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The Dynamics of Flight The Equations
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The Dynamics of Flight The Equations
Jean-Luc Boiffier
S U P A ~ R(Ecole O Nationale Supkrieure de 1 'Akronautique et de I 'Espace)
and
ONERA-CERT (Centre d'Etudes et de Recherche de Toulouse)
JOHN WLEY & SONS
Chichester New York Weinheim Brisbane Singapore Toronto
Copyright
@ 1998 by John Wiley & Sons Ltd,
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Reprinted October 1998 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under t h e terms of a licence issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London, UK W1P 9HE, without the permission in writing of the publisher. Other
Wdq Editorial Ofices
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Library of Congress Cataloging-in-PublicationData Boiffier, Jean-Luc Dynamics of flight : equations / Jean-Luc Boiffier. p. cm. Includes bibliographical references and index. ISBN 0-471-96737-8. - ISBN 0-471-94237-5 (pbk.) 1. Aerodynamics - Mathematics. 2. Equations. 3. Engineering mathematics - Formulae. I. Title. TL570.B585 1998 98-17337 629.132’3’0151 - dc21 CIP
British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0 471 94237 5 Produced from Postscript files supplied by t h e author Printed and bound in Great Britain by Bookcraft (Bath) Ltd This book is printed on acid-free paper responsibly manufactured from sustainable forestry, in which at least two trees are planted for each one used for paper production
to Christine, GaeZle, Matthieu and Xauier
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Preface The study of aircraft flight is based upon the model formed by the flight dynamics equations developed in this document in their most generalized form for a rigid flight vehicle. These equations and the associated hypotheses are the preliminary prerequisite of any study of flight dynamics. In this work, the equations are adapted to the study of the atmospheric flight of an aircraft, but their application could be extended t o transatmospheric or space flight of aircraft by extending the model of external efforts t o take into account rarefied gas and solar radiation dynamics. The deductive approach used in this document is in accordance with the tradition perpetrated by the mechanical engineers. Starting with the most generalized equations, the objective is to progressively obtain simplified equations. Thanks to this approach, the consequences of the various simplifying hypotheses can be clearly evaluated, and the context for the utilization of these simplified equations can be consciously appreciated. On the other hand, within the framework of a linear reading, at first the formalism used could appear quite superfluous. It must be mentioned that the reason for this formalism is to avoid possible confusion between one parameter and another. Thus, each time the notations are logical and necessary. The inductive approach, starting from the simplest towards the most complicated, would avoid this pitfall at least at first. However, to reach the same level of accuracy, it is inevitable t o join the deductive formalism by a way which is finally longer than the first one. In order t o overcome this slight difficulty of the deductive approach, a reading guide is available so that the reader may immediately obtain the simplest form of the equations. At each step, the results are given with clear reference t o the previous results. Furthermore, the detailed nomenclature and the list of the various hypotheses are useful and available, t o facilitate an oriented reading in order t o study a specific problem. Several representations of the equations are developed with a precise formulation of the influence of atmospheric perturbation. Following the establishment of the general equations, flat and fixed Earth hypotheses are made and the decoupled and linearized equations for longitudinal and lateral flight are established. After the definition of the equilibrium and pseudo-equilibrium notions, analytical and numerical solutions are proposed for research of equilibrium and linearization operations. The decoupling operation, fundamental for the analytical process of the equations, is developed in order to highlight its limits. In this document, students and engineers will find the definition of the numerous flight dynamics notations such as frame and angle, the definition of notions of equilibrium and the presentation of the decoupling and linearization operations. All these issues are fundamental points for a sound understanding of flight dynamics. Naturally, vii
viii the readers will also find the general equations which are often used in a simplified form. It is for this reason that, if they wish to find the frequently used simple models, they are advised to consult the Reading guide proposed below. In order to help the reader who wishes to find the results by himself or to exploit the methods of calculation for his own needs, the calculations needed to establish the equations are given in the appendices. The main text contains only the results needed to exploit the equations.
Reading guide This guide suggests a limited list of paragraphs which should be read in order t o have a reasonable understanding of the equations of flight dynamics leading to the knowledge of a frequently used model. It is primarily intended for students and engineers unfamiliar with flight dynamics. In the first reading of this document, it is not necessary to read the appendices as they contain detailed calculations, the essential results of which appear in the main text. These appendices represent approximately half of this publication. The first chapter, Presentation, explains the process leading up to the equations. It is essential reading for an overall view of the problem. This chapter is not very long and contains no equations. The equations obtained with the flat and fixed Earth hypotheses are sufficient to deal with most problems of flight dynamics where the Mach number is less than 2. These equations can be found in section (5.1), p. 104 and in an even simpler form in section (5.3), p. 130 if the wind velocity is constant. To understand the meaning of the terms of these equations, readers already acquainted with flight dynamics may refer to the nomenclature; others may find it necessary to refer the frame definitions with section (2.1.3), p. 16 and section (2.1.6)) p. 17, angles between frames (Section 2.2.5, p. 27) and velocities (Section 3.2, p. 45). The definition of external efforts in section (4.3), p. 82 is also important. To understand how the equations are derived, the paragraph on the fundamental theorems of mechanics will be useful (Section 4.1, p. 71). But this understanding is not necessary for the use of the equations. Finally, it would be beneficial to read the beginning of section (5.4), p. 132 on the decoupling of the equations, section (6.1), p. 158 on linearization and the definition of equilibrium in section (7.1), p. 180. These three notions are fundamental to flight dynamics. Decoupling is an operation which consists in processing a problem with a limited number of equations extracted from the complete system. This extraction qualifies the decoupling and the procedure is frequently used for example, when the flat and fixed Earth hypotheses are employed or during a study of lateral or longitudinal movement. The linearization of equations is a fundamental operation in the study of the dynamics of an aircraft. It may be carried out in a numerical or analytical way but, in this case, the point of equilibrium must be known. It is for this reason that the notion of equilibrium is not only defined but a numerical method for research of equilibrium is proposed. When the notions of decoupling and equilibrium are associated, the notion of pseudo-equili brium is obtained.
ix
Acknowledgments My most sincerely thanks goes to Philippe Mouyon for his precious advice, Marc Hillebrand for writing and testing the research of equilibrium programs and Ersin Eraydin for his careful rereading of the text and for his works on the consequences of the flat and fixed Earth hypotheses, and on linearization. I also wish t o thank my colleagues of the Toulouse Research Center and SupAero for their scientific and friendly support and in particular Pierre Vacher, Marc Labarrere, Alain Bucharles, Manuel Samuelides, Didier Bellet and Andre Fossard. I owe special thanks to Professor Piero Morelli for inviting me to share the fruits of his long and rich experience in flight dynamics and for his early support. Special thanks t o Edith Roques who translated this document from French with a particular competence and with a warm perseverance. Finally, my tender thanks t o Christine for her domestic contribution which was essential t o the completion of this work and t o Matthieu for his carefully-made figures. Thanks as well t o Xavier and Gaelle for the efficient support of their fond encouragements. I am grateful t o Ms Annie Bouchet who typed the first version of the text, with her usual capability and determination.
Toulouse, 1997
Jean-Luc BOIFFIER equationsQcert .fr
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Short table of contents 1
1 Presentation
I
11
General equations
2 Ekames
13
3 Kinematics
43
4 Equations
71
I1
101
Simplified equations
5 Simplified equations
103
6 Linearized equations
157
7 Equations for equilibrium
179
I11
193
Appendices
A Transformation matrices between frames
195
B Angular relationships
205
C Relationships between angles and velocities
215
D Kinematic relationships
225
E Accelerations
241
F State representation and decoupling
265
G Linearized equations
275
xi
xii
SHORT TABLE OF CONTENTS
H Software for the calculation of the equilibrium
321
Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Presentation 1.1 Presentation
I 2
.................................
1 1
11
General equations Frames 2.1 Frame definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The inertial frame FI (A. XI. y ~ 2.1 ) . . . . . . . . . . . . . . 2.1.2 Normal Earth-fixed frame FE (0.XE. YE. Z E ) . . . . . . . . . 2.1.3 Vehicle-carried normal Earth frame F, (0.x,. yo. z, ) . . . . . 2.1.4 Body frame Fb (G. Xb. Yb. Zb) . . . . . . . . . . . . . . . . . . . 2.1.5 Aerodynamic or air-path frame Fa (G. xa. ya. z a ) . . . . . . . 2.1.6 Kinematic or flight-path frame Fk (G. xk. yk. zk) . . . . . . . 2.2 Definition of angles between frames . . . . . . . . . . . . . . . . . . . . 2.2.1 Matrix of transformation from one frame to another . . . . . . 2.2.2 Transformation from inertial frame Fl to Normal Earth-fixed frame F E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Transformation from the inertial frame FI to vehicle-carried normal Earth frame FE . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Transformation from the normal Earth-fixed frame FE to the vehicle-carried normal Earth frame F, . . . . . . . . . . . . . . 2.2.5 Transformation from the vehicle-carried normal Earth frame F, to the body frame Fb . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Transformation from the vehicle-carried normal Earth frame F, t o the aerodynamic frame Fa . . . . . . . . . . . . . . . . . . . 2.2.7 Transformation from the body frame Fb to the aerodynamic frame Fa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.8 Transformation from the body frame Fb to the kinematic frame Fk
vii xii xix
..................................
2.2.9 Transformation from the kinematic frame F k to the aerodynamic frame Fa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...
Xlll
13 13 14 15 16 16
17 17 18 19 21 26 26 27 31 32 33 35
xiv
CONTENTS 2.2.10 Transformation from the normal Earth-fixed frame F, to the kinematic frame Fk . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Angular relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Angle of attack, slideslip angle - Relationships between the frames Fb.Fa. Fk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Angles of attack. inclination angle. climb angle - Relationship between the frames Fb. F,. Fa or Fk . . . . . . . . . . . . . . .
40
3 Kinematics 3.1 The fundamental relationship of kinematics . . . . . . . . . . . . . . . 3.2 Angular and linear velocities . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The definition of velocities . . . . . . . . . . . . . . . . . . . . . 3.2.2 The field of wind velocity . . . . . . . . . . . . . . . . . . . . . 3.2.3 Angular velocity expression . . . . . . . . . . . . . . . . . . . . 3.3 Relationships between angles and velocities . . . . . . . . . . . . . . . 3.3.1 Aerodynamic angle of attack and sideslip angle (&a, ,0a) . . . . 3.3.2 Aerodynamic climb, bank, and azimuth angle (?a. p a . x a ) . . 3.3.3 “Wind” angle of attack and sideslip angle (aw.pw) . . . . . . . 3.3.4 Aerodynamic angle of attack and sideslip angle measurements . 3.3.5 Kinematic climb angle and azimuth (yk. x k ) . . . . . . . . . . 3.4 Kinematic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Kinematic equations of velocity V, . . . . . . . . . . . . . . . . 3.4.2 Kinematic equations of angular velocity R . . . . . . . . . . . .
43 43 45 45 49 58 61 62 62 63 64 66 66 67 69
4 Equations 4.1 Fundamental equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Inertial acceleration of the aircraft’s center of mass . . . . . . 4.2 Inertial angular momentum derivative . . . . . . . . . . . . . . . . . 4.3 External efforts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Gravity - gravitation . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Standard atmosphere . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Aerodynamic efforts . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Propulsion efforts . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Flight Dynamics equations . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Equations of efforts . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Kinematic equations . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Exploitation of the equations . . . . . . . . . . . . . . . . . .
71 71 77 79 82 82 87 89 92 94 95 98 98
I1
. .
.
Simplified equations
5 Simplified equations 5.1 Flat and fixed Earth equations . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Force equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Moment equations . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Kinematic equations . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 The consequences of flat and fixed Earth hypotheses . . . .
36 38 39
101
..
103 104 105 114 116 118
CONTENTS
xv
5.2 Rotating wind velocity field equations . . . . . . . . . . . . . . . . . . 5.2.1 Force equations .second form . . . . . . . . . . . . . . . . . . . 5.2.2 Moment equations .second form . . . . . . . . . . . . . . . . . 5.2.3 Kinematic equations . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Uniform wind velocity field equations . . . . . . . . . . . . . . . . . . . 5.3.1 Force equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Moment equations . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Kinematic equations . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Decoupled equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The decoupling of navigational equations . . . . . . . . . . . . 5.4.2 Decoupled longitudinal equations . . . . . . . . . . . . . . . . . 5.4.3 Decoupled lateral equations . . . . . . . . . . . . . . . . . . . . 5.4.4 The consequence of lateral and longitudinal decoupling . . . . 6 Linearized equations 6.1 Linearization method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Numerical linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Longitudinal linearized equations . . . . . . . . . . . . . . . . . . . . . 6.3.1 Preliminary linearizations . . . . . . . . . . . . . . . . . . . . 6.3.2 Linearization of longitudinal equations . . . . . . . . . . . . . 6.4 Lateral linearized equations . . . . . . . . . . . . . . . . . . . . . . . .
7 Equations for equilibrium 7.1 Equilibrium notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Definition of equilibrium . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Pseudo-equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 The conditions of equilibrium . . . . . . . . . . . . . . . . . . 7.2 Numerical research of equilibrium . . . . . . . . . . . . . . . . . . . . . 7.3 General equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Longitudinal equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Lateral equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I11
.
.
.
Appendices
125 127 128 129 130 130 131 132 132 135 137 145 149 157 158 160 161 161 165 170 179 180 180 182 182 186 188 188 190
193
A Transformation matrices between frames A.l Transformation matrices from frames FZ to FE and from FZ to F. . . A.2 Transformation matrix from frames FE to F. . . . . . . . . . . . . . . A.3 Transformation matrix from frames F. to Fb . . . . . . . . . . . . . . A.3.1 First angular system . . . . . . . . . . . . . . . . . . . . . . . . A.3.2 Second angular system . . . . . . . . . . . . . . . . . . . . . . . A.4 Transformation matrix from frames F, t o Fa and from F. to F k . . . . A.4.1 Transformation matrix from frames F, to Fa . . . . . . . . . . A.4.2 Transformation matrix from F. to Fk . . . . . . . . . . . . . . A.5 Transformation matrix from frames Fb to Fa and from Fb to F k . . . . A.5.1 Transformation matrix from Fb to Fa . . . . . . . . . . . . . . A.5.2 Transformation matrix from Fb t o Fk . . . . . . . . . . . . . .
195 195 196 198 198 199 200 200 200 200 200 201
Dynamics of Flight: Equations
xvi
CONTENTS
.............. ..............
201 202
B Angular relationships B.l Relations between angles of attack and sideslip angles . . . . . . . . . B.2 Relationship between the angles of attack, inclination. climb. bank. sideslip and azimuth . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.1 Determination of inclination. bank and azimuth angles . . . . . B.2.2 Determination of angle of attack c y a . sideslip pa and bank pa angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.3 Third determination of the bank angle . . . . . . . . . . . . . .
205 205
C Relationships between angles and velocities C.l Velocity components of V. V k . V, . . . . . . . . . . . . . . . . . . . . C.2 Aerodynamic angle of attack cya and sideslip angle Pa . . . . . . . . . C.3 Aerodynamic climb and azimuth angles ya and xa . . . . . . . . . . . C.4 “Wind” angle of attack cyw and sideslip angle P, . . . . . . . . . . . . C.5 Measurement of angle of attack and sideslip angle with an aerodynamic probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5.1 Aerodynamic velocity of the probe . . . . . . . . . . . . . . . .
215 215 216 217 218
A.6 Transformation matrix from frames Fk to Fa A.7 Probe angle of attack and sideslip angle . . .
207 208 210 213
220 220
D Kinematic relationships 225 D.l Fundamental kinematic relation . . . . . . . . . . . . . . . . . . . . . . 225 229 D.2 Inertial linear velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3 Angular velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 D.3.1 Determination of the Earth angular velocity i 2 ~ 1. . . . . . . . 231 D.3.2 Determination of the angular velocities f l o and ~ fl, I . . . . . 233 D.3.3 Determination of fib, . . . . . . . . . . . . . . . . . . . . . . . . 233 D.3.4 Determination of n a b and f l k b . . . . . . . . . . . . . . . . . . . 234 D.3.5 Determination of f l k o . . . . . . . . . . . . . . . . . . . . . . . 235 236 D.4 Geographic position relationship . . . . . . . . . . . . . . . . . . . . . D.5 Velocity field of the aircraft . . . . . . . . . . . . . . . . . . . . . . . . 236 238 D.6 Wind velocity field, GRMDVG. . . . . . . . . . . . . . . . . . . . . . . E Accelerations 241 E.l Inertial acceleration of the center of mass G . . . . . . . . . . . . . . . 241 E.2 Two forms for the derivative of the kinematic velocity V k . . . . . . . . 244 E.3 Inertial angular momentum derivative . . . . . . . . . . . . . . . . . . 245 E.4 Derivation of the aerodynamic velocity . . . . . . . . . . . . . . . . . . 251 E.4.1 Wind velocity variation V, . . . . . . . . . . . . . . . . . . . . 251 E.4.2 Calculation of the aerodynamic velocity derivative . . . . . . . 252 253 E.5 Probe acceleration .load factor . . . . . . . . . . . . . . . . . . . . . . E.6 Relative accelerations .consequences of flat and fixed Earth hypotheses 257
CONTENTS
F State representation and decoupling F.l
xvii
Decoupling conditions for the longitudinal equations . . . . . . . . F . l . l Lateral force equation . . . . . . . . . . . . . . . . . . . . . . . F.1.2 Yaw and roll moment equations . . . . . . . . . . . . . . .
265
. . 265
..
265 272
G . l Numerical linearization . . . . . . . . . . . . . . . . . . . . . . . . . . G.2 Wind velocity field linearization . . . . . . . . . . . . . . . . . . . . . G.3 Linearization of the longitudinal equations . . . . . . . . . . . . . . . G.3.1 Linearization of the propulsion equation . . . . . . . . . . . . . G.3.2 Linearization of the sustentation equation . . . . . . . . . . . . G.3.3 Linearization of the moment equation . . . . . . . . . . . . . . G.3.4 Linearization of the kinematic equations . . . . . . . . . . . . . G.4 Linearization of the lateral equations . . . . . . . . . . . . . . . . . . . G.4.1 Linearization of the lateral force equation . . . . . . . . . . . . G.4.2 Linearization of the roll moment equation . . . . . . . . . . . . G.4.3 Linearization of the yaw moment equation . . . . . . . . . . . . G.4.4 Linearization of the bank kinematic equation . . . . . . . . . . G.4.5 Linearization of heading kinematic equation . . . . . . . . . . .
275 275 283 295 295 299 302 305 305 305 312 316 319 320
G Linearized equations
H Software for the calculation of the equilibrium 321 H.l Software for the calculation of the equilibrium . . . . . . . . . . . . . 321 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
Dynamics of Flight: Equations
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Nomenclature ’,
As far as is possible the notations used in this book respect the international I S 0 standard . The alphabetical nomenclature is included within the index. In the index, the Greek symbols can be found under the entry “Greek symbols”.
Reference notations Notations used to refer to other part of the document are explained below:
171 (Section 4.1.2) (Section C.2.3) (Figure 4.7) (Equation 4.18) (Hypothesis 23)
Reference number 7. The pages where references appear, can be found under the term “References” of the index.
Section 1.2 of chapter 4 Section 2.3 of appendix C Figure number 7 of chapter 4 Equation number 18 of section 4 Hypothesis number 23. The list of all the hypotheses is at the end of the book.
Vectors All the vectors in this document, printed in boldface X,are defined in three dimensional space R3. The superscript on a vector denotes the projection frame, that is to say the frame in which the vector is expressed :
X i represents the X vector projected on frame Fi. The notation in ordinary characters represents the modulus of the vector as X . The vector is representented by its components as a matrix with one column and three rows.
lAs far as is possible means that sometimes the definitions were not precise enough or that the recommendations did not seem compatible with the coherence and clarity of the notations and that it was necessary to adapt them. The AIAA (American Institute of Aeronautics and Astronautics) takes I S 0 1151 as a base for its recommendations in its report ANSI/AIAA R004 1992.
xix
Vectors are qualified by their subscripts In the symbol V,, “a” qualifies this velocity as aerodynamic. In principle, for velocity,
two subscripts are needed. For V,,G the first “a))indicates the type of velocity and the second the point concerned, in this case G the aircraft’s center of mass. Thus V,,S denotes probe S, etc. For the mass center of an aircraft (G), this second subscript is often omitted. For example, the simplified notation V, for V a ,can ~ be accepted if no confusion is possible. A third italic type of subscript can appear, in order t o indicate an initial value (for example, Vai ) particularly for a linearization operation (Section 6, p. 157).
Vector components The components of position vectors are 2, y, z. Linear velocity vector components are U, U , w and angular velocity components are p , q, r. These components are generally completed by a qualifying subscript and a superscript characteristic of the projection frame (see at the end of the nomenclature for some vector component notation comments). For example, the symbol p i represents the first component p of the angular kinematic velocity i l k , projected on the aircraft’s body frame Fb. The symbol ys represents the second component y of the position of point S projected on the vehicle-carried normal Earth frame Fo. Subscripts can be attached t o velocity components in order to indicate the parameter from which they are derived. For example, the symbol UZOV, represents the second component (v) of the wind velocity (w) projected on the vehicle-carried normal Earth frame Fo (0)and derived with respect t o z .
Operations on matrices and vectors Derivation of a vector is meaningful only when the projection frame is defined (Section X with respect to the frame Fi
3, p. 43)) thus the derivation of the vector
will be written
dXi dt
-
which seems simpler. Please note the temporal derivative of a function
Thesymbol x represents the cross product Operation between two vectors or vector product. The dot product operation is represented by “.‘I, or nothing, thus
X.Y or XY represents the dot product of vectors X and Y. The matrices are not printed boldface in this document but in outline type, and the matrix product has no special notation. The transposition is denoted h@. The superscript “t“ cannot be confused with the superscript of a frame projection as there is no frame Ft. Thus, the dot product can also be represented by XtY. The inverse
Nomenclature
-~
xxi
matrix is denoted M-'.The identity matrix is denoted by 1 1 . For the angular velocity vector 0, there is an associated skew-symmetric matrix & so that
nxx
=
with
n and
m =
&X
(f)
=
(E
-Q
-0'
:p)
P
Frames
Ff
Geocentric inertial frame Normal Earth-fixed frame Vehicle-carried normal Earth frame Body (aircraft) frame Aerodynamic (air-path) frame Kinematic (flight-path) frame Vehicle-carried normal Earth frame (Fo), aircraft azimuth oriented or aircraft course oriented Vehicle-carried normal Earth frame(F,), aircraft fuselage oriented
(Section (Section (Section (Section (Section (Section (Section
2.1.1, p. 14) 2.1.2, p. 15) 2.1.3, p. 16) 2.1.4, p. 16) 2.1.5, p. 17) 2.1.6, p. 17) A.3.1, p. 198)
(Section A.3.1, p. 198)
Frame origins Once a vector has been defined, with or without reference to a frame origin, its derivation with respect t o time is independent of the frame origin from which it was derived. Only the angular velocity of the frame intervenes in the fundamental kinematic relationship. Practically speaking, this means that, most of the time, the use of frames parallel t o those defined here would not change the calculation processes and their formulation. Thus, the importance of the definition of the origin is purely relative. A frame origin becomes important for position vectors if the position of the point concerned is referenced to this origin.
A 0
G
Origin of frame F I , Earth center Origin of frames FE and F,, an Earth surface point Origin of frames Fb, Fa, Fk, aircraft center of mass
(Section 2.1.1, p. 14) (Section 2.1.2, p. 15) (Section 2.1.4, p. 16)
Dynamics of Flight: Equations
xxii
Nomenclature
The Earth The subscript t is used as terrestrial.
