AE3-302

Flight DynamicsLecture Notes J.A. Mulder / W.H.J.J. van Staveren / J.C. van der Vaart / E. de Weerdt

March 7, 2007

Flight DynamicsLecture Notes J.A. Mulder / W.H.J.J. van Staveren / J.C. van der Vaart / E. de Weerdt March 7, 2007

Faculty of Aerospace Engineering

Delft University of Technology

Delft University of Technology

Copyright c Control and Simulation division, Delft University of Technology All rights reserved.

Table of Contents

Nomenclature 1 Introduction 1-1 Introduction to ight dynamics and control . . . . . . . . . . . . . . . . . . . . 1-1-1 1-1-2 Flight control surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flight control systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxiii 1 2 2 6 16 18

1-2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 Book outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

Equations of Motion

1921 22 22 24 25 26 27 30 30 32 33 33 38 42 43

2 Reference Frames 2-1 Overview of reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1-1 Inertial reference frame, FI . . . . . . . . . . . . . . . . . . . . . . . . . 2-1-2 2-1-3 2-1-4 2-1-5 2-1-6 2-1-7 Earth-centered, Earth-xed reference frame (FC ) . . . . . . . . . . . . . Normal Earth-xed reference frame, FE . . . . . . . . . . . . . . . . . . Vehicle carried normal Earth reference frame, FO . . . . . . . . . . . . . Body-xed reference frame, Fb . . . . . . . . . . . . . . . . . . . . . . . Aerodynamic (air-path) reference frame, Fa . . . . . . . . . . . . . . . . Kinematic (ight-path) reference frame, Fk . . . . . . . . . . . . . . . .

2-1-8 Vehicle reference frame, Fr . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 Transformations between reference frames . . . . . . . . . . . . . . . . . . . . . 2-2-1 Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2-2 2-2-3 2-2-4 Transformation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . Transformation from FI to FE . . . . . . . . . . . . . . . . . . . . . . . Transformation from FI to FO . . . . . . . . . . . . . . . . . . . . . . .

Flight Dynamics

iv 2-2-5 2-2-6 2-2-7 2-2-8 2-2-9

Table of Contents Transformation from FE to FO . . . . . . . . . . . . . . . . . . . . . . . Transformation from FO to Fb . . . . . . . . . . . . . . . . . . . . . . . Transformation from FO to Fa . . . . . . . . . . . . . . . . . . . . . . . Transformation from FO to Fk . . . . . . . . . . . . . . . . . . . . . . . Transformation form Fb to Fa . . . . . . . . . . . . . . . . . . . . . . . 45 46 49 50 52 53 55 56 63 66 66 67 68 68 69 70 70 72 77 83 83 91 95 99 99 102 102 103 104 105 106 111 112 112 113

2-2-10 Transformation form Fb to Fk . . . . . . . . . . . . . . . . . . . . . . . 2-2-11 Transformation from Fk to Fa . . . . . . . . . . . . . . . . . . . . . . . 2-3 Rotating reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3-1 2-3-2 2-3-3 2-3-4 2-3-5 2-3-6 2-3-7 2-4-1 2-4-2 2-4-3 Derivation of the angular velocity vector . . . . . . . . . . . . . . . . . . Derivation of E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EI Derivation of O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OI Derivation of O OE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivation of b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . bO Derivation of a and k aO kO . . . . . . . . . . . . . . . . . . . . . . . . Derivation of a and k . . . . . . . . . . . . . . . . . . . . . . . . . ab kb Euler angle singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quaternion transformation matrix . . . . . . . . . . . . . . . . . . . . . Quaternion propagation through time . . . . . . . . . . . . . . . . . . .

2-4 Quaternions, a brief introduction . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Derivation of the Equations of Motion 3-1 Newtons laws of physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2 Derivation of body translational acceleration . . . . . . . . . . . . . . . . . . . . 3-3 Derivation of body angular momentum derivative . . . . . . . . . . . . . . . . . 3-4 External forces and moments . . . . 3-4-1 External forces . . . . . . . . 3-4-2 External moments . . . . . . 3-5 Composition of equations of motion 3-5-1 3-5-2 Moment equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Force equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-5-3 Kinematic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6 Rotating masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7 Simplications of the equations of motion . . . . . . . . . . . . . . . . . . . . . 3-7-1 3-7-2 3-7-3 Non-rotating Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flat Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flat and Non-rotating Earth . . . . . . . . . . . . . . . . . . . . . . . .

Table of Contents 4 Linearized Equations of Motion

v 117

4-1 Linearization about arbitrary ight condition in arbitrary body-xed reference frame 119 4-1-1 4-1-2 4-1-3 4-2-1 4-2-2 Linearization of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linearization of forces and moments . . . . . . . . . . . . . . . . . . . . Linearization of kinematic relations . . . . . . . . . . . . . . . . . . . . . Linearized set of equations . . . . . . . . . . . . . . . . . . . . . . . . . Moments and products of inertia . . . . . . . . . . . . . . . . . . . . . . 121 122 128 130 131 132 134 135 138 148 151

4-2 Linearization about steady, straight, symmetric ight condition . . . . . . . . . .

4-3 Equations of motion in non-dimensional form . . . . . . . . . . . . . . . . . . . 4-3-1 4-3-2 Symmetric Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymmetric Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-4 Symmetric equations of motion in state-space Form . . . . . . . . . . . . . . . . 4-5 Asymmetric equations of motion in state-space form . . . . . . . . . . . . . . .

II

Static Stability Analysis

153155 156 156 169 175 184 195 206 219 220 226 233 238 241 255 258 262

5 Analysis of Steady Symmetric Flight 5-1 Aerodynamic center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1-1 5-1-2 5-1-3 5-1-4 5-1-5 5-1-6 5-1-7 Aerodynamic forces and moments acting on a wing . . . . . . . . . . . . A compact description for the aerodynamic moment characteristics . . .

The role of the aerodynamic center and Cmac in static stability . . . . . . The position of the aerodynamic center and Cmac of a wing . . . . . . . Inuence of wing shape on the aerodynamic center and Cmac . . . . . . . Characteristics of wing-fuselage-nacelle congurations . . . . . . . . . . . Aerodynamic eects of nacelles . . . . . . . . . . . . . . . . . . . . . . .

5-2 Equilibrium in steady, straight, symmetric Flight . . . . . . . . . . . . . . . . . . 5-2-1 5-2-2 5-2-3 5-2-4 5-2-5 5-2-6 5-2-7 Conditions for equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . Normal force on the horizontal tailplane . . . . . . . . . . . . . . . . . . Hinge moment of the elevator . . . . . . . . . . . . . . . . . . . . . . . Flow direction and dynamic pressure at the horizontal tailplane . . . . . . Eect of airspeed and center of gravity on tail load . . . . . . . . . . . . Elevator deection required for moment equilibrium . . . . . . . . . . . . Stick forces in steady ight . . . . . . . . . . . . . . . . . . . . . . . . .

Flight Dynamics

vi 6 Longitudinal Stability and Control Derivatives

Table of Contents 267 268 270 270 271 272 276 276 278 279 280 291 295 295 295 295 296 299 299 303 303 307 312 324 328 329 329 331 336 339 340 343 345 347 348 348

6-1 Aerodynamic forces in the nominal ight condition . . . . . . . . . . . . . . . . 6-2 Derivatives with respect to airspeed . . . . . . . . . . . . . . . . . . . . . . . . 6-2-1 6-2-2 6-2-3 6-3-1 6-3-2 6-3-3 Stability derivative CXu . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative CZu . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative Cmu . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative CX . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative CZ . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative Cm . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-3 Derivatives with respect to angle of attack . . . . . . . . . . . . . . . . . . . . .

6-4 Derivatives with respect to pitching velocity . . . . . . . . . . . . . . . . . . . . 6-5 Derivatives with respect to the acceleration along the top axis . . . . . . . . . . 6-6 Derivatives with respect to the elevator angle . . . . . . . . . . . . . . . . . . . 6-6-1 6-6-2 6-6-3 Control derivative CXe . . . . . . . . . . . . . . . . . . . . . . . . . . . Control derivative CZe . . . . . . . . . . . . . . . . . . . . . . . . . . . Control derivative Cme . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-7 Symmetric inertial parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Lateral Stability and Control Derivatives 7-1 Aerodynamic force and moments due to side slipping, rolling and yawing . . . . . 7-2 Stability derivatives with respect to the sideslip angle . . . . . . . . . . . . . . . 7-2-1 7-2-2 7-2-3 7-2-4 7-2-5 7-3-1 7-3-2 7-3-3 7-4-1 7-4-2 7-4-3 Stability derivative CY . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative C . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative Cm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative CYp . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative Cp . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative Cnp . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative CYr . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative Cr . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability derivative Cnr . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-3 Stability derivatives with respect to roll rate . . . . . . . . . . . . . . . . . . . .

7-4 Stability derivatives with respect to yaw rate . . . . . . . . . . . . . . . . . . . .

