Aircraft Flight Dynamics Robert F. Stengel Lecture11 Longitudinal Dynamics

Linearized Longitudinal Equations of Motion 
 Robert Stengel, Aircraft Flight Dynamics 
 MAE 331, 2012! •  6 th -order -> 4 th -order -> hybrid equations" •  Dynamic stability derivatives " •  Phugoid mode" •  Short-period mode" 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 ! 6 th -Order Longitudinal Equations of Motion! •  Symmetric aircraft" •  Motions in the vertical plane" •  Flat earth " x 1 x 2 x 3 x 4 x 5 x 6 ! " # # # # # # # # $ % & & & & & & & & = x Lon 6 = u w x z q θ ! " # # # # # # # $ % & & & & & & & = Axial Velocity Vertical Velocity Range Altitude(–) Pitch Rate Pitch Angle ! " # # # # # # # # $ % & & & & & & & &  u = X / m − gsin θ − qw  w = Z / m + g cos θ + qu  x I = cos θ ( ) u + sin θ ( ) w  z I = − sin θ ( ) u + cos θ ( ) w  q = M / I yy  θ = q State Vector, 6 components!Nonlinear Dynamic Equations! Fairchild-Republic A-10! 4 th -Order Longitudinal Equations of Motion!  u = f 1 = X / m − gsin θ − qw  w = f 2 = Z / m + gcos θ + qu  q = f 3 = M / I yy  θ = f 4 = q x 1 x 2 x 3 x 4 ! " # # # # # $ % & & & & & = x Lon 4 = u w q θ ! " # # # # $ % & & & & = Axial Velocity, m/s Vertical Velocity, m/s Pitch Rate, rad/s Pitch Angle, rad ! " # # # # # $ % & & & & & State Vector, 4 components! Nonlinear Dynamic Equations, neglecting range and altitude! Fourth-Order Hybrid Equations of Motion Transform Longitudinal Velocity Components"  u = f 1 = X / m − gsin θ − qw  w = f 2 = Z / m + gcos θ + qu  q = f 3 = M / I yy  θ = f 4 = q  V = f 1 = T cos α + i ( ) − D − mgsin γ $ % & ' m  γ = f 2 = T sin α + i ( ) + L − mgcos γ $ % & ' mV  q = f 3 = M / I yy  θ = f 4 = q x 1 x 2 x 3 x 4 ! " # # # # # $ % & & & & & = u w q θ ! " # # # # $ % & & & & = Axial Velocity Vertical Velocity Pitch Rate Pitch Angle ! " # # # # # $ % & & & & & x 1 x 2 x 3 x 4 ! " # # # # # $ % & & & & & = V γ q θ ! " # # # # # $ % & & & & & = Velocity Flight Path Angle Pitch Rate Pitch Angle ! " # # # # # $ % & & & & & i = Incidence angle of the thrust vector with respect to the centerline •  Replace Cartesian body components of velocity by polar inertial components" •  Replace X and Z by T, D, and L" Hybrid Longitudinal Equations of Motion"  V = f 1 = T cos α + i ( ) − D − mgsin γ $ % & ' m  γ = f 2 = T sin α + i ( ) + L − mgcos γ $ % & ' mV  q = f 3 = M / I yy  θ = f 4 = q  V = f 1 = T cos α + i ( ) − D − mgsin γ $ % & ' m  γ = f 2 = T sin α + i ( ) + L −mgcos γ $ % & ' mV  q = f 3 = M / I yy  α = f 4 =  θ −  γ = q − 1 mV T sin α + i ( ) + L −mgcos γ $ % & ' x 1 x 2 x 3 x 4 ! " # # # # # $ % & & & & & = V γ q θ ! " # # # # # $ % & & & & & = Velocity Flight Path Angle Pitch Rate Pitch Angle ! " # # # # # $ % & & & & & x 1 x 2 x 3 x 