Earth mass Mean Earth radius Semi Earth major axis Semi Earth minor axis Earth oblation or flattening Earth eccentricity Point 0 stellar time Point G stellar time Latitude of 0 Latitude of G Longitude of G with respect to Normal Earth-fixed frame FE Latitude of G with respect to Normal Earth-fixed frame FE Geocentric latitude Gravitation latitude Geodesic latitude Astronomic latitude
(Section 4.3.1, p. 82) (Section 2.1.2, p. 15) (Section 2.2.2, p. 21) (Section 2.2.2, p. 21) (Equation 2.25, p. 25) (Equation 2.26, p. 25) (Section 2.2.2, p. 21) (Section 2.2.3, p. 26) (Section 2.2.2, p. 21) (Section 2.2.3, p. 26) (Section 2.2.7, p. 32) (Section 2.2.7, p. 32) (Section 2.2.2, p. (Section 2.2.2, p. (Section 2.2.2, p. (Section 2.2.2, p.
21) 21) 21) 21)
The latitude of G, A L t , is equal t o
ALt = LtG
- Lto
Transformation matrix Transformation matrices (Section 2.2, p. 18) are denoted T with two suffices for the frames concerned. For example, the vector X" expressed in the frame F, is equal to the product of the transformation matrix To"from Fo to Fa , by the vector X" expressed in the frame Fa . The vectors X" and X" are the same vector X expressed in two different frames.
X" = T,"X" with the properties of composed matrices and the properties of inverse matrices
Nomenclature
TIE Tzo TEo Tob Toa Tba Tbk Tka Tok
Transformation Transformation Transformation Transformation Transformation Transformation Transformation Transformation Transformation
xxiii matrix from matrix from matrix from matrix from matrix from matrix from matrix from matrix from matrix from
FI to FE FI to F, FE to Fo F, to Fb F, to Fa Fb to Fa Fb to Fk Fk to Fa F, to Fk
(Section (Section (Section (Section (Section (Section (Section (Section (Section
2.2.2, p. 21) 2.2.3, p. 26) 2.2.4, p. 26) 2.2.5, p. 27) 2.2.6, p. 31) 2.2.7, p. 32) 2.2.8, p. 33) 2.2.9, p. 35) 2.2.10, p. 36)
(Section (Section (Section (Section (Section (Section
2.2.5, p. 27) 2.2.5, p. 27) 2.2.5, p. 27) 2.2.5, p. 27) 2.2.5, p. 27) 2.2.6, p. 31)
Angles between frames Azimuth angle (yaw angle, heading) Inclination angle (pitch angle or elevation angle) Bank angle (roll angle) True aircraft azimuth Magnetic aircraft azimuth Aerodynamic azimuth (air-path azimuth or airpath track angle) Aerodynamic climb angle (air-path climb or airpath inclination angle) Aerodynamic bank angle (air-path bank angle) Aerodynamic angle of attack (aerodynamic incidence) Aerodynamic sideslip angle Aerodynamic angle of attack at probe (or sonde) station Aerodynamic sideslip angle at probe (or sonde) stat ion Kinematic angle of attack (kinematic incidence) Kinematic sideslip angle Kinematic azimuth (flight-path azimuth or flightpath track angle) Kinematic climb angle (flight-path climb or flightpath inclination angle) Kinematic bank angle (flight-path bank angle) Wind angle of attack (wind incidence) Wind sideslip angle Wind bank angle
(Section 2.2.6, p. 31) (Section 2.2.6, p. 31) (Section 2.2.7, p. 32) (Section 2.2.7, p. 32) (Section C.5, p. 220) (Section C.5, p. 220) (Section 2.2.8, p. 33) (Section 2.2.8, p. 33) (Section 2.2.10, p. 36) (Section 2.2.10, p. 36) (Section (Section (Section (Section
2.2.10, p. 36) 2.2.9, p. 35) 2.2.9, p. 35) 2.2.9, p. 35)
Dynamics of Flight: Equations
Nomenclature
xxiv
Positions The position vector components are denoted 2, y, z with subscript and superscript. The subscript indicates the concerned point and the superscript indicates the projection frame. For example
xi
x position of the probe S in the body frame Fb
9: zs
z position of the probe S in the body frame
y position of the probe S in the body frame Fb
Engine position in the body frame
Fb
x position of the engine in the body frame
x&
yL
Fb
Fb
y position of the engine in the body frame Fb
z position of the engine in the body frame Fb Equivalent position tfo ZM for the pitching moment
zb
z!
(Section 3.3.4, p. 64) (Section 3.3.4, p. 64) (Section 3.3.4, p. 64)
(Section 4.3.4, p. 92) (Section 4.3.4, p. 92) (Section 4.3.4, p. 92) (Equation G.152, p. 303)
Altitudes h h, = H
Altitude Geopotential altitude
(Section 2.2.2, p. 21) (Equation 3.118, p. 69)
Linear velocities Linear velocities are denoted V and their components (U,w, U)). For the aircraft center of mass (G) velocity, it is possible to omit the G subscript. The term linear is justified by the nature of these velocities and characterizes them with respect to the angular velocities. As, however, there is little risk of confusion, they are usually simply called velocities.
VI = V I , G v k
=Vk,G
v k p
= Vkp,G
Va = V a , G
Inertial velocity of the aircraft center of mass or Velocity relative to the inertial frame FI Kinematic velocity of the aircraft’s center of mass Flat kinematic velocity of the aircraft’s center of mass Aerodynamic velocity of the aircraft center of mass
(Section 3.2.1, p. 45)
(Section 3.2.1, p. 45) (Section 5.1.4, p. 118) (Section 3.2.1, p. 45)
xxv
No mencl at ure
V , = V,,G M
Wind velocity (velocity of an atmospheric particle which could have been located at the center of mass) Mach number
Linear velocity components
(Section 3.2.1, p. 45)
(Equation 6.20, p. 162)
(U, w, w)
The linear velocity components are denoted U , U , w with subscript and superscript. The first indicates the concerned velocity and the second the projection frame. For example, U: means the projection on the y axis of the v k velocity on the body frame Fb. U;
= VN
U:
= VE
w; = -Vz
v~
x component of v k on PO, North velocity y component of V k on F,, VE East velocity z component of Vk on F,, V’ Vertical velocity
(Section 3.2.1, p. 45) (Section 3.2.1, p. 45) (Section 3.2.1, p. 45)
Angular velocities Angular velocities are denoted Cl and their components ( p , q, r )
0 = ak = O b E Kinematic angular velocity of the aircraft (Section 3.2.1, p. 45) (b) relative t o the Earth (E) a, = a a t m E Local wind angular velocity relative t o the (Section 3.2.1, p. 45) Earth (atmosphere angular velocity (atm) relative to the Earth (E)) a a Aerodynamic angular velocity (atmosphere (Section 3.2.1, p. 45) angular velocity relative to the aircraft) at = ~ E I Earth angular velocity relative to the iner- (Section 3.2.3, p. 58) tial frame aoI Angular velocity of the frame Fo relative (Section 3.2.3, p. 58) to the frame FI aoE Angular velocity of the frame Fo relative (Section 3.2.3, p. 58) to the frame FE abo Angular velocity of the frame Fb relative (Section 3.2.3, p. 58) t o the frame F, nab Angular velocity of the frame Fa relative (Section 3.2.3, p. 58) t o the frame Fb ako Angular velocity of the frame Fk relative (Section 3.2.3, p. 58) t o the frame F, akb Angular velocity of the frame Fk relative (Section 3.2.3, p. 58) t o the frame Fb
Dynamics of Flight: Equations
xxvi
Nomenclature Skew-symmetric matrix associated to the angular velocity vector s2
m-l
Angular velocity components ( p , q,
(Equation 3.2, p. 44)
T)
The angular velocity components are denoted p, q, r with subscript and superscript. The subscript indicates the concerned velocity and the superscript, the projection frame. For example, q t indicates the y projection of kinematic angular velocity h2k on the body frame Fb. Simplified notations p , q, r without subscript and superscript for the components of Oh on body frame Fb are generally accepted. Components on body frame Fb of the kinematic angular velocity s 2 h of the aircraft p = p$ Roll velocity (or roll rate) of the aircraft relative to
the Earth q = q i Pitch velocity (or pitch rate) of the aircraft relative to the Earth r = r i Yaw velocity (or yaw rate) of the aircraft relative to the Earth
(Section 3.2.1, p. 45) (Section 3.2.1, p. 45) (Section 3.2.1, p. 45)
Components on body frame Fb of the Earth angular velocity h2t (Section 4.2, p. 79) pt = pf Component on xb of the Earth angular velocity O , qt = q! Component on Yb of the Earth angular velocity rt = r: Component on z b of the Earth angular velocity 62,
(Equation 4.56, p. 81) (Equation 4.56, p. 81) (Equation 4.56, p. 81)
Wind gradient @mV;
Jacobian matrix of V, or wind gradient, that is to say the spatial derivative of V, relative to the vehicle-carried normal Earth frame F,
(Section 3.2.1, p. 45)
Each row of the &AD matrix is composed of successive derivatives with respect x, of one of the components of velocity V,. Another possible notation of &mV, could be V,V = (VtV:l,)t.The spatial derivation operator, 0, takes the form of a row vector to
2, y,
I
Components of (GRADVE)~
(cswmv;)o =
(
uxo,
vxo, wxo,
uyo,
uzo,
vy; wy;
vz; wz;
1
xxvii
Nomenclature
Here, the components are expressed in the vehicle-carried normal Earth frame F,, indicated by the superscript “0”. This projection frame can be modified. Some spatial derivatives are denoted p, q, r in order to recall their physical meaning that is associated with angular velocity. Spatial Spatial Spatial Spatial Spatial Spatial Spatial Spatial Spatial
derivative along x of the first component of V, derivative along y of the second component of V, derivative along z of the third component of V, derivative along y of the first component of V, derivative along z of the first component of V, derivative along x of the second component of V, derivative along z of the second component of V, derivative along x of the third component of V, derivative along y of the third component of V,
The p, q, r notations above are purely symbolic so as to recall the physical sense of the spatial derivatives of V,. Abbreviated notations (Section G.2, p. 283)
@;
(Equation G.38, p. (Equation G.39, p. (Equation G.40, p. (Equation G.41, p. (Equation G.42, p. (Equation G.43, p.
=Py;-P; - qx;
(jg = qz; ?g =
- rYL - UyL 6; = uy; - wz; ti3; = wz; - ux; -fX;
ii; = ux;
These six quantities denoted tilde pothesis 10)
DV,
286) 286) 286) 286) 286) 286)
((‘”, are null if the wind is modelled by a vortex (Hy-
Complementary acceleration peculiar to wind
(Equation 5.47, p. 112)
The components of the vector DV,, for example projected on the vehicle-carried normal Earth frame F, are du;, dug, dwg. The vector DV, is equal to &mV;V,.
Kinetics IIG
Inertial matrix of the aircraft with respect to G
(Section 4.1, p. 71)
The components of the matrix IG expressed in the body frame the components of 1;
1L
=
(3
-F B -D
Fb,
that is to say
-E - D ) C Dynamics of Flight: Equations
Nomenclature
xxviii
A = 1x2 B = Iyy c = Izz D = Iyz E = 1x2 F = Ixy
Aircraft Aircraft Aircraft Aircraft Aircraft Aircraft
moment of inertia with respect t o x b moment of inertia with respect t o Y b moment of inertia with respect to z b product of inertia with respect to x b product of inertia with respect to Y b product of inertia with respect to zb
HI,G Inertial angular momentum with respect to the aircraft center of mass G
(Section (Section (Section (Section (Section (Section
4.1, p. 4.1, p. 4.1, p. 4.1, p. 4.1, p. 4.1, p.
71) 71) 71) 71) 71) 71)
(Section 4.2, p. 79)
Accelerations Inertial acceleration of the aircraft center of mass
(Section 4.1.1, p. 77)
Pseudo relative acceleration Complementary acceleration related to Earth sphericity Complementary acceleration related to Earth rotation Complementary acceleration related to distance Complementary acceleration related to distance Complementary acceleration related to acceleration Wind complementary acceleration (null if wind is
(Equation 4.41, p. 78) (Equation 4.43, p. 78)
G
(Equation 4.45, p. 78) (Equation 5.94, p. (Equation 5.97, p. (Equation 5.93, p. (Equation 5.43, p.
121) 122) 121) 112)
modelled by a vortex)
Wind complementary acceleration related to vortex Wind complementary acceleration related to translation Complementary acceleration peculiar to wind
(Equation 5.44, p. 112) (Equation 5.45, p. 112) (Equation 5.47, p. 112)
The components of the vector DV,, for example projected on the vehicle-carried normal Earth frame F' are du;, dw;, dw;. The vector DV, is equal t o GRADV~V,.
Standard atmosphere The three thermodynamic air states p, p, T, depend on, Q priori, the spatial position of point M and of time t. Generally these air states only depend on altitude h. P P
T Th
ph
Static pressure Air density Static temperature Temperature gradient with respect to altitude Air density gradient with respect to altitude
(Section (Section (Section (Section (Section
4.3.2, p. 87) 4.3.2, p. 87) 4.3.2, p. 87) 4.3.2, p. 87) 4.3.2, p. 87)
Nomenclature
xxix
Ideal gas constant Altitude
7Z h
(Section 4.3.2, p. 87) (Section 2.2.2, p. 21)
The gradients are respectively equal t o
dT dh
T h = -
and
dp ph= -
dh
External efforts2
s
Gravitational constant Acceleration due t o gravity (free fall direction) Acceleration due t o gravitational attraction Aircraft reference area Aircraft reference length Dynamic pressure (Although the standard doesn't
Q g,
S
e
qp
(Section 4.3.1, p. (Section 4.3.1, p. (Section 4.3.1, p. (Section 4.3.3, p. (Section 4.3.3, p.
82) 82) 82) 89) 89)
give a subscript to q the subscript "p" as pressure is used to avoid confusion with the pitch velocity q )
The dynamic pressure is equal t o =0 . 7M ~2 qp = ZpV; 1
It is common American practice t o use for the length of reference k' the wing span k' = b for roll and yaw aerodynamic coefficients and the mean aerodynamic chord k' = C for pitch aerodynamic coefficients. In Europe, usually the length of reference k' is the mean aerodynamic chord 4! = C.
Aerodynamic coefficients expressed in aerodynamic frame Fa CO = Ca: = -Cx"
cc = c y = CY"
Drag coefficient Cross stream or lateral force coefficient (Cy is more often used than
C L = C z = -Cz"
Lift coefficient
CZ" Cm" Cn"
Cc)
Rolling moment coefficient Pitching moment coefficient Yawing moment coefficient
(Section 4.3.3, p. 89) (Section 4.3.3, p. 89)
(Section 4.3.3, p. 89) (Section 4.3.3, p. 89) (Section 4.3.3, p. 89) (Section 4.3.3, p. 89)
Aerodynamic coefficients expressed in body frame Fb 2Efforts means forces or moments. Dynamics of Flight: Equations
Nomenclature
XXX
C A= -Cxb CY = C y b C N = -Czb
Axial force coefficient Side force or transverse force coefficient Normal force coefficient
(Section 4.3.3, p. 89) (Section 4.3.3, p. 89) (Section 4.3.3, p. 89)
c1= Clb C m = Cmb Cn = Cnb
Rolling moment coefficient Pitching moment coefficient Yawing moment coefficient
(Section 4.3.3, p. 89) (Section 4.3.3, p. 89) (Section 4.3.3, p. 89)
Coefficient notations without superscript are commonly used. Equivalent notations with superscript, given above, agree with the general standard logic of notation, as for example Cx",CY", Cz" which are equivalent to CO, CC, C L . However, compared with this general logic of notation, there is one of the rare, if not the only, exceptions. It concerns the sign for the drag and lift coefficients CO and C L . This change comes from the very natural notation used by the aerodynamicians and which is respected by the flight dynamicists. Thus, there is positive lift and drag in most cases. The problem comes from the choice, in flight dynamics, of the positive downward z direction and positive forward x direction.
Aerodynamic coefficient derivatives The usual model for the aerodynamic coefficient is linear, based on the derivative with respect to the aircraft states. In this book the coefficient derivatives appear in the linearized equation (Section 6, p. 157). The whole list of the derivatives is not given here, but two derivatives are shown from which the others can be deduced.
CL^
gradient of lift with respect to the angle of attack
(Section 4.3.3, p. 89)
CY
CLq
gradient of lift with respect to the pitch velocity q
(Section 4.3.3, p. 89)
The gradient relative to angles are equal to
dCL CLa, = da, and the gradient relative to angular velocity are equal to
dCL v CLq= --
& e
Then the model for the aerodynamic coefficient is CL
= CLa CY
+ CLq + ... V
For the normalized angular velocity as qt!/V,it is common American practice to use for the length of reference e the wing semi-span e = b / 2 for roll and yaw derivatives and the mean aerodynamic semi-chord l = i?/2 for pitch derivatives. In Europe, usually the length of reference e is the mean aerodynamic chord l! = i?.
Nomenclature
xxxi
Contr01s 61
,6 6, 6,
Roll control Pitch control Yaw control Thrust control
Propulsive efforts F
x
km
am Pm
MF
Propulsive force Characteristic parameter of the engine Engine constant Pitch setting of the engine Yaw setting of the engine Propulsive moment relative to the aircraft center of mass G
(Section 4.3.4, p. (Section 4.3.4, p. (Section 4.3.4, p. (Section 4.3.4, p. (Section 4.3.4, p. (Section 4.3.4, p.
92) 92) 92) 92) 92) 92)
The components of the propulsive force and its moment are denoted F z , F y , F z and M F ~MQ,, , M F respectively, ~ with a superscript for the projection frame. This frame is often the body frame FJ,and components are denoted F:, Fy”, F: and AI:,, Mky, Mk,.
Engine position in the body frame
XL yk
zM z$
Fb
x position of the engine in the body frame FJ,
y position of the engine in the body frame FJ, z position of the engine in the body frame FJ,
Equivalent position to X M for the pitching moment
(Section 4.3.4, p. 92) (Section 4.3.4, p. 92) (Section 4.3.4, p. 92) (Equation G.152, p. 303)
Notations associated with the linearization operation The linearization operation in section (6), p. 157 uses a certain number of special not at ions. Subscript: “i” as initial, defines the parameter value around which linearization of the equations is accomplished. Example: V, aerodynamic velocity modulus, Vai initial aerodynamic velocity. Prefix A: This prefix indicates the difference between the parameter value and its initial value. It corresponds to the increment relative to this initial value, obtained through the differentiation of the non linear equations. Example: AV, = V, - Vai The reduced states give nondimensional parameters. These reduced states are obtained from the division of the state by his initial value. Reduced velocities
Dynamics of Flight: Equations
xxxii
Nomenclature
Va
Reduced velocity Component of reduced wind velocity, etc
-
U;
(Section 6.3.1, p. 161) (Section 6.3.1, p. 161)
The reduced velocities are obtained from the division of the velocities by his initial value of the aerodynamic velocity. So -
Va v, = Va
and
i
Reduced atmosphere states -
Reduced temperature gradient Reduced air density gradient
T h
-
ph
(Section 6.3.1, p. 161) (Section 6.3.1, p. 161)
And -
Th
Th
=T,
and
Matrices of the linearized system The linearization of longitudinal equations (Section 6.3.2, p. 165) takes on the form3 (11 -
GRADFXL~)AXL = (GWADFXL~AXL + GRADFuL~AUL + GRADFWGL~AWGL + GRADFWRL~AWRL + (GWADFWLL~AWLL
Or in a simpler form
AXL = AXLAXL + BULAUL+ BWLAWL with the wind participation
BWLAWL= B W L L A ~ + L BWGLAWGL L + BwRLAWRL
X L= ~ [Va, q , h, r,] Longitudinal state vector Longitudinal control vector ULt = a[, 6,] Wind linear velocities (L) vector W L L=~[gW,Pw] Gradients of the wind linear velocitW G L=~[U&, w.;] ies (G) vector Angular or rotational wind velocitW R L=~[q&,q&] ies (R) vector
(Section (Section (Section (Section
6.3.2, p. 6.3.2, p. 6.3.2, p. 6.3.2, p.
165) 165) 165) 165)
(Section 6.3.2, p. 165)
The matrix due to the state derivative influence A ~ =L1 1 - GRADFXL~ A X L = A~L-~GRADFxL, State matrix of the longitudinal system BUL= A~L-~GRADFuL~Matrix of controls of the longitudinal system Matrix of perturbation due to the wind translation BWLL= A~L-~GRADFwLL~ velocities 3The gradient & A D
is defined with the "wind gradient", p. xxvi
xxxiii
No m enc1at ure
Matrix of perturbation due t o the wind translation of BWGL= A&L-~(GWADFWGL~ velocity gradients Matrix of perturbation due t o the wind rotations BWRL= %cL-’(GWA.DFWRL~
REMARK 0.1 In a first approximation the matrix Ajc~could be taken equal to the identity matrix 111 equation (6.47), p. 165.
The matrices @mF are the Jacobian matrices of the vector F, see the “wind gradient” p. xxvi. The components of vector F are the longitudinal equations and according t o the longitudinal state vector XL components, the first component is the propulsion equation (6.50), p. 166, the second component is the kinematic angular equation (6.54), p. 168, the third component is the moment equation (6.52)) p. 167, the fourth component is the kinematic altitude translation equation (6.54)) p. 167 and the last component is the sustentation equation (6.54), p. 168. Each row of the GRAB matrices is composed of successive derivatives with respect to components of the state vector XL, control vector UL and wind vector WL,of one of the components of the vector F . The whole list of the components of the matrices (GWmF is not given here, but the general logic underlying the component notation is explained with examples. The two prefix letters of the components as a x , bu or bw are respectively associated t o the state matrix AX, the control matrix Bu and the wind perturbation matrices Bw . The first subscript is linked to the equation which is linearized, so this subscript is taken in the components of the state vector XL. The second subcript indicat,ed the derivation parameter and this is the reason why it is denoted with italic letters. For example, axqa is a component of the state matrix GRADFXL~ or AXL since “ax”, and in the third row which is linked t o the moment equation since the first subscript is q which is the third component of the state vector XL. The derivation is relative t o the angle of attack a , since the second subscript is a . As a is the second component of the state vector, the coefficient axqa is located in the second column of the state matrix. or BULsince In a same way, bUqm is a component of the control matrix GRADFUL~ ‘‘bu”, and in the third row which is linked t o the moment equation since the first subscript is q. The derivation is relative t o the pitch control 6m, since the second subscript is m. As an example, the longitudinal state matrix G A D F XisL given ~ as
axq,
bqm
Derivative of the pitch moment equation (4) with respect to the angle of attack cy Derivative of the pitch moment equation ( q ) with respect to the pitch control Sm
(Equation 6.52, p. 167) (Equation 6.58, p. 168)
Dynamzcs of Flight: Eguatzons
xxxiv
Nomenclature ~~
etc. The linearization of lateral equations (Section 6.4, p. 170) takes on the form
(11 - GADFX~JAXI= @mFxliAXi + (GWmFuliAUi+ G A D F x L ~ ~ A X L + &.ADFWLI,AWLI + GRADFWGI~AWGI + GRMI~FWRI~AWR Or in a simpler form
AXi
+
+
+
= AXIAXI AXLIAXL BUIAUI BWIAWI
with the wind participation
BWIAWI= B w L ~ A W+LBWGIAWGI ~ + BWRIAWRI Xit = [p,, p , T , 4, $1 X L= ~ [V,, a,, q , 191
Ult = [ S l , 6711 W L I=~[U:,,U:,,2 4 1 WGlt = [ U x ; , , Uy:,
Lateral state vector Longitudinal state vector which influence the lateral states Lateral control vector Wind linear velocities (L) vector
, Wz;]
W R I=~[PYL,pz;,, q x : , , ,
qz:,
,r x ; , , TYO,,]
Gradients of the wind linear velocities (G) vector Angular wind velocities (R) vector
(Section 6.4, p. 170) (Section 6.4, p. 170) (Section 6.4, p. 170) (Section 6.4, p. 170) (Section 6.4, p. 170)
(Section 6.4, p. 170)
Matrix due t o the inertial products or sideslipe derivat ive State matrix of the lateral system Matrix of the influence of the longitudinal states on the lateral system The matrix of controls of the lateral system Matrix of perturbation due t o the wind velocities of translation Matrix of perturbation due to the wind translation velocities gradients Matrix of perturbation due to the wind angular velocities (In a first approximation the matrix Ail could be taken equal to the identy matrix 11 (Equation 6.80, p. 172).)
