Dynamics of atmospheric flight

Front Matter......Page 1
Preface......Page 3
Table of Contents......Page 6
1. Introduction......Page 10
1.1 Types of Problems......Page 14
1.2 The Tools of Flight Dynamicists......Page 15
2.1 Introduction......Page 17
2.3 Laplace and Fourier Transforms......Page 18
2.3.1 Definitions......Page 20
2.3.3 Transforms of Derivatives......Page 23
2.3.5 Extreme Value Theorems......Page 25
2.4 Application to Differential Equations......Page 26
2.5.2 The Method of Partial Fractions......Page 27
2.5.3 Heaviside Expansion Theorem......Page 28
2.6 Random Process Theory......Page 29
2.6.1 Stationary Random Variable......Page 30
2.6.2 Ensemble Average......Page 31
2.6.3 Harmonic Analysis of vt......Page 32
2.6.4 Correlation Function......Page 33
2.6.5 Spectrum Function......Page 34
2.6.7 Probability Properties of Random Variables......Page 37
2.6.8 Mean Value of a Function of v......Page 38
2.6.9 Joint Probability......Page 39
2.6.10 Joint Distribution of a Function and its Slope......Page 41
2.6.11 Ordinate-Crossing Return Period......Page 43
2.6.12 Distribution of Peaks......Page 44
2.6.13 Probability of a Positive Event during Time t_1......Page 45
2.6.14 Discussion of Condition ii......Page 48
2.7 Machine Computation......Page 49
3.1 Concepts and Terminology......Page 51
3.1.1 Example......Page 52
3.1.2 Block Diagram......Page 54
3.1.4 Equilibrium, Control, and Stability......Page 55
3.2 Transfer Functions......Page 59
3.2.3 Systems in Series......Page 61
3.2.4 System with Feedback......Page 62
3.2.5 Transfer Functions of General Linear/Invariant System......Page 63
3.3 Autonomous Linear/Invariant Systems......Page 65
3.3.2 Eigenvalues and Eigenvectors......Page 67
3.3.3 Orthogonal Eigenvectors......Page 68
3.3.4 Computation of Eigenvalues and Eigenvectors......Page 69
3.3.6 Equations in Nonstandard Form......Page 70
3.3.7 Characteristic or Natural Modes......Page 71
3.3.8 Characteristic Coordinates......Page 75
3.3.9 Stability Criteria......Page 77
3.3.12 Test Functions for a Quintic......Page 79
3.4 Response of Linear/Invariant Systems......Page 80
3.4.1 Interpretation of High-Order System as a Chain......Page 81
3.4.2 Impulse Response......Page 82
3.4.3 Relation between Impulse Response and Autonomous Solution......Page 85
3.4.4 Step-Function Response......Page 86
3.4.5 Frequency Response......Page 87
3.4.6 Effect of Poles and Zeros on Frequency Response......Page 90
3.4.7 Frequency Response of First-Order System......Page 91
3.4.8 Frequency Response of a Second-Order System......Page 93
3.4.9 Relation between Impulse Response and Frequency Response......Page 97
3.4.10 Superposition Theorem Convolution Integral, Duhamel\'s Integral......Page 98
3.4.11 Solution Including Initial Conditions......Page 101
3.4.12 Response to a Set of Stationary Random Inputs......Page 102
3.4.13 A Useful Theorem Concerning Mean-Square Response......Page 104
3.5 Time-Varying and Nonlinear Systems......Page 105
3.5.2 Steady-State Describing Function......Page 106
3.5.4 Two-Input Describing Functions......Page 108
3.5.5 Lyapunov Stability Theory......Page 109
4.1 Notation......Page 113
4.2.1 Inertial Reference Frame F_I Inertial Axes, O_I x_I y_I z_I......Page 115
4.2.2 Earth-Fixed Reference Frame, F_E Earth Axes O_E x_E y_E Z_E......Page 116