7-5 The forces and moments due to aileron, rudder and spoiler deections . . . . . . 7-6 Aileron control derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6-1 Control derivative CYa . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table of Contents 7-6-2 7-6-3 Control derivative Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . Control derivative Cna . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii 350 351 352 355 355 355 357 359 359 360 365 368 377 379 381 385 388 390 392 403 408 410 411 412 417 420 426 431 437 442 445 445 445 448 448 450 450

7-7 Spoiler control derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8 Rudder control derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8-1 Control derivative CYr . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8-2 7-8-3 Control derivative Cr . . . . . . . . . . . . . . . . . . . . . . . . . . . Control derivative Cnr . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 Longitudinal Stability and Control in Steady Flight 8-1 Stick Fixed Static Longitudinal Stability . . . . . . . . . . . . . . . . . . . . . . 8-1-1 8-1-2 8-1-3 8-1-4 8-1-5 8-1-6 8-2-1 8-2-2 8-2-3 8-2-4 8-2-5 8-2-6 8-2-7 8-3-1 8-3-2 8-3-3 8-3-4 8-3-5 8-3-6 Stick xed static longitudinal stability in gliding ight . . . . . . . . . . . Neutral point, stick xed . . . . . . . . . . . . . . . . . . . . . . . . . . Elevator trim curve and elevator trim stability . . . . . . . . . . . . . . . Inuence of various parameters on elevator trim curve . . . . . . . . . . . Static longitudinal stability of tailless aircraft . . . . . . . . . . . . . . . Determination Cme from measurements in ight . . . . . . . . . . . . . Stick free static longitudinal stability in gliding ight . . . . . . . . . . . Neutral point, stick free . . . . . . . . . . . . . . . . . . . . . . . . . . . Elevator stick force curves and elevator stick force Stability . . . . . . . . Eect of center of gravity and mass unbalance on stick force stability . . Inuence of design variables on control forces . . . . . . . . . . . . . . . Control forces a pilot can exert . . . . . . . . . . . . . . . . . . . . . . . Airworthiness requirements for steady straight symmetric ight . . . . . . Characteristics of longitudinal control in turning ight . . . . . . . . . . . The stick displacement per g . . . . . . . . . . . . . . . . . . . . . . . . The manoeuvre point, stick xed . . . . . . . . . . . . . . . . . . . . . . The stick force per g . . . . . . . . . . . . . . . . . . . . . . . . . . . . The manoeuvre point, stick free . . . . . . . . . . . . . . . . . . . . . . Non-aerodynamic means to inuence the stick force Per g . . . . . . . .

8-2 Stick Free Static Longitudinal Stability . . . . . . . . . . . . . . . . . . . . . . .

8-3 Longitudinal Control in Pull-Up Manoeuvres and Steady Turns . . . . . . . . . .

9 Lateral Stability and Control in Steady Flight 9-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2 Equations of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3 Steady Horizontal Turns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3-1 9-3-2 9-3-3 Turns Using the Ailerons Only, r = 0 . . . . . . . . . . . . . . . . . . . Turns Using the Rudder Only, a = 0 . . . . . . . . . . . . . . . . . . . . Coordinated Turns, = 0 . . . . . . . . . . . . . . . . . . . . . . . . . .

Flight Dynamics

viii

Table of Contents 9-3-4 Flat Turns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-4 Steady, Straight, Sideslipping Flight . . . . . . . . . . . . . . . . . . . . . . . . 9-5 Steady Straight Flight with One or More Engines Inoperative . . . . . . . . . . . 9-6 Steady Rolling Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-7 Control Forces and Hinge Moments for Lateral Control . . . . . . . . . . . . . . 9-7-1 9-7-2 Roll Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rudder Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 453 458 463 470 470 472

III

Dynamic Stability Analysis

473475 477 489 490 494 503 503 507 513 517 519 535

10 Analysis of Symmetric Equations of Motion 10-1 Solution of the equations of motion . . . . . . . . . . . . . . . . . . . . . . . . 10-2 Stability criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3 The complete solution of the equations of motion . . . . . . . . . . . . . . . . . 10-4 Approximate solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Analysis of Asymmetric Equations of Motion 11-1 Solution of the equations of motion . . . . . . . . . . . . . . . . . . . . . . . . 11-2 General character of the asymmetric motions . . . . . . . . . . . . . . . . . . . 11-3 Routh-Hurwitz stability criteria for the asymmetric motions . . . . . . . . . . . . 11-4 Spiral and Dutch roll mode, the lateral stability diagram . . . . . . . . . . . . . 11-5 Approximate solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A List of Symbols

B Aircraft Parameters 547 B-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 B-2 Linearized Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 B-3 LTI-System Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

C MATLAB Files 559 C-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 C-2 Root Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 C-3 Eigenvalue Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 560

List of Figures

1 2 3

Denition of the mean aerodynamic cord and related parameters . . . . . . . . . Parameters dening the geometry of the wing . . . . . . . . . . . . . . . . . . . Examples of denitions of the vertical tailplane area (see also NACA TN 775) .

xxviii xxix xxx 3 5 6 6 7 7 9 10 11 12 14 15 21 23 23 24 25 26 27 28

1-1 The Wright Flyer I [136] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 Boeing 767 3D-view [113] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 Concorde 3D-view [113] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 X-29 in ight [35] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 737-400 Flight and ground spoiler deployment on landing [32] . . . . . . . . . . 1-6 Boeing 737 slats [32] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7 Mechanical ight control system example [132] . . . . . . . . . . . . . . . . . . 1-8 KC-135/707 Aileron Balance Panel Installation [132] . . . . . . . . . . . . . . . 1-9 Boeing 707 Flight Control Surfaces [132] . . . . . . . . . . . . . . . . . . . . . . 1-10 Boeing 707 Stabilizer-Actuated Tab Mechanism [132] . . . . . . . . . . . . . . . 1-11 Boeing 727 Aileron Control and Trim Systems [132] . . . . . . . . . . . . . . . . 1-12 Boeing 747 Elevator Control System [132] . . . . . . . . . . . . . . . . . . . . . 2-1 Aircraft velocity vector decomposition . . . . . . . . . . . . . . . . . . . . . . . 2-2 Denition of ecliptic plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3 Denition of earth-centered inertial reference frame . . . . . . . . . . . . . . . . 2-4 Polar motion - precession track . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5 Denition of earth-centered, earth-xed reference frame . . . . . . . . . . . . . . 2-6 Earth geoid and latitude denitions . . . . . . . . . . . . . . . . . . . . . . . . . 2-7 Normal earth-xed reference frame for spherical earth . . . . . . . . . . . . . . . 2-8 Vehicle carried normal earth reference frame for spherical earth . . . . . . . . . .

Flight Dynamics

x 2-9 Body-xed Reference Frame

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 29 31 32 33 36 37 37 38 40 40 41 44 48 50 53 57 59 61 64 71 73 73 74 85 86 101 107 120 123 125 127 133 141

2-10 Stability Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11 Aerodynamic Reference Frame in relation to Body-xed Reference Frame . . . . 2-12 Left-handed (!) Aircraft Reference Frame Fr . . . . . . . . . . . . . . . . . . . 2-13 Euler angles dene the attitude of Fb irrespectively of translation . . . . . . . . . 2-14 Euler angles tutorial: initial and end body orientation . . . . . . . . . . . . . . . 2-15 Euler angles tutorial: rotation sequence z y x . . . . . . . . . . . . . . 2-16 Euler angles tutorial: rotation sequence x y z . . . . . . . . . . . . . . 2-17 Simultaneous vector decomposition in two reference frames . . . . . . . . . . . 2-18 Vector decomposition for rotation about X-axis . . . . . . . . . . . . . . . . . . 2-19 Vector decomposition for rotation about Y -axis . . . . . . . . . . . . . . . . . . 2-20 Vector decomposition for rotation about Z-axis . . . . . . . . . . . . . . . . . . 2-21 Transformation from inertial reference frame to normal earth-xed reference frame 2-22 Transformation from vehicle carried normal earth reference frame FO to the bodyxed reference frame Fb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-23 Transformation from vehicle carried normal earth reference frame FO to the aerodynamic reference frame Fa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-24 Transformation from the body-xed reference frame Fb to the aerodynamic reference frame Fa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-25 General situation for vector dierentiation . . . . . . . . . . . . . . . . . . . . . 2-26 Change of a vector due to rotation vector . . . . . . . . . . . . . . . . . . . . 2-27 Example - derivation of pilot velocity with respect to FO . . . . . . . . . . . . . 2-28 Angular velocity vector E EI explanation . . . . . . . . . . . . . . . . . . . . . . 2-29 Euler angle singularity: Aircraft in vertical climb . . . . . . . . . . . . . . . . . . 2-30 Direction cosines equality for Euler axis rotations . . . . . . . . . . . . . . . . . 2-31 Rotating a reference frame about the Euler axis . . . . . . . . . . . . . . . . . . 2-32 Rotating an arbitrary vector about the Euler axis . . . . . . . . . . . . . . . . . 3-1 Vector denition for point mass P in a body. . . . . . . . . . . . . . . . . . . . 3-2 Moment about center of mass due to force on point mass . . . . . . . . . . . . . 3-3 Earth gravity vector denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4 Engine rotation axis denition . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-1 Linearization about a single point and a trajectory . . . . . . . . . . . . . . . . . 4-2 Example of a time-function: u(.) . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3 Airow tutorial: example of time-dependent aerodynamic characteristics . . . . . 4-4 The impossibility of a force in the XB -direction arising from a variation in velocity dv along the YB -axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 The angles between the principal inertial axes and the aircraft reference axes . . 4-6 Relation between moment of inertia Iy and the mass . . . . . . . . . . . . . . .