As for the longitudinal linearization the same logic is used. The matrices G R ~ F are the Jacobian matrices of the vector F. The components of vector F are the lateral equations and according to the lateral state vector Xi components, the first component is the lateral force equation (6.85), p. 173, the second component is the roll moment equation (6.87), p. 174, the third component is the yaw moment equation (6.89),
p. 174, and the fourth and fifth components are the kinematic angular inclination angle equation (6.91), p. 175 and azimuth equation (6.93), p. 175. The whole list of the components of the matrices &mF is not given here, but the general logic underlying the component notation is explained with examples. See the longitudinal linearization for detailed explanations. As an example, the lateral state matrix & m ~ F xis~ ~ given (Section 6.4, p. 170).
Derivative of the roll moment equation (lj) with respect to the sideslip angle P Derivative of the roll moment equation (lj) with respect to the yaw control Sn
________
~
~
_
_
_
_
(Equation 6.87, p. 174) (Equation 6.97, p. 176)
_
Dynamics of Flight: Equations
Some remarks on the logic behind the component notation of the vectors Position components are denoted 2, y, z ; their derivatives, the velocities, are denoted w. Logically, the acceleration components should be denoted r , s, t. However, r introduces a notation conflict with the third component of the angular velocity. Thus, a notation commonly adopted for the acceleration components is a z , a y , a z . The corresponding forces are denoted X, Y , 2, or CX, C Y , CZ for the aerodynamic force coefficients and F z , F y , F z for the propulsive force components. For the angular parameters, the previous linear parameter logic is not found. Angular position components itre denoted +, 0,4 or x,y, p or a,0, p depending on the circumstances. Angular velocity components are denoted p , q, r and moment L , M, N . To be coherent with the notation p , q, T , the angular position should have been denoted s, t , U , but in this case, a conflict would be created with the U of the linear velocity component. A solution could be to shift one letter towards the left: the angular position r , s, t , the angular velocity 0,p , q and the moment R , S, 2'. Another way would be to use the Greek alphabet, or a mixed solution: positions using the Greek alphabet and the remainder the Roman alphabet. In this case, the linear components will have to be redefined. A homogeneous notation could be obtained for the definition of angles between frames by replacing the notation t+h, 8, 4 by the notations x, y, p . On the other hand, it would be possible to give up the notations x, y , p and replace them by $, 8, U, U,
4.
The proposed notations are between parentheses; ( A B C) could be in conflict with the inertia.
1
Present at ion
1.1
Presentat ion
The purpose of this chapter is t o comment on the procedure developed in order t o obtain aircraft flight dynamics equations for a rigid aircraft. The creation of these equations is the goal of this book. The study of flight dynamics can be applied t o various aircraft capable of leaving the Earth’s surface. External efforts have been chosen to give the model suitable for an aircraft. Following the presentation of this machine, a short definition of flight dynamics is given and then the objectives of this discipline which is a branch of applied mechanics will be defined. Finally, the process to elaborate these equations will be commented on in detail.
The airplane[5I1 is an element of the aircraft family composed of aerostats and aerodynes. The “lighter than air” aerostats are opposed to the “heavier than air” aerodynes. This latter family is separated into two groups: the moving wing machines as rotorcraft and ornithopters, and the fixed wing machines with gliders and airplanes. The family of unmanned aerodynes are composed of missile and gnopter. Missile is a well known term; gnopter is a generic term2 which designates machines, from older to newer terms, as RPV (remotly piloted vehicule), drone and UAV (unmanned air vehicle). The term gnopter comes from the Greek roots gnosis for knowledge or gnome for intelligence, and pteron for wing. Most of the time, the gnopter is used for reconnaissance missions and this aerial platform is no more than a flying sensor. The sensor drives knowledge towards the ground operator. The gnopter is an aerial robot and it is becoming more and more autonomous. Thus gnopter can be understood as knowledge and intelligence equipped with a wing. The difference lLangley had named the airplane the “aerodrome”,literally “travelling through the air”. Over the years it has become the term known today. Lanchester used the word “aerodone”, litterally “tossed in the air”, for the glider. At the beginning of aeronautics, the word “flying machine” was reserved for the ornithopter. This use irritated Lanchester who was convinced that the flying principle on a machine could not be reached with flapping of the wings. 2The term is due to Pierre Vacher and Laurent Chaudron.
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1 - Presentation
2 Aircraft :
Aerostat and Aerodyne Aerostat “lighter than air“ Aircraft
Balloon - Airship
Aerodyne : “heavier than air“ Aircraft
MOVING WING Rotorcraft : Revolving wing Ornithopter : Flapping wing Helicopter Autogyro
f
I
FIXED WING Airplane : Motorised aerodyne with fixed wing
Glider : Airplane without engine
(Aeroplane)
i
Missile and Gnopter : Unmanned aerodyne \
Figure 1.1: Aircraft between missile and gnopter mostly lies in a higher endurance (flying time) and a higher rate of information transmitted to the manned operator by the gnopter than by the missile. The consequence of endurance on the gnopter configuration yields an aircraft-like vehicle, often with low speed performance. The aircraft is a motorized, “heavier than air” machine, ensuring its sustentation by fixed wings. The three external efforts - thrust, weight and aerodynamic - t o which the aircraft is submitted during flight appear in this definition (Figure 1.2). It should
Figure 1.2: External efforts be mentioned that this machine is piloted by the aerodynamic and propulsive efforts, which can be governed by the control systems. Generally, weight is not piloted3.
Flight dynamics: Dynamics is the analysis of the motion of a material system by a study of the efforts t o which it is submitted. Flight characterizes the airplane’s motion, a motion that the dynamic definition will demonstrate t o be connected t o the efforts defined previously. ~ _ _ _
~
3Certain modern transport aircraft, however, can displace their center of gravity by fuel transfers in order to improve their performance at cruising speed. There is also the case of hand gliders where the only means of piloting them is a displacement, longitudinally and laterally, of the center of gravity. Certain gliders and fire-fighting aircraft drop water and thus vary their mass.
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1.1 Presentation
Flight represents both the trajectory of the aircraft and the means which allow it t o stay in the air, two notions linked to the study of flight dynamics. Etymologically, the word flight is associated t o a movement in the atmosphere: the expression atmospheric flight, in this case, is a pleonasm. However, usage has extended this notion to flight outside the atmosphere. Etkin [2] uses the expression “flight of angels” t o justify this extension. In its widest sense, “flight” indicates the trajectory and the means of permitting an escape from the bidimensional world associated with the Earth’s surface. This is not in contradiction with another meaning of the word flight: soaring. The fluid in which flight takes place can be the air (atmospheric flight), water (subaquatic flight) or vacuum (space flight). Flight is thus characterized b y a non-null height trajectory which is, in most cases, three-dimenszonal. Therefore, flight could be considered as an escape from terrestrial contingencies.
Performances and flying qualities: The trajectory is representative of the performance of the aircraft during its mission: range, velocity, altitude, etc. This study of performance is carried out with the equations written for equilibrium, and it translates the airplane’s capacity t o accomplish its mission. The equilibrium corresponds to the specific case where the general dynamic equations, or some of them, are written with acceleration equal t o zero. A more precise definition of equilibrium is given in (Section 7.1, p. 180).
I PERFORMANCE e EQUILIBRIUM I Exploitation of the complete equations with non-null acceleration leads to the study of the aircraft dynamics. These dynamics are associated with the notion of flying qualities4, the translation of the pilot’s ability to accomplish his mission with 4Performance is sometimes characterized as the motion of the center of mass G, and the flying qualities as the motion around G. The guiding idea is Motion of G: derivative of position of G = forces Motion around G: derivative of angular position = moment However, there is no decoupling between these force and moment equations which depend on the same parameters. It is, therefore, illusory to try and separate these two vectorial expressions. Two contradictory examples show the limit of this classification: 0
0
steady state turning flight permits the definition of turning performance such as maximum load factor, radius curvature and the turning rate but with a non-null angular yaw velocity of the plane due to the turn itself. Thus, there is performance with an angular motion around G. during the longitudinal short period mode, fundamental for the flying qualities, the airplane is turning around G, but the center of mass also has a vertical motion which, with the rotation, is a large part of the damping phenomenon. The simplest pattern of this mode is composed of the sustentation force equation and the pitching moment equation. Flying qualities are concerned with the motion of the center of mass and not only with the rotational motion around G, as has been shown with the short period mode. It is also true for the other modes such as the phugoid, dutch roll, spiral mode, etc.
As a further example, take-off, rightly considered as a performance case, would seem to escape this rule as longitudinal acceleration is not null. In fact, if it is admitted that pseudo-equilibrium (Section 7.1, p. 180) is one of the different types of equilibrium, then take-off, which presents only one non-steady-state equation, is a pseudo-equilibrium and, thus, legitimately becomes a performance case. Dynamics of Flight: Equations
4
1-
-___
Presentation
the aircraft, that is to say aircraft pilotability. This specific meaning for the word dynamics should be understood for the study of non steady-state flight.
FLYING QUALITIES
DYNAMICS 1
Flight analysis is made with a mathematical model: the flight dynamics equations, the subject of this book. Applied mechanics are needed as well as some notions of aerodynamics and propulsion in order to establish the equations. The mechanics equations are fed by the external efforts5 models. There are three: aerodynamics, propulsive and weight. All these three efforts depend on the Earth environment. The Earth’s proximity influences the gravity which influences the weight, whereas the atmosphere influences the aerodynamic and propulsive efforts through the t hermodynamic state of the air. The definition of the gravity and atmospheric models constitute the terrestrial model which, in this book, is adapted to high velocity transatmospheric flight. The external efforts model will be general; more precise forms will be used for specific situations with reference to specialized work in aerodynamics or propulsion. The validity range of the equations presented in this work will essentially depend on the modeling of these external efforts. Emphasis is placed on acceleration modeling. Thus, as with aerostats and submarines, buoyancy forces due t o Archimedes’ thrust should be added and the aerodynamic or hydrodynamic efforts model should be adapted. Gliders correspond well to the equations proposed. The only difference is the absence of an engine. Only the thrust effort has to be cancelled in order to adapt gliders to the model. The models’ limits will appear where the aerodynamic efforts change due to the rarefaction of the atmosphere and the very high velocities. Solar radiation pressures are no longer negligible. These pressures are not modelized and the aerodynamic model does not use rarefied gas dynamics.
Aircraft design and control: Performance and flying qualities are the two major fields of study in flight dynamics. Two other objectives, aircraft design and control, appear when information about the motion is looped towards efforts or aircraft definition. The pilot can act on the aerodynamic and propulsive efforts; he will pilot as a result of his perception of the motion of the aircraft. This observation of motion is obtained through measurement. Measurement feedback on the external efforts constitutes the control loop. A lower frequency loop of the same form could be operated, not directly on the effort but on the definition of the aircraft. In this case, a modification of the external A rationale can be produced for this usual approach of performances. For equilibrium, a simple process of the moment equations gives a direct relation between the angle of incidence and the pitch control, the sideslip angle and the yaw control, the roll rate and the roll control. These relations are then implicitly used for the exploitation of the forces equation and therefore the moment equations can be forgotten. Sometimes the notion of performances, associated with equilibrium, implies a flight parameter at its maximum value; for example, the maximun lift coefficient, the maximum throttle position, etc. That is another meaning of the term performance. 5By efforts is meant forces and moments.
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1.1 Presentation
geometry or the engine affects the external efforts. The external geometry defines the aerodynamic effort in principle, just as the engine defines the propulsive effort. This is the design loop which is generally used on the ground by the aircraft design team. The border between control and design loops is not really clear. Some airplanes can perceptibly change their geometry during flight. This is the case of the variable sweep wing aircraft and some gliders, which can change their aspect ratio. The flap deflecting, a device that is often called a configuration control surface, is representative of an operation at the limits of control and design. This design stage needs models which are able to pass from the aircraft definition to the effort model; for example, from the external geometry of the aircraft to its aerodynamics or from the characteristics of an engine to its thrust. These models are essential in the design process. The figure (1.3) summarizes the functional organization of flight dynamics.
Figure 1.3: Functional organization of flight dynamics At this stage, a definition of flight dynamics can be attempted. The purpose of the flight dynamicist is to manage the various knowledge coming from the aerodynamics, the propulsion, the strength of materials, in order to analyse and optimize the behaviour of the aircraft. Flight dynamics is a synthesis science based on a closed loop process. The two essential “engines” which drive this analysis process are the mechanical and automatic control sciences. For example, if a wing planform is optimized to reduce drag, this is applied aerodynamics. If this planform is optimized to reduce drag and weight of the wing, this becomes flight dynamics because at least two disciplines are balanced in order to optimize the aircraft. Equations: The interest in flight dynamics (performance, flying qualities, control, design) having been established, the process which leads to the final flight dynamics equations will be studied. Dynamics of Flight: Equations
1 - Presentation
6
The creation of the equations is based upon well-known relationships in applied mechanics and frequently leads t o very simple results. Thus, in most cases (Section 7.4, p. 188)
lift is equal to weight and
drag is equal t o thrust The process is initialized by writing the two fundamental mechanical laws (Section
4.1, p. 71).
m a s s . Acceleration = C Forces derivative of Angular momentum = C Moment
of f o r c e s
(1.1) (1.2)
The mass m multiplied by inertial acceleration is equal t o the sum of the external forces, and the temporal derivative of angular momentum is equal to the sum of moments of the external forces. This angular momentum is supposed t o be an inertial momentum. During the procedure, the equations may seem complicated. However, t o obtain the general equations, only these two fundamental mechanical laws are needed. To do this, external forces and accelerations have to be modelized. External aerodynamic, propulsive and gravity effort models are simple. For the first two, the closer the projection frames are to the body frame, the simpler are the expressions. Thus, acceleration in these frames is expressed in order t o keep the simple analytic form of external efforts. Acceleration comes from two temporal derivations of the position. The use of the fundamental kinematic relationship is needed to derive vectors in any frame, with respect t o time (Section 3.1, p. 43).
dXo dt
-
dX1 dt
+ 52,oxx
g,
The derivative with respect to time of a vector X in a frame Fo, is equal t o the derivative of this vector X in a frame F l , $, t o which the cross-product of angular velocity 5210 of the frame F1 relative to the frame Fo by the vector X must be added. REMARK1 . 1 In practice, to express the temporal derivative of a vector in a given frame, the components in this projection frame are derived. Thus this vector derivative expressed or projected in this same frame is obtained. The notion of the derivation of a vector is meaningful only when associated with its derivation frame. Thus is an incomplete expression as it lacks the subscript for the derivation frame.
Knowledge of the two fundamental mechanical relationships (Equation 1.1) and (Equation 1.2) and that of kinematics (Equation 1.3) is suficient to establish the general equations. Moreover, it must be remembered that any vectorial relation can be projected o n any frame. The projection operation of a vectorial relation is, then, absolutely free of any constraint.
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1.1 Presentation
These two notions of derivation with respect t o a relative frame and projection in a frame are essential in order to establish the general equations of flight dynamics.
Frames: In order to apply the equations of mechanics, a Galilean frame called t'he inertial frame must first be defined. Then the relative frames allows the simplification of the writing of the fundamental relationships, thanks to relative derivations by approaching the material system itself. In frames close to the material system, external efforts take on a simpler form. The first step is to define useful frames in order to put the equations in a userfriendly form (Section 2.1, p. 13). Once the frames have been defined, the transformation from one to another gives angles of particular importance in the external effortss expression (Section 2.2, p. 18). Between two frames, angles define their relative position. If a third frame is introduced between the first two, a second path appears in order for the first frame t'o reach the second, thus creating two families of angles. The equivalence of the two paths gives angular relationships (Section 2.3, p. 38) which have a simple form only in certain specific situations. Derivations: Once the frames have been defined, the derivative operation can begin in order t o obtain velocity and acceleration. Representation of equation (1.1) is put into a first-order form by the derivation of velocity instead of a second derivation of the position. These equation (1.4) are completed by the kinematic equation (1.5) and allow the change of the six second-order equations into twelve first-order equations m
d( Velocity) = C Forces dt d(Position) = Velocity dt
instead of writing
m
d2 (Position) dt2
=
C Forces
This representation naturally leads to the explicit appearance of the velocity components which intervene in the external forces models. Thus, the equations are easier t o manage. It will be shown later how this demand leads to state representation, rich in possibilities and well formalized.
Position derivation - Kinematics: The first stage in derivation consists in deriving the position in order to obtain velocity. This is the objective of the chapter Kinematics (Section 3, p. 43).
It begins with Velocity (Section 3.2, p. 45) calculation and definition. Wind velocity notions are developed, particularly t o define the wind velocity field around the aircraft. Next, two velocity representations are given, either by their components in a given frame or by the velocity modulus and vector angular position relative t o a frame. Relationships between these two representational modes are established in the chapter Angles- Velocities relationship (Section 3.3, p. 61). Dynamics of Flight: Equations
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1 - Presentation
Finally, the six kinematic relationships themselves:
d( Position) = Velocity dt are defined in the chapter Kinematic equations (Section 3.4, p. 66). These six relationships form half of the general equations of flight dynamics. Three of them are associated with the position and linear velocity and the three others are associated with angular position and angular velocity.
Velocity derivation - Fundamental mechanical equations: The second stage consists in deriving velocity in order to obtain acceleration. These accelerations multiplied by masses are equal to external efforts. These are the two fundamental mechanical laws (Section 4.1, p. 71).
i
For the first law, mass m multiplied by the inertial acceleration AI,G(subscript I) of the center of mass (subscript G), is equal to the sum of the three external forces Fest: aerodynamic, propulsive and weight. For the second law, the derivative in an inertial frame of the inertial (subscript I) angular momentum HI,Gabout the center of mass (subscript G), is equal to the aerodynamic plus propulsive moment about the center of mass of the three external forces M F ~ ~Inertial ~ ~ ,angular ~ . momentum is equal to the product of the inertia matrix IIG, calculated about the center of mass of the aircraft, and the angular velocity of the aircraft relative to the inertial frame S2 (Equation 4.18, p. 75). These two theorems are written in three stages: the first and second in order to calculate Inertial acceleration (Section 4.1.1, p. 77) and Inertial angular m o m e n t u m derivative (Section 4.2, p. 79) and the third to define External eflorts (Section 4.3, p. 82) models. Finally, these results are grouped so as to formulate the Flight equations (Section 4.4, p. 94) themselves. At this level, the objective has been reached since the flight dynamics equations are now at the reader's disposal. In the rest of this book, these equations will now be exploited, thanks to simplifying hypotheses. These simplified forms are of great practical use and constitute the second part of this book. Simplified equations (Section 5, p. 103): The first simplification consists in assuming that the Earth is flat and fized (Section 5.1, p. 104). This hypothesis leads to simple equation forms whose validity range is more or less defined by a Mach number lower than two for an acceleration error criterion of approximately one per cent of the weight. For a precise estimation of the price of the flat and fixed Earth hypothesis, the equations must be rewritten so as to take into account the different simplification hypotheses. Four different approaches have been examined to evaluate the error caused by these flat, fixed Earth hypotheses (Section 5.1.4, p. 118).
1.1 Presentation
9
With these simpler equations, it is possible to propose various ways of writing the equations, in particular for the force equations. These forms are linked t o the choice of the derivative frames (relative frame) and the projection frames. The second form of the force equations is attractive because of its simplicity (Section 5.1.1, p. 105). Among other possibilities, the wind terms could be made t o appear not among the external forces but among acceleration terms. This second form results from the use of the body frame Fb as the derivative frame and the aerodynamic frame Fa as the projection frame. The number of forms is limited for moment equations due t o the difficulty of defining an aerodynamic angular velocity 61,. This difficulty vanishes if there is a “Vortex” wind velocity field (Section 5.2, p. 125). This comes down t o assuming that, around the aircraft, wind velocity is modelized by a velocity at a given point and angular velocity. The wind would then be a kind of vortex whose rotation axis and angular velocity would be known. Another simplification, but more advanced, consists in assuming a Uniform wind velocity field (Section 5.3, p. 130). This hypothesis embraces the case of flight in a no wind situation. When the wind velocity field is uniform, the wind velocity is the same whatever the aircraft position is and whatever the time is. A different form of simplification consists in decoupling the equations. This leads to an analysis of aircraft flight with a reduced number of equations. It involves “isolating” certain equations out of the total of twelve, without affecting the quality of the results with this reduced system. The chapter Decoupling equations (Section 5.4, p. 132) deals with three examples of decoupling: 0
0
0
The navigation equations (Section 5.4.1, p. 135), that is t o say geographical position x , y and azimuth $ with respect t o the other equations. This decoupling is absolutely rigorous in the case of the flat and fixed Earth hypotheses. Without them, only the longitude can be decoupled rigorously. The lateral equations (Section 5.4.2, p. 137), which then allow the use of only the five decoupled longitudinal equations. This decoupling is rigorous with certain hypotheses which have often been proved. It corresponds to flight within a vertical plane, with horizontal wings. The longitudinal equations (Section 5.4.3, p. 145), which then allow the use of only the four decoupled lateral equations. This decoupling is not as easy to justify as the previous one and the lateral equations should be handled with care. It is really rigorous only for level horizontal flight without wind and with weak sideslip and bank angle.
Linearixed equations (Section 6, p. 157), are part of the simplified equations, but their major importance in the analysis of flying qualities justifies a special chapter for them. With linearization, it is a matter of finding a simple equation model generally around a steady state flight situation. This model is suitable for a dynamic analysis of the aircraft but with a validity range which could, in some cases, be reduced. The Linearization method (Section 6.1, p. 158) is first expounded, then a numerical and analytical process is proposed t o exploit this method. Numerical linearization (Section 6.2, p. 160) can be implemented for every flight situation with non-decoupled Dynamics of Flight: Equations
10
~
1
- Presentation
equations and non-analytical external effort models. The analytical linearization carried out on the decoupled Longitudinal (Section 6.3, p. 161) and Lateral (Section 6.4, p. 170) equations is more limited due t o its long processing, but it gives the opportunity for an explicit parametric study of the dynamic. It should be noted that linearization around a steady state flight with wind modifies the state matrix with respect t o steady state flight without wind. This means that modes and, therefore, flying qualities can depend on the wind. With linearization around straight horizontal steady state flight without wind, it appears that lateral equations are really decoupled. As the wind is defined in the Earth reference frame, linearization of the wind terms expressed in a frame close t o that of the aircraft are rather long (Section G.2, p. 283). Finally, the last simplification case, Equilibrium (Section 7, p. 179), is a specific case of general flight dynamics equations. These equilibrium equations correspond t o a study of aircraft performance. As a first step, Equilibrium notions (Section 7.1, p. 180) or pseudo-equilibrium (Section 7.1.2, p. 182) are defined. An equilibrium definition linked to the state representation is chosen: the aircraft will be in steady state when the state vector derivative of the major system is null. Physically, this is equivalent t o considering that there is a steady state when all the state variables with any influence on external efforts or complementary acceleration terms are constant. In order to obtain steady state conditions in a linear system (Section 7.1.3, p. 182), it is necessary t o complete this equation system with a number of independant equations equal to the number of control devices. Pratically speaking, these conditions can be extended t o the non linear system representative of the aircraft. It is preferable, in order t o avoid numerical problems in the research of equilibrium, to take into account the decoupling phenomenon between lateral and longitudinal motion, in the choice of supplementary equations. Once the equilibrium conditions have been defined, a method for Numerical research of equilibrium (Section 7.2, p. 186) is proposed, based on the linearization of the equations system around any flight situation. This numerical method has been implemented in Fortran (Section H, p. 321). It allows a search for any type of steady or pseudo steady state without specific initialization and the free choice of supplementary steady state conditions. Equilibrium can thus be generally defined by any four conditions concerning the state of the aircraft. The four equations which these conditions generate, however, must be independent of the general equations. This method also allows for the detection of a false formulation of steady state conditions, such as non-independence. General equilibrium (Section 7.3, p. 188) is evoked when flat and fixed Earth hypotheses are not made. Within the decoupling frame, Longitudinal equilibrium (Section 7.4, p. 188) and Lateral equilibrium (Section 7.5, p. 190) equations are given. These are the simplest equations of the document but rich with multiple practical information. The exploitation of these equations is not one of the aims of this book, but that is another problem!