4 ! " # # # # # $ % & & & & & = V γ q α ! " # # # # # $ % & & & & & = Velocity Flight Path Angle Pitch Rate Angle of Attack ! " # # # # # $ % & & & & & •  Replace pitch angle by angle of attack! α = θ − γ θ = α + γ Why Transform Equations and State Vector?" •  Phugoid (long-period) mode is primarily described by velocity and flight path angle" •  Short-period mode is primarily described by pitch rate and angle of attack" x 1 x 2 x 3 x 4 ! " # # # # # $ % & & & & & = V γ q α ! " # # # # $ % & & & & = Velocity Flight Path Angle Pitch Rate Angle of Attack ! " # # # # # $ % & & & & & Why Transform Equations and State Vector?" •  Hybrid linearized equations allow the two modes to be examined separately" F Lon = F Ph F SP Ph F Ph SP F SP ! " # # $ % & & Effects of phugoid perturbations on phugoid motion" Effects of phugoid perturbations on short-period motion" Effects of short- period perturbations on phugoid motion" Effects of short-period perturbations on short- period motion" = F Ph small small F SP ! " # # $ % & & ≈ F Ph 0 0 F SP ! " # # $ % & & Nominal Equations of Motion in Equilibrium (Trimmed Condition)"  x N (t ) = 0 = f[x N (t ),u N (t ),w N (t ),t ]  V N = 0 = f 1 = T cos α N + i ( ) − D − mgsin γ N $ % & ' m  γ N = 0 = f 2 = T sin α N + i ( ) + L − mg cos γ N $ % & ' mV  q N = 0 = f 3 = M I yy  α N = 0 = f 4 = q − 1 mV T sin α N + i ( ) + L − mg cos γ N $ % & ' •  T, D, L, and M contain state, control, and disturbance effects" x N T = V N γ N 0 α N # $ % & T = constant Linearized  Equations of Motion 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 = ∂ f 1 ∂ V ∂ f 1 ∂γ ∂ f 1 ∂ q ∂ f 1 ∂α ∂ f 2 ∂ V ∂ f 2 ∂γ ∂ f 2 ∂ q ∂ f 2 ∂α ∂ f 3 ∂ V ∂ f 3 ∂γ ∂ f 3 ∂ q ∂ f 3 ∂α ∂ f 4 ∂ V ∂ f 4 ∂γ ∂ f 4 ∂ q ∂ f 4 ∂α $ % & & & & & & & & & & & ' ( ) ) ) ) ) ) ) ) ) ) ) G = ∂ f 1 ∂δ E ∂ f 1 ∂δ T ∂ f 1 ∂δ F ∂ f 2 ∂δ E ∂ f 2 ∂δ T ∂ f 2 ∂δ F ∂ f 3 ∂δ E ∂ f 3 ∂δ T ∂ f 3 ∂δ F ∂ f 4 ∂δ E ∂ f 4 ∂δ T ∂ f 4 ∂δ F # $ % % % % % % % % % % & ' ( ( ( ( ( ( ( ( ( ( L = ∂ f 1 ∂ V wind ∂ f 1 ∂α wind ∂ f 2 ∂ V wind ∂ f 2 ∂α wind ∂ f 3 ∂ V wind ∂ f 3 ∂α wind ∂ f 4 ∂ V wind ∂ f 4 ∂α wind # $ % % % % % % % % % % % & ' ( ( ( ( ( ( ( ( ( ( ( Velocity Dynamics"  V = f 1 = 1 m T cos α − D − mgsin γ [ ] = 1 m C T cos α ρ V 2 2 S − C D ρ V 2 2 S − mgsin γ % & ' ( ) * •  Nonlinear equation" Thrust incidence angle neglected! •  First row of linearized dynamic equation" Δ  V (t) = ∂ f 1 ∂ V ΔV(t) + ∂ f 1 ∂γ Δ γ (t)+ ∂ f 1 ∂ q Δq(t)+ ∂ f 1 ∂α Δ α (t) % & ' ( ) * + ∂ f 1 ∂δ E Δ δ E(t) + ∂ f 1 ∂δ T Δ δ T (t)+ ∂ f 1 ∂δ F Δ δ F(t) % & ' ( ) * + ∂ f 1 ∂ V wind ΔV wind + ∂ f 1 ∂α wind Δ α wind % & ' ( ) * ∂ f 1 ∂ V = 1 m C T V cos α N − C D V ( ) ρ V N 2 2 S + C T N cos α N − C D N ( ) ρ V N S % & ' ( ) * ∂ f 1 ∂γ = −1 m mg cos γ N [ ] = −g cos γ N ∂ f 1 ∂ q = −1 m C D q ρ V N 2 2 S % & ' ( ) * ∂ f 1 ∂α = −1 m C T N sin α N + C D α ( ) ρ V N 2 2 S % & ' ( ) * •  Coefficients in first row of F" Sensitivity of Velocity Dynamics to State Perturbations " C T V ≡ ∂ C T ∂ V C D V ≡ ∂ C D ∂ V C D q ≡ ∂ C D ∂ q C D α ≡ ∂ C D ∂α  V = C T cos α −C D ( ) ρ V 2 2 S − mgsin γ % & ' ( ) * m Sensitivity of Velocity Dynamics to Control and Disturbance Perturbations " ∂ f 1 ∂δ E = −1 m C D δ E ρ V N 2 2 S % & ' ( ) * ∂ f 1 ∂δ T = 1 m C T δ T cos α N ρ V N 2 2 S % & ' ( ) * ∂ f 1 ∂δ F = −1 m C D δ F ρ V N 2 2 S % & ' ( ) * •  Coefficients in first rows of G and L" ∂ f 1 ∂ V wind = − ∂ f 1 ∂ V ∂ f 1 ∂α wind = − ∂ f 1 ∂α C T δ T ≡ ∂ C T ∂δ T C D δ E ≡ ∂ C D ∂δ E C D δ F ≡ ∂ C D ∂δ F ∂ f 2 ∂ V = 1 mV N C T V sin α N + C L V ( ) ρ V N 2 2 S + C T N sin α N + C L N ( ) ρ V N S $ % & ' ( ) − 1 mV N 2 C T N sin α N + C L N ( ) ρ V N 2 2 S − mg cos γ N $ % & ' ( ) ∂ f 2 ∂γ = 1 mV N mgsin γ N [ ] = gsin γ N V N ∂ f 2 ∂ q = 1 mV N C L q ρ V N 2 2 S $ % & ' ( ) ∂ f 2 ∂α = 1 mV N C T N cos α N + C L α ( ) ρ V N 2 2 S $ % & ' ( ) •  Coefficients in second row of F" Sensitivity of Flight Path Angle Dynamics to State Perturbations " •  Coefficients in second row of G and L in Supplemental Slide! C T V ≡ ∂ C T ∂ V C L V ≡ ∂ C L ∂ V C L q ≡ ∂ C L ∂ q C L α ≡ ∂ C L ∂α  γ = C T sin α + C L ( ) ρ V 2 2 S − mg cos γ % & ' ( ) * mV ∂ f 3 ∂ V = 1 I yy C m V ρ V N 2 2 Sc + C m N ρ V N Sc # $ % & ' ( ∂ f 3 ∂γ = 0 ∂ f 3 ∂ q = 1 I yy C m q ρ V N 2 2 Sc # $ % & ' ( ∂ f 3 ∂α = 1 I yy C m α ρ V N 2 2 Sc # $ % & ' ( •  Coefficients in third row of F" Sensitivity of Pitch Rate Dynamics to State Perturbations " C m V ≡ ∂ C m ∂ V C m q ≡ ∂ C m ∂ q C m α ≡ ∂ C m ∂α  q = C m ρ V 2 2 ( ) Sc I yy ∂ f 4 ∂ V = − ∂ f 2 ∂ V ∂ f 4 ∂γ = − ∂ f 2 ∂γ •  Coefficients in fourth row of F" Sensitivity of Angle of Attack Dynamics to State Perturbations " ∂ f 4 ∂ q = 1− ∂ f 2 ∂ q ∂ f 4 ∂α = − ∂ f 2 ∂α  α =  θ −  γ = q −  γ Dimensional Stability and Control Derivatives 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 Lift mass ⇒ L ∝V  γ Moment moment of inertia (I yy ) ⇒ M ∝  q Dimensional Stability-Derivative Notation" ∂ f 1 ∂ V ≡ −D V  1 m C T V cos α N −C D V ( ) ρ V N 2 2 S + C T N cos α N −C D N ( ) ρ V N S & ' ( ) * + ∂ f 2 ∂α ≡ L α V N  1 mV N C T N cos α N + C L α ( ) ρ V N 2 2 S % & ' ( ) * ∂ f 3 ∂α ≡ M α  1 I yy C m α ρ V N 2 2 Sc % & ' ( ) * Thrust and drag effects are combined and represented by one symbol! Thrust and lift effects are combined and represented by one symbol! Longitudinal Stability Matrix" F Lon = F Ph F SP Ph F Ph SP F SP ! " # # $ % & & = −D V −g cos γ N −D q −D α L V V N g V N sin γ N L q V N L α V N M V 0 M q M α − L V V N − g V N sin γ N 1− L q V