Elementary Flight Dynamics with an Introduction to Bifurcation and Continuation Methods

390 Pages·2021·33.54 MB·English

by  Nandan K. Sinha, N. Ananthkrishnan

Elementary Flight Dynamics
with an Introduction
to Bifurcation and
Continuation Methods

Elementary Flight Dynamics
with an Introduction
to Bifurcation and
Continuation Methods
Second Edition
Nandan K. Sinha
N. Ananthkrishnan
Second edition published 2022
by CRC Press
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© 2022 Nandan K Sinha and N. Ananthkrishnan
First edition published by CRC Press 2014
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ISBN: 978-0-367-56207-6 (hbk)
ISBN: 978-0-367-56211-3 (pbk)
ISBN: 978-1-003-09680-1 (ebk)
DOI: 10.1201/9781003096801
Typeset in Times
by KnowledgeWorks Global Ltd.
Contents
Preface.......................................................................................................................xi
Authors ...................................................................................................................xiii
Chapter 1 Introduction ..........................................................................................1
1.1 What, Why and How? ...............................................................1
1.2 Aircraft as a Rigid Body ...........................................................2
1.3 Six Degrees of Freedom ............................................................7
1.4 Position, Velocity and Angles ..................................................10
1.5 Aircraft Motion in Wind .........................................................14
1.6 Longitudinal Flight Dynamics ................................................18
1.7 Longitudinal Dynamics Equations ..........................................20
1.8 A Question of Timescales .......................................................22
1.9 Longitudinal Trim ...................................................................24
1.10 Aerodynamic Coefficients C , C , C .....................................27
D L m
1.10.1 Aerodynamic Coefficients with Angle
of Attack (α) ...............................................................29
1.10.2 Aerodynamic Coefficients with Mach
Number (Ma) ..............................................................31
1.11 Wing–Body Trim .....................................................................33
Exercise Problems ..............................................................................39
References ..........................................................................................41
Chapter 2 Stability Concept ................................................................................43
2.1 Linear First-Order System .......................................................43
2.2 Linear Second-Order System ..................................................46
2.3 Non-Linear Second-Order System ..........................................55
2.4 Pitch Dynamics about Level Flight Trim ................................56
2.5 Modelling Small-Perturbation Aerodynamics ........................58
2.6 Pitch Dynamics about Level Flight Trim (Contd.) ..................62
2.6.1 Numerical Example ....................................................63
2.7 Short-Period Frequency and Damping ....................................67
2.8 Forced Response ......................................................................67
2.8.1 First-Order System .....................................................68
2.8.2 Second-Order System .................................................71
2.9 Response to Pitch Control .......................................................75
2.9.1 Pitch Dynamics about Level Flight Trim with
Elevator Control .........................................................77
Exercise Problems ..............................................................................78
v
vi Contents
Chapter 3 Longitudinal Trim and Stability.........................................................81
3.1 Wing–Body Trim and Stability ...............................................81
3.2 Wing–Body Plus Tail: Physical Arguments ............................85
3.3 Wing–Body Plus Tail: Math Model.........................................86
3.3.1 Airplane Lift ..............................................................90
3.3.2 Airplane Pitching Moment .........................................92
3.4 Role of Downwash ...................................................................98
3.5 Neutral Point ............................................................................99
3.5.1 Static Margin ............................................................101
3.5.2 NP as Aerodynamic Centre of Entire Airplane .......101
3.6 Replacing V with V/ ............................................................103
H H
3.6.1 Revised Expressions for NP .....................................105
3.6.2 NP as Aerodynamic Centre of the Entire
Airplane ....................................................................106
3.6.3 Trim and Stability, Again! ........................................107
3.7 Effect of CG Movement.........................................................109
3.8 Rear CG Limit due to Airplane Loading and
Configuration at Take-Off .....................................................111
3.9 C , C Curves–Non-Linearities ............................................112
m L
Exercise Problems ............................................................................114
Appendix 3.1 ....................................................................................118