Part I
General equations
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2
Frames This chapter is t o define the frames useful for the establishment of equations for flight dynamics. Mechanical equations, in order t o be applied, at first, need the definition of a Galilean frame which is called inertial (Section 2.1.1, p. 14). Afterwards, relative frames will allow the simplification of the writing of fundamental relationships thanks t o relative derivatives coming close to the material system. In frames near the material system, external efforts take on a much simpler form. The first step, therefore, consists in defining the useful frames in order to give the general equations a more attractive form (Section 2.1, p. 13). Once these frames have been defined, the transformation from one t o the other will make angles appear. These angles often determine the modelisation of the external efforts (Section 2.2, p. 18). Between two frames, angles are defined because of their relative positions. If a third frame is introduced between the first two, a second path from the first frame t o join the second frame appears and includes two sets of angles. The equivalence of these two paths gives birth t o angular relationships (Section 2.3, p. 38) which have a simple form only in certain specific cases. The last page of this chapter is a summary of the principle definitions, notations and results in the form of a table. Thus this whole chapter is written to establish the frames, the associated angles and the angular relationship.
2.1
Frame definitions
All frames are three-dimensional orthogonal and right-handed: Pi (Oi, xi, y i , zi) The unitary vectors of each axis are marked as follows: xi, y i , zi. The origin of the frame Fi is Oi. All of the vectors of this document, printed in boldface such as X,are defined in three-dimensional space R3. 13
Figure 2.1: Orthogonal and right-handed frame
2.1.1
The inertial frame FI ( A , XI, yI,
zI)
The inertial frame F I , which is a Galilean frame, is a geocentric inertial axis system. The origin of the frame A being the center of the Earth, the axis “south-north’’ ZI is carried by the axis of the Earth’s rotation, axes XI and y~ keeping a fixed direction in space. The angular velocity of the Earth relative to FI is noted at.
Figure 2.2: Inertial frame FI This frame is only Galilean (Hypothesis 1) (Section 4.1, p. 71) when used in relationship with the accuracy searched for in Flight Dynamics. It can be assumed that in the time scale used in Flight Dynamics, the point A moves at a constant velocity modulus and direction
VI.A = Constant = 30 k m / s on a polar orbit
(2.1)
REMARK 2 .1 Taking the center of the Earth as the origin of the inertial frame FI is consistent with the application of astronomical positioning [7]. The orientation of XI and y~ is able to be oriented in a direction defined by the stars. The frame reference is called kinematic and its precision of orientation is of the order of 1 second of arc per year, which corresponds to the observed relative motion of 2000 stars. The inertial directions can also be defined such as when the principle of inertia is verified in this frame F = mA. This reference is called dynamic and the precision is around 50 times better than for the kinematic reference. Thanks to measures effected by satellites such as Hipparcos, precision could still be improved by a factor of 10. XI is sometimes oriented towards the vernal point or “point y intersection of the equator and the ecliptic. The ecliptic is the intersection of the trajectory plane of the sun with respect to “the stars” and the Earth’s surface. There is, therefore, a reference of sideral time which is not exactly stellar time because the vernal point y depends on the sun trajectory.
REMARK 2.2 The axis It
REMARK 2.3 The axis ZI still called the “polar axis” or the “World’s axis” is animated by a movement of precession and nutation of a long period with respect to a stellar frame. The axis ZI traces a cone of a +23”26’ half-vertex angle due to the precession movement, over a period of 25,800 years. This movement is because of the solar and lunar attraction forces. The axis ZI traces also another cone of a +10” half-vertex angle due to the nutation movement, over a period of 19 years. This movement is principally due to Bradley’s nutation linked to the evolution of the lunar orbit characteristics. The World’s axis ZI also has a oscillatory movement with respect to the Earth itself. This movement is actually composed of two oscillation movements of a 12 or 14 month period, of which the amplitude of the arc is around 0.3”. The pole moves about in a positive rotation, following an approximative circular trajectory which could be drawn within a 20 meter square. Essentially the yearly oscillation is a consequence of the redistribution of the air mass in the heart of the atmosphere which provokes a modification of the atmosphere moment of inertia. However, the origin of the oscillation period of 14 months or “Chandlerian” nutation, still remains obscure. It might be provoked by the movements of the Earth’s core a t the edge of this core, 2900 km deep.
2.1.2 Normal Earth-fixed frame
FE
(0,xE,
YE, zE)
The normal Earth-fixed frame, FE, is linked to the Earth. The origin 0 is a fixed point relative t o the Earth and the axis Z E is oriented following the descending direct,ion of gravitational attraction g, located (Section 4.3.1, p. 82) on 0. This frame is therefore fixed relative t o the Earth (Figure 2.7, p. 22). The plane ( XE,YE) is tangent to the Earth’s surface. The origin 0 and the orientation of the axis XE is, a priori, arbitrary. But, from now on in this document, the point 0 will be placed at the surface of the Earth’s geoid and the axis XE will be directed towards the geographical North (Section 2.2.2, p. 21) (Hypothesis 23) Thus the altitude h of point 0 is zero. For the definition of the Normal Earth-fixed frame, t8heEartlh’s shape is rather important. In this document, it is assumed that the Earth is spherical (Hypothesis 2). Therefore A 0 = Rt, with Rt as the Earth radius assumed constant. Some enlightenment is brought to these hypotheses when defining the latit,ude (Section 2.2.2, p. 21). The Earth’s geoid is well represented by an ellipsoid of a revolution flatten at the poles. __
~__
~~~
~~~
~
....
Dynamics of Flight: Equations
2 - Frames
16
REMARK 2.4 The choice of the axis Z E directed towards the bottom might seem strange with respect to the notion of altitude. This choice is somewhat justified by the positive sense of rotation around x , in accordance with the positive sense of rotation used with the headings, North, East, etc. REMARK 2.5 The Earth’s crust is a solid that can change its shape [7]. It can change its shape by a few decimeters under the influence of either internal geophysical phenomenas or external stimulations such as tides and ocean loading. The principle deformations are periodical, thus establishing the existence of the average Earth’s crust. If the Normal Earth-fixed frame is physically materialized on the Earth crust, it is not absolutely fixed relative to a frame that will be defined, for example by the axis of the Earth’s inertia. The Earth’s tides are the response of the whole Earth’s land masses which are considered as elastic (excluding the oceans) to the external perturbation potential of the ocean’s tides.
2.1.3
Vehicle-carried normal Earth frame F, (0,x,, y,, z,)
The axis z, of the vehicle-carried normal Earth frame F, is oriented towards the descending direction of the local gravitational attraction g, (Section 4.3.1, p. 82) in G, the center of mass of a aircraft. The axis zo is therefore the direction of the gravitation as viewed by the aircraft (Figure 2.7, p. 22). The vehicle-carried normal Earth frame F,, has the same origin 0 as the Normal Earth-fixed frame FE, but contrary to the latter, it follows the local gravity as seen by the aircraft. The axis, xo,is oriented in the direction of the aircraft’s geographical North (Hypothesis 24)) therefore North of point G and not of point 0. Thus x, is not parallel to XE. Later on, the utilization of this specific frame clearly brings out the terms that were neglected when the Earth was assumed flat (Hypothesis 3). As a matter of fact, with this hypothesis, the frame FE no longer turns with respect to F,. REMARK 2.6 All this is under the hypothesis of a spherical Earth (Hypothesis 2). In case there is the supposition that the Earth is close to a geoid, z, will have the direction of vertical, that is to say the acceleration due to gravity g (Section 2.2.2, p. 21). In both cases, the argument is to give a direction perpendicular to the Earth’s surface to the axis so.
2.1.4
Body frame
Fb
(G, xb, yb, zb)
This frame is linked to the aircraft body (subscript “b” as body). The origin is the point of reference of the aircraft, which in general, is the center of mass G (Hypothesis 18). If the gravity field is assumed constant, then the center of mass joins the center of gravity. The fuselage axis xb, oriented towards the front, belongs to the symmetrical plane of the aircraft. Its definition in this plane remains arbitrary, and thus renders the angle of attack arbitrary. It is generally linked to a geometric definition of the fuselage. If the fuselage is a cylinder, xb is parallel to a generatrix. The axis zb is in the symmetrical plane of the aircraft and oriented downward relative to the aircraft (Figure 2.3).
The axis Yb is perpendicular to the symmetrical plane and oriented towards the right “pilot’s side” of the aircraft.
2.1 Frame definitions
17
This definition assumes that there exists a symmetrical plane for the aircraft (Hypothesis 4). This is a classical hypothesis and well justified for Flight Dynamics. But
this hypothesis is not necessary for the establishment of the equations. It will be recalled to mind later on when this hypothesis might intervene. Several symmetrical plans, inertial, geometrical and propulsion, could exist. The body frame Fb could therefore be chosen differently with the choice depending on the form that would be given to the equations. The definition must be stated even more precisely if the aircraft is going to be characterized as flexible. REMARK 2.7 Established standards have not been given a subscript for this frame. This would notably complicate the writing of the equations because of the noncoherence of the notation with other frames, particularly with the derivation operation when the name of the derivative frame is essential. The lack of subscript is a lack of information and a potential source of confusion. REMARK 2.8 It must be remembered that, in reality, the center of mass is not absolutely fixed relative to the aircraft on account of the consumption of fuel (Hypothesis 14) and the relative deformation of the aircraft (Hypothesis 13).
2.1.5
Aerodynamic or air-path frame Fa (G, x,, y,, z,)
With the same origin G as the body frame Fb, the aerodynamic frame or air-path frame Fa, is defined by 0
The axis xa carried and oriented by the aerodynamic velocity of the aircraft V a (Equation 3.13, p. 47)
0
Two rotation angles a a and f l a y named as the aerodynamic angle of attack and sideslip angle, for transformation from the body frame Fb to the aerodynamic frame Fa (Section 2.2.7, p. 32) (Figure 2.13, p. 34)
It is shown that the axis Za is in the symmetrical plane of the aircraft (xb, yb) and the intermediate axis xi, is the projection of xa in the symmetrical plane of the aircraft (xb, yb).
2.1.6
Kinematic or flight-path frame
Fk
(G, xk, yk, zk)
With the same origin G as the body frame Fb, the kinematic frame or flight-path frame Fk is defined by 0
The axis
Xk
carried and oriented by the kinematic velocity of the aircraft V k
(Equation 3.10, p. 46)
0
Two rotation angles a k and p k , named as the kinematic angle of attack and sideslip angle, for transformation from the body frame Fb to kinematic frame Fk (Section 2.2.8, p. 33) and (Figure 2.14, p. 37).
The kinematic frame Fk is for
v k
the equivalent of the aerodynamic frame Fa for
VaDynamics of Flight: Equations
18
2 - Frames
Figure 2.3: Body frame
2.2
Fb
Definition of angles between frames
Several frames having been established, it is useful to define their relative positions by means of angles. Several rotations about the axes of frames are necessary t o join one frame t o another. There exists a certain number of solutions which are functions of the rotation axes and the order of these rotations. The most widely used ones will be reviewed. These angles often represent a capital interest. They intervene in the modeling of applied efforts for an aircraft and in the definition of its trajectory. In the first stage, the method used for modeling the transformation of one frame to another through the medium of the matrix of transformation is presented.
2.2 Definition of angles between frames
2.2.1
19
Matrix of transformation from one frame to another
The transformation of one frame t o another is modelled by a matrix of transformation T.The projections of a vector X in the frames Fi and F' are therefore connected by
The matrix Tij is the tranformation matrix of the frame Fi to the frame F'. The supercript of the vector indicates the projection frame of this vector. The term Xi represents the vector X projected or expressed in the frame Fi. It is shown that the matrix associated with the rotation around an axis passing through the origin of the orthogonal right-handed frame is a real orthogonal matrix. This matrix has two remarkable characteristics, its transpose Tfj is equal t o its inverse Tzl and its determinant is equal t o the unit.
This characteristic will be of great use later on to simplify the calculations. It can be justified by the invariability of the modulus of vector X in the change of the projection frame.
and the relationship (Equation 2.2, p. 19) allows t o write
therefore
'f'TZj
= 11
The elementary rotations about the axis x, y, z are modelled by the following matrices. Rotation about the x axis (Figure 2.4) 1 0 0 coscu, 0 sincu,
--ha, COSQ,
Rotation about the y axis (Figure 2.5) coscu, I[.Ol(cuy)
=
--sin&,
0 sincu, 0
COSQ,
Dynamics of Flight: Equation,s
2 - Frames
20
Figure 2.4: Elementary rotation about x axis
Figure 2.5: Elementary rotation about y axis Rotation about the z axis (Figure 2.6) TOl(cy,)
=
coscy, sina,
-sina, coscy, 0
0) 1
Generally speaking, the transformation of the frame Fi into the frame Fj is done by three rotations (cy1, cy2, cy3) about the three axes. The first rotation is a 1 , the second a 2 and the third a 3 . Therefore Tij will be equal to
and the inverse transformation of Fj to Fi is modelled by
therefore (2.12)
21
2.2 Definition of angles between frames ~~~
f
zo=zl
Figure 2.6: Elementary rotation about z axis The result of the rotation product is a rotation. The matrices of transformation between the different frames defined in (Section 2.1, p. 13) will be constructed following this method. REMARK 2.9 It is useful to note that when a matrix of transformation of this type is calculated, this matrix must have the characteristic of being equal to the identity matrix (11) for the angles of rotation that are null and of having their determinant equal to +l. This is a means of controlling the final result.
2.2.2
Transformation from inertial frame FI to Normal Earthfixed frame FE
The appendix which corresponds to this paragraph is (Section A . l , p. 195).
The change in orientation for the transformation of an inertial frame FZ t o a Normal Earth-fixed frame F E , in the case where the axis XE is directed towards the geographical north (Hypothesis 23), is done by two rotations (Figure 2.7, p. 22) lst rotation -wto about the axis zz 2nd rotation -Lto about the axis YE During the first rotation, the axis y~ is led to the axis Y E of the frame F E , with Lto geographical latitude of 0, origin of the frame F E , and wto the stellar time
2
wto
= -Rtto
(2.13)
The term Rt is the angular velocity of the Earth (Section 3.2.3, p. 58). And t o is the stellar time of point 0. The term wto is sometimes called stellar time as well, even though it is homogenous t o an angle and not t o a time. The stellar time of point 0, w t o , is positive if 0 is to the west of the plane (XI,ZI). The latitude of point 0, L t o , is positive if 0 is in the northern hemisphere; a positive latitude is therefore a northern latitude. According to convention, the ranges of angles can be stated 7r
--<
7r
52 0 5 wt < 27r
2 -
Lt
The axes of the frame FL are parallel t o those of the frame F,. Dynamics of Flight: Equations
22
2 - Frames
Figure 2.7: Angles between the Normal Earth-fixed frame F E , Vehicle-carried normal Earth frame Fo and Inertial frame FI The longitude of Navigator LgG can be found with reference to point 0 by taking the difference of stellar time between the center of mass G and 0 the origin of the normal Earth-fixed frame FE.
and the equivalent notation for the latitude
and it can be remembered that (Equation 2.13, p. 21) (2.16) (2.17)
It can be noted that the latitude of point 0, L t o , is constant as a definition, so
Lto Matrix of transformation
=
0
(2.18)
2.2 Definition of angles between frames
23
.
(2.19)
- sin L t o cos wto
TIE =
sinLto sinwto cos L t o
sin wto coswto 0
cos w t o cos L t o sinwto - sin L t o
- cos L t o
(2.20)
(2.21) REMARK 2 . 1 0 Taking the axis X E directed towards the geographical north (Hypothesis 23) is not a very restrictive hypothesis but it simplifies the matrix. To introduce whatever kind of orientation, it is sufficient to add a third rotation of heading around the axis Z E . REMARK 2.11 For the accuracy generally looked for in Flight Dynamics, there is no room to make the distinction between stellar time, sideral time or solar time [7]. REMARK 2.12 The longitudes and latitudes are positive in the figure 2.7. The longitudes west and the latitudes north itre therefore considered as positive. REMARK 2.13 Even though w t is an angle, it is often called time because of its proportionality to time equations 2.16 and equations 2.17, p. 22.
Definition of the latitude and Earth geometry The definition of the latitude, above defined, is simple in the framework of the spherical Earth hypothesis (Hypothesis 2). In reality, the Earth is an ellipsoid and this fact leads to different definitions of latitude. This fact makes it necessary to state the problem of the geometry of the Earth's surface. In this document, the hypothesis of the spherical Earth has been made (Hypothesis 2). This is only an approximation. Under the influence of the Earth's angular velocity (Equation 3.62, p. 58) carried by the axis ZZ,the Earth ''flattens" at the pole axes and thus resembles a revolution ellipsoid. Thus several notions of latitude appear [7]. Assuming that point 0 is on the Earth's surface, the axis zz is the Earth's axis of rotation (axis south-north) and the axis xi belongs to the equatorial plane, then t8he plane (xi,zz) is the meridian plane containing 0.
Geocentric Latitude Lt,, is defined by the angle between A 0 and x;. It indicates the direction of the center of the Earth. Gravitation Latitude Lt,, indicates the direction of the gravitational attraction
g,(Section 4.3.1, p. 82). Because of the flattening of the Earth, the gravitation vector is
no longer directed towards the center of mass of the Earth A , for the middle latitudes. When the latitudes reunite with the zero values or $, all of the latitudes are confounded. Gravitation being proportional t o the vector g, deflects from the center of the Earth towards the equator.
5,
Dynamics of Flight: Equations
2 - Frames
24
NorthA
ZI
Figure 2.8: Latitudes Astronomical Latitude Lt,, indicates the direction of the measured vertical. It is the direction of the plumb line or free fall. Geodesic Latitude Lt,d indicates the direction of the perpendicular to the tangent plane at the Earth’s surface at the point 0, the Earth having been modelled by the revolution ellipsoid. The geodesic latitude is a model of the astronomical latitude. A difference exists between these two latitudes but in a matter of importance that remains inferior to the detailed accuracy searched for in practical studies of Flight Dynamics. The deviation is of the order of 10-4 rad. Finally, the latitude previously defined Lto is called geographical latitude. It is the geocentric latitude when the Earth is assumed to be spherical. The vector g of acceleration due to gravity defining the vertical in 0 , has to all intents and purposes the definite direction by the geodesic latitude. The gravity g is the sum of the gravitation g,. and the inertial centrifugal force equal to the opposite of the inertial acceleration A, of point 0 (Equation 4.63, p. 83), considered as fixed relative t o the Earth (Section 4.3.1, p. 82)) A, = AI,ofize (Equation 4.73, p. 85). In looking for the surface of equilibrium of the ellipsoid under the effect of the Earth’s rotation with a gravitational vector g,. headed for the Earth’s center A , half of the Earth’s “flattening” can be explained. The other half is due to the rotation of gravity towards the equator, under the influence of a new distribution of the masses. The geoid represents the average surface of the seas which is extended under the continents in such a way that each point is perpendicular to the direction of a plumbline. This geoid representing the Earth is similar to an revolution ellipsoid. In certain regions, the maximum variation between the two can be up to 100m of difference in a1t it ude .
2.2 Definition of angles between frames
25
_____--
The standard modeling for the Earth in an revolution ellipsoid is the following: Semi Earth major axis a Semi Earth minor axis b a Flattening f Eccentricity
e
=
1.0034 b
=
1 a-b ) = 0.003353 a -(298.25
=
Jm
cos Lt,d
4mSinLtgd a(1 - e 2 )
=
(2.25)
BY
a
zo
(2.22) (2.23) (2.24)
(2.26)
The coordinates of 0 on the axis xi and " =
= 6378.2 km = 6356.8 km
(2.27) (2.28)
The relation between the geocentric latitude and the geodesic latitude is given by tan Lt,, with
=
b2 a2
- tanLt,d
(2.29)
b2 a2 = 1 - e 2 = 0.9932
(2.30)
If it can be considered that f = 3 10-3 is infinitely small with regard of 1 (Hypothesis 5), then with
The preceding equation (2.29) could be approximate by [8] Ltgd
=
Lt,,
+f
sin(2Ltg,)
(2.31)
The same goes for O A , the distance from point 0 to the center of the Earth A , and could be established as
O A = a (1 - f sin2 Lt,,)
(2.32)
REMARK 2.14 With the hypothesis of the spherical Earth (Hypothesis a ) , the gravitational vector g, is oriented towards the center of the Earth A . Thus it can be admitted that Lt,, is equal to Lt,, and to Lto the geographical latitude. But with the Earth in rotation S l t , the inertial acceleration of 0 fixed with respect to the Earth is not zero and the gravity vector g is not aimed towards the center of the Earth. The vertical is not perpendicular to the Earth's surface (Section 5.4.1, p. 135)(Figure 2.8,
p. 24).
Dynamics of Flight: Equations
REMARK 2.15 With the hypothesis of the spherical Earth (Hypothesis 2) the distance O A is constant and equal to the mean Earth radius Rt. REMARK 2.16 The altitude h of the point M is defined as the distance O M between the surface of the ellipsoid and the point M , carried by the measured vertical (astronomical latitude). With the hypothesis of the spherical Earth, the altitude is taken carried by an Earth radius and this leads to a simplification because it is the direction of gravitation g,. and not gravity g.
2.2.3
Transformation from the inertial frame FI to vehiclecarried normal Earth frame FE
The appendix which corresponds to this paragraph is (Section A . l , p. 195).
The transformation of the inertial frame FZ to the vehicle-carried normal Earth frame F,, is done in a manner analogue t o the transformation of the frame FI t o FE (Equation 2.2, p. 19)
-WtG about the axis X I l S trotation 2'"' rotation -LtG about the axis yo In the course of the first rotation, the axis y~ is led to the axis y, of the frame F,. The term LtG is the geographical latitude of G center of mass of the aircraft (Section 2.2.2, p. 21).
with Rt angular velocity of the Earth and tG stellar time of point G (Section 2.2.2, p. 21).
Matrix of transformation By analogy with TIE(Equation 2.20, p. 23)
x'
=
- sin L t c cos LdtG
Tz,
2.2.4
=
sin LtG sin WtG cos LtG
TroXO sin WtG cos LdtG 0
(2.34) - cos L t c cos WtG
cosLtc sin wtG - sin L t c
(2.35)
Transformation from the normal Earth-fixed frame FE to the vehicle-carried normal Earth frame Fo
The appendix which corresponds to this paragraph is (Section A.2, p. 196).
The transformation of the vehicle-carried normal Earth frame Fo t o the normal Earth-fixed frame FE is built from the two matrices in the preceding transformation TEI and Tro. Later, this transformation will be used t o determine the angular velocity of F, relative to FE and in bringing t o light the terms that were forgotten while using
2.2 Definition of angles between frames
~
_
_
_
_
_
_
~
_
~
.
~-
27
~
the hypothesis of the flat ground. In fact, in the framework of the hypothesis of a flat ground, the vehicle-carried normal Earth frame F, stays parallel t o the normal Earth-fixed frame FE and TE,is equal t o the identity matrix II1 (Hypothesis 3).
TEo =
TEo =
i
TEZ
TZo
(2.37)
sin Lto sin Ltc cos Lgc cos Ltc cos Lto
- sin Lto sin Lgc
cos Ltc sin Lto cos Lgc - sin Ltc cos Lto
sin Lt G sin Lgc
cos Lgc
cos Ltc sin Lgc
cos Lto sin Ltc cos Lgc - sin Lto cos Ltc
- cos Lto sin Lgc
cos Lto cos Ltc cos Lgc sin Lto sin Ltc
+
+
(2.38)
A reminder of the notation (Equation 2.14, p. 22)
2.2.5
Transformation from the vehicle-carried normal Earth frame Fo to the body frame Fb
The appendix which corresponds to this paragraph is (Section A.3, p. 198).