N * + , - . / − L α V N ! " # # # # # # # # $ % & & & & & & & & Effects of phugoid perturbations on phugoid motion" Effects of phugoid perturbations on short-period motion" Effects of short-period perturbations on phugoid motion" Effects of short-period perturbations on short- period motion" Primary Longitudinal Stability Derivatives" D V  −1 m C T V −C D V ( ) ρ V N 2 2 S + C T N −C D N ( ) ρ V N S # $ % & ' ( Assuming γ N  α N  0 L V V N  1 mV N C L V ρ V N 2 2 S + C L N ρ V N S " # $ % & ' − 1 mV N 2 C L N ρ V N 2 2 S − mg " # $ % & ' M q = 1 I yy C m q ρ V N 2 2 Sc " # $ % & ' M α = 1 I yy C m α ρ V N 2 2 Sc # $ % & ' ( L α V N  1 mV N C T N + C L α ( ) ρ V N 2 2 S # $ % & ' ( Origins of Stability Effects Velocity-Dependent Derivative Definitions" •  Air compressibility effects are a principal source of velocity dependence" C D M ≡ ∂ C D ∂ M = ∂ C D ∂ V / a ( ) = a ∂ C D ∂ V C D V ≡ ∂ C D ∂ V = 1 a # $ % & ' ( C D M C L V ≡ ∂ C L ∂ V = 1 a # $ % & ' ( C L M C m V ≡ ∂ C m ∂ V = 1 a # $ % & ' ( C m M C D M ≈ 0 C D M > 0 C D M < 0 a = Speed of Sound M = Mach number = V a Pitch-Moment Coefficient Sensitivity to Angle of Attack" M B = C m q Sc ≈ C m o + C m q q + C m α α ( ) q Sc C m α ≈ −C N α net h cm − h cp net ( ) ≈ −C L α net h cm − h cp net ( ) = −C L α net x cm − x cp net c $ % & ' ( ) = C m α wing + C m α ht Pitch-Rate Derivative Definitions" •  Pitch rate derivatives are often expressed in terms of a normalized pitch rate" C m q = ∂ C m ∂ q = c 2V N " # $ % & ' C m ˆ q C m ˆ q = ∂ C m ∂ ˆ q = ∂ C m ∂ qc 2V N ( ) = 2V N c " # $ % & ' C m q ˆ q = q c 2V N M q = ∂ M ∂ q = C m q ρ V N 2 2 ( ) Sc = C m ˆ q c 2V N # $ % & ' ( ρ V N 2 2 # $ % & ' ( S c = C m ˆ q ρ V N Sc 2 4 # $ % & ' ( often tabulated! used in pitch-rate equation! M B = C m q Sc ≈ C m o + C m q q + C m α α ( ) q Sc ≈ C m o + ∂C m ∂q q + C m α α $ % & ' ( ) q Sc •  Pitch acceleration sensitivity to pitch rate" Pitch-Rate Derivative Definitions" •  Pitch rate derivatives are often expressed in terms of a normalized pitch rate" C m q = ∂ C m ∂ q = c 2V N " # $ % & ' C m ˆ q C m ˆ q = ∂ C m ∂ ˆ q = ∂ C m ∂ qc 2V N ( ) = 2V N c " # $ % & ' C m q ˆ q = q c 2V N M B = C m q Sc ≈ C m o + C m q q + C m α α ( ) q Sc ≈ C m o + ∂C m ∂q q + C m α α $ % & ' ( ) q Sc •  Then" •  But dynamic equations require ∂C m /∂q " Angle of Attack Distribution Due to Pitch Rate" •  Aircraft pitching at a constant rate, q rad/s, produces a normal velocity distribution along x" Δw = −qΔx Δ α = Δw V N = −qΔx V N •  Corresponding angle of attack distribution" Δ α ht = ql ht V N •  Angle of attack perturbation at tail center of pressure" € l ht = horizontal tail distance from c.m. Horizontal Tail Lift Due to Pitch Rate" •  Incremental tail lift due