Linearized Lateral-Directional Equations of Motion!

Transcription

1 Linearized Lateral-Directional Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 16 Learning Objectives 6 th -order -> 4 th -order -> hybrid equations Dynamic stability derivatives Dutch roll mode Roll and spiral modes Reading: Flight Dynamics Copyright 16 by Robert Stengel. All rights reserved. For educational use only Review Questions Why can we normally reduce the 6 th -order longitudinal equations to 4 th -order without compromising accuracy? What are the longitudinal modes of motion? Why are the hybrid 4 th -order equations useful? What are dimensional stability and control derivatives? Under what circumstances can the 4 th -order set be replaced by two nd -order sets? What benefits accrue from the separation, and what are the limitations? What is normal load factor? Would you rather have one so-so flying car or a good airplane and a good car?

2 6-Component Lateral-Directional Equations of Motion v Y B / m + gsin cos ru + pw y I cos sin u + cos cos + sin sin sin v + sin cos + cos sin sin w p I zz L B + I xz N B { I xz I yy I xx I zz p + I xz + I zz I zz I yy r }q I xxi zz I xz r I xz L B +I xx N B { I xz I yy I xx I zz r + I xz + I xx I xx I yy p }q I xxi zz I xz p + qsin + r cos tan qsin + r cos sec x 1 x x 3 x 4 x 5 x 6 x LD6 v y p r Side Velocity Crossrange Body-Axis Roll Rate Body-Axis Yaw Rate Roll Angle about Body x Axis Yaw Angle about Inertial x Axis 3 Eurofighter Typhoon 4- Component Lateral-Directional Equations of Motion Nonlinear Dynamic Equations, neglecting crossrange and yaw angle v Y B / m + gsin cos ru + pw p I zz L B + I xz N B { I xz I yy I xx I zz p + I xz + I zz I zz I yy r }q r I xz L B + I xx N B { I xz I yy I xx I zz r + I xz + I xx I xx I yy p }q p + qsin + r cos tan I xxi zz I xz I I I xx zz xz x 1 x x 3 x 4 x LD4 v p r Side Velocity Body-Axis Roll Rate Body-Axis Yaw Rate Roll Angle about Body x Axis 4

3 Lateral-Directional Equations of Motion Assuming Steady, Level Longitudinal Flight Longitudinal variables are constant v Y B / m + gsin cos N ru N + pw N I xx I zz I xz I xx I zz I xz p I zz L B + I xz N B r I xz L B + I xx N B q N p + r cos tan N N N N 5 Lateral-Directional Force and Moments in Steady, Level Flight Dynamic pressure is constant 1 Y B C YB NV N S 1 L B C lb NV N Sb 1 N B C nb NV N Sb Body-Axis Side Force Body-Axis Rolling Moment Body-Axis Yawing Moment 6

4 Linearized Lateral-Directional Equations of Motion in Steady, Level Flight 7 Body-Axis Perturbation Equations of Motion vt pt rt t F 11 F 1 F 13 F 14 vt F 1 F F 3 F 4 pt F 31 F 3 F 33 F 34 rt F 41 F 4 F 43 F 44 t + [ Control]+ [ Disturbance] 8

5 Body-Axis Perturbation Variables v p r Side Velocity Perturbation Body-Axis Roll Rate Perturbation Body-Axis Yaw Rate Perturbation Roll Angle about Body x Axis Perturbation u 1 u A R Aileron Perturbation Rudder Perturbation w 1 w v wind p wind Side Wind Perturbation Vortical Wind Perturbation 9 Vortical Wind Perturbations Wind Rotor Wake Vortex Tangential Velocity, ft/s Wake Vortex Radius, ft Wind Rotor 1

6 Side Velocity Dynamics Nonlinear equation v Y B / m + gsin cos N ru N + pw N First row of linearized dynamic equation [ ] + [ G 11 At + G 1 Rt ] + [ L 11 v wind + L 1 p wind ] vt F 11 vt + F 1 pt + F 13 rt + F 14 t 11 Side Velocity Sensitivity to State Perturbations v Y B / m + gsin cos N ru N + pw N Coefficients in first row of F F 11 1 m C N Y v S Y v F 1 1 m C N Y p S + w N Y p + w N F 13 1 m C N Y r S u N Y r u N F 14 gcos cos N g 1