Chapter 4 Longitudinal Control ........................................................................119
4.1 All-Moving Tail .....................................................................119
4.2 Elevator ..................................................................................121
4.3 Tail Lift with Elevator ...........................................................122
4.4 Airplane Lift Coefficient with Elevator.................................125
4.5 Airplane Pitching Moment Coefficient with Elevator ...........129
4.6 Elevator Influence on Trim and Stability ..............................131
4.6.1 Change in Trim Lift Coefficient ...............................132
4.6.2 Another Viewpoint of Stability ................................134
4.7 Longitudinal Manoeuvres with the Elevator .........................135
4.8 Most Forward CG Limit ........................................................141
4.8.1 Using Elevator to Compensate for CG Shift ............142
4.8.2 Typical Elevator Deflection Limits ..........................143
4.8.3 Forward-Most CG Limit due to Elevator
Up-Deflection Limit .................................................145
4.9 Np Determination from Flight Tests .....................................145
4.10 Effect of NP Shift with Mach Number ..................................147
Exercise Problems ............................................................................150
References ........................................................................................152
Contents vii
Chapter 5 Long-Period (Phugoid) Dynamics ....................................................153
5.1 Phugoid Mode Equations .......................................................153
5.2 Energy....................................................................................153
5.2.1 Normal Acceleration ................................................156
5.3 Phugoid Mode Physics ...........................................................157
5.4 Phugoid Small-Perturbation Equations .................................158
5.5 Aerodynamic Modelling with Mach Number .......................159
5.6 Phugoid Dynamics .................................................................161
5.7 Phugoid Mode Frequency and Damping ...............................163
5.8 Accurate Short-Period and Phugoid Approximations ...........165
5.8.1 Short-Period Mode Dynamics ..................................166
5.8.2 Phugoid Mode Dynamics .........................................167
5.9 Derivative C .....................................................................168
mMa
5.10 Derivative C in Pitching Motion .......................................169
mq1
5.11 Derivative C in Phugoid Motion ........................................172
mq1
5.12 Flow Curvature Effects .........................................................173
Exercise Problems ............................................................................174
References ........................................................................................176
Chapter 6 Lateral-Directional Motion ..............................................................177
6.1 Review ...................................................................................177
6.2 Directional Disturbance Angles ............................................177
6.3 Directional versus Longitudinal Flight .................................179
6.4 Lateral Disturbance Angles ...................................................181
6.5 Lateral-Directional Rate Variables ........................................183
6.6 Small-Perturbation Lateral-Directional Equations ...............184
6.7 Lateral-Directional Timescales .............................................187
6.8 Lateral-Directional Aerodynamic Derivatives ......................189
6.9 Lateral-Directional Small-Perturbation
Equations (Contd.) .................................................................190
6.10 Lateral-Directional Dynamic Modes ....................................194
6.10.1 Roll (Rate) Mode ......................................................195
6.10.2 Dutch Roll Mode ......................................................196
6.10.3 Spiral Mode ..............................................................199
Exercise Problems ............................................................................201
Reference ..........................................................................................202
Chapter 7 Lateral-Directional Dynamic Modes ...............................................203
7.1 Roll (Rate) Mode ...................................................................203
7.2 Roll Damping Derivative C ................................................203
lp2
7.2.1 Special Case of Trapezoidal Wing ...........................205
7.2.2 Owing to Vertical Tail ..............................................207
viii Contents
7.3 Roll Control ...........................................................................208
7.4 Aileron Control Derivative, C ............................................212
lδa
7.4.1 Other Roll Control Devices ......................................215
7.4.1.1 Roll Control with Spoilers ........................216
7.4.1.2 Roll Control by Differential Tail ..............217
7.4.1.3 Roll Control by Rudder .............................217
7.5 Yaw Due To Roll Control ......................................................219
7.5.1 Yaw due to Aileron ...................................................219