4.2.3 Vehicle-Carried Vertical Frame, F_V Axes O_V y_V z_V......Page 117
4.2.6 Body-Fixed Reference Frame F_B Body Axes, Oxyz......Page 118
4.2.7 Stability Axes F_S O_S x_S y_S z_S......Page 119
4.3.1 The Vehicle Euler Angles......Page 121
4.4 Transformation of a Vector......Page 123
4.4.1 Properties of the L Matrix......Page 124
4.5 The L Matrix in Terms of Rotation Angles......Page 125
4.6 Transformation of the Derivative of a Vector......Page 127
4.7 Transformation of a Matrix......Page 129
5. General Equations of Unsteady Motion......Page 130
5.1 Velocity and Acceleration in an Arbitrarily Moving Frame......Page 131
5.2.2 Angular Velocity omega^V of F_V......Page 133
5.2.3 Angular Velocities omega^W , omega of F_W , F_B......Page 134
5.2.4 The Direction Cosine Rates......Page 136
5.2.5 The Aerodynamic Angle Rates......Page 137
5.3.1 Position and Velocity Relative to the Earth......Page 138
5.3.2 Inertial Acceleration......Page 139
5.4 Equations of Motion of an Arbitrary System......Page 143
5.4.1 The Angular Momentum......Page 145
5.4.3 The Remainder of h*......Page 148
5.5.1 The Force Vector......Page 150
5.6 Force and Moment Equations in Body Axes Euler\'s Equations......Page 152
5.7 Discussion of the System of Equations......Page 154
5.8 The Flat-Earth Approximation......Page 157
5.9 Steady States......Page 161
5.10.2 Notation for Small Disturbances......Page 163
5.10.3 Further Assumptions......Page 164
5.10.4 The Linear Equations......Page 165
6. Longitudinal Aerodynamic Characteristics - Part 1......Page 167
6.1 The Basic Longitudinal Forces......Page 168
6.2 Pitch Stiffness and Possible Configurations for Flight......Page 170
6.3.1 Pitching Moment of a Wing......Page 173
6.3.2 Pitching Moment of a Body and Nacelles......Page 175
6.3.3 Pitching Moment of a Tail......Page 176
6.3.4 Pitching Moment of a Propulsive System......Page 178
6.3.5 Total Pitching Moment......Page 179
6.3.6 Linear Lift and Moment......Page 180
6.4 Longitudinal Control......Page 183
6.4.1 The Derivatives C_L _delta and C_m _delta......Page 185
6.4.2 Elevator Angle to Trim......Page 186
6.4.3 Variation of delta_e _trim with Speed......Page 189
6.4.5 Flight Determination of h_n and h_s......Page 192
6.5 Control Hinge Moment......Page 193
6.6 Influence of a Free Elevator on Lift and Moment......Page 197
6.6.1 Free-Elevator Factor......Page 198
6.6.2 Control-Free Neutral Point......Page 199
6.7.1 Trim Tabs......Page 200
6.7.2 Geared Tabs......Page 201
6.7.3 Spring Tabs......Page 202
6.7.4 Servo Tabs......Page 203
6.8 Control Force to Trim......Page 204
6.9 Control Force Gradient......Page 208
6.10 Maneuverability - Elevator Angle and Control Force per g......Page 209
6.10.1 Control Force per g......Page 212
7.1 Bob Weights and Springs......Page 215
7.1.1 Effect upon Control Force to Trim and h\'_n......Page 216
7.1.2 Effect upon Stick Force per g and h\'_m......Page 217
7.2 Influence of High-Lift Devices on Trim and Pitch Stiffness......Page 218
7.3 Influence of the Propulsive System on Trim and Pitch Stiffness......Page 220
7.3.1 The Influence of Running Propellers......Page 223
7.3.4 Example of Propeller Effect......Page 224
7.3.5 The Influence of Jet Engines......Page 226
7.4 Effect of Structural Flexibility......Page 229
7.5 Ground Effect......Page 230
7.6.1 The AFT Limit......Page 231