to pitch rate, referenced to tail area, S ht " •  Lift coefficient sensitivity to pitch rate referenced to wing area" ΔL ht = ΔC L ht ( ) ht 1 2 ρ V N 2 S ht C L q ht ≡ ∂ ΔC L ht ( ) aircraft ∂ q = ∂ C L ht ∂α % & ' ( ) * aircraft l ht V N % & ' ( ) * ΔC L ht ( ) aircraft = ΔC L ht ( ) ht S ht S " # $ % & ' = ∂ C L ht ∂α " # $ % & ' aircraft Δ α * + , , - . / / = ∂ C L ht ∂α " # $ % & ' aircraft ql ht V N " # $ % & ' •  Incremental tail lift coefficient due to pitch rate, referenced to wing area, S" Moment Coefficient Sensitivity to Pitch Rate of the Horizontal Tail" •  Differential pitch moment due to pitch rate" ∂ ΔM ht ∂ q = C m q ht 1 2 ρ V N 2 Sc = −C L q ht l ht V N % & ' ( ) * 1 2 ρ V N 2 Sc = − ∂ C L ht ∂α % & ' ( ) * aircraft l ht V N % & ' ( ) * , - . . / 0 1 1 l ht c % & ' ( ) * 1 2 ρ V N 2 Sc C m q ht = − ∂ C L ht ∂α l ht V N $ % & ' ( ) l ht c $ % & ' ( ) = − ∂ C L ht ∂α l ht c $ % & ' ( ) 2 c V N $ % & ' ( ) •  Coefficient derivative with respect to pitch rate" •  Coefficient derivative with respect to normalized pitch rate is insensitive to velocity" C m ˆ q ht = ∂ C m ht ∂ ˆ q = ∂ C m ht ∂ qc 2V N ( ) = −2 ∂ C L ht ∂α l ht c $ % & ' ( ) 2 Comparison of Fourth- and Second-Order Dynamic Models •  0 - 100 sec" •  Reveals Phugoid Mode" 4 th -Order Initial-Condition Responses of Business Jet at Two Time Scales" •  0 - 6 sec" •  Reveals Short-Period Mode" •  Plotted over different periods of time" –  4 initial conditions" Second-Order Models of Longitudinal Motion" •  2 nd -Order Approximate Phugoid Equation" Δ  x Ph = Δ  V Δ  γ # $ % % & ' ( ( ≈ −D V −gcos γ N L V V N g V N sin γ N # $ % % % & ' ( ( ( ΔV Δ γ # $ % % & ' ( ( + T δ T L δ T V N # $ % % % & ' ( ( ( Δ δ T + −D V L V V N # $ % % % & ' ( ( ( ΔV wind Δ  x SP = Δ  q Δ  α # $ % % & ' ( ( ≈ M q M α 1 − L q V N + , - . / 0 − L α V N # $ % % % & ' ( ( ( Δq Δ α # $ % % & ' ( ( + M δ E −L δ E V N # $ % % % & ' ( ( ( Δ δ E + M α −L α V N # $ % % % & ' ( ( ( Δ α wind •  2 nd -Order Approximate Short-Period Equation" •  Assume off-diagonal blocks of (4 x 4) stability matrix are negligible" F Lon = F Ph ~ 0 ~ 0 F SP ! " # # $ % & & •  Phugoid Time Scale" •  Short-Period Time Scale" •  Full and approximate linear models" Comparison of Bizjet Fourth- and Second-Order Model Responses" •  Fourth Order" •  Second Order" •  Approximations are very close to 4 th -order values because natural frequencies are widely separated" Comparison of Bizjet Fourth- and Second-Order Models and Eigenvalues" Fourth-Order Model F = G = Eigenvalue Damping Freq. (rad/s) -0.0185 -9.8067 0 0 0 4.6645 -8.43e-03 + 1.24e-01j 6.78E-02 1.24E-01 0.0019 0 0 1.2709 0 0 -8.43e-03 - 1.24e-01j 6.78E-02 1.24E-01 0 0 -1.2794 -7.9856 -9.069 0 -1.28e+00 + 2.83e+00j 