7 Side Velocity Sensitivity to Control and Disturbance Perturbations Coefficients in first rows of G and L G 11 1 m C N Y A G 1 1 m C N Y R S Y A S Y R L 11 F 11 L 1 F 1 13 Roll Rate Dynamics Nonlinear equation I xx I zz I xz p I zz L B + I xz N B Second row of linearized dynamic equation [ ] + [ G 1 At + G Rt ] + [ L 1 v wind + L p wind ] pt F 1 vt + F pt + F 3 rt + F 4 t 14

8 Roll Rate Sensitivity to State Perturbations I xx I zz I xz p I zz L B + I xz N B Coefficients in second row of F F 1 F F 3 L I B zz v + I N B xz v L I B zz p + I N B xz p L I B zz r + I N B xz r I xx I zz I xz L v I xx I zz I xz L p I xx I zz I xz L r F 4 15 Yaw Rate Dynamics Nonlinear equation I xx I zz I xz r I xz L B + I xx N B Third row of linearized dynamic equation [ ] + [ G 31 At + G 3 Rt ] + [ L 31 v wind + L 3 p wind ] rt F 31 vt + F 3 pt + F 33 rt + F 34 t 16

9 Yaw Rate Sensitivity to State Perturbations r I xz L B + I xx N B I xx I zz I xz Coefficients in third row of F F 31 F 3 F 33 L I B xz v + I N B xx v L I B xz p + I N B xx p L I B xz r + I N B xx r I xx I zz I xz N v I xx I zz I xz N p I xx I zz I xz N r F Roll Angle Dynamics Nonlinear equation p + r cos tan N Fourth row of linearized dynamic equation [ ] + [ G 41 At + G 4 Rt ] + [ L 41 v wind + L 4 p wind ] t F 41 vt + F 4 pt + F 43 rt + F 44 t 18

10 Roll Angle Sensitivity to State Perturbations p + r cos tan N Coefficients in fourth row of F F 41 F 43 cos N tan N tan N F 4 1 F 44 r N sin N tan N 19 Linearized Lateral-Directional Response to Initial Yaw Rate ~Dutch-roll response of side velocity ~Spiral response of crossrange ~Dutch-roll response of roll and yaw rates ~Spiral response of yaw angle ~Roll response of roll angle

11 Dimensional Stability Derivatives Stability Matrix 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 Y v Y p + w N Y r u N gcos N L v L p L r N v N p N r 1 tan N 1 Dimensional Control- and Disturbance-Effect Derivatives Control Effect Matrix G 11 G 1 G 1 G G 31 G 3 G 41 G 4 Y A Y R L A L R N A N R Disturbance Effect Matrix L 11 L 1 L 1 L L 31 L 3 L 41 L 4 Y v Y p L v L p N v N p

12 LTI Body-Axis Perturbation Equations of Motion Rolling and yawing motions Y p + w N Y r u N gcos* N vt Y v pt L rt v L p L r N v N p N r t 1 tan* N + Y + A L + A N + A Y + R L + R N + R + At + Rt + Y v L v N v Y p L p N p v wind p wind vt pt rt t 3 Asymmetrical Aircraft: DC--1/ DC-3 with DC- right wing Quick fix to fly aircraft out of harm s way during WWII 4

13 Asymmetric Aircraft - WWII Blohm und Voss, BV 141 B + V 141 derivatives B + V P. Recent Asymmetric Aircraft Scaled Composites Ares Scaled Composites Boomerang

14 Oblique Wing Concepts High-speed benefits of wing sweep without the heavy structure and complex mechanism required for symmetric sweep Blohm und Voss, R. T. Jones, Handley-Page concepts Improved supersonic L/D by reduction of shockwave interference and elimination of the fuselage in flying-wing version 7 Handley-Page Oblique Wing Concepts Advantages 1- higher supersonic speed compared to delta planform Flying wing: no fuselage Issues Which way do the passengers face? Where is the cockpit? How are the engines and vertical surfaces swiveled? What does asymmetry do to stability and control? 8

15 NASA Oblique Wing Test Vehicles Stability and control issues abound: The fact that birds and insects are symmetric should give us a clue though they use huge asymmetry for control Strong aerodynamic and inertial longitudinal-lateral-directional coupling High side force at zero sideslip angle Torsional divergence of the leading wing Test vehicles: Various model airplanes, NASA AD-1, and NASA DFBW F-8 below, not built 9 Stability Axis Representation of Dynamics 3

16 Stability Axes Stability axes are an alternative set of body axes Nominal x axis is offset from the body centerline by the nominal angle of attack, N 31 Transformation from Original Body Axes to Stability Axes H S B cos N sin N 1 sin N cos N u v w S H B S u v w B p q r S S H B p q r B Side velocity v and pitch rate q are unchanged by the transformation 3

17 Stability-Axis State Vector Rotate body axes to stability axes vt pt rt t BodyAxis * + N * vt pt rt t StabilityAxis 33 vt pt rt t Stability-Axis State Vector Replace side velocity by sideslip angle StabilityAxis * +, v * V N +t pt rt t StabilityAxis 34

18 Stability-Axis State Vector t pt rt t Revise state order Stability*Axis + rt t pt t Stability*Axis Stability-Axis Yaw Rate Perturbation Sideslip Angle Perturbation Stability-Axis Roll Rate Perturbation Stability-Axis Roll Angle Perturbation 35 Stability-Axis Lateral-Directional Equations + rt t pt t N 1 A Y 1 A L 1 A S N 1 R Y 1 R L 1 R N r N N p + Y r. *1, - / S Y Y p g L r L L p 1 1 At 1 Rt + N Y L N p Y p L p S S rt t pt t wind p wind S 36

19 Why Modify the Equations? Dutch-roll motion is primarily described by stabilityaxis yaw rate and sideslip angle Dutch Roll, top Dutch Roll, front Roll and spiral motions are primarily described by stability-axis roll rate and roll angle Stable Spiral Roll Unstable Spiral 37 Why Modify the Equations? Effects of Dutch roll perturbations on Dutch roll motion Effects of roll-spiral perturbations on Dutch roll motion F LD F DR RS F DR DR F RS F RS F DR small small F RS F DR F RS Effects of Dutch roll perturbations on roll-spiral motion Effects of roll-spiral perturbations on roll-spiral motion... but are the off-diagonal blocks really small? 38

20 Stability, Control, and Disturbance Matrices F LD F DR RS F DR DR F RS F RS N r N N p Y r, 1 * + -. Y Y p g L r L L p 1 x 1 x x 3 x 4 r p G LD N A Y A L A N R Y R L R u 1 u A R L LD N N p Y Y p L L p w 1 w 39 A R nd -Order Approximate Modes of Lateral-Directional Motion 4

21 nd -Order Approximations in System Matrices F LD F DR F RS N r N Y r, 1 * + -. Y L p 1 G LD N R Y R L A L LD N Y L p 41 nd -Order Models of Lateral- Directional Motion Approximate Dutch Roll Equation x DR r N r N + Y r. *1, - / Y r + N 1 R Y 1 R 1 R + N Y wind Approximate Spiral-Roll Equation x RS p L p 1 p + L * A * A + L p p wind 4

22 Comparison of 4 th - and nd - Order Dynamic Models 43 Comparison of nd - and 4 th -Order Initial- Condition Responses of Business Jet 4 initial conditions [r,, p, ] Fourth-Order Response Second-Order Response Speed and damping of responses is adequately portrayed by nd -order models Roll-spiral modes have little effect on yaw rate and sideslip angle responses BUT Dutch roll mode has large effect on roll rate and roll angle responses 44

23 Next Time: Analysis of Time Response Reading: Flight Dynamics , Learning Objectives Methods of time-domain analysis Continuous- and discrete-time models Transient response to initial conditions and inputs Steady-state equilibrium response Phase-plane plots Response to sinusoidal input 45 Supplemental Material 46

24 Primary Lateral-Directional Control Derivatives L A C N l A Sb I xx N R C N n R Sb I zz 47 Initial Condition Response of Approximate Dutch Roll Mode r t t 48

25 Dutch roll Video Calspan Variable-Stability Learjet Dutch roll Oscillation 49