7.7 Longitudinal Aerodynamic Derivatives......Page 233
7.8.1 The Derivative C_T _V......Page 234
7.8.2 The Derivative C_D _V......Page 235
7.8.3 The Derivatives C_L _v , C_m _v , C_h _ev......Page 236
7.9 The q Derivatives C_L _q , C_m _q , C_he _q......Page 238
7.9.1 C_L _q of a Tail......Page 239
7.9.3 Contributions of a Wing......Page 240
7.9.4 Dependence on h......Page 242
7.9.5 Pitch Damping of Propulsive Jets......Page 244
7.10.1 Contributions of a Wing......Page 247
7.10.2 Contributions of a Tail......Page 254
7.10.5 C_he _alpha of a Tail......Page 255
7.12 The z Derivatives C_T _z , C_D _z , C_L _z , C_m _z......Page 256
7.13 Aeroelastic Derivatives......Page 257
7.13.1 Forces on the Wing......Page 258
7.13.4 The Derivative A_n _alpha......Page 259
7.13.5 The Derivative b_ll......Page 260
7.14 Summary of the Formulae......Page 262
8. Lateral Aerodynamic Characteristics......Page 263
8.1 Yaw Stiffness Weathercock Stability......Page 264
8.2 Yaw Control......Page 267
8.3 Roll Stiffness......Page 269
8.4 Rolling Control......Page 271
8.4.1 Aileron Reversal......Page 273
8.5.1 The Derivative C_y _beta......Page 274
8.5.2 The Derivative C_l _beta......Page 275
8.5.4 Influence of Sweep on C_l _beta......Page 277
8.5.5 Influence of Fin on C_l _beta......Page 278
8.6 The p Derivatives C_y _p , C_l _p , C_n _p , C_ha _p , C_hr _p......Page 279
8.6.1 The Derivative C_y _p......Page 280
8.6.2 The Derivative C_l _p......Page 281
8.6.3 The Derivative C_n _p......Page 282
8.7 The r Derivatives C_v _r , C_l _r , C_n _r , C_ha _r , C_hr _r......Page 284
8.7.3 The Derivative C_n _r......Page 286
8.7.4 The Derivative C_ha _r......Page 287
8.8 Summary of the Formulae......Page 289
9. Stability of Steady Flight......Page 290
9.1 Longitudinal Modes; Flat-Earth Approximation......Page 291
9.1.1 Eigenvalues......Page 293
9.1.2 Eigenvectors......Page 295
9.1.3 Flight Paths in the Characteristic Modes......Page 297
9.2 Approximate Equations for the Longitudinal Modes......Page 299
9.2.1 Phugoid Mode......Page 300
9.2.2 Short-Period Mode......Page 303
9.3 General Theory of Static Longitudinal Stability......Page 305
9.4.1 Effect of Speed......Page 308
9.4.2 Effect of Altitude......Page 312
9.4.3 Effect of Flight-Path Angle......Page 315
9.4.4 Effect of Vertical Density-Gradient......Page 317
9.4.5 Effect of Static Margin......Page 322
9.4.6 Effect of Speed Derivatives......Page 325
9.5 Longitudinal Characteristics of a STOL Airplane......Page 327
10.1 Introduction......Page 332
10.1.1 Longitudinal Control......Page 333
10.1.2 Lateral Control......Page 334
10.2 Response to Elevator Input......Page 335
10.2.1 Short-Period Approximation......Page 337
10.2.2 Phugoid Approximation......Page 338
10.2.3 Numerical Example - Frequency Response......Page 341
10.2.4 Numerical Example - Step Response......Page 346
10.3 Response to the Throttle......Page 349
10.3.1 Approximate Transfer Functions......Page 350
10.3.2 Numerical Example - Step Response......Page 352
10.4.1 The Steady Sideslip......Page 354
10.4.2 The Steady Turn......Page 355
10.4.3 Numerical Example......Page 359
10.5 Lateral Frequency Response......Page 360
10.5.2 Approximate Lateral Transfer Functions......Page 361
10.5.3 Dutch-Roll Approximation......Page 368
10.5.4 Spiral/Roll Approximation......Page 369