List of Figures 5-1 The aerodynamic forces and the moment acting on a wing in symmetric ight .

xi 156 158 159 159 160 161 162

5-2 CN , CL and CT , CD as functions of for the case of a Fokker F-27 wing (from reference [149]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3 CN as a function of CT for a Fokker F-27 wing (from reference [149]) . . . . . 5-4 Rectangular wing conguration with zero-twist, = 1 , as used for linear potential 3 ow simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5 Simulated CN , CL and CT , CD as functions of for the wing conguration depicted in gure 5-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6 Simulated CN as a function of CT and CL as a function of CD for the wing conguration depicted in gure 5-4 . . . . . . . . . . . . . . . . . . . . . . . . 5-7 The variation of the moment with changes in the reference point . . . . . . . .

5-8 Location of point (x, z) with respect to the position of the mean aerodynamic chord 162 5-9 Moment curves for various positions of the reference point for the Fokker F-27 wing (from reference [149]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10 Simulated moment curves for various positions of the reference point for the wing conguration of gure 5-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-11 Denition of the angle 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 165 166

5-12 Denition of the center of pressure relative on the mean aerodynamic chord (mac) 167 5-13 The magnitude of C R and the position of the line of action of C R as a function of the angle of attack for Fokker F-27 wing (from reference [149]) . . . . . . . . 5-14 The position of the center of pressure as a function of the angle of attack for Fokker F-27 wing (from reference [149]) . . . . . . . . . . . . . . . . . . . . . . 5-15 The aerodynamic forces and moment about an arbitrary point at two adjacent values of the angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-16 The line of action of the dierence between aerodynamic force vectors at adjacent values of the angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-17 The position of the aerodynamic center . . . . . . . . . . . . . . . . . . . . . . 5-19 The moment about an arbitrary point (x, z) if the position of the ac and the Cmac are known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20 Equilibrium and stability of a wing, suspension point ahead of the ac, Cmac negative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21 Equilibrium and stability of a wing, suspension point ahead of the ac, Cmac positive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22 Equilibrium and stability of a wing, suspension point in the aerodynamic center, Cmac equals 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-23 Equilibrium and stability of a wing, suspension point in the aerodynamic center, Cmac negative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 168 169 170 172

5-18 The aerodynamic forces and moment, using the ac as the moment reference point 174 176 178 179 180 181

5-24 Equilibrium and stability of a wing, suspension point behind the ac, Cmac negative 182 5-25 Equilibrium and stability of a wing, suspension point behind the ac, Cmac positive 183 5-26 The basic, additional and total lift distribution of a wing with negative twist . . 185

Flight Dynamics

xii

List of Figures 5-27 Numerical simulation of the basic lift distribution (cb c at = CL =0 ), additional lift distribution (ca c) and total lift distribution (c c) for a wing without wingtwist (left, = 0o ) and a wing including wing-twist (right, = 3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c = 2 m, = 0o , = 0o , = 1, r = 0o , = 3o , = 5o . Results from a source/doublet singularity panelmethod, linearized potential ow. . . . . . . . . . . . . . . . . . . . . . . . . . 5-28 Numerical simulation of the basic lift distribution (cb c at = CL =0 ), additional lift distribution (ca c) and total lift distribution (c c) for a wing without wingtwist (left, = 0o ) and a wing including wing-twist (right, = 3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c = 2 m, = 0o , = 37o , = 1, r = 0o , = 3o , = 5o . Results from a source/doublet singularity panelmethod, linearized potential ow. . . . . . . . . . . . . . . . . . . . . . . . . . 5-29 Numerical simulation of the basic lift distribution (cb c at = CL =0 ), additional lift distribution (ca c) and total lift distribution (c c) for a wing without wingtwist (left, = 0o ) and a wing including wing-twist (right, = 3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c = 2 m, = 0o , = 0o , = 1 , 3 r = 0o , = 3o , = 5o . Results from a source/doublet singularity panelmethod, linearized potential ow. . . . . . . . . . . . . . . . . . . . . . . . . . 5-30 Numerical simulation of the basic lift distribution (cb c at = CL =0 ), additional lift distribution (ca c) and total lift distribution (c c) for a wing without wingtwist (left, = 0o ) and a wing including wing-twist (right, = 3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c = 2 m, = 0o , = 37o , = 1 , 3 r = 0o , = 3o , = 5o . Results from a source/doublet singularity panelmethod, linearized potential ow. . . . . . . . . . . . . . . . . . . . . . . . . . 5-31 Numerical simulation of the basic lift distribution (cb c at = CL =0 ), additional lift distribution (ca c) and total lift distribution (c c) for a wing without wingtwist (left, = 0o ) and a wing including wing-twist (right, = 3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c = 2 m, = 5o , = 0o , = 1 , 3 o , = 3o , = 5o . Results from a source/doublet singularity panelr = 0 method, linearized potential ow. . . . . . . . . . . . . . . . . . . . . . . . . . 5-32 Numerical simulation of the basic lift distribution (cb c at = CL =0 ), additional lift distribution (ca c) and total lift distribution (c c) for a wing without wingtwist (left, = 0o ) and a wing including wing-twist (right, = 3o ); NACA 0012 airfoil, b = 16 m, S = 32 m2 , c = 2 m, = 5o , = 37o , = 1 , 3 r = 0o , = 3o , = 5o . Results from a source/doublet singularity panelmethod, linearized potential ow. . . . . . . . . . . . . . . . . . . . . . . . . . 5-33 The calculation of the position of the ac and Cmac for a wing with arbitrary geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-34 The inuence of camber on the moment curves (from references [119, 118]) . .

187

188

189

190

191

192 193 196 198 199 201 202

5-35 The variation of Cmac with wing sweep and angle of twist (from reference [49])197 5-36 The positions of the local acs of swept wings and delta wings . . . . . . . . . . 5-37 The inuence of wing sweep on the moment curves of wings of aspect ratio A = 4, t = 0.6, c = 0.06, M = 0.40 (from reference [169]) . . . . . . . . . . . . . . . 5-38 The inuence of taper ratio on the moment curves of swept wings, A = 4, = t c = 0.06, M = 0.40 (from reference [87]) . . . . . . . . . . . . . . . . . . . . . 45o ,

5-39 The position of the ac of delta wings and concept wings as a function of aspect ratio, = 45o (from reference [159]) . . . . . . . . . . . . . . . . . . . . . . .

List of Figures 5-40 The inuence of aspect ratio on the moment curve of swept wings, = 45o , t = 0.6, c = 0.06, M = 0.40 (from reference [97]) . . . . . . . . . . . . . . . . 5-41 The moment curve of a straight wing and a swept wing, A = 5, = 1, Re = 1106 (DUT measurements) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-42 Boundaries for the combinations of aspect ratio and sweep angle for which destabilizing changes in the moment curves may be expected (from references [58, 75]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-43 Pressure distribution over a fuselage in inviscid ow . . . . . . . . . . . . . . . 5-44 Numerical simulation of particle lines (top) and pressure distribution over a fuselage in inviscid ow, = 10o , linearized potential ow . . . . . . . . . . . . . . . . . 5-45 The variation of the angle of attack and the normal force along the fuselage axis in a wing induced ow eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-46 Wing-fuselage conguration as used in the ow eld simulations depicted in gures 5-47 and 5-48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-47 Numerical simulation of the velocity eld of a wing-fuselage conguration, = 10o 5-48 Numerical simulation of the velocity eld of a wing-fuselage conguration, = 10o , detail of gure 5-47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .f 5-49 The inuence of aspect ratio on the variation of d along the fuselage axis ( = 0) (from reference [143]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f 5-50 The inuence of sweep angle of wings with A = on d along the fuselage () (from reference [143]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

203 204

205 208 209 211 212 .212 213 214 214

d

d

5-51 Fuselage induced variations of the local angle of attack along the wing span . . 215 5-52 Numerical simulation of the fuselage induced upwash along the wing span, = 5o 215 5-53 The c c-distribution along the span of a wing with and without a fuselage (from reference [54]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-54 The change in wing moment due to loss in lift over the wing center part . . . . 5-55 Shift of ac due to fuselage eects, as a function of wing sweep angle(from reference [141]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-56 Flow simulations over a nacelle, = computed with a source/doublet singularity panel-method, linearized potential ow . . . . . . . . . . . . . . . . . . . 5-57 Generic Large Transport Aircraft (GLTA), no nacelles . . . . . . . . . . . . . . . 5-58 Particle lines in an XB OZB -plane (top) and non-dimensional sectional pressure distribution Cp (bottom) for the left wing of a generic transport aircraft model, no nacelles, = 10o , computed with a source/doublet singularity panel-method, linearized potential ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-59 Particle lines in an XB OZB -plane (top) and non-dimensional sectional pressure distribution Cp (bottom) for the left wing of a twin-engined transport aircraft , = 10o , computed with a source/doublet singularity panel-method, linearized potential ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-60 Spanwise lift distribution c c at an angle-of-attack = 10o twin-engined transport aircraft conguration , source/doublet singularity panel-method, linearized potential ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-61 Particle lines in an XB OZB -plane and non-dimensional sectional pressure distribution coecients Cp for the left wing of a four-engined transport aircraft , = 10o , source/doublet singularity panel-method, linearized potential ow . . . . . . . . 15o , 216 217 218 219 220