4.11E-01 3.10E+00 -0.0019 0 1 -1.2709 0 0 -1.28e+00 - 2.83e+00j 4.11E-01 3.10E+00 Phugoid Approximation F = G = Eigenvalue Damping Freq. (rad/s) -0.0185 -9.8067 4.6645 -9.25e-03 + 1.36e-01j 6.78E-02 1.37E-01 0.0019 0 0 -9.25e-03 - 1.36e-01j 6.78E-02 1.37E-01 Short-Period Approximation F = G = Eigenvalue Damping Freq. (rad/s) -1.2794 -7.9856 -9.069 -1.28e+00 + 2.83e+00j 4.11E-01 3.10E+00 1 -1.2709 0 -1.28e+00 - 2.83e+00j 4.11E-01 3.10E+00 Approximate Phugoid Roots " •  Approximate Phugoid Equation ( ! N = 0)" Δ  x Ph = Δ  V Δ  γ # $ % % & ' ( ( ≈ −D V −g L V V N 0 # $ % % % & ' ( ( ( ΔV Δ γ # $ % % & ' ( ( + T δ T L δ T V N # $ % % % & ' ( ( ( Δ δ T •  Characteristic polynomial" sI − F Ph = det sI − F Ph ( ) ≡ Δ(s) = s 2 + D V s + gL V / V N = s 2 + 2 ζω n s + ω n 2 ω n = gL V / V N ζ = D V 2 gL V / V N •  Natural frequency and damping ratio" ω n ≈ 2 g V N ; T = 2 π / ω n ζ ≈ 1 2 L / D ( ) N •  Neglecting compressibility effects" Effect of Airspeed and L/D on Approximate Phugoid Natural Frequency, Period, and Damping Ratio " ω n ≈ 2 g V N ≈ 13.87 /V N (m / s) ζ ≈ 1 2 L / D ( ) N Velocity Natural Frequency Period L/D Damping Ratio m/s rad/s sec 50 0.28 23 5 0.14 100 0.14 45 10 0.07 200 0.07 90 20 0.035 400 0.035 180 40 0.018 Neglecting compressibility effects" Period, T = 2 π / ω n ≈ 0.45V N sec Approximate Phugoid Response to a 10% Thrust Increase " •  What is the steady-state response?" Approximate Short-Period Roots " •  Approximate Short-Period Equation (L q = 0)" •  Characteristic polynomial" •  Natural frequency and damping ratio" Δ  x SP = Δ  q Δ  α # $ % % & ' ( ( ≈ M q M α 1 − L α V N # $ % % % & ' ( ( ( Δq Δ α # $ % % & ' ( ( + M δ E −L δ E V N # $ % % % & ' ( ( ( Δ δ E Δ(s) = s 2 + L α V N − M q $ % & ' ( ) s − M α + M q L α V N $ % & ' ( ) = s 2 + 2 ζω n s + ω n 2 ω n = − M α + M q L α V N $ % & ' ( ) ; ζ = L α V N − M q $ % & ' ( ) 2 − M α + M q L α V N $ % & ' ( ) Generally, L α > 0 M α < 0 M q < 0 Approximate Short-Period Response to a 0.1-Rad Pitch Control Step Input " Pitch Rate, rad/s! Angle of Attack, rad! [...]... Load Factor, g s at c.m.! Aft Pitch Control (Elevator)! Normal Load Factor, g s at c.m.! Forward Pitch Control (Canard)! Next Time: Lateral-Directional Dynamics Reading Flight Dynamics, 96-101, 574-582, 587-591 Virtual Textbook, Part 12 Flight Path Angle Dynamics " •  Nonlinear equation" γ = f2 = Supplemental Material ( ρV 2 1 1 % ρV 2 [T sin α + L − mg cos γ ] = mV 'CT sin α 2 S + CL 2 S − mg cos γ... + 2 Δγ (t) + 2 Δq(t) + 2 Δα (t) * ∂γ ∂q ∂α & ∂V ) ∂ f2 ∂ f2 ∂ f2 % ( Δδ E(t) + Δδ T (t) + Δδ F(t) * +' ∂δ T ∂δ F & ∂δ E ) % ∂ f2 ( ∂ f2 ΔVwind + Δα wind * +' ∂α wind & ∂Vwind ) Pitch Rate Dynamics " Angle of Attack Dynamics " •  Nonlinear equation" ( ) 2 M Cm ρV 2 