7.5.2 Yaw due to Spoilers ..................................................220
7.5.3 Yaw due to Differential Tail .....................................221
7.5.4 Yaw due to Rudder ...................................................221
7.6 Aileron Input for a Bank Angle .............................................222
7.7 Dutch Roll Mode ...................................................................223
7.8 Directional Derivatives C and C ......................................228
Yβ nβ
7.8.1 Other Contributors to Yaw Stiffness ........................230
7.8.2 Loss of Vertical Tail Effectiveness...........................234
7.9 Lateral Derivative: C ...........................................................235
lβ
7.9.1 Wing Dihedral ..........................................................235
7.9.2 Other Sources of C .................................................238
lβ
7.9.2.1 Wing Sweep ..............................................239
7.9.2.2 Wing Vertical Position on Fuselage ..........241
7.9.2.3 Vertical Tail ..............................................242
7.10 Damping Derivatives: C and C ........................................244
nr1 lr1
7.10.1 Wing Contribution to C and C ...........................244
nr1 lr1
7.10.2 Vertical Tail Contribution to
C and C ...............................................................246
nr1 lr1
7.11 Rudder Control ......................................................................248
7.11.1 Crosswind Landing ..................................................250
7.11.2 Other Rudder Trim Cases .........................................252
7.12 Spiral Mode ...........................................................................252
7.12.1 C and C Derivatives ...........................................254
nr2 lr2
7.12.2 Spiral Mode Stability ...............................................255
7.13 Real-Life Airplane Data ........................................................256
Exercise Problems ............................................................................257
References ........................................................................................267
Chapter 8 Computational Flight Dynamics ......................................................269
8.1 Aircraft Equations of Motion ................................................269
8.2 Derivation of Aircraft Equations of Motion ..........................269
8.2.1 Equations of the Translational Motion .....................270
8.3 3–2–1 Rule .............................................................................273
8.3.1 Euler Angles and Transformation ............................274
8.3.2 Kinematic Equations (Attitude and Position
Dynamics) ................................................................278
Contents ix
8.3.2.1 Relation between Body Rates (p, q, r)
and Euler Rates (ϕ˙ , θ˙, ψ˙) ..........................278
8.3.2.2 Relation between Inertial Velocity and
Body-Axis Velocity Components .............280
8.3.2.3 Relation between Body-Fixed and
Wind-Fixed Coordinates ..........................280
8.3.2.4 Relation between the Body-Axis and
Wind-Axis Euler Angles ..........................282
8.3.2.5 Relation between the Body-Axis and
Wind-Axis Angular Rates ........................283
8.3.3 Force Equations Summed Up ...................................285
8.3.3.1 Derivation of Force Equations in
Wind-Fixed Axis System ..........................285
8.4 Derivation of Aircraft Equations of Motion (Contd.) ............289
8.4.1 Equations for the Rotational Motion ........................289
8.4.2 Symmetry of Aircraft ...............................................291
8.4.3 Sources of Nonlinearity ...........................................293
8.5 Numerical Analysis of Aircraft Motions ...............................295
8.5.1 Generalized Airplane Trim and Stability
Analysis ....................................................................296
8.5.1.1 Local Dynamic Behaviour: Trim and
Stability Analysis ......................................297
8.6 Standard Bifurcation Analysis ..............................................299
8.6.1 Application of SBA to F-18/HARV Dynamics ........302
8.6.1.1 Stall and Post-Stall Solutions ....................303
8.6.1.2 Roll Manoeuvres .......................................305
8.7 Extended Bifurcation Analysis (EBA) ..................................307
8.7.1 Straight and Level Flight Trim .................................308
8.7.2 Coordinated (Zero Sideslip) Level Turn Trim ..........313
8.7.3 Performance and Stability Analysis .........................315
8.7.3.1 Straight and Level Flight Trim .................316
8.7.3.2 Level Turn Manoeuvre .............................318
8.7.3.3 Maximum Roll Rate in a Roll
Manoeuvre ................................................319
Exercise Problems ............................................................................322
Appendix 8.1: Small-Perturbation Equations
(At Longitudinal Trim State) .................................................323
Appendix 8.2: F-18 Data ..................................................................325
Aerodynamic Data .................................................................325
Geometric and Other Data for F-18 .......................................326
Appendix 8.3: Equations and Aircraft Data Used for Roll
Manoeuvre .............................................................................326
References ........................................................................................327