221

222

223

224

Flight Dynamics

xiv

List of Figures 5-62 Four-engined transport aircraft with spanwise lift distribution c c at an angle-ofattack = 10o , source/doublet singularity panel-method, linearized potential ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-63 The equilibrium in steady, symmetric ight . . . . . . . . . . . . . . . . . . . . 5-64 The forces and moments in steady, straight, symmetric ight . . . . . . . . . . 5-65 Geometry parameters of the horizontal tailplane, elevator and trim tab, and their positive deections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-66 Simplied version of the equilibrium of forces and moments . . . . . . . . . . . 5-67 Pressure distributions over the tailplane chord due to h , e and te . . . . . . . 5-68 The normal force coecient CNh as a function of h , e and te for the tailplane of the Fokker F-27 (Wind tunnel measurements from reference [17]) . . . . . . 5-69 The quotient reference [21])CNh CNh

225 227 229 231 234 236 237

as a function of

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ce ch

for various trailing edge angles (from 238 239 242 243 244 244

5-70 The hinge moment coecient Che as a function of h , e and te for a tailplane of the Fokker F-27 (wind tunnel measurements from reference [17]) . . . . . . . 5-71 Overbalance with respect to angle of attack and elevator angle. . . . . . . . . . 5-72 The angle of attack h of the horizontal tailplane, h = + ih . . . . . . . 5-73 The contribution of the lifting and free vortices to the induced vertical velocities in front of and behind a wing . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-74 The position of the vortex plane behind the wing . . . . . . . . . . . . . . . . . 5-75 The vertical displacement of the wake and the tailplane relative to the ow eldxh

due to an increase in angle of attack ; zh =xh xw

dx = (xh xw ),

zwake =xw

(x) dx, zwake < zh as (x) is smaller than . With 245 246 247

increasing angle of attack the horizontal stabilizer moves downwards through the wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-76 Numerical simulation of the deformation and wake roll-up of the vortex sheet behind a wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .d 5-77 Computed values of d behind a wing of elliptical lift distribution, as a function of aspect ratio and distance behind the wing . . . . . . . . . . . . . . . . . . . . .

5-78 Numerical simulation of for wings with varying taper ratio (CL = 1.0, = 0o , A = 6), computed with a source/doublet singularity method, also see reference [143] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c 5-79 Numerical simulation of c2b for wings with varying sweep angle (CL = 1.0, = 1, A = 6), computed with a source/doublet singularity method, also see reference [143] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

c c 2b

248

249

5-80 Numerical simulation of downwash angles in the plane of symmetry behind a wing ( = 13o , NACA 23014 airfoil, A = 6, = 0.5 and CL = 1.1615, , computed with a source/doublet singularity method, see also reference [146]) . . . . . . . 5-81 Cessna Ce550 Citation II conguration with and without nacelles . . . . . . . . .d 5-82 The eect of nacelles on d (at X = 5.0 m, Z = 1.5 m) aft of the wing of a Cessna Ce550 Citation II, computed with a source/doublet singularity panelmethod, linearized potential ow . . . . . . . . . . . . . . . . . . . . . . . . . .

250 252

253

List of Figures 5-83 Calculation of ground eect . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv 254 256 257 258

5-84 The ground eect on the downwash angle and the location of the wake relative to the horizontal tailplane of a Siebel 204 D-1 aircraft . . . . . . . . . . . . . . 5-85 Simplied picture of the equilibrium of moments . . . . . . . . . . . . . . . . . 5-86 The variation of the tail toad with airspeed at dierent c.g. positions and values of Cmac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-87 The tail load as a function of airspeed at two c.g. positions for De Havilland Mosquito II F (from reference [26]), W = 6800 kg, S = 41.8 m2 , lh = 8.0 m and c = 2.81 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-88 The positive direction of control deections, control forces, control surface deections and hinge moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-89 The relation between the control force and the hinge moment . . . . . . . . . . 5-90 Contributions to the hinge moment . . . . . . . . . . . . . . . . . . . . . . . .

259 263 263 264 268

6-1 The attitude of the stability reference frame relative to the Xr - and Zr -axis, after a disturbance from the equilibrium ight condition . . . . . . . . . . . . . . . . 6-2 The denition of aircraft parts wing, horizontal stabilizer, pylon, nacelles, vertical n and fuselage for the Cessna Ce550 Citation II model (starting from the left top gure, in clockwise direction) . . . . . . . . . . . . . . . . . . . . . 6-3 Calculated contributions of various aircraft parts to the force curve CX versus for the Cessna Ce550 Citation II . . . . . . . . . . . . . . . . . . . . . . . . . 6-4 Calculated contributions of various aircraft parts to the force curve CZ versus for the Cessna Ce550 Citation II . . . . . . . . . . . . . . . . . . . . . . . . . 6-5 Calculated contributions of various aircraft parts to the moment curve Cm versus for the Cessna Ce550 Citation II . . . . . . . . . . . . . . . . . . . . . . . . 6-6 The pure q-motion, with =xxc.g. R

277 278 280 281 282 283 284 285 286 286 287 287 289 291 292 300 301 302

=

xxc.g. q c c V

. . . . . . . . . . . . . . .

6-7 Harmonic q-motion, = constant, = . . . . . . . . . . . . . . . . . . . . . 6-8 Harmonic -motion, q = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9 Combined harmonic q- and -motion, q = , = constant 6-10 Aircraft in a curved ow eld . . . . . . . . . . . . 6-11 Equivalent curved aircraft in a straight ow eld with Fr . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12 Equivalent curved aircraft in a straight ow eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . aircraft frame of reference, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .q c V q c V

6-13 Equivalent curved aircraft in a straight ow eld, 3-dimensional view

6-14 Calculated contributions of various aircraft parts to the force curve CX versus for the Cessna Ce550 Citation II . . . . . . . . . . . . . . . . . . . . . . . . . 6-15 Calculated contributions of various aircraft parts to the force curve CZ versus for the Cessna Ce550 Citation II . . . . . . . . . . . . . . . . . . . . . . . . . 6-16 Calculated contributions of various aircraft parts to the moment curve Cm versus q c V for the Cessna Ce550 Citation II . . . . . . . . . . . . . . . . . . . . . . . . 7-1 The six degrees of freedom of a rigid aircraft 7-3 Asymmetric force and moments . . . . . . . . . . . . . . . . . . .

7-2 Denition of the angle of attack and sideslip angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Flight Dynamics

xvi

List of Figures 7-4 CY as a function of measured on a model of the Fokker F-27 in gliding ight (from references [148, 150, 24]) . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5 Relation between the sideslip angle , the sidewash angle and the angle of attack v at the vertical tailplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6 Aerodynamic force coecient CY as a function of for the Cessna Ce550 Citation II, = 0o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7 The origin of the dierence in angles of attack at the left and right wing for a wing with dihedral in sideslipping ight . . . . . . . . . . . . . . . . . . . . . . . . . 7-8 The origin of the dierence in the velocities over the left and right wing for a swept wing in sideslipping ight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9 The variation of C with CL for swept wings . . . . . . . . . . . . . . . . . . . 7-10 High-wing-fuselage conguration with the calculated velocity eld in sideslipping ight ( = 10o , = 0o ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-11 Low-wing-fuselage conguration with the calculated velocity eld in sideslipping ight ( = 10o , = 0o ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-12 Calculated particle traces around the bottom half of the Ce550 Citation II, sideslipping ight = 10o , = 0o related to the magnitude of the airows velocity along traces) . . . . . . . . . . . . . . . . . . . . . . . . . . . fuselage of the Cessna (note: the speedbar is the calculated particle . . . . . . . . . . . . .

304 306 307 309 310 310 313 314

315 316 316 317 319 320 321

7-13 Origin of a rolling moment caused by wing-fuselage interactions in sideslipping ight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-14 Aerodynamic moment coecient C as a function of for the Cessna Ce550 Citation II, = 0o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-15 The position of the point of action of (CYv )v relative to the X- and Z-axis in the stability reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16 The contribution of the slipstream to C . . . . . . . . . . . . . . . . . . . . . 7-17 Cn as a function of measured on a model of the Fokker F-27 in gliding ight (from references [148, 150, 24]) . . . . . . . . . . . . . . . . . . . . . . . . . . 7-18 The sidewash induced by the fuselage at the vertical tailplane . . . . . . . . . . 7-19 Calculated particle traces around the fuselage of the Cessna Ce550 Citation II, sideslipping ight = 10o , = 0o (note: the speedbar is related to the magnitude of the airows velocity along the calculated particle traces) . . . . . . . . . . . 7-20 Calculated particle traces around the fuselage of a large 4-engined transport aircraft, sideslipping ight = 10o , = 0o (note: the speedbar is related to the magnitude of the airows velocity along the calculated particle traces) . . . . . 7-21 The eect of wing-fuselage interactions on the sidewash at the vertical tailplane of a low-wing aircraft in sideslipping ight . . . . . . . . . . . . . . . . . . . . . 7-22 The eect of wing-fuselage interactions on the sidewash at the tailplanes and the derivative Cn (from reference [142]) . . . . . . . . . . . . . . . . . . . . . . . 7-23 The contribution of the sideforce on the propeller to Cn . . . . . . . . . . . . 7-24 Aerodynamic moment coecient Cn as a function of for the Cessna Ce550 Citation II, = 0o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-25 The pitching moment as a function of sideslip angle (from reference [126]) . . . 7-26 The inuence of the vertical position of the horizontal tailplane on the variation of Cm in sideslipping ight for a fuselage with tailplanes (from reference [126]) . .