The rotation which allows the transformation of the vehicle-carried normal Earth frame Fo t o the body frame Fb corresponds to the transformation of the frame determining the orientation of one solid to another. Three angles are necessary: lStrotation 1c, azimuth angle about axis z, Znd rotation 8 inclination angle about axis y, 3‘d rotation 4 bank angle about axis x b In the transformation of the vehicle-carried normal Earth frame F, to the body frame Fb, the three transformations are associated with two intermediate frames Fc and F f (Figure 2.9). These frames will be useful later on, for simplification, particularly with the linearization process.
Figure 2.9: Intermediate frames The frame F, is deduced from the vehicle-carried normal Earth frame F, by a rotation of the azimuth 11 of the aircraft. The frame F, represents the vehicle-carried normal Earth frame whose axis x, is aligned with the heading of the aircraft. The index ‘(,” stands for the course or heading oriented frame. The frame F f is deduced from the course oriented frame F, by a rotation of the inclination angle 8. The index“f” stands for the fuselage oriented frame. The intermediate axis y, is in the horizontal plane, and it is obtained from the axis yo by the azimuth rotation 1c, which leads x , to x,, itself in the vert>icalplane Dynamics of Flight: Equations
2 - Frames
28
containing the fuselage axis xb. The inclination rotation 8 leads xc t o xb. Finally the bank angle rotation 4, around the fuselage axis xb, will rejoin the axis Yb. According t o convention, the ranges of angles can be stated: --n< ?r
,L L
.21)
I n
8
5 L,
(2.39)
-n
(2.40) (2.41)
REMARK2.17 The convention in the range of the variation of 1c, is a ANSI recommendation; the standard IS0 1151 does not give any particular indication. The positive directions are those of figure (2.10), p. 29, in accordance with the definition of a right-handed axis system. In this figure, the inclination angle and bank angle are positive, when the azimuth angle is negative in order t o improve the visualization of the figure. The azimuth angle is sometime named yaw angle, as the inclination angle could be named elevation angle, and the bank angle, roll angle. Depending on the orientation of the axis x,, the azimuth uses the following name: 0
xo in the direction of the geographic North:
0
xo in direction of the magnetic North:
11,
true heading
qrnmagnetic heading
The difference between these two headings is called the magnetic declination dm. It is positive if the magnetic North is east of the geographic North. The transformation matrices of the two intermediate frames Fc and Ff are given as (2.42) (2.43) (2.44)
cos@ -sin+ L c
=
0 cos8 0
I f c f = (
-sin8
0
sin4
= Ifcb
=
1
01 sin8 0 )
0 cos8
cos+
+
sin 8 sin sin 8 cos 4 cos 4 - sin 4 - sin 8 cos 8 sin 4 cos 8 cos 4 cos 8
TcfTfb
0
(2.45)
(2.46)
(2.47)
(2.48)
2.2 Definition of angles between frames
zo=
29
._____
vertical plane containing the fuselage axis Xb
Lc
Figure 2.10: Body frame Fb relative to the vehicle-carried normal Earth frame F, Matrix of transformation
cos 8 cos $J sin $J cos 8 - sin 8
sin 8 sin 4 cos $J - sin $J cos 4 sin 8 sin 4 sin 2c, cos $J cos 4 cos 8 sin 4
+
+
cos $J sin 8 cos 4 sin 4 sin $ sin 8 cos 4 sin $J - sin 4 cos cos e cos
+
$J
(2.49) and (2.50)
Second system of angles There exists another system of rotation that is sometimes used: Dynamics of Flight: Equations
Zf
zf is a vertical axis only if the inclination angle 8 is equal to zero
Figure 2.11: Bank angle
lStrotation $ transversal azimuth about the axis zo 2nd rotation Cbl lateral inclination about the axis x, 3rd rotation 81 pitch angle about the axis Y b The intermediate axis x, belongs t o the horizontal plane and is obtained from the axis x, by the transversal azimuth rotation $I. This rotation $I is the same as the first one of the previous system. The rotation $I leads the axis yo t o yc and t o the vertical plane containing the right wing axis Yb. The lateral inclination rotation $1 leads y , to the right wing axis Y b . Finally, the pitch angle rotation 01 leads x, to the fuselage axis xb. Matrix of transformation
X"
=
T,b,Xb
(2.51)
The matrix Tobl is available in the Appendix A.3.2. The equality of the two matrices of transformation, Tobl and Tab, leads to the following relationships. sin41 = cos0 sin$ sin8 = cos$1 sine1
(2.52)
It can be noted that in the small angles, the angles of the second system are nearly equal to those of the first system and therefore, no difference is found between the two systems of angles.
41 = d
and
O1 = 8
REMARK 2.18 Homogeneity of notation could be obtained for the system of angles between frames F, and Fb, and between the frames F, and Fa by using the symbol X b for the azimuth angle instead of $, and the symbol yb for the inclination angle of the aircraft, instead of 8, and finally the symbol i.$b for the bank angle of the aircraft, instead of 4; or, on the contrary, forgetting the notations >i, y,L./ and using the notations $, 8, 4.
2.2.6
Transformation from the vehicle-carried normal Earth frame F, to the aerodynamic frame Fa
The appendix which corresponds to this paragraph is (Section A.4, p. 200).
The transformation of the vehicle-carried normal Earth frame F, to the aerodynamic frame F, is defined by three angles: lStrotation xU aerodynamic azimuth angle about the axis zo 2'Id rotation Y~ aerodynamic climb angle about the axis yoia 3'd rotation pu aerodynamic bank angle about the axis x, The intermediate axis yoia is in the horizontal plane and is obtained from yo by the azimuth rotation 2,. The rotation X, leads the axis x, to the vertical plane containing the aerodynamic velocity axis x,. The climb angle rotation 7, leads the intermediate axis x,iU into a vertical plane, on the aerodynamic velocity axis x,. In the same transformation, the axis z, joins the axis z,iU. Finally the bank angle rotation pu leads the intermediate axis yoiu to y u , or axis z,ia to axis z u , that is to say p, is used to re-establish the axis z, of the vehicle-carried normal Earth frame F, into the symmetrical plane of the aircraft. The positive orientation are those of (Figure 2.12, p. 32), in agreement with the definition of a right-handed axis system. The aerodynamic azimuth angle X, is sometime called the air-path azimuth or air-path track angle, as the aerodynamic climb angle could be named the air-path climb angle or air-path inclination angle, and the aerodynamic bank angle pu, the air-path bank angle. According t o convention, the ranges of angles can be stated as (2.53) (2.54) (2.55) REMARK 2.19 Normal standards do not state if 70 is the aerodynamic climb angle with respect to the local vertical (Hypothesis 12) or relative to the vertical of the Normal Earth-fixed frame F E ; that is to say, if the defined angles above are those of the transformation from FE to Fa or from F, to Fa.It seems more natural to take this last definition. REMARK 2.20 The convention in the range of the variation of xa is an ANSI recommendation, the standard IS0 1151 does not give any particular indication.
Matrix of transformation
X" = To,X"
(2.56) Dynamics of Flight: Equations
(2.57) \
-sinra
COS ya
sin
COS p a COS 70
(2.58) and
(2.59)
Figure 2.12: Aerodynamic frame Fa relative to the vehicle-carried normal Earth frame F O
2.2.7
Transformation from the body frame namic frame Fa
Fb
to the aerody-
The appendix which corresponds to this paragraph is (Section A.5, p. 200).
2.2 Definition of angles between frames
33
The transformation of the body frame Fb t o the aerodynamic frame F, is, in reality, the transformation of one vector t o another, from the fuselage axis vector xb, to the aerodynamic velocity vector x,. The axis x, is carried by the aerodynamic velocity V, (Equation 3.13, p. 47). Thus only two rotations will be necessary: l S t rotation -a, about the right wing axis Yb Znd rotation p, about the axis z, = zi The intermediate axis zi is in the symmetrical plane (xb,zb) of the aircraft and is obtained from zb by the angle of attack rotation -a,. Just as the second and last rotation is made around zi, this axis is kept and is equal t o z, which belongs t o the aircraft symmetrical plane. The aerodynamic frame Fa is therefore linked t o the body frame Fb. The angle of attack rotation -aa leads the fuselage axis xb to the projection of the aerodynamic velocity V, (or x,), in the aircraft symmetrical plane (xb, zb), that is to say xi. The rotation 0, leads the intermediate axis xi to the aerodynamic velocity V, (or x,). The aerodynamic angle of attack a, is positive when the fuselage axis a, xb is above the plane (y,, x,), in other words if “the pilot is above the aerodynamic velocity vector”. The aerodynamic sideslip angle 3/, is positive if “the wind blows on the p, right cheek of the pilot”. According to convention, the ranges of angles can be stated as (2.60) (2.61)
Matrix of transformation (2.62)
Tba
=
cos a, cos p, sin Pa sin a, cos p,
- cos aa sin pa
cos P a - sin a, sin p,
- sin a ,
cos a,
(2.63)
(2.64)
2.2.8
Transformation from the body frame Fb to the kinematic frame F k
The appendix which corresponds to this paragraph is (Section A.5, p. 200).
The transformation of the body frame Fb t o the kinematic frame Fk is a transformation of one vector to another, from the fuselage axis vector xb to the kinematic velocity vector xk or Vk (Equation 3.10, p. 46). By analogy t o the transformation of the body frame Fb t o the aerodynamic frame F, (Section 2.2.7, p. 32), it can be determined that Dynamics of Flight: Equations
2 - Frames
34
symmetrical plane of the aircraft
\
Figure 2.13: Aerodynamic frame Fa relative to the Body Frame
Fb
lYtrotation - a k kinematic angle of attack about the right wing axis yb about the axis zk = zi 271drotation P b kinematic sideslip angle The intermediate axis z, is in the aircraft symmetrical plane (xb,zb) and is obtained from the axis zb by the angle of attack rotation - a k . Therefore zk belongs to the symmetrical plane of the aircraft and the kinematic frame Fk is thus linked to the body frame Fb. The angle of attack rotation - a k leads the fuselage axis xb to the projection of the kinenatic velocity Vk (Equation 3.10, p. 46) (or xk), in the aircraft symmetrical plane (xb,zb), that is to say xi. The rotation P k leads the intermediate axis xi t o the kinematic velocity VI,(or xk). According to convention, the ranges of angles can be stated as -T< T
--<
2 -
cyk Pk
57r
(2.65)
7r
(2.66)
SZ
35
2.2 Definition of angles between frames
Matrix of transformation
Defined in a manner similar to Tb, (Equation 2.63, p. 33), it is found that (2.67)
2.2.9
Transformation from the kinematic frame dynamic frame Fa
Fk
to the aero-
The appendix which corresponds to this paragraph is (Section A.6, p. 201).
The transformation of the kinematic frame Fk to the aerodynamic frame F, will allow the kinematic velocity Vk to be connected t o the aerodynamic velocity V,. Kinematics will demonstrate later on (Equation 3.17, p. 48) that these two velocities are made up of the wind velocity V,. Therefore, it is not surprising to see the angles for “wind” indication appear. This transformation will be performed in the same manner as the transformation of the body frame Fb to the aerodynamic frame Fa (Section 2.2.8, p. 33). However, two angles are not sufficient, even though one vector Vk is connected to anot,her, V,. Because the two frames Fk and F ‘ are linked in an independant manner to the body frame Fb, this connection must be respected by a third angle t o lead frame Fk to frame Fa. In order for the reader t o be convinced, it can be noted that in the rotations a , 0, the axis z stays in the plane (x,z) of the initial frame, and thus 0 0
axes z, and zk belong t o the symmetrical plane of the aircraft (xb,zb), if the transformation of Fk to F, is solely done by cy and 0,then the axis z, belongs t o the plane (xk,zk), and thus xk should belong to the symmetrical plane of the aircraft (xb, zb) whatever a, is, and this is not the case.
REMARK 2.21 The transformation of Fk to F a is therefore defined by aw and Pw, wind angle of attack and wind sideslip angle. These angles could have been defined by the inverse transformation Fa to F k , or by the inversion of the order of rotations (aw and Pw). It is a question of convention. Appendix A.6, gives the matrix of transformation when the order of rotations is inverted.
The transformation from Fk to F, is accomplished by three rotations lStrotation -a, wind angle of attack about the axis yk 2nd rotation p, wind sideslip angle about the axis zki, 3rd rotation p, wind bank angle about the axis x, The intermediate axis zki, is in the plane ( x k , z k ) and is obtained from the axis zk by the angle of attack rotation -a,. The rotation -0, leads the kinematic velocity Vk (Equation 3.10, p. 46) (or xk) on to the projection of the aerodynamic velocity V, Dynamics of Flight: Equations
36
2 - Frames
(Equation 3.13, p. 47) (or x,) on the plane (xk, Z k ) , that is t o say the intermediate axis
The sideslip angle rotation pw leads the intermediate axis Xkia t o the aerodynamic velocity Va (or x,). Finally, the bank angle rotation pw leads the intermediate axis Z k i a t o za.
xkio.
Matrix of transformation
Xk cos a, cos p,
- cos a, sin p, cos pw - sin a, sin p,
cos a, sin p, sin p, - sin a, cos pw
sin p,
cos pw cos p,
- sin p, cos p,
sin a , cos pw - sin awsin ,%, cos p, cos a, sin p,
+
(2.68)
sin a , sin p, sin p, cos a, cos pw
+
REMARK2.22 From a practical point of view, the angle p, remains small. In the case of Figure 2.14, it can be observed that: a, = 14.31", p, = 10.32", p, = 2.6", with a k = 20" p k = 10" and = 35", pa = 20". The statement can be made that the relationships (Equation 2.79, p. 39) and (Equation 2.82, p. 40) = a k +a, and = p k p, are pretty much confirmed even in general circumstances.
+
2.2.10
Transformation from the normal Earth-fixed frame F, to the kinematic frame Fk
The appendix which corresponds to this paragraph is (Section A.4, p. 200).
The transformation of the normal Earth-fixed frame F, to the kinematic frame F k is defined by three angles and is a procedure similar t o the transformation of the normal Earth-fixed frame F, t o the aerodynamic frame F, (Section 2.2.6, p. 31). It can be noted that Fk is linked to the aircraft by zk which belongs t o the symmetrical plane of the aircraft (xb, zb). lStrotation Xk kinematic azimuth angle about the vertical axis z, Znd rotation ~k kinematic climb angle about the axis yi 3'd rotation p k kinematic bank angle about the kinematic velocity axis x k The intermediate axis yi is in the horizontal plane and is obtained from yo by the azimuth rotation x k . The azimuth rotation x k leads x, t o the vertical plane containing x k (or Vk), The rotation ~k leads into the vertical plane, xi on x k (or vk). Finally the bank angle rotation p h leads yi t o yk and zi t o zk. The kinematic azimuth angle x k is sometime called the flight-path azimuth or flight-path track angle, as the kinematic climb angle yk could be named the flight-path climb angle or flight-path inclination angle, and kinematic bank angle pk, the flightpath bank angle. The kinematic azimuth XA: is called the true course if x, is orientated
2.2 Definition of angles between frames
37
Figure 2.14: Relative positions of the aerodynamic Fa, kinematic Fk and body Fb frames
towards the geographical north and the magnetic course if x, is orientated towards the magnetic north. The angle between the azimuth and the course corresponds to the drift. According t o convention, the ranges of angles can be stated as (2.69)
(2.70) (2.71) REMARK 2 .2 3 The convention about the range of variation of x l ~is an ANSI recommendation, the standard I S 0 1151 does not give any particular indication.
Matrix of transformation In a transformation similar to the transformation of the normal Earth-fixed frame F, t o the aerodynamic frame Fa (Section 2.2.6, p. 31), the Dynamics of Flight: Equations
38
2-
Frames
Figure 2.15: Enlargement of the zones of summation of the angle of attack and sideslip angle
matrix TOais recovered by changing the subscript.
(2.72)
X"
Tok
(2.73)
=
,
-sinyk
cos yk sin pk
(2.74)
2.3
Angular relationships
As soon as there are three frames, two paths are possible to join one to another. The equivalence between these paths is the origin of angular relationships. The definition of the relationship between three frames leads to the relationship between three
2.3 Angular relationships
39
matrices of transformation. This will allow for the establishment of essential relationship between the angle of attack, the sideslip angle, the inclination angle and the bank angle. T13
=
(2.75)
T12T23
Figure 2.16: Angular relationships
2.3.1
Angle of attack, slideslip angle the frames Fb, Fa, Fk
-
Relationships between
The appendix which corresponds to this paragraph is (Section B . l , p. 205).
The matrices of transformation between the aircraft body frame Fb and the aerodynamic and kinematic frames Fa and Fk, are linked by the following relationship: (2.76) This equality, term by term, allows the establishment of the relationship between angles a a , a k , a,, P a , P k , P w and C l W . The aerodynamic sideslip angle Pa can be determined by the relationship
with
-
lr
5P5
7r
(2.78)
If the wind velocity a , is zero (Hypothesis 7)) the following relationship is obtained:
Pfl
=
,& +P,
when
a, = O
(2.79)
The aerodynamic angle of attack a , can be determined by the relationship sin aflcosp, = sin a k cos P k cos a, cos P, - sin P, sin a k sin P k cos cyk sin a,, cos Pu, cos a, cos pfl = cos cyk cos P k cos a , cos P, - sin p, cos cyk sin P k - sin a k sin a , cos P,
+
(2.80) (2.81)
-~
Dynamacs of Flzght: Equataons
2 - Frames
40
If it can be assumed that the sideslip angles P are zero (Hypothesis 6), the following relationship is established:
The sideslip angles are zero in the framework of the pure longitudinal movement
(Hypothesis 27) (Equation 5.139, p. 133).
REMARK 2.24 The two simplified relationships (Equation 2.79, p. 39) and (Equation 2.82, p. 40) are true at the first order for weak angles of attack and sideslip angles. They are relatively proven in general, see Figures 2.14, section (2.2.9), p. 35. The complete trigonometrical relationships, generally used, are available in Appendix B. 1. REMARK 2.25 It is shown that if the wind angle of attack is zero (aw= 0) or if the sideslip angles are zero ( P k = Pw = PO = 0) then the wind bank angle pw is zero ( p w = 0).
2.3.2 Angles of attack, inclination angle, climb angle tionship between the frames Fb, F,, Fa or Fk
-
Rela-
The appendix which corresponds to this paragraph is (Section B.2, p. 207).
The matrices of transformation between the Normal Earth-fixed frame Fo and the body frame Fb and aerodynamic frame Fa, are linked by the following relationship:
This equality of terms establishes the following relationship: 8 = cya
+ Ya
when
Pa = 4 = 0
(2.85)
The absence of aerodynamic sideslip Pa and bank angle 4 is part of the hypothesis of pure longitudinal movement (Hypothesis 27) (Equation 5.139, p. 133). This relation is therefore directly applicable in the case of pure longitudinal flight, that is to say, flight in the vertical plane.
Other useful relationships are given
If the aerodynamic sideslip angle Pa is null, then this proposition and its converse are true: a null aerodynamic bank angle pa gives a null bank angle 4
If the aerodynamic sideslip angle Pa and the aerodynamic bank angle pa are null, then the azimuth angle 11 is equal to the aerodynamic azimuth angle x a
The aerodynamic bank angle pa can be determined through three kinds of relation (Section B.2, p. 207). The relation which seems the more useful is given (Equation B.34, p. 210), then
____ 2.3 Angular relationships
-.
41
(2.88)
for
Pa = 0 then (2.89)
(2.90)
For a small value of the aerodynamic climb angle and sideslip angle between the aerodynamic bank angle pa and the bank angle 4 yields
cos e
sin pa
COS
COS /3a
the relation
sin 4 x cos 8 sin 4
(2.91)
Relationships between the frames Fb, Fo, Fk The same type of relationship can be established with the kinematic frame Fk instead of the aerodynamic frame Fa. The matrices of transformation are linked by Tob
=
(2.92)
Toklfkb
The same kind of relationship can be found here as before by changing the subscript “a” to “k”
8 = cllk
+ ~k
when
Pk
=4 =0
Another way of writing this relationship establishes the determination of and pk in terms of 8, 4, @ and ~ k X,k . The equations are obtained from
(2.93) a k , Pk
(2.94)
and a method of calculation is suggested in Appendix B.2.2. In the simple cases previously described with the inclination angle 8 = 0 and kinematic climb angle yk = 0, the following relationship can be obtained: sin /3k and if the bank angle
= cos 4 sin(Xk - $)
4 = 0 then
This relationship can be found intuitively between the azimuth II, (or heading) and the kinematic azimuth x k . The principle definitions, notations and results of this chapter on frames are grouped together on the following page. Dynamics of Flight: Equations
2 - Frames
42 Frames and Angles
I I
DEFINITIONS O F ANGLES
DEFINITION OF F R A M E S
NORM A L E A RTH- F I x ED F R A M E FO
Transformation
zo vertical downward BODYF R A M E Fb x b fuselage aircraft 8 6 belongs to the aircraft
syriiinetrical plane and downward
Euler angles: 11, azimuth 8 inclination angle 4 bank angle 1 11,lZ" 2 e/ Yi 3 41 z b
Transformat ion
Transformat ion
F, to F, Euler aerodynamic: xa azimuth y, climb angle p, aero bank angle 1 x a / 20 2 Yu/ Ya 3 P u l xa
aerodynamic:
a, angle of attack Pa sideslip angle
X" = T , m X a Transformat ion Fk to Fa
Transformation Fb to Fk
Transformat ion Fo to Fk
wind: angle of attack a, sideslip angle Pw bank angle p, 1 -awl Yk
kinematic: angle of attack a k sideslip angle P k
kinematic : azimuth ~k climb angle yk bank angle P k Xk
/
zo
2 Pw 1 % 3 PZLI 1 xu
2 Y k l Yi 3 - P k I Xk
xk = T k , x a
xh = T,kXa
3
Kinematics The beginning of chapter 3 will deal with the definition of the linear and angular velocity (Section 3.2, p. 45). Kinematic equations (Section 3.4, p. 66) are established by the relationship between these velocities and the linear or angular positions.
d(Position) dt In order to establish kinematic equations, the first question t o answer is the derivation of the aircraft's positions. To do that, the fundamental relationship of kinematics will be established at the beginning of chapter 3. This relationship will allow a vector t o be derived in any kind of frame and will be essential during the procedures leading to the equations of Flight Dynamics. The establishment of kinematic equations corresponds to the first step of derivation. The second step will express the accelerations by deriving the velocities and thus will allow the two fundamental laws of Mechanics (Section 4.1, p. 71) t o be developed. Furthermore, the equivalence of the representation of velocities is furnished by the intervention of the relationship between angles and the components of the different velocities (Section 3.3, p. 61). This equivalence will be used later to write the equations so that their solution will be easier to find. The manifestation of the angles between the velocities and the aircraft axis is essential in order t o communicate the aerodynamic effort. Velocity =
3.1
The fundamental relationship of kinematics
In order t o calculate the velocities, as stated above, the first question to answer is the derivation of positions. This derivation is obtained thanks to the fundamental relationship of kinematics, established in (Section D.1, p. 225), a relationship useful as well for the calculation of accelerations.
