Aircraft Flight Dynamics Robert F. Stengel Lecture18 Advanced Longitudinal Dynamics

Advanced Problems of Longitudinal Dynamics 
 Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012 " •  Angle-of-attack-rate aero effects" •  Fourth-order dynamics" –  Steady-state response to control" –  Transfer functions" –  Frequency response" –  Root locus analysis of parameter variations" •  Numerical solution for trimmed flight condition" •  Nichols chart" •  Pilot-aircraft interactions" 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 ! Distinction Between Angle-of- Attack Rate and Pitch Rate"  α = q  α ≠ 0 ≠ q; q = 0 !  With no vertical motion of the c.m., pitch rate and angle-of- attack rate are the same " !  With no pitching, vertical heaving (or plunging) motion of the c.m., produces angle-of-attack rate but no pitch rate" Vertical velocity distribution induced by pitch rate! Angle-of-Attack Rate Has Two Effects " !  Pressure variations at wing convect downstream, arriving at tail Δt sec later" !  Lag of the downwash" !  Delayed tail-lift/pitch- moment effect" !  Vertical force opposed by a mass of air (apparent mass) as well as airplane mass" !  Vertical acceleration produces added lift and moment " Flight Dynamics, pp. 204-206, 284-285" Δ  q = M q Δq + M α Δ α + M δ E Δ δ E + M  α Δ  α Δ  α = 1− L q V N % & ' ( ) * Δq − L α V N ( ) Δ α − L δ E V N ( ) Δ δ E − L  α V N ( ) Δ  α Δ  q − M  α Δ  α = M q Δq + M α Δ α + M δ E Δ δ E Δ  α + L  α V N ( ) Δ  α = 1− L q V N % & ' ( ) * Δq − L α V N ( ) Δ α − L δ E V N ( ) Δ δ E 1 −M  α 0 1+ L  α V N ( ) # $ % & ' ( # $ % % % & ' ( ( ( Δ  q Δ  α # $ % % & ' ( ( = M q M α 1− L q V N * + , - . / − L α V N ( ) # $ % % % & ' ( ( ( Δq Δ α # $ % % & ' ( ( + M δ E − L δ E V N ( ) # $ % % % & ' ( ( ( Δ δ E Angle-of-Attack-Rate Effects Principally Affect the Short-Period Mode" !  Lift and pitching moment proportional to angle-of-attack rate" !  Bring effects to left side" !  Vector-matrix form" 1 −M  α 0 1 + L  α V N ( ) # $ % & ' ( # $ % % % & ' ( ( ( −1 = 1 + L  α V N ( ) # $ % & ' ( M  α 0 1 # $ % % % & ' ( ( ( 1 + L  α V N ( ) # $ % & ' ( Δ  q Δ  α # $ % % & ' ( ( = 1 −M  α 0 1+ L  α V N ( ) # $ % & ' ( # $ % % % & ' ( ( ( −1 M q M α 1− L q V N * + , - . / − L α V N ( ) # $ % % % & ' ( ( ( Δq Δ α # $ % % & ' ( ( + M δ E − L δ E V N ( ) # $ % % % & ' ( ( ( Δ δ E 1 2 3 3 4 3 3 5 6 3 3 7 3 3 Angle-of-Attack-Rate Effects" !  Inverse of the apparent mass matrix" !  Pre-multiply both sides by inverse" Δ  q Δ  α # $ % % & ' ( ( = 1+ L  α V N ( ) # $ % & ' ( M  α 0 1 # $ % % % & ' ( ( ( 1+ L  α V N ( ) # $ % & ' ( M q M α 1− L q V N * + , - . / − L α V N # $ % % % & ' ( ( ( Δq Δ α # $ % % & ' ( ( + 0 1 2 2 3 2 2 4 5 2 2 6 2 2 Angle-of-Attack-Rate Effects" Δ  q Δ  α # $ % % & ' ( ( = 1 1+ L  α V N ( ) # $ % & ' ( 1+ L  α V N ( ) # $ % & ' ( M q + M  α 1− L q V N * + , - . / 0 1 2 3 4 5 1+ L  α V N ( ) # $ % & ' ( M α − M  α L α V N ( ) 0 1 2 3 4 5 1− L q V N * + , - . / − L α V N # $ % % % % % & ' ( ( ( ( ( Δq Δ α # $ % % & ' ( ( + 1+ L  α V N ( ) # $ % & ' ( M δ E − M  α L δ E V N ( ) − L δ E V N # $ % % % % & ' ( ( ( ( Δ δ E 0 1 7 7 7 7 7 2 7 7 7 7 7 3 4 7 7 7 7 7 5 7 7 7 7 7 !  