322

323 325 326 327 327 328 330

List of Figures 7-27 The variation of the local geometric angle of attack at the wing and the tailplane of a rolling aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-28 Aerodynamic force coecient CY as a function of for the Cessna Ce550 Citation II, = 0o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-29 The variation of the local geometric and eective angle of attack along the span of a rolling wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-30 The roll-damping as a function of aspect ratio and sweep for various wing taper ratios (from reference [34]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-31 The ow at the tailplanes of a rolling aircraft . . . . . . . . . . . . . . . . . . .pb 7-32 Aerodynamic moment coecient C as a function of 2V for the Cessna Ce550 o Citation II, = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-33 The origin of a negative yawing moment about the Z-axis of the stability reference frame for a wing having a positive rate of roll (attached ow) . . . . . . . . . . pb 2V

xvii

330 331 332 333 335 336 337 338 339 341 342 342 343 344 345 346 347 349 350 351 353 354 356 358 362 363 364

7-34 The side force and yawing moment on a rolling, swept back wing

. . . . . . . .

pb 7-35 Aerodynamic moment coecient Cn as a function of 2V for the Cessna Ce550 o Citation II, = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-36 The pure r- motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-37 The change in geometric ow direction at an arbitrary point of the aircraft due to an r-motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-38 The variation in airspeed in spanwise direction due to an r-motion . . . . . . . 7-39 The side forces on the vertical tailplane and the propeller due to an r-motion. . rb 7-40 Aerodynamic force coecient CY as a function of 2V for the Cessna Ce550 Citao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tion II, = 0

7-41 Aerodynamic moment coecient C as a function of for the Citation II, = 0o . . . . . . . . . . . . . . . . . . . . . . . . 7-42 The eects of the size of the vertical tailplane and the taillength reference [88]) . . . . . . . . . . . . . . . . . . . . . . . . . . .

rb 2V

Cessna Ce550 . . . . . . . . on Cnr (from . . . . . . . .

rb 7-43 Aerodynamic moment coecient Cn as a function of 2V for the Cessna Ce550 o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Citation II, = 0 7-44 The positive direction of control deections, control forces, control surface deections and hinge moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-45 The eect of an aileron deection on the lift distribution in spanwise direction . 7-46 The Frise aileron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-47 Spoiler types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-48 The local wing twist of a wing cross-section due to elastic deformation caused by an aileron deection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-49 The side force and yawing moment due to a rudder deection . . . . . . . . . . 7-50 Asymmetric stability and control derivatives . . . . . . . . . . . . . . . . . . . .

8-1 The contribution of various parts of the aircraft to the moment curve Cm . 8-2 Measured contributions of various aircraft parts to the moment curve of the Fokker F- 27, reference point at 0.346 c (from reference [147]) . . . . . . . . . . . . . 8-3 Calculated contributions of various aircraft parts to the moment curve of the Cessna Ce550 Citation II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Flight Dynamics

xviii

List of Figures

8-4 Inuence of the tailplane incidence on the moment curve of the Fokker F-27, reference point at 0.346 c (from reference [147]) . . . . . . . . . . . . . . . . . 8-5 Inuence of the position of the reference point (center of gravity) on the moment curves of the Fokker F-27, (from reference [147]) . . . . . . . . . . . . . . . . . 8-6 Change in the moment dCm due to a change in the angle of attack 8-7 Moment curves and corresponding trim curves . . . . . . . . . . . . . . . . . . . . . .

365 367 368 370 372 373 374 375 378 380 382 384 386 387 390 392 394 397 398 399 404 406 407 407 408 411 415 416 419

8-8 Initial control surface deections . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9 Initial and ultimate control displacement and control surface deection for the transition to a lower airspeed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-10 Aircraft responses to a control deection . . . . . . . . . . . . . . . . . . . . . 8-11 Inuence of cg position on the trim curve . . . . . . . . . . . . . . . . . . . . . 8-12 Inuence of the tailplane angle of incidence on the trim curve (aircraft has control position stability) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13 A positive Cma.c is obtained by combining sweepback and negative wing twist . 8-14 The determination of elevator eciency from ight tests . . . . . . . . . . . . . 8-15 Trim curves for two cg positions of the Fokker F-27 (from reference [51]) . . . . 8-16 Some stability characteristics derived from the measured trim curves shown in gure 8-15 (Fokker F-27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-17 The equilibrium of the free elevator . . . . . . . . . . . . . . . . . . . . . . . . 8-18 The change in the moment dCmf ree due to a change in angle of attack . . . 8-19 The behaviour of the free elevator after a change of the angle of attack . . . . . 8-20 Schematic form of the elevator control force curve, Fe as a function of (a) dynamic pressure 1 V 2 and (b) Fe as a function of equivalent airspeed Ve . . . . . . . . 2 8-21 Measured trim curves and elevator control force curves for the De Havilland D.H.98 Mosquito M II F in gliding ight (from reference [27]) . . . . . . . . . . . . . 8-22 Schematic representation of the concept of positive feel,dFe dse

. . . . . . . . . . . . . . . . .

8-23 Uncertainty in trim speed due to friction in the control mechanism

8-24 Measured elevator control force curves for the North American Harvard II B in gliding ight (from reference [15]) . . . . . . . . . . . . . . . . . . . . . . . . . 8-25 Trim curves, hinge moment coecients and required trim tab angles as functions of CL for the Siebel 204-D-1 aircraft (from reference [16]) . . . . . . . . . . . . 8-26 The inuence of a bobweight in the control mechanism . . . . . . . . . . . . . 8-27 The inuence of a spring in the control mechanism . . . . . . . . . . . . . . . . 8-28 The inuence of a spring or bobweight on the elevator control force curve . . . . 8-29 Maximum control forces as a function of the duration (from reference [140]) . . 8-30 Response curves due to an elevator step deection, Lockheed 1049 C Super Constellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-31 Response curves due to an elevator step deection, Auster J-5B Autocar (from reference [23]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-32 Description of the character of the control feel for various ratios of the control force per g to the control displacement per g (from reference [170]). . . . . . .

List of Figures 8-33 The inuence of the stick displacement per g and the stick force per g on the pilots opinion of the control characteristics at constant airspeed (from reference [86]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-34 The forces and angular velocity q in the idealized pull-up manoeuvre . . . . . . 8-35 The forces and angular velocity in a steady horizontal turn . . . . . . . . . . . . 8-36 The incremental elevator deection e as a function of the incremental load factor n in pull-up manoeuvres, Auster J-5B Autocar (from reference [23]). . 8-37 The incremental elevator angle e as a function of the incremental load factor in turns, Auster J-5B Autocar (from reference [23]) . . . . . . . . . . . . . . . . 8-38 Calculated positions of the manoeuvre point, stick xed, and the stick displacement per g of the Fokker F-27 Friendship . . . . . . . . . . . . . . . . . . . . . . . 8-39 The variation of the angle of attack in an arbitrary point of the aircraft caused by a pure q-motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-40 The incremental control force Fe as a function of the incremental load factor n in pull-up manoeuvres, Auster J-5B Autocar (from reference [23]) . . . . . 8-41 The incremental control force Fe as a function of the incremental load factor n in turns, Auster J-5B Autocar (from reference [23]) . . . . . . . . . . . . . 8-42 Calculated positions of the manoeuvre point, stick free, and the stick force per g of the Fokker F-27 Friendship . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-43 Inuence of altitude and magnitude and sign of Ch on the position of the manoeuvre point, stick free, and the stick force per g . . . . . . . . . . . . . . . . 8-44 Increasing the range of permissible cg positions by decreasing |Ch | . . . . . . . 8-45 The incremental hinge moment due to a bobweight in the control mechanism in a pull-up manoeuvre at a load factor n . . . . . . . . . . . . . . . . . . . . . . .

xix

419 423 424 427 428 430 432 438 438 440 441 443 444

9-1 The forces along the YB -axis of an aircraft in steady, horizontal, asymmetric ight 446 9-2 Steady turns using ailerons only, North American Harvard II B, gliding ight, CL = 0.31, xc.g. = 0.304 c, V = 78 m/sec (from reference [14]) . . . . . . . . . 9-3 Steady turns using rudder only, North American Harvard II B, gliding ight, CL = 0.31, xc.g. = 0.304 c, V = 78 m/sec (from reference [14]) . . . . . . . . . 9-4 Steady coordinated turns, North American Harvard II B, gliding ight, CL = 0.31, xc.g. = 0.304 c, V = 78 m/sec (from reference [14]) . . . . . . . . . . . . . . . 9-5 Steady at turns, North American Harvard II B, gliding ight, CL = 0.31, xc.g. = 0.304 c, V = 78 m/sec (from reference [14]) . . . . . . . . . . . . . . . . . . . 9-6 Steady, straight, sideslipping ight, North American Harvard II B, gliding ight, CL = 0.31, xc.g. = 0.304 c, V = 78 m/sec (from reference [14]) . . . . . . . . . 9-7 The magnitude of Cne due to an engine failure, as a function of the location and the sense of rotation of the propeller and the vertical position of the wing, measured on a model of a twin-engined propeller-driven aircraft (from reference [105]). . . 9-8 The magnitude of Cne due to an engine failure, as a function of the location and the sense of rotation of the propeller and the vertical position of the wing, measured on a model of a twin-engined propeller-driven aircraft (from reference [105]). . . 9-9 Steady, straight, single-engined ight at = 0 9-10 Steady, straight, single-engined ight at = 0 9-11 Minimum control speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 451 454 456 457

459

461 462 463 464

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Flight Dynamics

xx

List of Figures 9-12 Steady, straight, sideslipping ight with the right propeller feathered, Fokker F-27, h = 1850 m, CL = 1.92, xc.g. = 0.28 c, V = 46.3 m/sec (from reference [51]) . 9-14 Variation of the wing twist angle along the wing span due to aileron deection for two locations of the ailerons . . . . . . . . . . . . . . . . . . . . . . . . . . 9-15 The average value of along the aileron span in steady, rolling ight . . . . . 9-16 Roll-rate p as a function of airspeed V for various values of a and Fa . . . . .