+
dXo - dX1 - - n1oxx dt dt 43
44
3 - Kinematics
5)
The derivative with respect to the time t of a vector X in the frame Fo is equal to the derivative of this vector X with respect to the time t in the frame Fl t o which must be added the cross product of angular velocity of the frame F1 relative t o frame FO f'I10) by the vector X. Other useful relationships to calculate the derivative of the angular momentum are shown in (Section D.l, p. 225)) with II a matrix (3.3) which could be a matrix of inertia
F)
(Equation D.9, p. 227)
The matrix & is a skew-symmetric matrix associated with the vector 52 such as
&X
and
f'I =
=
nxx
(a)
(3.4)
Again as in any kind of Y vector
d (It OYO ) dt
~
=
f21OXlIY --IISzlOXY
dI1 + -Y dt
+It-d Y o dt
(3.5)
This relationship will be useful in working on the derivations of the angular momentum H which is represented by the term It Y. The characteristics of angular velocity vectors between frames are stated below The vector nij defines the angular velocity of frame Fi relative to the frame F'. This velocity can be calculated thanks to the matrix of transformation Tij (Equation 2.2, p. 19) between the frames Fi and F' such as
These angular velocities between frames are defined during the demonstration of the fundamental kinematic relationship (Equation D.6, p. 226) and can be written in two formulas thanks to this relationship and to (Equation 2.3, p. 19)
45
3.2 Anoular and linear velocities
Thus the components of the angular velocities of the frame Fi relative to the frame F', aij, expressed in the frame Fi can be calculated thanks t o the associated skew-symmetric matrix. This skew-symmetric matrix is equal to the product of the matrix of transformation Tij between the frame Fi and the frame F' multiplied by the temporal derivative of its transpose. Moreover, these angular velocities have the following characteristics (Equation D.11, p. 228) and (Equation D.12, p. 228): The angular velocity of frame Fi relative t o frame Fj, is equal to opposite of the angular velocity of frame F' relative to the frame Fi. f2ij
=
-slji
(3.8)
A summation property allows the establishment of the velocity between two frames as the sum of two angular velocities relative to a third frame. aij
3.2
=
Oil,
+n k j
(3.9)
Angular and linear velocities
In this paragraph, the linear and angular velocities are defined (Section 3.2.1, p. 45). In the following section, the angular velocities are calculated from the matrix of transformation between frames (Section 3.2.3, p. 58). The three linear velocities, the kinematic velocity V k , the aerodynamic velocity V a and the wind velocity V , , are defined by the derivatives of the positions with respect to the normal Earth-fixed frame FE. The kinematic velocity V k , or the velocity of the aircraft with respect t o the Earth, will be used to define the trajectory of the aircraft. The aerodynamic velocity V , , or the velocity of the aircraft with respect t o air, determines the external aerodynamic efforts and propulsion. The wind velocity V , , or the atmospheric velocity with respect t o the Earth, completes these first two velocities because there is a simple relationship between these three velocities (Equation 3.17, p. 48) v k
=
v a + v w
The inertial velocity of the aircraft VI,G,fundamental for the calculation of acceleration, is equally defined. The three angular velocities equivalent to the linear velocities are defined: the kinematic angular velocity ak, the aerodynamic angular velocity aa and the wind angular velocity aW.With the last two velocities is also associated the notion of the field of wind velocity developed in this paragraph. The kinematic angular velocity ak represents the angular velocity of the aircraft relative to the Earth and it can also be written
3.2.1
The definition of velocities
Kinematic velocity
v k
Dynamics of Flight: Equations
The kinematic velocity of the center of mass G of the aircraft Vk,is defined relative t o the normal Earth-fixed frame FE (0, XE, Y E , ZE). In general, the center of mass of the aircraft is the origin of the aircraft body frame Fb. (3.10)
The kinematic velocity is equal t o the derivative of the position of the aircraft center of mass OG relative to the Earth FE. This velocity is sometimes called flight-path velocity, or ground velocity, that is t o say velocity relative to the ground. Sometimes the term ground velocity is used t o characterize the projection of Vk in the horizontal plane. The subscript “IC” is used for kinematic. It is thus the velocity of the aircraft relative t o the Earth. The vector Vk has for components in the vehicle-carried normal Earth frame F,
(3.11)
The components U ; , U& wg are the conventional notations, whereas VN “North velocity”, VE “East velocity”, and Vz “vertical velocity”, correspond to the notations often used and have a much more physical sense. The classical vertical velocity V z , corresponds to a positive vertical velocity when the aircraft climbs, which justifies the minus sign. The North velocity VN is positive when the aircraft is heading North and VE, the East velocity, is positive when the aircraft is heading East. These components are therefore the components of the velocity of the aircraft relative t o the Earth as expressed in the local Earth axis, linked t o the position of the aircraft. These local references explain the heading notations North, East and Vertical . REMARK 3 . 1 This specific appellation (North, East, Vertical) of the components of
Vk in F, is introduced here because of the “physical significance” of its components.
These notations will be used later to show inertial acceleration in the aircraft. It must also be pointed out that these appellations are allowed only if the axis xo of F, is directed towards the geographic North direction as viewed by the aircraft (Hypothesis 24).
REMARK 3.2 A specific kinematic velocity could be defined, by not taking the normal Earth-fixed frame FE as the derivation frame but by taking the vehicle-carried normal Earth frame F,. This kinematic velocity is denoted Vkp, and it is introduced in section
5.1.4 (Equation 5.74, p. 118).
According to the definition of the kinematic frame F k , the component, of Vk in this frame is (3.12)
Angular and linear ~ velocities _ _ -
_ 3.2 __________
-
~-
47
_.
~
Aerodynamic velocity V, The velocity of the center of mass of the aircraft G (in general, the origin of the body frame Fb) relative to the air not influenced by the aerodynamic field of the aircraft is denoted Va, the aerodynamic velocity, air-path velocity, or “air velocity”. At the time t , G has a certain position in the atmosphere; if there is no aircraft, this position would have been occupied by the particle of air W , thus
REMARK 3.3 The subscript “a” a t V is suggested here, even though normal standards do not include this subscript. This will later avoid confusions with notations provoked by the absence of this subscript, in particular with the components of other velocities with a subscript assigned to them. REMARK 3.4 It is much more precise to note v a , G instead of V a , in order to convey the aerodynamic velocity of the center of mass G and not that of another point in the aircraft where velocity could be very different. As it is rare to calculate other velocities then that of the center of mass G, there is very little risk of confusion and the simplified notation V, instead of V a , is~ usually admitted. In the same way, it would have been more precise to note WG instead of W , to define the position of the particle of air occupying the place of G and not that of any other point in the aircraft.
According t o the definition of the aerodynamic frame F a , the component of V,1 in this frame is
v:
=
(!)
(3.14)
Wind velocity V, The velocity of the particle W situated in G, relative to the normal Earth-fixed frame F E , is defined as the wind velocity V,, or the velocity of air (3.15) In general, it can be stated that the wind V, is defined in the vehicle-carried normal Earth frame F,. The known wind data is therefore
v;
=
(q
(3.16)
w:,
That can be in agreement with the notation of Vi (Equation 3.11, p. 46)
Dynamics of Flight: Equations
3 - Kinematics
48
Fundamental relationship between the Earth velocities The definitions of the velocities v k , V , and V , lead to the fundamental relationship, since OG = O W + W G
The kinematic velocity dynamic velocity Va.
v k
is equal to the sum of the wind velocity V , and aero-
Inertial velocity VI,G The inertial velocity of the aircraft center of mass G , V Z , Gis, a velocity relative to the inertial frame F z , and is defined by dAGz (3.18) dt It is the temporal derivative of the aircraft center of mass position AG with respect to the inertial frame F I . This velocity is developed again by (Equation D.14, p. 229) and (Equation D.15, p. 229) VI,G =
V Z , G = Vk + n , y ~ x A G= dAGo + OOzxAG dt
(3.19)
and is expressed in the vehicle-carried normal Earth frame F, (Equation D.17, p. 230) VN
(3.20)
with Slt Earth angular velocity (Section 3.2.3, p. 58).
Kinematic angular velocity nk By definition, the kinematic angular velocity of the aircraft nk is the angular velocity of the aircraft relative t o the normal Earth-fixed frame FE. It is often simply denoted SZ. The subscript “k” is omitted when there is no possible confusion with the wind angular velocity O,, or the aerodynamic angular velocity a,.The body frame Fb is linked to the vehicle, then the kinematic angular velocity of the aircraft is also the angular velocity of Fb relative to FE. Thus, there is an equivalence of the notation between
The definition equivalent to that of vk (Equation 3.10, p. 46) for n k is obtained thanks to (Equation 3.7, p. 44). The angular velocity is then obtained, like v k , by the derivation of a position, but this time the position is an angular one. This angular positions are given by the transformation matrix TEb
49
3.2 Anqular and linear velocities
By definition, this vector has components in the body frame Fb
(3.22)
These components are called roll rate p , pitch rate q, and yaw rate r . Roll rate p is supported by the fuselage axis xb and the pitch rate is supported by the right wing axis Yb. REMARK 3.5 Here again, if no ambiguity is possible, the notations could be simplified by omitting the subscript “lc” and the superscript “g” on the components p , q, r of
n: .
3.2.2
The field of wind velocity
It is useful t o specify the notion of the atmosphere in the vicinity of the aircraft which leads t o the model of the field of wind velocity around the aircraft [2] [4] [3]. This modeling is done in three stages. The first one, thanks t o a simple example, starts the definition of the gradient of wind (Cr;wmV,, which will be generalized in the second step. Finally, the third step uses this notion of gradient of wind t o define the field of wind velocity in the vicinity of the aircraft itself. The movements in the atmosphere are characterizied generally speaking by the turbulence. This turbulence comes in “waves”. The aircraft “experiences” turbulent zones of limited dimensions all along its trajectory. The probability of the occurrence of these zones is modeled as a function of the intensity of the turbulence. In a zone, the turbulence is characterized by a random spatial distribution of atmospheric particle velocity. This distribution is assumed “frozen” in space. This is Taylor’s hypothesis (Hypothesis 8). This hypothesis is due to the weak value of the temporal evolution velocity of the field of atmospheric velocity with respect t o the aircraft velocity in a turbulent zone, for a spatial distribution of the given atmosphere. Thus the aircraft “experiences” a field of constant wind velocity in space, that is to say a velocity independent of time. From a given point in space, the velocity of a particle of the atmosphere is constant The random spatial distribution of velocity is modeled by a Gaussian process of a given spectral distribution. The intensity of velocity (standard deviation) and the bandwith characterize this model. What this means is that well below the wavelength of the order of the turbulence scale L, the energy diminishes quite noticeably. This turbulence scale is defined by the integration of a correlation function of the atmospheric velocities. The greatest part of the energy is given by the wavelengths superior t o L . The dominant wavelength giving the most energy is around 5L, so that the influence of the turbulence whose wavelengths are below L can be ignored. The turbulence scale L grows with the altitude. Around 2500m high, L is around 1OOOm depending on the turbulent models suggested. At low altitudes, L is around 10 to 100m depending on the models.
’.
lThe reader must not mistake the notion of a field of constant wind for the notion of uniform wind velocity field. For the second notion, the wind velocity is the same whatever the aircraft position and whatever the time (Section 5.3, p. 130) ~ _ _ _ _ _ _ _ _ _ _ ~ ~
~
Dynamzcs of Flaght: Equataons
3 - Kinematics
50
Figure 3.1: Wind field of velocity When L is a multiple of the length of the aircraft, in the vicinity of the aircraft, the field of velocity of the atmospheric particles is a linear function of the distance (Hypothesis 9). Thus, the spatial derivative of Vw is noticeably constant in the different directions of space. This hypothesis will be retained for the rest of the document.
A simple example of the gradient of wind Before arriving at a general model of the field of velocity in the vicinity of the aircraft, it is easier to begin with a simple specific case that can later be extended t o
the final definition. It can be assumed that in the vicinity of the aircraft, the field of velocity of the atmosphere can be locally modeled by a vortex (Hypothesis 10) whose relative t o the Earth FE. local wind angular velocity is at,,, a w
=
a2atrnE
(3.23)
Therefore between two particles W of atmosphere positioned as X1 and X 2 (Figure
3.1, p. 50), the velocities are linked by the following relationship vw,x2
=
vw.x1+ a w x x 1 x 2
(3.24)
This relationship is shown in (Equation 3.1, p. 43) and translates the links that exist between the velocities of a rigid system. By going t o the limit, when X 2 joins X1 in the vehicle-carried normal Earth frame F,, the following equation can be obtained
NO,,, = n W x d X o
(3.25)
REMARK 3.6 The transformation to the limit linked to the notion of derivation causes a frame of differentiation to appear such as here, the vehicle-carried normal Earth frame F,. In a strict sense, the frame of reference of the wind angle, in this case F E , and the frame of differentiation, in this case F,, must be the same. But in practice in this document, these relationships are only used with the hypothesis of a flat and fixed Earth for which there is no difference between these two frames. Furthermore, given the nature of modeled atmospheric phenomena, this slight difference really does not have a practical sense (Hypothesis 3) (Hypothesis 17).
By noting (3.26)
3.2 Angular and linear velocities
51
according t o (Equation 3.25, p. 50) projected in the vehicle-carried normal Earth frame F O
with (3.27)
( g!)
(dX")" =
(3.28)
The specific case in Figure 3.2 will be studied simply to illustrate this result. The vortex modeling the wind has an angular velocity flw carried by the axis y, and the result is
Therefore it can be concluded that
that is t o say (3.29) With this last relation, it is clear that the value of q; gives the variation of the vertical velocity w , along the horizontal axis x, and the variation of the horizontal velocity uw along the vertical axis z. The result is what has been suggested in Figure 3.2
-duo, > o dz
and
dWo,
dX
< o
Modeling of the gradient of wind &m>V, The simple preceding example gives a physical idea of the method used for modeling a field of wind velocity in the vicinity of the aircraft. This introduces the more general formulation of this field of velocity modeled by
( 2; ) ( dw;
=
uxo,
vxo, wx;
uyo, uzo, vyo, vzo, wy;, wzo,
) ( g;)
(3.30)
Dynamics of Flight: Equations
3 - Kinematics
52
20
Figure 3.2: Vortex wind field of velocity that is to say
(3.31) with
(GRmVO,)” =
(
uz;
uxo,
uy;
vx;
vyo,
212;
wx;
wyo,
wzo,
(3.32)
The matrix &mVz is the Jacobian matrix of V,, made up for each row of the spatial derivatives of one of the components of V, with respect to the variables of the positions 2, y, z , in the vehicle-carried normal Earth frame F,. The Jacobian matrix &mVg is defined with respect to the frame of derivation, in this case Fo. The notation associated with the components of GRADV;aids in establishing various kinds of information. If UX: is taken as an example, “uW”designates the first component “u”ofthe wind “w” velocity, the exponent “0” designates the projection frame, in this case the vehicle-carried normal Earth frame F,, and finally the wind velocity component “U,” is derived with respect to the spatial variable “x” indicated by the letter “x”. REMARK 3.7 In a strict sense, a supplementary letter should appear to designate the derivation frame. In this document, the only derivation frame will be the vehiclecarried normal Earth frame F,. In order to simplify the writing of these formula, this letter has been omitted. This omitted supplementary letter on the components of (GRADVE)~, corresponds to the first “0” in exponent of VG in (CkmVG)”.
In light of the preceding specific case (Equation 3.29, p. 51), a clearer physical representation can be established that recalls the notion of angular wind uy;
=
-ry;
uz; = q z ; vzo,
=
-pz;
vx;
=rX0,
wx;
= -qx;
WY;
= PYL
(3.33) (3.34)
3.2 Anqular and linear velocities
53
(3.35) REMARK 3.8 These p w , q w , rw axe the "wind angular velocity" as seen by the aircraft following the axes x, y, z. As the simply illustrated example suggests in Figure 3.2 p. 52, it is clear that ('x", ((y", '(z" of (Equation 3.35) does not strictly signify the derivative of p , q, r with respect to x, y, z. They are there, so that the original notation will be remembered (Equation 3.34, p. 52) and to allow the two versions of p , q, r to be distinguished. For example, p , and p , are two types of gradient of the wind as seen by the aircraft such as a roll rate velocity but these gradients stem from two different directions.
The breakdown of CswmV, The gradient of the wind @mV; can be broken down into three elements in order t o obtain a much more physical approach
with
( C s w m ~ V ~=) ~
( "+
0
0
0
w$,
WZ;
)
(3.37)
The matrix Cswrn~VLrepresents the variation of the linear or translational velocity (2') of the wind along the radial axis, for example, the horizontal velocity of the wind uw which varies along the radial axis x which supports this velocity u x ; .
( :;; 0
=
(GUUDRV;)"
-?)
-rE
qE
(3.38)
The matrix GRADRVO, represents the variation of the velocity perpendicular t o the radial axis ( R ) along this axis. This variation of velocity is associated with a vortex wind with an angular velocity ClW.This is the case of Figure 3.2
n; and
=
(;)
(3.39)
C~~MDRV;~XO = St,xdX"
Finally
(
0 ( Q r n ~ p ~ V=~ ) ~-drE dq;
-drG 0 -d&
dq;
-d& 0
(3.40)
This symmetric matrix CGWADNVL represents the variation of the velocity perpendicular t o the radial axis during the rotation of this axis, that is to say along the normal axis ( N ) perpendicular t o the radial axis, as is the specific case in Figure 3.4. ~
~
~~
Dynamics of Flight: Equations
3 - Kinematics
54
By adding these two terms (GRNDRVG)" + (GRADNV;)O,the result obtained is
For example, in the following case represented by the Figure 3.3 which represents a horizontal velocity gradient with respect to the altitude, this yields uz;
> 0
wx;
=
0
From equation (3.34), p. 52 it is found that uz;
=
qz;
wxo,
=
-qx;
>0
=o
This fact is modeled by the relationship (Equation 3.42, p. 54) and the two preceding results
Figure 3.3: Horizontal wind velocity gradient
Therefore, the result obtained is (3.42)
represented by Figure 3.2 page 52. In addition (3.43)
55
3.2 Angular and linear velocities
f XO
Figure 3.4: Normal wind velocity gradient represented by the figure below Moreover, to simplify further calculations, it can be stated that (Section G.2, p. 283)(Equation G.38, p. 286) to (Equation G.43, p. 286) @;
= py; - p z ;
g; FE
= qz; - qx; = rx; -ry;
ii;
=
ux;
-vy;
6;
=
vy;
-wz;
6; = wz; -ux;,
(3.44) (3.45) (3.46) (3.47) (3.48) (3.49)
These components with a tilde “-)’ are equal to zero if the wind is locally modeled by a vortex (Hypothesis 10). This corresponds in (Equation 3.36, p. 53) to have
GRmTv; = GRmINv; = 0
(3.50)
The field of wind velocity in the vicinity of the aircraft The preceding development has allowed the modeling of the gradient of the wind. In the framework of the hypothesis of a linear field of velocity (Hypothesis 9) of which the limits will be specified, the results will be used t o define the field of the wind velocity in the vicinity of the aircraft. When the scale of turbulence L is clearly larger than the length of the aircraft, the length of the shortest wave containing a significant energy is still slightly larger that the largest dimensions of the aircraft (the length of the fuselage or wing span). The aircraft “experiences” a spatial evolution of the atmospheric velocity which is linear with respect t o the distance. Therefore &mVw is a matrix with constant components, and this allows the prediction of the field of wind velocity in the vicinity of the aircraft. When L is as great as the length of the aircraft, it is no longer possible to determine the field of wind velocity in the vicinity of the aircraft by knowing Gx.mV, at the center of mass of the aircraft. When L becomes the size of the mean aerodynamic chord of the wing, this time it is the notion of the wind velocity at G which is to be Dynamics of Flight: Equations
3 - Kinematics
56
questioned. The aircraft can no longer be considered as a point with respect t o the turbulence. The encountered excitations no longer deal with a rigid aircraft but with a flexible aircraft and the associated structural modes. In the following part of this document, the assumption framework has been set up where the turbulence scale is clearly greater than the length of the aircraft. Thus in the vicinity of the aircraft, the wind velocity V w can be calculated as a function of the wind velocity of an atmospheric particle at the center of mass of the aircraft Vw,c by the relationship
(3.51) This relationship is the generalization of the equation (3.24))p. 50 or the integration of the equation (3.31))p. 52. The particular atmospheric perturbations, such as wind gusts or a constant gradient wind, are the particular cases which are included in the framework of this modeling. If it can be assumed that wind velocity can be considered in the framework of an atmosphere comparable t o an incompressible fluid, in order t o satisfy the equation of continuity, the following result is obtained
diuV, = 0 That is t o say, the trace of
&mV; is zero or
tr(GRmVL) = 0 uxW + w W+ w z w = 0
(3.52)
The notion of aerodynamic angular velocity na It is often admitted [3] that because of the flat configuration of the aircraft, only the angular velocities representing the evolution of along the axis xb and Yb in the body frame Fb, have an influence on the aerodynamic efforts, that is t o say r y L , r& and q&, py:. This is the opportunity t o introduce the notion of aerodynamic angular velocity Qa equivalent to the linear velocity Va.The components of 52, will have an influence on the aerodynamic coefficients and can be defined as follows
v,
b Pa
!l: ryt
= r - r y b,
=
= and
b
P-pYw
Q - Qxw b
(3.53)
r x b, = r - rx,b
The difficulty of introducing the notion of aerodynamic angular velocity is in the fact that, with the yaw motion, two wind gradients, a priori distinct, have an influence on the aerodynamics of the aircraft
Nevertheless, for a given aircraft, the aerodynamic coefficients of the type Cnr, CZr result in the influence of the fuselage and the fin (U&) and the influence of the wing
3.2 Angular and linear velocities
57
( u y & ) in the proportions linked t o the geometry of the aircraft. Thus there exists an effective yaw wind angular velocity which represents this distributed influence b rk = k , ry& + (1 - k r ) rxw
(3.54)
The coefficient k r is a coefficient of “weight” belonging to each aircraft and representative of the relative influences between the fuselage and fin and the wing subjected to the yaw rate r. If k, is equal t o 1, it means the yaw rate r has an influence only on the wing, and for k r equal t o zero, the yaw rate r has an influence only on the fuselage and fin. Thus an effective aerodynamic angular velocity appears 0, such as P-PYW
(3.55)
that is t o say (3.56) The effective wind angular velocity, in other words an angular velocity that represents the influence of wind on the aerodynamic efforts of the aircraft, is therefore defined as
( ;;) b
=
(3.57)
with b
b
Pw
=
PYW
Qw
=
QX,
b
b
r& = k r
ryb,
+ (1 - k,) rx&
(3.58)
However, to establish the general equations, this effective angular velocity is only exploitable if it is possible to define an expression of the wind gradient @.mVz, uniquely as a function of the wind angular velocity nw.To make ClL appear in terms of (GRADVL)~, it is necessary t o choose k, equal to 1 or 0 and thus ( G R A D V L ) ~ is written, for example for k, = 0 and under the conditions that b
b
dPb, = Pzw - P w b b d4; = Q z , - Q w drk = r y ; - r ;
(3.59)
According t o equation (3.42)) p. 54
D y n a m i c s of Flight: Equations
In this relationship, the term containing the effective wind angular velocity 0% is associated with the matrix & r n ~ V , and the matrix t o the right is equal to the sum and GRAJDNV~ of GRADTV,
( G R m v y = (GRmRv;)b + (CuIRmTV;,
+&mNV;)b
(3.61)
+
This matrix (GRADTV; Ghrn~V;) to the right, in the first rough estimate, could be omitted in the term &mV; if all the terms having an influence on the aerodynamic efforts of the aircraft were contained in CGrsrn~V,. In general, this is not the case since neither ryL or r x : are represented, depending on the value of k,.. Here, with k,. = 0, it is r y b , which is not represented. This problem disappears if r y ; = r&, in this case drk = 0 and a wind frame could be introduced by using the preceding definition of 0,. The aerodynamic angular velocity 0, could normally be exploited in the equations and takes on a clear physical signification. Thus, this is the framework of the wind vortex hypothesis (Hypothesis 10) (Section 5.2, p. 125) and GRADTV;, = GRAIDNV; = 0 (Equation 3.50, p. 55). In this specific case, the relation 0: = 06,can be directly exploited in the equations of the moment of which a parallel appears with the force equations and the aerodynamic velocity V,. Another specific case can be treated in this way, in the framework of a pure longitudinal flight (Hypothesis 27), if only the influence of a vertical wind is considered.