Multiply matrices" !  Substitute" Simplification of Angle-of-Attack- Rate Effects" Δ  q Δ  α # $ % % & ' ( (  M q + M  α { } M α − M  α L α V N ( ) { } 1 − L α V N ( ) # $ % % % % % & ' ( ( ( ( ( Δq Δ α # $ % % & ' ( ( + M δ E − M  α L δ E V N ( ) − L δ E V N ( ) # $ % % % % & ' ( ( ( ( Δ δ E !  Typically" L q and L  α have small effects for large aircraft* M q and M  α are same order of magnitude and have more significant effects !  Neglecting" L q and L  α * but not for small aircraft, e.g., R/C models and micro-UAVs" 2 nd -Degree Characteristic Polynomial with" !  Short-period characteristic polynomial" !  Damping is increased" !  Natural frequency is unaffected" Δ s ( ) = s − M q + M  α ( ) $ % & ' − M α − M  α L α V N ( ) $ % ( & ' ) −1 s + L α V N ( ) $ % ( & ' ) = s − M q + M  α ( ) $ % & ' s + L α V N ( ) $ % ( & ' ) − M α − M  α L α V N ( ) $ % ( & ' ) Δ s ( ) = s 2 + L α V N ( ) − M q + M  α ( ) $ % & ' ( ) s + M α − M q L α V N ( ) $ % & ' ( ) * + , - . / = s 2 + 2 ζω n s + ω n 2 = 0 L q and L  α  0 = s 2 + L α V N ( ) − M q + M  α ( ) # $ % & ' ( s + M α − M q + M  α ( ) L α V N ( ) # $ % & ' ( + M  α L α V N ( ) ) * + , - . Linear, Time-Invariant Fourth-Order Longitudinal Model " (Neglecting angle-of-attack-rate aero) " Δ  V (t) Δ  γ (t) Δ  q(t) Δ  α (t) $ % & & & & & ' ( ) ) ) ) ) = −D V −g 0 −D α L V V N 0 0 L α V N M V 0 M q M α − L V V N 0 1 − L α V N $ % & & & & & & & ' ( ) ) ) ) ) ) ) ΔV(t) Δ γ (t) Δq(t) Δ α (t) $ % & & & & & ' ( ) ) ) ) ) + 0 T δ T 0 0 0 L δ F / V N M δ E 0 0 0 0 −L δ F / V N $ % & & & & & ' ( ) ) ) ) ) Δ δ E(t) Δ δ T (t) Δ δ F(t) $ % & & & ' ( ) ) ) •  Stability and control derivatives are defined at a trimmed (equilibrium) flight condition " Perturbations to the Trimmed Condition" •  Initial pitch rate [Δq(0)] = 0.1 rad/s" •  Elevator step input [Δ δ E(0)] = 1 deg" •  Small linear and nonlinear perturbations are virtually identical " Steady-State Response Steady-State Response of the 4 th -Order LTI Longitudinal Model" ΔV SS Δ γ SS Δq SS Δ α SS $ % & & & & & ' ( ) ) ) ) ) = − −D V −g 0 −D α L V V N 0 0 L α V N M V 0 M q M α − L V V N 0 1 − L α V N $ % & & & & & & & ' ( ) ) ) ) ) ) ) −1 0 T δ T 0 0 0 L δ F / V N M δ E 0 0 0 0 −L δ F / V N $ % & & & & & ' ( ) ) ) ) ) Δ δ E SS Δ δ T SS Δ δ F SS $ % & & & ' ( ) ) ) Δx SS = −F −1 G Δu SS •  How do we calculate the equilibrium response to control? " Δ  x(t ) = FΔx(t ) + GΔu(t ) •  For the longitudinal model " Algebraic Equation for Equilibrium Response " ΔV SS Δ γ SS Δq