V a = Speed of Sound M = Mach number = V a C DM 0 C DM > 0 C DM < 0 C DV C LV C mv C D V = # % 1 $ a ' C D M C L V = # % 1 $ a ' C L M C m V = # % 1 $ a ' C m M 51 Wing Lift and Moment Coefficient Sensitivity to Pitch Rate Straight-wing incompressible flow estimate Etkin) C L ˆqwing = C Lαwing h cm 0.75) C m ˆqwing = C Lαwing h cm 0.5) Straight-wing supersonic flow estimate Etkin) C m ˆqwing C L ˆqwing = C Lαwing h cm 0.5) = 3 M 1 C L αwing h cm 0.5 ) Triangular-wing estimate Bryson, Nielsen) C L ˆqwing = π 3 C L αwing C m ˆqwing = π 3AR 5 6
Control- and Disturbance- Effect Matrices Control-effect derivatives portray acceleration sensitivities to control input perturbations # D δ E T δt D δ F % % L δ E / L δt / L δ F / G Lon = % M δ E M δt M % δ F % L δ E / L δt / L δ F / V $ N Disturbance-effect derivatives portray acceleration sensitivities to disturbance input perturbations # D Vwind D αwind % % L Vwind / L αwind / L Lon = % % M Vwind M αwind % L Vwind / L αwind / V $ % N ' ' 53 Primary Longitudinal Stability Derivatives D V 1 # % m C T V C DV $ ) ρ S + C T N C DN )ρ S ' L V 1 " ρv LV m S + C % $ L N ρ S' # 1 " ρv LN mv $ N # S mg % ' M q = 1 " ρv mq I $ yy # Sc % ' M α = 1 # ρv mα I % yy $ Sc ' ) ρv N L α 1 # C TN + C Lα mv % N $ S ' Small angle assumptions 54 7
Primary Phugoid Control Derivatives D δt! 1 m C T δt S ρ L δ F! 1 ρv Lδ mv F N S 55 Primary Short-Period Control Derivatives ρ M δ E = mδe Sc L δ E V = C Lδ E I yy ρ N m S 56 8
Flight Motions Dornier Do-18D Dornier Do-18 Short-Period Demonstration http://www.youtube.com/watch?v=3hdlxe0rc9q Dornier Do-18 Phugoid Demonstration http://www.youtube.com/watch?v=jzxtpq30nlgfeature=related 57 9