3.2.3
Angular velocity expression
It is possible t o calculate the angular velocity of certain frames, relative to others, in terms of the derivatives of angles between these frames (Section D.3, p. 231). This work will be necessary t o define the kinematic relationships (Section 3.4, p. 66). The equation (3.7), p. 44 allows these calculations t o be carried out. In order to collect the information on the angular velocity, it is given here a little in advance, the kinematic relationships which will be determined in section (3.4), p. 66.
The Earth’s angular velocity
The angular velocity of the normal Earth-fixed frame FE relative t o the inertial frame F I , or the Earth’s angular velocity, is denoted as ~
E
I=
at
(3.62)
The vector 0 t is the Earth angular velocity relative to a fixed frame in space. Its modulus is equal to the value of the rotation linked to the Sideral day, either 23 h56‘4” or 86,164s.
27r Rt = -= 7.292 10-5 radls 86,164
(3.63)
The angular velocity vector 0, is carried by ZI the “world’s axis” (South-North), and expressed in the normal Earth-fixed frame FE (Equation D.28, p. 232), gives cos L t o - sin L t o
(3.64)
59
3.2 Angular and linear velocities
______. -~
With the associated kinematic relationship (Equation D.27, p. 232). and The same vector
p. 232), is
flt
Lto wto
= 0 =
-Rt
(3.65) (3.66)
expressed in the normal Earth-fixed frame F, (Equation D.29,
(3.67)
Angular velocity of the vehicle-carried normal Earth frame relative to the inertial frame The angular velocity s t , ~of the vehicle-carried normal Earth frame F, relative to the inertial frame FI (Equation D.30, p. 233) is expressed in the normal Earth vehicle carrried frame F, in terms of the angles between the two frames.
The derivative of the stellar time of point G , b t G , is equal t o the derivative of the , the derivative of the stellar time of point 0, w t o , which, in longitude of G L ~ Gplus turn, is equal to the opposite of the Earth angular velocity Rt (Equation 2.14, p. 22) (Equation 2.13, p. 21)
&tG
=
LSG-flt
(3.69)
The derivative of the latitude of G L t G , is equal to the derivative of the difference of the latitude between G and 0, ALt, (Equation 2.15, p . 22) and (Equation 2.16, p. 22)
with the navigational kinematic relationships (Equation 3.118, p. 68) (Equation D.19, p. 231)
ALt
=
VN Rt h
+
(3.71) (3.72)
Thanks to (Equation 3.72, p. 59), a new expression is obtained of the angular velocity of the normal Earth-fixed frame F, relative t o the inertial frame FI in terms of the aircraft’s velocity and the Earth angular velocity (3.73)
Dynamics of Flight: Equations
3 - Kinematics
60
Angular velocity of the vehicle-carried normal Earth frame relative to the normal Earth-fixed frame The angular velocity n , ~ of the vehicle-carried normal Earth frame F, relative t o the normal Earth-fixed frame FE is expressed in the vehicle-carried normal Earth frame F, in terms of the angles between the two frames (Equation D.33, p. 233).
The second relationship in terms of the aircraft’s velocity is obtained thanks t o the equation (3.72), p. 59.
Angular velocity of the aircraft relative to the Earth The angular velocity s l b , of the body frame Fb relative to the vehicle-carried normal Earth frame Fo, is expressed in the body frame Fb in terms of the angles between two frames (Equation D.35, p. 234)
sag
=
(
-4sine + 4 +
Bcos4 tjcosBsin4 -Bsin$+$cosBcos+
(3.75)
Expressed in the vehicle-carried normal Earth frame F,, it takes the form (Equation D.36, p. 234)
(3.76)
Angular velocity of the aerodynamic velocity relative to the aircraft The angular velocity f t Q b of the aerodynamic frame Fa relative t o the body frame Fb, is expressed in the aerodynamic frame Fa in terms of the angles between the two frames (Equation D.37, p. 235) (3.77)
Angular velocity of the kinematic velocity relative to the aircraft By the same procedure as before for the aerodynamic velocity, the angular velocity Q k b of the kinematic frame Fk relative to the body frame Fb, can be obtained and expressed in the kinematic frame Fk (Equation D.38, p. 235) (3.78)
61
3.3 Relationships between angles and velocities -
_ . .
-
Angular velocity of the kinematic velocity relative to the Earth The angular velocity nk of the kinematic frame Fk with respect to the vehiclecarried normal Earth frame F,, is expressed in the kinematic frame F k in terms of the angles between the two frames (Equation D.39, p. 235) (3.79)
Angular velocity of the aerodynamic velocity relative to the Earth In a similar manner as the preceding case,the angular velocity i l a o of the aerodynamic frame Fa relative t o the vehicle-carried normal Earth frame F,, is expressed in the aerodynamic frame Fa in terms of angles between the two frames (3.80)
3.3 Relationships between angles and velocities What is looked for in this section is an equivalence between two representations of velocity, their components in the frame and their modulus with its angular position relative t o this frame. The components will be naturally useful for establishing certain kinematic relationships and for calculating the accelerations that appear in the general equations under their first form. The angular positions of velocity are useful for defining the aerodynamic efforts (angle of attack and sideslip angle) or the trajectory of the aircraft (climb angle, azimuth angle). This equivalence is characterized by the relationship between the angles and the components of velocity. The angles useful for the model of the efforts are those between the aerodynamic velocity Va and the aircraft or the Earth, that is to say between the aerodynamic frame Fa and the body frame Fb or the vehicle-carried normal Earth frame F,. These angles will appear in terms of the components of V , . For the acceleration terms, it is rather the kinematic velocity v k which will appear. Thus it will be necessary t o express these angles in terms of the components of Vk. To do this, the equation (3.17), p. 48 is used
Va = V k - V ,
(3.81)
which allows the connection of the components of Va t o the components of Vk and V,. Expressed in the body frame Fb, the following relationship is obtained
vg
= V;-TboV;
(3.82)
The wind is defined in the vehicle-carried normal Earth frame F, which explains the presence of V ; . The general results dealing with the expression of these velocities Dynamics of Flight: Equations
3 - Kinematics
62 are established in (Section C.1, p. 215). In the kinematic frame velocity Va is expressed by
Fk,
the aerodynamic (3.83)
(3.84)
3.3.1
Aerodynamic angle of attack and sideslip angle (a,,p,)
The calculations associated with this paragraph are carried out in (Section C.2, p. 216).
The angles are defined between the aerodynamic frames Fa and body frame The relationships looked for are thus obtained from the vectorial relationship
Fb.
which will give for the modulus of the aerodynamic velocity (3.86) Thus the relationship is obtained between the aerodynamic angle of attack the aerodynamic sideslip @a and the components in the body frame Fb of the aerodynamic velocity Va @a
= arcsin
($)
7r
with - 2
a - 2
(3.87)
Either if
U:
>0
or if
u:
<0
&a
= arctan = arctan
($-) ($)+ sign(wt)
(3.88) 7r
(3.89)
This last case is improbable.
3.3.2
Aerodynamic climb, bank, and azimuth angle
(ra,pa, xa)
The calculations associated with this paragraph are carried out in (Section C.3, p. 217)
The angles Y ~ , X a are defined between the aerodynamic frame Fa and the vehicle-carried normal Earth frame Fo. The relationships looked for will thus be obtained from the vectorial relationship
63
3.3 Relationships between angles and velocities which will give for the modulus
(3.91) The relationship between the aerodynamic climb angle the aerodynamic azimuth Xa and the components in the vehicle-carried normal Earth frame Fo of the aerodynamic velocity Va can be obtained in the following manner = arcsin
?a
-Wa” (yo)
- -
7r
(3.92)
3.3.3
“Wind” angle of attack and sideslip angle (a,, p,)
The calculations associated with this paragraph are carried out in (Section C.4, p. 218) The “wind” angle of attack and sideslip angle are angles that connect the kinematic frame F k and the aerodynamic frame F a . The relationships looked for will thus be obtained from the vectorial relationship
With a procedure similar to that of (Section 3.3.1, p. 62), this will give the modulus of the aerodynamic velocity
(3.94) The relationship between the “wind” angle of attack cyw, the “wind” sideslip the components in the body frame F b of the aerodynamic velocity Va Pw
cosa,
= arcsin =
4
(2)
Va COS P w
k
Pw
and
with - - < Pw 5 2 2 IT
7r
with
- 7r
< aw< 7r
(3.95)
By utilizing the equation (3.84), p. 62 between Va, Vk and V,, the following relationships are obtained, which allows the connection of the “wind” angle of attack a, and “wind” sideslip Pw with the components of wind and the modulus of kinematic velocity v,
(3.96) Dynamics of Flight: Equations
tanp,
=
tana,
=
-21,
k
v k - ‘ui -W,
vk -
cos a,
(3.97)
k
(3.98)
‘uh
REMARK3.9 It must be stated that the wind V, is not generally defined in the kinematic frame Fk but in the vehicle-carried normal Earth frame F,. Thus U:, vk, I& are not directly known quantities but depend on kinematic angles of climb, bank and azimuth ( r k , p k , xk). 3.3.4
Aerodynamic angle of attack and sideslip angle measurements
The relationships previously defined are generally used t o exploit the equations of Flight Dynamics, for example, to make simulations and controls. Because of this, it would turn out to be useful t o measure these aerodynamic angles of attack and sideslip angles at the center of mass G of the aircraft. However in general, because of the local nature of the aerodynamic field, the measurement probe is not situated at the center of mass of the aircraft. Thus the measurement will be disturbed by the rotations of the aircraft and the field of wind velocity. This paragraph will define the new relationships in order t o reconstitute the angles of attack and the sideslip angles at the center of mass in terms of those measured on any point whatsoever of the aircraft. The calculations associated with this paragraph are in section ( C . 5 ) ,p. 220. The measurement probe, located on any point whatsoever of the aircraft, measures a local aerodynamic angle of attack and a sideslip angle, ass, Pas. A relationship between ass, pas and the components of kinematic velocity v k and “wind” V , will be looked for. At the point S, where the probe is situated, the kinematic velocity v k , S and aerodynamic velocity Va,s are calculated in terms of the components of kinematic velocity at G V i , of the aircraft angular velocity f2: and the position of the probe GSb by the relationship between the velocities of a solid (Equation D.49, p. 237).
(3.99) with xi, &, z i , the coordinates of the probe S in the body frame
GSb =
(q)
Fb.
(3.100)
2.S
The aerodynamic velocity of the probe V,,S is deducted from the kinematic velocity of the probe V k , S by the relationship
65
3.3 Relationships between angles and velocities where the velocity of “wind” appears at the level of the probe
(Equation D.56,
V,,S
p. 238)
v:,s
(
=
U;
- U b,
- u x ,bx s b
U;
- U:,
-
b b uy,ys b w; - w, - wz;z;
+ ( q - qz;)z; + ( r Tx;).; + ( p - pY&)& -
- ( T - TyL)y; - ( p - pz:,.; - ( q - qx;,.;
(3.101) The “wind” velocity at the level of the probe V,,S also makes the components of wind at the center of mass G appear, as well as the components of the gradient of wind GmVG established in the body frame F b (Equation D.50, p. 237). Next, the angle of attack and sideslip probe a,, and Pas are expressed in terms of the components of V k and V, thanks t o the preceding relationship (Equation 3.101) and by writing (Equation C.31, p. 220)
The frame Fa,s is the equivalent of the aerodynamic frame F, for which the axis x a is parallel t o the local aerodynamic velocity V,,S and not t o the aerodynamic velocity of the aircraft V a . Thus the result (Equation C.34, p. 221) and (Equation C.36, p. 221) sinp,,
=
sina,,
=
1
-
V,,S
(41
K,s cos P a s
b
U,
b
b
- VYWYS
(Wi- w;
+ ( T - Tx,).sb - wz:z;
b
b
b
- ( P - pzw)zs)
+ ( p - pY;)y;
-
( q - qx:).;)
(3.103) The relationship between the aircraft’s aerodynamic angle of attack and sideslip a,, P a and that of the probe a,, and P a s can be expressed thanks t o the components of the aircraft aerodynamic velocity (Equation C.42, p. 222) and (Equation C.43, p. 222)
Measurement of the sideslip with a probe linked to the fuselage For practical reasons, the sideslip probe often has a rotation axis linked to the
fuselage that is supposed to be parallel to z b . In this case, the sideslip measured is not the classical slideslip used in the preceding relationships. The classical slideslip corresponds t o a rotation around Z a and not z b . The preceding formulas can be corrected of this influence thanks to the relationships available at the end of the appendix (Section C.5, p. 220)) (Equation C.47, p. 223) and (Equation C.48, p. 223). Dynamics of Flight: Equations
3 - Kinematics
66
3.3.5
Kinematic climb angle and azimuth
( ~ k ,x k )
The angles of kinematic climb and azimuth yk and X k , are defined between the vehiclecarried normal Earth frame F, and kinematic frame F', thanks to the same procedure as in section (3.3.2). The relationships looked for are obtained from the relationship
v;
= Tokv;
(3.105)
Thus, there is a relationship between the components North, East and vertical of the kinematic velocity Vk and these angles, which gives
(3.106) or still yk
vz
= arcsin vk
(3.107) (3.108) The table summarizes the principal angular relationships obtained in this paragraph.
A N G L E s- VELo cITIEs
v;
RELATIo NS HIP s
= T,,V,.
I
3.4
Kinematic equations
These relationships link the derivatives of the parameters of position and the velocities. They are important because they complete the six equations of dynamics (F = mA ) to obtain a harmony of twelve first order differential equations which constitute the equations of Flight Dynamics. The acceleration is no longer the second derivative of
67
3.4 Kinematic equations
a position but the first derivative of a velocity. This makes up part of the process of putting equations under the state form. The method used consists of looking for the expression of these velocities (V and a)in terms of derivatives of the parameters of position, thanks to the very definition of velocity
Velocity =
d(Position) dt
REMARK 3.10 The kinematic equations associated with kinematic velocity VI,will assure the link between the “external forces” and the Earth environment on which they depend. In this manner, the position relative to this environment, (altitude, latitude and longitude) will be connnected to the velocity V k . External forces depend on this position, aerodynamic and propulsion forces depend on altitude, gravitational forces depend on altitude and latitude. REMARK 3.11 The kinematic equations associated with the kinematic angular velocity 0 will make the angles of the position of the aircraft relative to the Earth appear and thanks to the angular relationships (Section 2.3, p. 38)) the angles of attack and sideslip between the velocity and the aircraft can also explicitly appear in the general equations. External forces depend on these angles of attack and sideslip. REMARK 3.12 On the whole, kinematic equations allow for states having an influence on the external efforts to manifest themselves explicitly within the general equations.
3.4.1
Kinematic equations of velocity
Vk
The calculations associated with this paragraph are carried out in (Section D.2, p. 229) The kinematic velocity VI,(Equation 3.10, p. 46) is expressed directly in the kinematic frame Fk (3.109)
By definition VI, represents the velocity of the center of mass of the aircraft G with respect t o the normal Earth-fixed frame FE, thus (Equation 3.10, p. 46) (3.110)
The kinematic velocity VI,is equally expressed in the vehicle-carried normal Earth frame F, by (Equation 3.11, p. 46) (3.111)
Dynamics of Flight: Equations
3 - Kinematics
68
The velocity components of tionship (Equation 3.105, p. 66)
v k
can be expressed, moreover, thanks to the rela-
(3.112)
The notation VN, VE, Vz, besides the very clear physical signification of the parameters, has the advantage of simplifying the writing of the kinematic equations as long as the hypothesis of the fiat and immobile Earth (Hypothesis 3)) (Hypothesis 17) is not made. Nevertheless, it is possible to retrieve the positions of G in the normal Earth-fixed frame by the following relationship (3.113)
This relationship is not explicit here; it is rather complicated! Besides, this relationship (Equation 3.113, p. 68) does not have a great usefulness because it does not make a link with the altitude which directly influences the efforts applied to the aircraft. To obtain this result, the inertial velocity of the aircraft VI,G (Equation 3.18, p. 48) has to be expressed directly in two different manners, by taking as relative frames the vehicle-carried normal Earth frame F, and the normal Earth-fixed frame F E , thus (3.114)
VI,G =
and
dAG" dt
+ O0rxAG
(3.115)
where (Equation D.16, p. 229) v k
=
dAGo + 0 , x~ AG dt
( 3.1 16)
This last vectorial relationship (Equation 3.116, p. 68) constitutes the kinematic velocity relationship. Expressed in the vehicle-carried normal Earth frame F', the following results are obtained (Equation D.19, p. 231) (3.117)
where h is the altitude of G, thus the distance of G to the surface of the Earth, following the direction of gravitational attraction g,. This direction is not exactly vertical (Section 2.2.2, p. 21). The radius of the Earth Rt is around 6400 km (Section 2.2.2, p. 21). The altitude h is positive when G is above the Earth.
3.4 Kinematic eauations
69
REMARK 3 . 1 3 It is assumed here that the radius of the qarth is constant (Hypothesis 2) (Rt = constant) in compliance to the hypothesis of the spherical Earth (Section 2.1.2, p. 15), in the opposite case VZ = Rt + h REMARK 3 . 1 4 The standard symbols h or z g for the altitude and symbols H for the geopotential altitude are defined by (3.118) The altitude h is the geometric altitude.
The equation (3.113), p. 68 and equation (3.118), p. 68 establish the writing of the kinematic relationships associated with the kinematic velocity vk in comparison to its modulus and its orientation relative t o the Earth
The geographic positions are linked to the latitude and longitude by (Equation D.46, p. 236)
x z = Rt (cos Lto sin LtG cos LgG - sin L t o cos L ~ G ) y ; = -Rt cos Lto sin LgG (3.120) = - (Rt + h ) + Rt (cos Lto cos LtG cos LgG + sin Lto sin LtG) 2; 3.4.2
Kinematic equations of angular velocity fl
The kinematic angular velocity 0 (Section 3.2.1, p. 45) is the angular velocity of the aircraft relative t o the normal Earth-fixed frame FE which is denoted indifferently
By definition
It is found, thanks t o the relationship of the summation of the angular velocity (Equa-
tion 3.9, p. 45), that
The angular velocity no^ between the vehicle-carried normal Earth frame Fo and the normal Earth-fixed frame FE has already been calculated, expressed in the vehiclecarried normal Earth frame Fo from the derivatives of the longitude and latitude (Equation 3.74, p. 60)
(3.122)
Dynamics of Flight: Equations
70
3 - Kinematics
The kinematic angular velocity Obo has already been calculated, expressed in the body frame Fb from the angular derivatives (Equation 3.75, p. 60). The looked for kinematic relationship is obtained by projecting the relationship (Equation 3.121, p. 69) in the body frame Fb, thus
(8)
=
(i
coos+ 0 -sin+
- sin8
cosOsin+) cos8cos+
(1)
+Tbo
(
-LgG cos LtG A Lt .-ALt sin LtG
)
(3.123) The second part is not explicit because the writing is rather complicated. The transformation matrix Tb, depends on inclination angle 8, bank angle 4 and azimuth angle $J (Section 2.2.5, p. 27). REMARK 3.15 The vector CkzE could be expressed as a function of V N ,V E ,Vz (Equation 3.74, p. 60). REMARK 3.16 The vector C k o ~is zero if it is assumed that the Earth is flat (Hypothesis 3).
It is possible t o express 8 as a function of p, q, relationship (Equation 3.123, p. 70).
4,4,
T
thanks to the preceding
With the kinematic equations (Equation 3.120, p. 69) linked to the kinematic velocity ~ and G A& as a function of Vk and the angles X k , ~ k Accordingly, in the general framework of a spherical, rotating Earth (Hypothesis 2), (Hypothesis 17)) the two families of kinematic equations (Equation 3.120, p. 69) and (Equation 3.123, p. 70) of linear velocity and angular velocity are coupled. This will not be the case with the hypothesis of a flat and fixed Earth. The coupling comes from the relationship (Equation 3.120, p. 69) which can be put in this new form to be exploitable in the relationship (Equation 3.124, p. 70)
vk,it is possible to express L
.
4
Equations In the preceding chapter, the first step of derivation led t o the definition and calculation of the velocities thanks to the derivation of the positions. The second step of derivation, studied in this chapter, consists in deriving the velocities t o obtain the accelerations. These accelerations are one of the two terms in the fundamental theorem of Mechanics (Section 4.1, p. 71). The second term is made up of external efforts which will be developed at the end of this chapter (Section 4.3, p. 82). The aircraft in flight is subjected to three categories of effort: its weight, the aerodynamic efforts and propulsion efforts. The weight model will be precisely developed in this chapter. The models of aerodynamic and propulsion efforts will be presented in a general manner. The detailed treatment of these models, which can be found in works on aerodynamic and propulsion, are not the objectives of this document. Accelerations are calculated in two stages: the inertial acceleration of the aircraft (Section 4.1.1, p. 77) and the derivative of the angular moment (Section 4.2, p. 79). The first step is associated with the force equations and the second with the momentum equations. Finally, all the results are combined to constitute the properly so-called equations of flight (Section 4.4, p. 94). The objective sought in this document is obtained with the equations developed in this chapter. So the following chapters will exploit a certain number of simplified forms of these equations.
4.1
Fundamental equations
In classical Newtonian mechanics, the fundamental law which leads to the establishment of the equations necessary for studying the movement of a system is expressed as the following: the combination of the acceleration quantities, dynamic resultant and dynamic moment, is equivalent to the combination of external efforts applied t o the system, that is t o say the external forces resultant and the moment of external forces. The moment of the external forces depends on the external forces resultant and the 71
72
_
_
~
_
_
_
_
4 - Equations
point of reference. The combination of the acceleration quantities is composed of the dynamic resultant
and the dynamic moment
The acceleration AI,Mis the inertial acceleration (subscript I) of point M. The dynamic resultant RD equal to the external forces resultant is
The dynamic moment M D is~equal to the moment of external forces is
xi
The combination of external efforts is composed of the external forces Fexti and the moment of external forces C i M t ~ ( ~ , , tat) ithe point of reference 0. If the moment is due to an external force Fexti applied to point Pi, this moment is written as follows
This fundamental law (Equation 4.3) and (Equation 4.4) assumes that the overall internal forces of the system (S) constitute a null combination of forces resultant and moment. This is especially the case of a solid that is rigid, such as the aircraft, which is assumed to have this characteristic (Hypothesis 13) in this document. The dynamic resultant RD is obtained by the integration relative to position, of inertial acceleration AI,Mon the overall system (S),at each point M of the system (S),affected by the elementary mass dm. The inertial acceleration of point M , AI,M, is the temporal derivative of the inertial velocity of M , VI,M,with respect to the inertial frame FI. The inertial velocity of M (Equation 3.18, p. 48) is itself the temporal derivative of the position of Ad, AM, with respect to the inertial frame F I . The origin A of the vector AM is the origin of the inertial frame F I .
n:,,
AI,M = dt dAM' VI,M= dt
(4.7)
The dynamic moment M D corresponds ~ to the moment of the dynamic resultant in 0. The combination of the acceleration quantities, dynamic resultant and dynamic
73
4.1 Fundamental equations
moment, results from the derivation of the combination of the temporal derivative of the momentum quantities, that is to say the momentum and the angular momentum. Galilean frame This combination of acceleration quantities is expressed with respect to a Galilean frame (Hypothesis 1). The Galilean frame is in constant rectilinear translation with respect to an inertial frame. A frame is in constant rectilinear translation if each point M of a solid linked to this frame is driven by the same velocity VM which itself is constant (modulus and direction).