SS Δ α SS $ % & & & & & ' ( ) ) ) ) ) = −gM δ E L α V N $ % & ' ( ) 0 gM α L δ F / V N [ ] D V L α V N − D α L V V N ( ) M δ E $ % & ' ( ) M V L α V N − M α L V V N ( ) T δ T $ % & ' ( ) D α M V − D V M α ( ) L δ F / V N $ % ' ( 0 0 0 −gM δ E L V V N $ % & ' ( ) 0 L δ F / V N [ ] $ % & & & & & & & & ' ( ) ) ) ) ) ) ) ) g M V L α V N − M α L V V N ( ) Δ δ E SS Δ δ T SS Δ δ F SS $ % & & & ' ( ) ) ) ΔV SS Δ γ SS Δq SS Δ α SS $ % & & & & & ' ( ) ) ) ) ) = a 0 b c d e 0 0 0 f 0 g $ % & & & & ' ( ) ) ) ) Δ δ E SS Δ δ T SS Δ δ F SS $ % & & & ' ( ) ) ) •  Roles of stability and control derivatives identified" •  Result is a simple equation relating input and output " 4 th -Order Steady-State Response May Be Counterintuitive" ΔV SS = aΔ δ E SS + 0 ( ) Δ δ T SS + bΔ δ F SS Δ γ SS = cΔ δ E SS + dΔ δ T SS + eΔ δ F SS Δq SS = 0 ( ) ΔE SS + 0 ( ) Δ δ T SS + 0 ( ) Δ δ F SS Δ α SS = f Δ δ E SS + 0 ( ) Δ δ T SS + gΔ δ F SS •  Observations" –  Thrust command" –  Elevator and flap commands" –  Steady-state pitch rate is zero! –  4 th -order model neglects air density gradient effects ! Δ θ SS = Δ γ SS + Δ α SS = c + f ( ) Δ δ E SS + dΔ δ T SS + e + g ( ) Δ δ F SS •  Steady-state pitch angle " Effects of Stability Derivative Variations on 4 th -Order Longitudinal Modes Primary and Coupling Blocks of the Fourth-Order Longitudinal Model" F Lon = −D V −g 0 −D α L V V N 0 0 L α V N M V 0 M q M α − L V V N 0 1 − L α V N # $ % % % % % % % & ' ( ( ( ( ( ( ( = F Ph F SP Ph F Ph SP F SP # $ % % & ' ( ( •  Some stability derivatives appear only in primary blocks (D V , M q , M α )" –  Effects are well-described by 2 nd -order models" •  Some stability derivatives appear only in coupling blocks (M V , D α )" –  Effects are ignored by 2 nd -order models" •  Some stability derivatives appear in both (L V , L α )" –  Require 4 th -order modeling" ΔM α Effect on 4 th -Order Roots! Short Period! Short Period! Phugoid! Phugoid! Δ Lon (s)= s 4 + D V + L α V N − M q ( ) s 3 + g − D α ( ) L V V N + D V L α V N − M q ( ) − M q L α V N − M α o $ % & ' ( ) s 2 + M q D α − g ( ) L V V N − D V L α V N $ % & ' ( ) + D α M V − D V M α o { } s + g M V L α V N − M α o L V V N ( ) − ΔM α s 2 + D V s + g L V V N ( ) ≡ d(s)+ kn(s) •  Group all terms multiplied by M α to form numerator for ΔM α# ΔM α Effect on 4 th -Order Roots! •  Primary effect: The same as in the approximate short- period model" •  Numerator zeros" –  The same as the approximate phugoid mode characteristic polynomial" –  Effect of M α variation on phugoid mode is small # Short Period! Short Period! Phugoid! Phugoid! Direct Thrust Effect on Speed Stability, T V " •  In steady, level flight, nominal thrust balances nominal drag" ∂ T ∂ V = < 0, for propeller aircraft ≈ 0, for turbojet aircraft > 0, for ramjet aircraft # $ % & % ∂ T ∂ V − ∂ D ∂ V > 0 T N − D N = C T N 