465

9-13 The limitation in the maximum attainable roll-rate due to elastic wing deformation 467 468 469 470 483 486 493 495 496 497

10-1 Aperiodic motions corresponding to a real eigenvalue . . . . . . . . . . . . . . 10-2 Periodic motion corresponding to a pair of complex, conjugate eigenvalues 1,2 = j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3 The location of the eigenvalues for the symmetric motions . . . . . . . . . . . . 10-4 Response curves for a step elevator deection (e = 0.005 [Rad]) for the Cessna Ce500 Citation, phugoid response . . . . . . . . . . . . . . . . . . . . . . . . 10-4 (Continued) Response curves for a step elevator deection (e = 0.005 [Rad]) for the Cessna Ce500 Citation, phugoid response . . . . . . . . . . . . . . . . 10-5 Response curves for a step elevator deection (e = 0.005 [Rad]) for the Cessna Ce500 Citation, magnication of gure 10-4, short period response . . . . . . .

10-5 (Continued) Response curves for a step elevator deection (e = 0.005 [Rad]) for the Cessna Ce500 Citation, magnication of gure 10-4, short period response 498 11-1 The location of the eigenvalues = b V for the asymmetric motions . . . . . . b 11-2 Response curves for a pulse-shaped rudder deection (r = +0.025 [Rad] during 1 second) for the Cessna Ce500 Citation . . . . . . . . . . . . . . . . . . . . . 11-2 (Continued) Response curves for a pulse-shaped rudder deection (r = +0.025 [Rad] during 1 second) for the Cessna Ce500 Citation . . . . . . . . . . . . . . 11-2 (Continued) Response curves for a pulse-shaped rudder deection (r = +0.025 [Rad] during 1 second) for the Cessna Ce500 Citation . . . . . . . . . . . . . . 11-3 Response curves for a pulse-shaped aileron deection (a = +0.025 [Rad] during 1 second) for the Cessna Ce500 Citation . . . . . . . . . . . . . . . . . . . . . 11-3 (Continued) Response curves for a pulse-shaped aileron deection (a = +0.025 [Rad] during 1 second) for the Cessna Ce500 Citation . . . . . . . . . . . . . . 11-3 (Continued) Response curves for a pulse-shaped aileron deection (a = +0.025 [Rad] during 1 second) for the Cessna Ce500 Citation . . . . . . . . . . . . . . 11-4 The spiral motion of an aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . 11-5 Characteristics of the Dutch Roll motion . . . . . . . . . . . . . . . . . . . . . 11-6 Roll- and yaw-rate characteristics of the Dutch Roll motion . . . . . . . . . . . 11-7 Lateral stability diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 508 509 510 511 512 513 514 515 516 518

List of Tables

4-1 Non-dimensional parameters in the equations of motion, symmetric motions

. .

142 143 144 144 145 145 146 147 147 150 152 235 240

4-2 Non-dimensional parameters in the equations of motion, asymmetric motions . . 4-2 (Continued) Non-dimensional parameters in the equations of motion, asymmetric motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3 Stability derivatives, symmetric motions . . . . . . . . . . . . . . . . . . . . . . 4-4 Control derivatives, symmetric motions . . . . . . . . . . . . . . . . . . . . . . . 4-5 Moments and products of inertia, symmetric motions . . . . . . . . . . . . . . . 4-6 Stability derivatives, asymmetric motions . . . . . . . . . . . . . . . . . . . . . 4-7 Control derivatives, asymmetric motions . . . . . . . . . . . . . . . . . . . . . . 4-8 Moments and products of inertia, asymmetric motions . . . . . . . . . . . . . . 4-9 Symbols appearing in the general state-space representation of equation (4-46) . 4-10 Symbols appearing in the general state-space representation of equation (4-50) . 5-1 Shorthand notation for the normal force derivatives on a tailplane . . . . . . . . 5-2 Abbreviated notation for the hinge moment derivatives . . . . . . . . . . . . . . 6-1 Simplied formulae for the calculation of stability and control derivatives and coecients in the initial, steady ight condition for the symmetric motions (without the eects of propellers and jets, aeroelasticity and compressibility of the air). . 6-1 (Continued) Simplied formulae for the calculation of stability and control derivatives and coecients in the initial, steady ight condition for the symmetric motions (without the eects of propellers and jets, aeroelasticity and compressibility of the air). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2 Inuence of the center of gravity position on some stability derivatives . . . . .

297

298 298 413 418

8-1 Maximum permissable values of the required control forces, according to the U.S. and British Civil Airworthiness Regulations . . . . . . . . . . . . . . . . . . . . 8-2 Permissable values of the stick force per g in kg . . . . . . . . . . . . . . . . .

Flight Dynamics

xxii

List of Tables 10-1 Symmetric stability and control derivatives for the Cessna Ce500 Citation . . . 11-1 Asymmetric stability and control derivatives for the Cessna Ce500 Citation . . . B-1 Symmetric and asymmetric stability and control derivatives for the Cessna Ce500 Citation, Cruise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Symmetric and asymmetric stability and control derivatives for the Fokker F-27 Friendship, Cruise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-4 Symmetric stability and control derivatives for the Learjet I, Approach . . . . . . B-5 Symmetric stability and control derivatives for the Beechcraft M99, Cruise . . . B-6 Symmetric stability and control derivatives for the Boeing 747-100, Approach . . B-7 Symmetric stability and control derivatives for the Boeing 747-100, Holding, aps up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-8 Symmetric stability and control derivatives for the Boeing 747-100, Approach, aps 33o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-9 Symmetric stability and control derivatives for the Boeing 747-100, Landing . . B-10 Asymmetric stability and control derivatives for the Lockheed L1049C stellation, Cruise . . . . . . . . . . . . . . . . . . . . . . . . . . . B-11 Asymmetric stability and control derivatives for the Lockheed L1049C stellation, Approach . . . . . . . . . . . . . . . . . . . . . . . . . Super Con. . . . . . . Super Con. . . . . . . 492 506 550 551 552 552 553 553 554 554 555 555 556 556 557

B-3 Symmetric stability and control derivatives for the Cessna Ce-172 Skyhawk, Cruise 551

B-12 Asymmetric stability and control derivatives for the BAC-Aerospatiale Concorde, Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-13 Asymmetric stability and control derivatives for the North-American X-15 experimental aircraft, Cruise (unspecied) . . . . . . . . . . . . . . . . . . . . . . . . B-14 Asymmetric stability and control derivatives for the De Havilland Canada DHC-2 Beaver, Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Nomenclature

Before reading the main chapters of this book, we advise the reader to start with this chapter. We will introduce several key notations and characters which will be used throughout the book. Knowing these notations and characters will facilitate the reading of the book and will improve the understanding of the text. We have tried to keep the reasoning behind the notations the same throughout the book. Only one exception exists and is with respect to the use of subscripts (see page xxv).

VectorsAll vectors in this book are in three dimensional space, i.e. R3 , unless dened otherwise. All vectors are in boldface. The superscript denotes the reference frame in which the vector is expressed while the subscript indicates the particular parameter in question. For example:b Vk

is the kinematic (subscript k) velocity (vector V) expressed in the body-xed reference frame (superscript b). The legend for the subscripts en superscripts are given in the section on page xxv. Sometimes a second subscript is added. In that case the subscript denotes the point of origin of the vector or to which point or body the vector properties belong. For example:b Vk,G

is the kinematic velocity of point G expressed in the body-xed reference frame. Four types of vectors have xed component notations. Position vectors have components x, y, z while velocity vectors have components u, v, w. Acceleration components are denoted by ax , ay , az . Finally, the rotational velocity vector has components p, q, r. Other vectors do not have a xed components notation. The same rules regarding the superscriptFlight Dynamics

xxiv and the subscript notation apply for the components of the vectors.

Nomenclature

A vector can be dierentiated with respect to time which is always coupled to a reference frame (without a reference frame time dierentiation only has a pure theoretical meaning). The time derivative of vector with respect to reference frame Fb is written as: dX dt

b

Note that the derivative of a vector is again a vector. Thus the previous expression can be appended by a superscript denoting the reference frame in which the time derivative is expressed. In cases where the meaning is obvious, the superscript and/or the subscripts are sometimes dropped, cleaning the text of any unnecessary notations which would only cloud the text. If any notation is simplied, it is explicitly mentioned so in the text.

Reference framesA crucial element in this book is the denition of reference frames. All reference frames are right-handed orthogonal unless stated otherwise and are dened by their origin and orientation of the axes. A reference frame is denoted by F and its axes by X, Y, Z. The subscript appended to these letters indicates the reference frame we are talking about. For example: Fb (GXb Yb Zb ) is the body-xed reference frame which has the origin in the vehicle center of mass G and has the axes Xb , Yb , Zb . A list of all reference frames used in this book is given next.