Sc  = q = f3 = I yy I yy Cm may include thrust as well as aerodynamic effects! •  Third row of linearized equation" %∂ f ( ∂f ∂f ∂f  Δq(t)... of the Stability Matrix " ∂ f1 ≡ −DV ; ∂V •  Nonlinear equation" ∂ f2 Lα ≡ V N ∂α ∂ f3 ≡ Mα ∂α ∂ f4 L ≡ 1− q V ; N ∂q ∂ f4 L ≡− αV N ∂α Control and Disturbance Sensitivities in Flight Path Angle, Pitch Rate, and Angle-ofAttack Dynamics " ∂ f3 1 $ ∂ f2 ρV 2 ' 1 $ = Cm = C Lδ E N S ) & ∂δ E mVN % 2 ( ∂δ E I yy & δ E % ∂ f2 ρV 2 ' 1 $ $ = CTδT sin α N N S ) ∂ f3 = 1 Cm ∂δ T mVN & 2 ( ∂δ T I & δT % yy %... wind −Lα wind / VN & ( ( ( ( ( ( ' 2 ρVN ' −1 $  CT S m & δT 2 ) % ( Dδ T Lδ F 1  VN mV N MδE 1 = I yy 2 $ ρVN ' &C Lδ F 2 S ) % ( 2 $ ρVN ' &Cmδ E 2 Sc ) % ( Flight Motions" Effects of Airspeed, Altitude, Mass, and Moment of Inertia on Fighter Aircraft Short Period " Airspeed variation at constant altitude! Airspeed m/s 91 152 213 274 Dynamic Pressure P 2540 7040 13790 22790 Angle of Attack deg 14.6... 2π CL 3 αwing π =− 3AR C Lqˆwing = − Cmqˆwing Control- and DisturbanceEffect Matrices " •  Control-effect derivatives portray acceleration sensitivities to control input perturbations" G Lon •  Primary Longitudinal Control Derivatives " # −Dδ E % % Lδ E / VN =% MδE % % −Lδ E / VN $ Tδ T −Dδ F Lδ T / VN Lδ F / VN M δT MδF −Lδ T / VN −Lδ F / VN & ( ( ( ( ( ' Disturbance-effect derivatives portray acceleration... ∂Vwind ∂V ∂ f3 ∂f = 2 ∂α wind ∂α Horizontal Tail Lift Sensitivity to Angle of Attack " Wing Lift and Moment Coefficient Sensitivity to Pitch Rate " •  C Lqˆwing = −2C Lαwing ( hcm − 0.75 ) 2 (C ) Lαht aircraft " V % " ∂ε % "S % = $ tail ' $1− 'ηelas $ ht ' C Lαht #S & # VN & # ∂α & ( ) Cmqˆwing = −2C Lαwing ( hcm − 0.5 ) •  2 Straight-wing supersonic flow estimate (Etkin)" C Lqˆwing = −2C Lαwing ( hcm... 1.64 Damping Ratio 0.3 0.31 0.3 0.3 Mass Variation % -50 0 50 Natural Frequency rad/s 2.4 2.3 2.26 Period sec 2.62 2.74 2.78 Damping Ratio 0.44 0.31 0.26 Rapid damping" Pitch angle and rate response" Flight path angle reoriented by difference between pitch angle and angle of attack" Dornier Do-128D! Altitude variation with constant dynamic pressure! Airspeed m/s 122 152 213 274 Altitude m 2235 6095 . X-29! Next Time: Lateral-Directional Dynamics  Reading Flight Dynamics, 96-101, 574-582, 587-591 Virtual Textbook, Part 12 Supplemental Material Flight Path Angle Dynamics& quot; •  Second row. only.! http://www.princeton.edu/~stengel/MAE331.html ! http://www.princeton.edu/~stengel/FlightDynamics.html ! 6 th -Order Longitudinal Equations of Motion! •  Symmetric aircraft& quot; •  Motions in the vertical plane" • . Linearized Longitudinal Equations of Motion 
 Robert Stengel, Aircraft Flight Dynamics 
 MAE 331, 2012! •  6 th -order -> 4 th -order ->