1 2 ρ V N 2 S − C D N 1 2 ρ V N 2 S = 0 •  Effect of velocity change" •  Small velocity perturbation grows if " •  Therefore" –  propeller is stabilizing for velocity change" –  turbojet has neutral effect" –  ramjet is destabilizing " Pitching Moment Due to Thrust, M V! •  Thrust line above or below center of mass induces a pitching moment" •  Aerodynamic and thrust pitching moments sensitive to velocity perturbation" •  Couples phugoid and short-period modes " Martin XB-51! McDonnell Douglas MD-11! Consolidated PBY! Fairchild-Republic A-10! •  Negative ∂M/∂V (Pitch-down effect) tends to increase velocity" •  Positive ∂M/∂V (Pitch-up effect) tends to decrease velocity" •  With propeller thrust line above the c.m., increased velocity decreases thrust, producing a pitch-up moment" •  Tilting the thrust line can have benefits" –  Up: Lake Amphibian, MD-11" –  Down : F6F, F8F, AD-1 " ∂ M thrust ∂ V ≈ ∂ T ∂ V × Moment Arm Douglas AD-1! Lake Amphibian! Grumman F8F! McDonnell Douglas ! MD-11! Pitching Moment Due to Thrust, M V! M V Effect on 4 th -Order Roots" •  Large positive value produces oscillatory phugoid instability" •  Large negative value produces real phugoid divergence" Δ Lon (s) = s 4 + D V + L α V N − M q ( ) s 3 + g −D α ( ) L V V N + D V L α V N − M q ( ) − M q L α V N − M α $ % & ' ( ) s 2 + M q D α − g ( ) L V V N − D V L α V N $ % & ' ( ) + D α M V − D V M α { } s gM α L V V N + M V D α s + g L α V N ( ) = 0 € D α = 0 Short Period! Phugoid! L α /V N and L V /V N Effects on Fourth-Order Roots! •  L V /V N : Damped natural frequency of the phugoid" •  Negligible effect on the short- period" •  L α /V N : Increased damping of the short-period" •  Small effect on the phugoid mode" Short Period! Phugoid! Pitch and Thrust Control Effects Longitudinal Model Transfer Function Matrix (H x = I, H u = 0)" H Lon (s) = n δ E V (s) n δ T V (s) n δ F V (s) n δ E γ (s) n δ T γ (s) n δ F γ (s) n δ E q (s) n δ T q (s) n δ F q (s) n δ E α (s) n δ T α (s) n δ F α (s) $ % & & & & & & ' ( ) ) ) ) ) ) s 2 + 2 ζ P ω n P s + ω n P 2 ( ) s 2 + 2 ζ SP ω n SP s + ω n SP 2 ( ) ΔV(s) Δ γ (s) Δq(s) Δ α (s) $ % & & & & & ' ( ) ) ) ) ) = H x A s ( ) G Δ δ E(s) Δ δ T (s) Δ δ F(s) $ % & & & ' ( ) ) ) = H Lon (s) Δ δ E(s) Δ δ T (s) Δ δ F(s) $ % & & & ' ( ) ) ) A Little More About Output Matrices" •  With H x = I and H u = 0! Δy = Δx = H x Δx; then H x = I 4 and Δy 1 Δy 2 Δy 3 Δy 4 " # $ $ $ $ $ % & ' ' ' ' ' = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 " # $ $ $ $ % & ' ' ' ' Δx 1 Δx 2 Δx 3 Δx 4 " # $ $ $ $ $ % & ' ' ' ' '  ΔV Δ γ Δq Δ α " # $ $ $ $ $ % & ' ' ' ' ' •  Only output is ΔV! Δy = ΔV = 1 0 0 0 " # $ % ΔV Δ γ Δq Δ α " # ( ( ( ( ( $ % ) ) ) ) ) •  ΔV and Δ α are measured" Δy = Δy 1 Δy 2 " # $ $ % & ' ' = ΔV Δ α " # $ % & ' = 1 0 0 0 0 0 0 1 " # $ % & ' ΔV Δ γ Δq Δ α " # $ $ $ $ $ % & ' ' ' ' ' A Little More About Output Matrices" •  Output (measurement) of body-axis velocity and pitch rate and angle" •  Transformation from [ΔV, Δ γ , Δq, Δ θ ] to [Δu, Δw, Δq, Δ α ]" Δu Δw Δq Δ θ # $ % % % % & ' ( ( ( ( = cos α N 0 0 −V N sin α N sin α N 0 0 V N cos α N 0 0 1 0 0 1 0 1 # $ % % % % % & ' ( ( ( ( ( ΔV Δ γ Δq Δ α # $ % % % % % & ' ( ( ( ( ( •  Separate measurement of state and control perturbations! Δy = Δx Δu " # $ % & ' = H x Δx + H u Δu Δy 1 Δy 2 Δy 3 Δy 4 Δy 5 Δy 6 " # $ $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' ' = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 " # $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' ΔV Δ γ Δq Δ α " # $ $ $ $ $ % & ' ' ' ' ' + 0 0 0 0 0 0 0 0 1 0 0 1 " # $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' Δ δ E Δ δ T " # $ % & ' Elevator-to-Normal-Velocity Numerator" (L  E = 0)" H x Adj sI − F Lon ( ) G = sin α N 0 0 V N cos α N # $ % & n V V (s) n γ V (s) n q V (s) n α V (s) n V γ (s) n γ γ (s) n q γ (s) n α γ (s) n V q (s) n γ q (s) n q q (s) n α q (s) n V α (s) n γ α (s) n q α (s) n α α (s) # $ ( ( ( ( ( ( % & ) ) ) ) ) ) 0 0 M δ E 0 # $ ( ( ( ( % & ) ) ) ) = n δ E w (s) •  Transform though α N back to body axes " n δ E w (s) = sin α N 0 0 V N cos α N # $ % & n q V (s) n q γ (s) n q q (s) n q α (s) # $ ( ( ( ( ( ( % & ) ) ) ) ) ) M δ E = M δ E sin α N ( ) n q V (s)+ V N cos α N ( ) n q α (s) # $ % & •  Scalar transfer function numerator " Elevator-to-Normal-Velocity Transfer Function" Δw(s) Δ δ E(s) = n δ E w (s) Δ Lon (s) = M δ E s 2 + 2 ζω n s + ω n 2 ( ) Approx Ph s − z 3 ( ) s 2 + 2 ζω n s + ω n 2 ( ) Ph s 2 + 2 ζω n s + ω n 2 ( ) SP • Normal velocity transfer function is analogous to angle of attack transfer function (Δ α ≈ Δw/V N )" • z 3 often neglected due to high frequency ! Elevator-to-Normal- Velocity Frequency Response" Δw(s) Δ δ E(s) = n δ E w (s) Δ Lon (s) ≈ M δ E s 2 + 2 ζω n s + ω n 2 ( ) Approx Ph s − z 3 ( ) s 2 + 2 ζω n s + ω n 2 ( ) Ph s 2 + 2 ζω n s + ω n 2 ( ) SP 0 dB/dec! +40 dB/dec! 0 dB/dec! –40 dB/dec! –20 dB/dec! • (n – q) = 1" • Complex zero almost (but not quite) cancels phugoid response " Elevator-to-Pitch-Rate " Numerator and Transfer Function" H x Adj sI − F Lon ( ) G = 0 0 1 0 " # $ % n V V (s) n γ V (s) n q V (s) n α V (s) n V γ (s) n γ γ (s) n q γ (s) n α γ (s) n V q (s) n γ q (s) n q q (s) n α q (s) n V α (s) n V α (s) n V α (s) n V α (s) " # ( ( ( ( ( ( $ % ) ) ) ) ) ) 0 0 M δ E 0 " # ( ( ( ( $ % ) ) ) ) = n δ E q (s) Δq(s) Δ δ E( s) = n δ E q (s) Δ Lon (s) ≈ M δ E s s − z 1 ( ) s − z 2 ( ) s 2 + 2 ζω n s + ω n 2 ( ) Ph s 2 + 2 ζω n s + ω n 2 ( ) SP • Free s in numerator differentiates pitch angle transfer function " Elevator-to-Pitch- Rate Frequency Response" +20 dB/dec! +20 dB/dec! +40 dB/dec! 0 dB/dec! –20 dB/dec! Δq SS = 0 ( ) Δ δ E SS + 0 ( ) Δ δ T SS + 0 ( ) Δ δ F SS •  (n – q) = 1" •  Negligible low- frequency response, except at phugoid natural frequency" •  High-frequency response well predicted by 2 nd - order model " Δq(s) Δ δ E(s) = n δ E q (s) Δ Lon (s) ≈ M δ E s s − z 1 ( ) s − z 2 ( ) s 2 + 2 ζω n s + ω n 2 ( ) Ph s 2 + 2 ζω n s + ω n 2 ( ) SP Transfer Functions of Elevator Input to Angle Output*" Δ θ (s) Δ δ E(s) = n δ E θ (s) Δ Lon (s) ; n δ E θ (s) = M δ E s + 1 T θ 1 $ % & ' ( ) s + 1 T θ 2 $ % & ' ( ) Δ α (s) Δ δ E(s) = n δ E α (s) Δ Lon (s) ; n δ E α (s) = M δ E s 2 + 2 ζω n s + ω n 2 ( ) Approx Ph Δ γ (s) Δ δ E(s) = n δ E γ (s) Δ Lon (s) ; n δ E γ (s) = M δ E L α V N s + 1 T γ 1 % & ' ( ) * •  Elevator-to-Flight Path Angle transfer function " •  Elevator-to-Angle of Attack transfer function " •  Elevator-to-Pitch Angle transfer function " * Flying qualities notation for zero time constants" Frequency Response of Angles to Elevator Input" •  Pitch angle frequency response (Δ θ = Δ γ + Δ α )" –  Similar to flight path angle near phugoid natural frequency" –  Similar to angle of attack near short- period natural frequency" Δ γ SS = cΔ δ E SS Δ α SS = f Δ δ E SS Δ θ SS = c − f ( ) Δ δ E SS Transfer Functions of Thrust Input to Angle Output" Δ θ (s) Δ δ T (s) = n δ T θ (s) Δ Lon (s) ; n δ T θ (s) = T δ T s + 1 T θ T $ % & ' ( ) Δ α (s) Δ δ T (s) = n δ T α (s) Δ Lon (s) ; n δ T α (s) = T δ T s s + 1 T α T $ % & ' ( ) Δ γ (s) Δ δ T (s) = n δ T γ (s) Δ Lon (s) ; n δ T γ (s) = T δ T L V V N s 2 + 2 ζω n s + ω n 2 ( ) Approx SP •  Thrust-to-Flight Path Angle transfer function " •  Thrust-to-Angle of Attack transfer function " •  Thrust-to-Pitch Angle transfer function " Frequency Response of Angles to Thrust Input" •  Primarily effects flight path angle and low-frequency pitch angle" Gain and Phase Margins: The Nichols Chart Nichols Chart: 
 Gain vs. Phase Angle " •  Bode Plot" –  Two plots" –  Open-Loop Gain (dB) vs. log 10 ω " –  Open-Loop Phase Angle vs. log 10 ω# •  Nichols Chart" –  Single crossplot; input frequency not shown" –  Open-Loop Gain (dB) vs. Open- Loop Phase Angle" Gain and Phase Margins" •  Gain Margin " –  At the input frequency, ω , for which ϕ (j ω ) = –180°" –  Difference between 0 dB and transfer function magnitude, 20 log 10 AR(j ω )" •  Phase Margin " –  At the input frequency, ω , for which 20 log 10 AR(j ω ) = 0 dB " –  Difference between the phase angle ϕ (j ω ), and –180°" •  Axis intercepts on the Nichols Chart identify GM and PM" Examples of Gain and Phase Margins" •  Bode Plot # •  Nichols Chart" H blue ( j ω ) = 10 j ω + 10 ( ) " # $ % & ' 100 2 j ω ( ) 2 + 2 0.1 ( ) 100 ( ) j ω ( ) + 100 2 " # $ $ % & ' ' H green ( j ω ) = 10 2 j ω ( ) 2 + 2 0.1 ( ) 10 ( ) j ω ( ) + 10 2 " # $ $ % & ' ' 100 j ω + 100 ( ) " # $ % & ' [...]