FI FE FO Fb Fs Fp Fz Fa Fk Fr

Inertial reference frame Normal Earth-xed reference frame Vehicle carried normal Earth reference frame Body-xed reference frame Stability reference frame Principle axis reference frame Zero-lift body axis reference frame Aerodynamic reference frame Kinematic reference frame Vehicle reference frame

(section (section (section (section (section (section (section (section (section (section

2-1-1) 2-1-3) 2-1-4) 2-1-5) 2-1-5) 2-1-5) 2-1-5) 2-1-6) 2-1-7) 2-1-8)

After chapter 2 the notations Fx , Fy , Fz are used to denote the total aerodynamic forces along the X, Y, Z-axis respectively. Although the context in which these parameters are used is dierent, care should be taken with the interpretation of the parameter F . Transformation matrix The orientation between reference frames is dened by a maximum of three Euler angles. The

Nomenclature

xxv

sequence of rotation combined with the set of angles enables us to transform any coordinate from one reference frame to another. A transformation matrix T is used to quickly transform a complete vector. The subscripts indicate the reference frames involved in the transformation. For example: Xb = Tba Xa where Xb is the vector X expressed in reference frame Fb , Tba is the matrix for the transformation from frame Fa to Fb , and Xa is the vector X expressed in reference frame Fa . Angular velocity vectors When the orientation between reference frames is time-variant, one can dene a rotation vector describing this change through time. The variable dening a rotation vector is . The subscripts indicate which reference frames are involved while the superscript denotes the reference frame in which the vector is expressed. For example: b ba is the rotation vector describing the angular velocity of reference frame Fb with respect to reference frame Fa (subscripts) expressed in frame Fb (superscript). One should be careful not to mix-up the interpretation of the subscripts for the angular velocity vectors and for the transformation matrices.

Superscripts and subscriptsAt this point a distinction must be made between chapters 1 to 3 and the remainder of the book. Chapters 1 to 3 For the rst three chapters the main subject is the derivation of the equations of motion for the most general case of a spherical, rotating Earth. In those chapters the superscripts appended to a vector indicate the reference frame in which the vector is expressed. The possible superscripts are:

I E O b s p z a k r

Inertial reference frame Normal Earth-xed reference frame Vehicle carried normal Earth reference frame Body-xed reference frame Stability reference frame Principle axis reference frame Zero-lift body axis reference frame Aerodynamic reference frame Kinematic reference frame Vehicle reference frameFlight Dynamics

xxvi

Nomenclature

The subscripts in chapters 1 to 3 denote either reference frames (in case of transformation matrices and angular velocity vectors) or indicate a particular form of the vector. The subscript for the reference frames are the same as for the superscripts given above. The other subscripts are: a k aerodynamic kinematic

Remaining chapters In the remainder of the book superscripts are not used often. One exception which occurs in chapter 4, is the superscript indicating the derivative with respect to parameter i, e.g. f = df . di The function of the subscript changes considerably. A subscript in chapter 4 and subsequent chapters indicates the partial derivative of the function with respect to the parameter indicated by the subscript. For example: Xu = X u

Geometric of aircraft parametersThe geometric aircraft parameters are used to determine the aerodynamic force and moments and to make the equations of motion dimensionless. Figure 2 illustrates the various parameters dening the geometry of the wing. The reference axes used in this gure are those of the Vehicle reference frame (see section 2-1-8). Wing area, S The area of the wing projection on the Xr OYr -plane. Often the wing is partially covered by the fuselage and the engine nacelles. The wing area is then calculated using straight line extensions of the wing leading and trailing edges through the fuselage and the nacelles. The wing area can be expressed as,b +2

S=b 2

c dy

where y is the coordinate in the Yr -direction. Wingspan, b The distance in Yr -direction between the wing tips. Mean aerodynamic chord, c (mac) is dened as,

Nomenclature

xxvii

1 c= S

b +2

c2 dyb 2

One can dene the location and orientation of the mean aerodynamic cord within the OXr Zr plane (see gure 1). Four coordinates, xo , xe , z0 , ze ,determine the location and orientation of the mac. There denition is similar that of the mean aerodynamic cord: xo = zo = = 1 S +b/ 2

xo (y) c (y) dy xe = b/ 2 +b/ 2

1 S

+b/ 2

xe (y) c (y) dyb/ 2 +b/ 2

1 S

zo (y) c (y) dyb/ 2 +b/ 2

ze =

1 S

ze (y) c (y) dyb/ 2

1 S

(y) c (y) dyb/ 2

Mean or geometric chord, cm Is very often used in the literature and is dened as: cm = S b

Taper ratio, A measure of the variation in chord length along the span. It is expressed by, = Aspect ratio, A Dened as, A= ct cr

b2 S

Wing sweep, The angle between the Yr -axis and the projection of the 1/4-chord line on the Xr OYr plane (gure 2). In some cases such as delta wings, only the angle between the Yr -axis and the projection of the wings leading edge on the Xr OYr -plane is given. Dihedral, The dihedral of a wing is the angle between the Yr -axis and the projection of the 1/4chord line on the Yr OZr -plane. Washout or wing-twist, Expresses the variation in direction of the local wing chord relative to the direction d of the chord line at the wing root. Neither the gradient of the washout ( d y ) nor the magnitude of wing sweep or dihedral need to be constant along the span. If necessary, these parameters are given as functions of the coordinate in span direction.Flight Dynamics

xxviii

Nomenclature

Zr c zo ze

xo

xe

Xr

c(y) y(< 0) xo (y) xe (y) Xr xo YrFigure 1: Denition of the mean aerodynamic cord and related parameters

c

xe

Nomenclature

xxix

Yr

O

1 4 -chord

line

cr

c dy ct

y Xr b Zr

Yr O

Figure 2: Parameters dening the geometry of the wing

Flight Dynamics

xxx

Nomenclature

Figure 3: Examples of denitions of the vertical tailplane area (see also NACA TN 775)

Wing airfoil The shape of the cross section of the wing parallel to the plane of symmetry. The above geometric parameters apply not only to wings, but to tailplanes as well. Denitions of the geometric parameters for the elevator, or control surfaces in general, are given in chapter 7. It proves to be dicult to dene the geometry of vertical tailplanes in a way applicable to all aircraft. In particular the distinction between fuselage and vertical tailplane is often hard to make. In many instances the division between vertical tailplane and dorsal n is also more or less arbitrary. Usually the surface of the dorsal n is not considered to be part of the vertical tailplane. Figure 3 shows examples of denitions of vertical tailplanes, see reference [11]. Similar and other denitions are given in references [8] and [101].

Aircraft CongurationsThe ight dynamic characteristics are dependent on the aircraft conguration. A full description of the aircraft conguration gives the aircrafts weight or mass, center of gravity position and internal as well as external loading, undercarriage position, control surface deections, ap angle, airbrake and spoiler deections. A description of the engine operating condition, such as throttle position, engine speed etcetera, is also required.

Nomenclature

xxxi

Some denitions as used in U.S. military requirements are briey described below. CR (Cruising Flight) Engine thrust or power for level ight at cruising speed, aps in the position for cruising ight, undercarriage retracted. L (Landing) Throttle closed, undercarriage down, aps in the position for landing. PA (Powered Approach) Undercarriage down, aps and airbrakes in the normal position for the powered approach, engine thrust or power for level ight at 1.15 VSL or the normal airspeed in the powered approach, if the latter is lower. NOTE: VSL is the stall speed (in the aircraft conguration for landing).

Flight ConditionsBy specifying a specic ight condition, we specify (a part of) the state of the aircraft, i.e. we indicate which aircraft states are varying. This information can be used to deduce the equations of motion for that ight condition. The following denitions of ight conditions will be used throughout this book. Steady ight An aircraft is in steady ight if the components of the aerodynamic velocity vector VG (u, v, w) and the components of the body rotation vector (p, q, r) in the body-xed reference frame Fb are constant. In this ight condition the aerodynamics force vector components in Fb and the aircrafts pitch and roll attitude (, ) are constant too. Straight ight An aircraft is in a straight ight condition, if the ightpath vector is straight. Symmetric ight An aircraft is in symmetric ight, if the velocity vector of any point of the aircraft is parallel to the plane of symmetry (roll angle = 0). Slipping ight An aircraft is in slipping ight, if the velocity vector of the aircrafts center of gravity is not parallel to the plane of symmetry of the aircraft.