10.6 Transient Response to Aileron and Rudder......Page 370
10.7 Inertial Coupling in Rapid Maneuvers......Page 375
10.7.2 When C_M _alpha < 0 Aerodynamically Stable Configuration......Page 377
10.7.3 Numerical Example - Pitch/Roll Coupling of a Small Airplane......Page 379
11.1 General Principles......Page 384
11.1.1 Sensors......Page 387
11.2 Example - Suppression of the Phugoid......Page 388
11.2.1 Change of Flight-Path Angle......Page 391
11.3 Equations of Motion of the Control Systems......Page 394
11.3.2 The Generalized Force F_n......Page 395
11.3.3 Equations of Motion......Page 398
11.3.5 The Servo Equation......Page 399
11.3.6 Coupling of Controls with Elastic Degrees of Freedom......Page 401
11.4 Example - Stability Augmentation System for STOL Airplane......Page 403
11.4.1 Elimination of Steady-State Rudder Angle......Page 406
11.5.1 Flight at Exactly Constant Height - Speed Stability......Page 410
11.5.2 Numerical Example......Page 412
11.5.3 Flight on ILS Glide Slope......Page 413
11.5.4 Numerical Example......Page 415
11.6 Stability of Closed-Loop Systems......Page 417
11.6.1 Characteristic Equation......Page 418
11.6.4 Sinusoidal Input - Nyquist Criterion......Page 419
12.1 The Human Pilot......Page 422
12.2 Mathematical Model of Human Pilots - Compensatory Display......Page 423
12.3 Mathematical Model of Human Pilots - Pursuit Display......Page 433
12.4 The Future Role of the Human Pilot......Page 438
12.5 Aircraft Handling Qualities......Page 439
12.5.1 The Rating of Handling Qualities......Page 440
12.6 Flight Simulators......Page 442
12.8 Longitudinal Handling Qualities......Page 443
12.8.1 Other Longitudinal Handling Qualities Parameters......Page 444
12.8.3 Longitudinal Control System Characteristics \"Feel\"......Page 449
12.9 Lateral-Directional Handling Qualities......Page 450
12.9.1 Spiral Mode......Page 451
12.9.3 Closed-Loop Roll Control......Page 452
12.9.4 Open-Loop Roll Control......Page 454
12.9.5 Dutch-Roll Characteristics......Page 455
12.9.6 omega_phi /omega_d as a Handling Qualities Parameter......Page 456
12.10 Handling Qualities Requirements......Page 458
12.10.3 Static Stability......Page 459
12.10.5 Stalling and Spinning......Page 460
13.1 Introduction......Page 461
13.2 Description of Atmospheric Turbulence......Page 463
13.2.1 Simplifying Assumptions......Page 464
13.2.2 Spectral Component of Turbulence......Page 468
13.2.3 Integral Scale......Page 469
13.2.4 Model of High-Altitude Turbulence......Page 470
13.2.5 Model of Low-Altitude Turbulence......Page 472
13.2.6 Velocity Profile, Mean Wind......Page 473
13.2.10 Integral Scales......Page 474
13.3 The Input to the Airplane......Page 475
13.3.1 The Airplane as a Point......Page 476
13.3.2 Range of Validity of the Point Approximation......Page 478
13.3.3 The Finite Airplane......Page 479
13.3.4 The \"Panel\" Method......Page 480
13.3.5 Skelton\'s Method......Page 482
13.3.6 The Powers Series Method......Page 483
13.3.7 Ribner\'s Method......Page 486
13.3.8 The Simulation Method......Page 487
13.4 An Example......Page 488
13.5 Gust Alleviation......Page 494
Bibliography......Page 496
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M......Page 517
P......Page 518
R......Page 520
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T......Page 523
Y......Page 524
Errata......Page 525