... Shortal-Maggin Longitudinal Stability Boundary for Swept Wings" •  Stable or unstable pitch break at the stall" •  Stability boundary is expressed as a function of " AR! –  Aspect ratio" –  Sweep angle of the quarter chord" –  Taper ratio" Λc/4! NACA TR-1339! Next Time: Fourth-Order LateralDirectional Dynamics Reading Flight Dynamics, 595-627 Virtual Textbook, Part 19 Supplemental Material Flight Conditions... Control " Effect of Pilot Dynamics on Pitch-Angle Control Task " •  Pilot introduces neuromuscular lag while closing the control loop" •  Example" –  Model the lag by a 1st-order time constant, TP, of 0.25 s" –  Pilot s gain, KP, is either 1 or 2" * p 421-425, Flight Dynamics" Pilot Transfer Function = Δu ( s ) 1 / TP 1 / 0.25 = KP = KP s +1 / 0.25 Δε ( s ) s +1 / TP Open-Loop Pilot -Aircraft Transfer Function... Supplemental Material Flight Conditions for Steady, Level Flight " •  Nonlinear longitudinal model" Trimmed Solution of the Equations of Motion 1  V = f1 = $T cos (α + i ) − D − mg sin γ & ' m% 1 γ = f2 = $T sin (α + i ) + L − mg cos γ & ' mV %  q = f3 = M / I yy 1   α = f4 = θ − γ = q − $T sin (α + i ) + L − mg cos γ & ' mV % •  Nonlinear longitudinal model in equilibrium" 1 $T cos (α + i ) − D... = K P # &# 2 2 2 2 " ( s + 1 / TP ) % # s + 2ζω n s + ω n Ph s + 2ζω n s + ω n # " ( ) ( Effect of Pilot Dynamics on Pitch-Angle Control Task" ) $ & & & SP & % •  Gain and phase margins become negative for pilot gain between 1 and 2" •  Then, pilot destabilizes the system (PIO)" Effect of Pilot Dynamics on Elevator/PitchAngle Control Root Locus " Configuration Effects •  Pilot transfer function changes... bowl, i.e., when " $ ∂J & % ∂δT ∂J ∂δ E ∂J ' )= 0 ∂θ ( •  Search to find the minimum value of J " J (δT , δ E,θ ) = a ( f12 ) + b ( f22 ) + c ( f32 ) Example of Search for Trimmed Condition (Fig 3.6-9, Flight Dynamics) " •  In MATLAB, use fminsearch [Nelder-Mead Downhill Simplex Method] to find trim settings" (δT *,δ E*,θ *) = fminsearch # J, (δT ,δ E,θ )% $ & Airspeed Frequency Response to Elevator and... f2 = $T sin (α + i ) + L − mg cos γ & ' mV % 0 = f3 = M / I yy 0 = f1 =  0 = f4 = θ − γ = q − 1 $T sin (α + i ) + L − mg cos γ & ' mV % Numerical Solution to Estimate the Trimmed Condition for Level Flight " •  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% &T (δT , δ E,θ ,h,V . Advanced Problems of Longitudinal Dynamics 
 Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012 " •  Angle-of-attack-rate aero effects" •  Fourth-order dynamics& quot; – . Lateral- Directional Dynamics  Reading Flight Dynamics, 595-627 Virtual Textbook, Part 19 Supplemental Material Trimmed Solution of the Equations of Motion Flight Conditions for Steady, Level Flight& quot; . dB" Pilot-Vehicle Interactions Pilot Inputs to Control" * p. 421-425, Flight Dynamics& quot; Effect of Pilot Dynamics on Pitch-Angle Control Task " Pilot Transfer Function = Δu s (