Flight Dynamics

xxxii

Nomenclature

Chapter 1 Introduction

This book discusses the theory of stability and control of aircraft at subsonic airspeeds. This book handles the theory of ight dynamics which describes the aircraft/spacecraft velocities (translational and rotational), position, and orientation in four dimensional space. Velocities, position, and orientation through time can be computed from the accelerations in time. The analysis of the ight dynamics is based on Newtons Second law: F = ma. If we know the forces acting on the aircraft/spacecraft and the mass distributions we can determine the accelerations. By integrating the accelerations with respect to time and by knowing the initial velocities, the velocities at each moment in time can be calculated. Likewise, by integrating the velocities with respect to time and knowing the initial position and orientation, the position and orientation at each time instant can be determined. The dynamics, i.e. the behavior through time, of the aircraft or spacecraft can be inuenced by using controls, e.g. aerodynamic surfaces or thrusters. There are two types of ight dynamics analysis. The rst concerns the stability. How does the aircraft react to movements of the controls or other types of disturbances? Does it have inherent stability? This type of analysis concerns the characteristics of the aircraft without the pilot aspect in it. The second type is about the aircraft/pilot connection. It is the analysis of the dynamics of the aircraft or spacecraft with respect to the ability of the pilot to control the craft. This is the eld of handling qualities. There are specic regulations that state the required handling qualities for each type of aircraft. These regulations are usually dened in terms of responses of the aircraft to control inputs, like elevator, rudder or throttle inputs. For aviation these regulations have matured trough the years and are described in detail in regulation documents, such as the Joint Aviation Regulations (JAR) set up by the Joint Aviation Authorities (Europe) and the Federal Aviation Regulations set up by the Federal Aviation Administration (USA). The handling qualities of an aircraft are related to the control surfaces and control mechanism. There is a rich history on the development of control surfaces and control mechanism. The following section gives an overview. After that a list of terms and denitions used in ight dynamics is given. The setup of the book is to keep everything as general as possible and to simplify things through the use of assumptionsFlight Dynamics

2

Introduction

where necessary. The assumptions which are used throughout the book are listed in section 1-2. Finally, the book outline will be given in section 1-3.

1-1

Introduction to ight dynamics and control

The Wright Flyer was the rst powered piloted aircraft which had full attitude control (see gure 1-1). It had a double rudder to control yaw, a double elevator to control pitch and used warping of the wings to control roll. The control technique of using mechanical links connected to aerodynamic control surfaces was the founding of modern controlled ight. Modern aircraft still use the idea of pitch, roll, and yaw control through means of deectable control surfaces. (Warping of wings requires exible wings therefore limiting its eld of application. Soon rigid wings with control surfaces where used.) However, over the years, many additional (control) surfaces have been developed. The location, shape, and purpose of each surface varies. Also the mechanisms to move the ight control surfaces, i.e. the ight control systems, have changed over the years. To fully comprehend the inuences these development have on the eld of ight dynamics and control, a short introduction is given in this section. First the control surfaces which have been developed throughout the years are addressed (section 1-1-1). Thereafter, in section 1-1-2, the mechanisms of translating pilot commands to the control surface deections are discussed.

1-1-1

Flight control surfaces

Flight control surfaces have been developed throughout the years and many variants exist. A classication has been made for control surfaces. Two types are dened: Primary ight control surfaces Flight-critical control surfaces. If control of these surfaces is lost, control over the aircraft is (partially) lost and the aircraft is likely to crash. Secondary ight control surfaces Non-ight-critical control surfaces. If control of these surfaces is lost, complete control over aircraft is still possible. The most obvious examples of primary ight control surfaces are the elevator, aileron, and rudder. These basic control surfaces are needed to control the aircraft about all three axis of the aircraft. Without any redundancy of these ight control surfaces, malfunction of one of these control surfaces would make the aircraft hard to control. Stable ight may still be possible if the pilot has thorough knowledge of the ight dynamics of the aircraft. Examples of secondary ight control surfaces are: speed brakes, lift dumpers, slats, aps, and trim surfaces. Control over the orientation of the aircraft is still possible when these surfaces malfunction. As said, many control surfaces have been developed. The dierence between surfaces are, apart from their dimensions, the location on the aircraft and their purpose. In the following, a list is presented in which dierent types of control surfaces are explained briey.

1-1 Introduction to ight dynamics and control

3

(a) First ight (Kittyhawk England, 10:35 AM, 17 Dec 1903)

Elevator

Rudder(b) Prole view

Figure 1-1: The Wright Flyer I [136]

Ailerons Purpose: provide roll control. Location: on the trailing edge of the main wing. Extra information: ailerons are placed either near the tip of the wing (Out-board ailerons) or near the root of the wing (In-board ailerons). Out-board ailerons are only active at low speeds. In-board ailerons are active at all speeds. Elevators Purpose: provide pitch control. Location: trailing edge of the horizontal stabilizer. Rudders Purpose: provide yaw control. Location: trailing edge of the vertical stabilizer. Canards Purpose: provide pitch control. Location: in front of the main wing. Canards can consist of a xed surface with trailing edge control surface or the whole canard surface can be rotated and controlled. ElevonsFlight Dynamics

4

Introduction Purpose: provide pitch and roll control. Location: trailing edge of the main wing. Extra information: to provide pitch control, the elevons are extended symmetrically. For roll control the elevons are extended asymmetrically. Elevons are typically used on tailless delta-wing aircraft.

Flaperons Purpose: provide roll control and additional lift. Location: trailing edge of main wing. Extra information: the aperons is a combination of the ailerons and trailing edge aps. The ap function is obtained by symmetrical extension of the aperons. Roll control is obtained through asymmetric extension. Flaps Purpose: increase lift. Location: either on the leading-edge of the wing or the trailing edge of the wing. Extra information: the leading-edge and trailing-edge aps are used to deform the shape of the wing cross-section (extending the chamber line). The increase in wing surface causes an increase in lift. Therefore slower take-o and landing speeds are possible. Slots Purpose: increase lift (at higher angles of attack). Location: trailing edge of the wing. Extra information: a slot is an alternative to the trailing-edge ap. Slots outperform trailing-edge aps by letting air ow through the wing such that, at high angles of attack, the collapse of airow on the upper surface of an airfoil is reduced, thus maintaining maximum lift at high angles of attack. Slats Purpose: increase lift (at higher angles of attack). Location: leading edge of the wing. Extra information: the slat is actually a leading-edge slot. It delays wing stall at higher angles of attack. Spoilers Purpose: spoil lift, increase drag. Location: on main wing surface. Extra information: Spoiler have two main eects. By symmetric extension on both wings in ight, the spoilers act as speed brakes. During landing, symmetric extension will lead to a smaller roll-out distance. Asymmetric extension in ight will provide additional roll control without inducing wing twist to cause roll-reversal. There are two types of spoiler known: ground spoilers and ight spoilers. Ground spoilers are used during landing and extend further than ight spoilers (which are used in ight). Speed brakes Purpose: add drag to decelerate aircraft. Location: on wing surface. Extra information: speed brakes are aerodynamic surfaces which pop out of the wing at a near perpendicular angle to the airow. This increases drag to the maximal possible extent. Stabilators, Stabilons, and Tailerons (names are synonymous) Purpose: provide pitch and roll control. Location: These surfaces are the two halves (left and right) of the horizontal stabilizer. Extra information: pitch control is achieved by rotating the horizontal stabilizer symmetrically. Roll control is achieved through asymmetric rotation. Stabilators are used to augment the surfaces which provide pitch (elevators, elevons, canards) and roll (ailerons, aperons, spoilers) control.

1-1 Introduction to ight dynamics and control

5

Trim surfaces Purpose: remove required control force exerted by the pilot. Location: trailing-edge of the vertical stabilizer (or part of the rudder) and/or trailing-edge of the horizontal stabilizer (or part of the elevator). Extra information: trim surfaces are used to balance the aerodynamic moment on control surfaces (during steady ight) such that the loads on the controls are eliminated . The pilot can then y hands-free. Two main possibilities exist for the horizontal trim surface. Either the entire horizontal stabilizer acts as a trim surface (so-called variable- incidence horizontal stabilizer) or a smaller surface is used (trailing-edge surface of part of the elevator, creating the so-called variable camber horizontal stabilizer).

Each aircraft can have a dierent conguration regarding its control surfaces. In gures 1-2 to 1-6, several aircraft are depicted. Each gure shows the location of several specic control surfaces.

Rudder

Aileron Outboard trailingedge ap Flight spoiler Slats

Elevator

Stabilizer Inboard trailingedge ap Ground spoiler

Leadingedge ap

Figure 1-2: Boeing 767 3D-view [113]

Flight Dynamics

6

Introduction

Elevons

Figure 1-3: Concorde 3D-view [113]

Carnard

Flaperon

Elevator

Figure 1-4: X-29 in ight [35]

1-1-2

Flight control systems

How do you move the control surfaces of an aircraft and what does a pilot feel when moving a control stick or wheel? Why have the ight control systems changed so much through time? These are just a few questions which one can ask when looking at the history of ight control systems. In the development of control systems their have been two main drives. The rst is the desire to build larger aircraft. Larger aircraft means larger aerodynamic surfaces which in turn means larger aerodynamic forces and thus larger control forces. The very rst control systems where purely mechanical consisting of wires, springs, and wheels, which relied on the pilot strength to move them. The control forces a pilot can generate (especially for larger periods of time) is limited, so when the aircraft size increased, something had to be done to the controls.

1-1 Introduction to ight dynamics and control

7

Aileron - Ground spoiler Outboard t railing-edge ap

- Flight spoiler

Inboard t railing-edge ap - Ground spoiler

Figure 1-5: 737-400 Flight and ground spoiler deployment on landing [32]

Figure 1-6: Boeing 737 slats [32]

Flight Dynamics

8

Introduction

The second drive in the development of control systems is the invention of the autopilot. With the development of new technical systems, like computers and electric actuators, it became possible to control the aircraft electrically. Looking at the past, several types of ight control systems can distinguished: Mechanical human powered system (reversible) Mechanical hydraulic powered system (irreversible/reversible) Fly-by-wire system (irreversible) These dierent t