VM = constant whatever M
(4.8)
Thus, the angular velocity of this frame with respect to the inertial frame is zero, and the velocity of the origin of the frame is constant. The inertial frame is fixed, but as the notion of velocity is relative, it is therefore necessary to establish a practical reference. The standard is to use the frames oriented by the stellar references as inertial frames (Section 2.1.1, p. 14). Property of the center of gravity G The position G of the center of mass of the system is defined by OG
is, is) dm=
OM d m
(4.9)
The elementary mass d m is situated in M and the mass m of the system is equal to =
J6)
(4.10)
dm
Figure 4.1: Center of mass of a system
If the field of gravity is constant (Hypothesis 15), the center of mass merges with the center of gravity. Thus the dynamic resultant RD is written (4.11) Dynamics of Flight: Equations
4 - Equations
74
The inertial velocity VZ,Gis defined in (Section 3.2, p. 45), and if the mass m of the system, in this case the aircraft, is constant (Hypothesis 14), then
dv;G
RD = m -= m AZ,G dt
(4.12)
REMARK4 . 1 For an aircraft using a fuel propulsion system, the constant mass is obviously an approximation, since the fuel is evacuated after combustion. For transport aircraft, the mass variation per minute can be from 0.2% to 0.02% of the aircraft mass. The mass of the aircraft will be assumed constant (Hypothesis 14) in this document. However, if a variable mass is desired, the term V;,G is easy to add to the force equations in order to take into account this variable mass. If the matrix of inertia expressed in the body frame Fb is no longer constant, due to this variable mass, the term 42, must be added to the moments equations (Equation D.8, p. 227).
9
2
Thus the well-known fundamental law is found again (4.13)
The dynamic moment MD is expressed as (4.14) The inertial angular momentum H is defined in equation (4.17), p. 74 and equation (4.18), p. 75. If the point of reference 0 is a fixed point, the preceding equation (4.14) leads to (4.15) if the point of reference 0 merges with the center of mass G, the relation (Equation
4.14, p. 74) leads to
(4.16) This last result is the second fundamental law of mechanics and is complementary to the first law (Equation 4.13, p. 74). These two laws (Equation 4.13, p. 74) and (Equation 4.16, p. 74) will be used as the basis for the writing of the equations for the Flight Dynamics of the aircraft. Definition of the angular momentum H The angular momentum of a system with respect to a point P anywhere, and defined relative to the inertial frame F I , is written
4.1 Fundamental equations
75
The symbol G is the center of gravity of the system (S) and
HI,G = I G ~ I
(4.18)
The matrix of inertia IIG of the system is calculated in G. The angular velocity of system (S) with respect to the inertial frame FI is written 0 1 . This angular velocity is the inertial angular velocity. It can be remarked that the notation associated with the angular momentum H specifies the point of reference of the calculation of the matrix of inertia, in this case G, and the frame with respect t o which is defined the rotation n, in this case F I . If the point P is fixed HI,P
If the point P is anywhere,
HI,^ =
IIp
(4.19)
= IPOI
fli
+ m PGxV1,p
(4.20)
or even taking equation (4.17), p. 74 and equation (4.18),p. 75 into account
H I , P = IIG
n~+ m PGxVI,G
(4.21)
Matrix of inertia The matrix of inertia of the system (S) with respect t o a point anywhere P is written I x x -Ixy -Ixz 1 ; = (4.22) -Ixy r y y -Iyz - I x x -ryx Ixz
(
)
The superscript “2” indicates the frame Fi, the frame of the projection of the matrix of inertia lip. The symbol P is the point of reference of the calculation of the matrix of inertia of the system (S) and therefore
The moment of inertia of S Ixx
ls)+ (y2
x2) dm with respect t o the axis xi
(4.23)
I y y = d s ) ( z 2+ x2) dm with respect t o the axis yi
(4.24)
(x2 + y2) dm with respect t o the axis zi
(4.25)
=
The product of inertia of S IXY=
Ixz =
dS) is,
xy dm with respect to the axis xi
(4.26)
xx dm with respect to the axis yi
(4.27)
yx dm with respect t o the axis xi
(4.28)
Dynamics of Flight: Equations
76
4 - Equations
The components 2, y, and z are the coordinates of the point M of the system (S) written in the frame Fi, and whose origin is P. The point M determines the position of the elementary mass dm (Figure 4.1, p. 73). To simplify these notations, the following classical writing for the inertia matrix calculated from the aircraft center of gravity G has been adopted
(4.29)
Point of reference of the inertia matrix To change the point of reference of the calculation of the inertia matrix, there are two solutions. 0
Change the frame Fi by changing the origin P and recalculating the inertia in this new frame with the preceding formulas.
0
Use Koenig’s theorem by staying in the frame Fi.
IIb
= IIf=+m
-bc
(i)
a2
+ b2
(4.30)
with m as the total mass of the system (S) and
PG2 =
(4.31)
with G the center of gravity of the system. From this theorem, it can be deduced that the inertia moment is minimum in the center of gravity G.
Projection frame of the inertia matrix To change the projection frame (rotation of the frame Fd), the transformation matrix can be used. In this case, Fi was the initial projection frame, and IIG can be projected into the new frame F’. The transformation matrix Tij has the characteristic (Equation 2.2, p. 19) Xi = TijXj. Thanks to the example of the angular momentum, it can be established that (4.32) (4.33) where
(4.34)
4.1 Fundamental equations
77
REMARK 4 . 2 It can be noted that the reference point, in that case G, is intrinsic to the inertia matrix 1, but not to the frame Fi, which is only a projection frame. REMARK 4.3 The system S can be broken down into sub-systems Sk and the inertia of S will be equal to the algebraic sum of the inertia of the sub-systems Sk, such as
b+qi
Ip(Sk)i
=
(4.35)
k
Obviously the elementary inertia IIp(Slc)*in order to be summed up, must be calculated relative to the same reference point P and projected in the same frame F,.
REMARK 4.4 The aircraft is assumed to be a rigid solid (Hypothesis 13), the elements of the inertia matrix IIG are therefore constant, projected in a frame linked to the aircraft. REMARK 4.5 If the plan ( x b , zb) of the body frame Fb is a geometrical plane of symmetry (Hypothesis 4) for the distribution of the mass, then the elements D and F of IIL are zero. This is most often the case. REMARK 4.6 In a strict sense, when the center of gravity position of the aircraft changes, that is to say when the position of the center of gravity varies with respect to the geometry of the aircraft, this is due to a new distribution of mass and the inertia matrix IIG must be recalculated (Hypothesis 16).
4.1.1
Inertial acceleration of the aircraft’s center of mass
The inertial acceleration (Section E.l, p. 241) of the center of mass of a solid is obtained by the derivative with respect t o the inertial frame FI of the inertial velocity VI.Gof the center of mass, (Equation 3.18, p. 48) thus (4.36) The inertial velocity VI,G(Equation 3.19, p. 48) is written
The vehicle-carried normal Earth frame F, is used as the relative frame t o derive the kinematic velocity VI,. This allows for the introduction of the relative pseudoacceleration (4.38) This relative pseudo-acceleration represents the variation of the aircraft velocity with respect to the Earth, as seen in a local normal Earth-fixed frame F,. In the notation A r , ~ ,the , subscript “Eo” represents the two successive relative frames used for derivation. These kinds of notations are used in (Section 5.1.4, p. 118), as various and true relative accelerations are developed. In this section, only one relative acceleration is taken into account, then the notation A’, is adopted to simplify the writing. The kinematic velocity VI, is the relative velocity of the center of mass G of the aircraft Dynamics of Flight: Equations
with respect to the normal Earth-fixed frame FE (Equation 3.10, p. 46). The relative acceleration associated with v k would have been
dv; A', = dt In the two successive derivations of the position of the center of mass G in order to obtain A'r, the same relative frame was not used. This choice simplifies the writing of inertial acceleration, in reference to the components of VI, (VN, VE, V'). Thus with the vehicle-carried normal Earth frame Fo as a relative frame, inertial acceleration is written and not
This relationship projected into the vehicle-carried normal Earth frame Fo (Equation E.10, p. 243) leads to
Ay.G = A':
+ AA: + AA;
(4.40)
The relative pseudo-acceleration (Equation E.8, p. 243)
(4.41) The complementary acceleration related to the Earth sphericity (Equation E.15, p. 244)
AAs
= noEXVk
(4.42)
(4.43) The complementary acceleration related to the Earth rotation (Equation E.13,
p. 243) and (Equation E.19, p. 244)
79
4.2 Inertial angular momentum derivative
The pseudo-acceleration A': can be expressed as function of V k , +k and x k , if a direct linkage t o the modulus vk is desired (Equation E.22, p. 245). The flat and fixed Earth hypothesis is now directly translated into the expression of inertial acceleration.
Fixed Earth Hypothesis (Hypothesis 17) This hypothesis goes back t o consider that the normal Earth-fixed frame FE is fixed with respect t o the inertial frame F I , thus the Earth angular velocity is null, 52, = f l = ~ 0. The ~ formula (Equation 4.39, p. 78) is written (4.46) In equation (4.40), p. 78 it is sufficient t o cancel AAR.
Flat Earth Hypothesis (Hypothesis 3) This hypothesis goes back t o consider that the vehicle-carried normal Earth frame F, is not in rotation with respect to the normal Earth-fixed frame F E , thus 0 , =~0. In the equation (4.39), p. 78, n , ~ is replaced by ~ E I This . operation comes back in equation (4.40), p. 78 to cancel the term AAs. Flat and fixed Earth Hypothesis (Hypothesis 17) and (Hypothesis 3) It is thus possible t o understand that the normal Earth-fixed frame FE is a Galilean frame (fixed Earth). The vehicle-carried normal Earth frame F, is not in rotation with respect t o FE. It is therefore Galilean as well and the derivative of the kinematic velocity with respect to this frame becomes an inertial acceleration. (4.47) This result is found in equation (4.39), p. 78 with ~ E =I 0 and f l , ~= 0 which leads O,I = 0. These conditions convey the hypothesis of the flat and fixed Earth and AAs = AAR = 0.
4.2
Inertial angular momentum derivative
Angular momentum of the aircraft The angular momentum of the aircraft with respect to the center of mass of the aircraft G and with respect t o the inertial frame FI (Equation 4.18, p. 75) is equal t o HI,G= IG %I
(4.48)
This angular momentum is the inertial angular momentum in G. The inertia matrix of the aircraft with respect t o G as expressed in Fb (Equation 4.29, p. 76) is writt,en
IL=(
-%
-B F
-E
-D
-ED ) C Dynamics of Flight: Equations
80
4
- Equations
In general, the aircraft has a massic plane of symmetry ( x b , Y b ) (Hypothesis 4) and F = D = 0. It is assumed that the aircraft is rigid (Hypothesis 13). The terms of IIG are therefore constant. The inertial angular velocity of the aircraft &,I, can be broken down into two terms
The angular velocity of the aircraft with respect to the Earth where angular velocity f l =~Ot ~
nbE
and the Earth
(4.50) It can be remarked that when making the hypothesis of the fixed Earth (Hypothesis 17), then at = O E I= 0 and
AHG = 0 Inertial derivation of the angular momentum of the aircraft (Section E.3, p. 245)
This derivation is expressed vectorially with respect to a relative frame, for HE,G the body frame Fb, and for AHG the normal Earth-fixed frame F E .
+ -dAHL Term C dt + ~ ~ E x A H Term G D
+
~ E I X A H GT e r m E
(4.51)
The preceding formula can be written in another way (Equation E.30, p. 246)
(4.52) Projected in the body frame Fb, the first term, or term A, is written as the formula (Equation E.36, p. 247)
Ap - E+ - Fq + rq(C - B ) - Epq + F r p + D ( r 2 - q 2 ) Bq - Fp - DI: r p ( A - C ) + E ( p 2 - r 2 )- Frq + Dpq (T)b = Cf - Ep - D q + p q ( B - A ) + E r q + F(q2 - p 2 ) - Dpr m E,G
(
+
(4.53)
A simplified notation will be used to express the angular velocity of the Earth nt, that is to say the angular velocity of the normal Earth-fixed frame FE with respect to
4.2 Inertial angular momentum derivative
81
the inertial frame Fz,previously calculated (Equation 3.67, p. 59)
\
-sinLtG
(4.54)
J
This angular velocity vector of the Earth, as expressed in the E.39, p. 247)
I
Fb
is written (Equation
(4.55)
The second term of (Equation 4.52) AIHi,G regroups the terms B , C and D (Equation E.55, p. 250)
The third term of (Equation 4.52, p. 80) A2Hi,G is established from the term E
(Equation E.51, p. 250)
REMARK 4.7 The term A (Equation 4.53, p. 80) represents the derivative of the inertial angular momentum of the aircraft when the Earth is fixed (Hypothesis ~ the terms B , C, D and E are zero or 17). Indeed, in this case f l =~ 0 and A ~ H ;= , ~A ~ H ; , , = o
In the majority of cases of the symmetrical aircraft (Hypothesis 4) ( D = F = 0), the following is obtained.
The inertial derivative of the aircraft's angular momentum for a symmetrical aircraft D = F = 0
Dynamics of Flight: Equations
82
4.3
4
- Equations
External efforts
The aircraft in flight is subject t o three kinds of efforts: its weight, the aerodynamic effort and the propulsion effort. The two last categories are the same in nature. They result from the effort of contact between the aircraft and the atmosphere. This is the pressure and friction force. The buoyancy or Archimedes force will be neglected in this case with respect t o the other efforts and especially the weight. The fields of pressure around the aircraft and in the vicinity of the propulsion system are characteristic of these efforts. It is understood that there always exists, more or less, an interaction between these two fields of pressure and the separation of this pressure effort into two categories is not always very clear. However, it is possible to model the interaction effort under the form of the retained models for the aerodynamic effort or for the propulsion effort. The proposition stated in this work is not t o detail the models of external efforts but to give a general formula and t o show how these models are inserted in the equations for flight dynamics. The aerodynamic and propulsion efforts depend upon the characteristics of the atmosphere: density, temperature and pressure. It is therefore useful, first of all, t o define the standard atmospheric model. The important result, in order t o exploit the equations, will be t o be able to describe the atmospheric data necessary t o calculate these efforts, as a function of a single kinematic parameter: the altitude h. It must be noted that on the ground, the aircraft is subjected to a fourth category of efforts: the ground’s reaction t o the aircraft.
4.3.1
Gravity
- gravitation
Figure 4.2: Weighing of a solid
4.3 External efforts
83
The weighing of a solid
The weight or the force due to gravity, marked m g, is accessible by the schematized weighing in this manner: the mass m is placed on a fixed scale with respect to the Earth, and the center of mass G of the mass is fixed with respect to the scale (Figure 4.2).
Then the mass m is subjected to two external forces: the reaction of the scale R and the force due t o the gravitational attraction m g r , G . By definition, the reaction of the scale is equal to the opposite of the weight:
R = -mg
(4.60)
REMARK 4.8 Archimedes buoyancy is ignored in this case and it is taken into account that there is no aerodynamic force, that is t o say that the air is calm (V, = 0 ) . This detail becomes important if the aircraft is t o be weighed externally.
The fundamental equation of Mechanics (Equation 4.13, p. 74) is written m AI,G
= m gr,G + R
(4.61)
with the vector AI,Gequal t o the inertial acceleration of point G which is considered as fixed with respect t o the Earth. Then
R = m ( A I , G - g r , G ) = -m
gG
(4.62)
thus the acceleration due to gravity gG
gr,G - A I , G
(4.63)
The acceleration due t o gravity g G defines the local vertical (Section 2.1.3, p. 16) (Section 2.2.2, p. 21) and it results from the sum of the gravitation g r , G and the centrifugal acceleration - A I , G , linked t o the Earth rotation equal to (Figure 2.8, p. 24)
-AI,G
(4.64)
This centrifugal acceleration (Equation 4.73, p. 85) is at the origin of an inertial force “felt” by the mass m.
Gravitation If it is assumed that the Earth is spherical and in rotation, the external force to which the aircraft is subjected to, is the gravitational force m gT,G and not the gravity force m g G , which, as shown beforehand, integrates the acceleration A r , G in which G is assumed fixed. In the framework of the spherical Earth hypothesis (Hypothesis 2)) the model of the field of Earth gravitation g r , G to point G , corresponds to the simple law of gravitation (4.65) (4.66) with Dynamics of Flight: Equations
4 - Equations
84 mt
G Rt
2,.
mass of the Earth the constant of gravitation mean Earth radius
5.983 1024kg 6.664 10-l1 SI 6368 103 m
The gravitation vector gF indicates the direction of the Earth radius, thus following A formulation of the gravitation function of the latitude is given later (Equation
4.83, p. 86).
REMARK 4.9 The gravitational field depending uniquely on altitude is a simplification. There are models of gravitational potential [7] that are more sophisticated. Even by ignoring the influence of the Moon and the Sun, the static part of the gravitational potential gives a high order series expansion (up to 36) functions of the latitude and the longitude. Recent models (1984 GRIM3B-Ll) give an accuracy level of 10-5 m / s 2 . An example of the development on these first terms of gravitational potential V , is given by Mac Cullogh, by assuming that the Earth is a revolving ellipsoid with ZI the principal inertial axis as well as XI and y~
This expression is simplified if it is assumed that I,, = I,, which is the case for the dynamic ellipsoid that represents the Earth. From the gravitational potential V , the gravitational vector g, is obtained by g,
=
GradV
Usually, flight dynamics does not call for this level of precision. However, a simplified form function of the latitude, could be useful. This will be given in the following paragraph.
REMARK 4.10 The gravitational potential V equally depends on time. This dependance is linked to the non-stationary distribution of the Earth masses such as the tides, the internal geophysical phenomenons, and the Earth elastic response to external perturbation potential. This non-stationary potential is considered as weak with respect to stationary terms.
Constant gravitational hypothesis (Hypothesis 19) In the framework of the spherical Earth hypothesis (Hypothesis 2), it has previously been shown that gravitation can be modeled under the form (Equation 4.65, p. 83)
(4.67) Gravitation is marked with an altitude of zero gF,o (4.68) where (4.69)
4.3 External efforts
85
To evaluate the sensibility of gr at the altitude, the altitude hl% which provokes a diminution of 1%of gr with respect t o g r , o , has t o be calculated
5Rt
= 5 6.368 k m = 32 k m 1000 Thus for the altitudes of classic flight, it is assumed that gr is constant and equal t o
hl% =
gr,o
(Hypothesis 19).
This fact signifies that the gravitational force in the vehicle-carried normal Earth frame F, is written
(4.70) With the flat and fixed Earth hypothesis (Hypothesis 3) and (Hypothesis 17)) there is no longer a distinction between the directions of gravitational force m gr and the force of gravity m g which is written
(4.71) The external force is no longer the gravitational force m gr but the force of gravity m g of which the modulus will be defined in equation (4.81), p. 86. The components of the force of gravity m g are calculated further on in the body aircraft frame Fb (Equation 5.11, p. 106), aerodynamic frame Fa (Equation 5.50, p. 113) and (Equation 5.51, p. 113) and kinematic frame
Fk
(Equation 5.24, p. 108).
Gravity The equation (4.63))p. 83 calculates the gravity. In the first step, it calculates the inertial acceleration of point G which is considered fixed. The weight of the aircraft is defined by a weighing operation, the aircraft remaining immobile with respect t o the Earth. Thus the inertial acceleration of G is given by the equation (4.40) to equation (4.45) with VN
= V E = vz = V N = VE = vz = 0
(4.72)
The point G is therefore considered as a fixed point with respect t o the Earth. There exists only the part of the acceleration due t o the Earth rotation Rt in AAR AY,G
= fl?(Ri?+ h )
(
cos L ~ sin G LtG 0 cos2 LtG
)
(4.73)
At the beginning of the paragraph with the weighing operation, the equation (4.63), p. 83 defines the acceleration of the gravity g& in the vehicle-carried normal Earth frame Fo which is given by equation (4.66), p. 83 and equation (4.73) Dynamics of Flight: Equations
4 - Equations
86
-a: (Rt + h ) cos L t G sin L t G 0
- Q:(&
s&
(4.74)
+ h ) cos2 L t G
REMARK 4 . 1 1 The weight is not exactly an external force. It includes an inertial force associated with the term in a:, of the equation (4.45), p. 78 or equation (4.74), p. 86. REMARK 4 . 1 2 The gravity vector g$ has a component on xo, or, by definition, it is a vector which defines the vertical. Here are the limits of the spherical Earth hypothesis (Hypothesis 2). In fact, the Earth is an ellipsoid and the vector g r , G is in an intermediate direction between the center of the Earth and the local vertical (Figure 2.8, p. 24). This contradiction would have disappeared if the hypothesis of the Earth its an ellipsoid form had been assumed (Section 2.2.2, p. 2 l ) , thus g$ would have had only one component on zo. REMARK 4 . 1 3 The moment in G of the gravitational force is zero, if the field of gravity is constant (Hypothesis 15). This is one of the characteristics of the center of gravity.
A simple model of gravity at zero altitude, a function of the latitude, was developed in 1929 by Somigliana (4.75) with the gravity at the equator
and the gravity at the pole
gEo
gpo
= 9.7803m/s2 = 9.8324m/s2
g~~ gpo
(4.76) (4.77)
geometric flattening (Equation 2.25, p. 25) f
a-b
=
-
a
= 0.003353
(4.78)
dynamic flattening fd
=
g P o - gEo
= 0.005302
(4.79)
Q: b 2.5 - = 0.008655
(4.80)
gEo
The relationship between these flattenings is
f + fd which numerically gives gGo
= 9.7803 (1
=
gEo
+ 0.0053 sin2 L t G - 5.8 I O - ~sin2 2 L t G )
(4.81)
The value of the gravity g~~ for the medium latitudes of 45" is around 9.806m/s2. Thanks t o this result and the equation (4.74), p. 86 with a series expansion, it is possible t o find an expression approaching gravitation at zero altitude %-,Go
thus, numerically, gr,Go
x
gEo
(1 + f -t (fd - f) sin2 L t G )
9.7803 (1.00335 + 0.00195 sin2 L t G )
(4.82) (4.83)
4.3 External efforts
87
4.3.2 Standard atmosphere The atmosphere is characterized by the state of the air and its velocity. The air is a gas whose state is defined by the characteristics of the standard atmosphere. Other hyphotheses govern the velocity of air V, (Section 3.2.1, p. 45). At each point M of the atmosphere, the state of the air is defined by three parameters
p ( M , t ) pressure p ( M , t ) density T ( M ,t ) temperature
Pa kg/m3
K
Pascal kilogram per cubic meter Kelvin
With the pressure units
1Pa = 1 N / m 2 = 0.01 mbar = 1.45 10-4 psi The standard atmosphere has well defined characteristics and, in general, it is close to the real atmosphere. It is characterized by the following four hypotheses. It is assumed that the atmosphere is “frozen” (Hypothesis S), that is to say the state does not depend on time with respect to the time it takes the aircraft to cross the concerned area. This hypothesis, named the Taylor hypothesis, is rich enough to extend to the velocity of the air (Section 3.2, p. 45). Air is assumed to be an ideal gas, therefore
(4.84) With R = 287 Joules/K.kg and 1 Joule = 1 m.N = 0.239calories The atmosphere can be weighed and is subjected to Laplace’s law where dp
=
-pgdh
(4.85)
Finally there is a law of temperature according to the altitude. This is certainly the hypothesis which leads to the most perceptible deviations with respect to the real atmosphere. This law is called the Toussaint law and it is characterized by gradient of the temperature, denoted Th
dT = dh
(4.86)
This gradient of the temperature is a function of altitude. With three laws for four parameters ( p , p, T , h ) , the knowledge of one parameter leads to the determination of the others. Thus, from the altitude h, it will be possible t o define the characteristics of the atmosphere necessary for the evaluation of the aerodynamic and propulsion efforts. This result is very important for the exploitation of flight equations. At sea level, the following data are found p , = 101,325P a
To = 288.16 K
p, = 1.225k g / m 3 Dynamics of Flight: Equations
88
4 - Equations
n
l
Altitude h Okm 5 h 5 l l k m l l k m < h 5 20km 20km < h 5 32km 32km < h 5 47km
I
Gradient of temperature dT dh
fl
-6.5 1 0 - 3 ~ / ~
Table 4.1: Atmosphere law of temperature Gravity g can be taken equal to gGo for a medium latitude (Equation 4.81, p. SS), g = 9.806m/s2. And when the altitude h increases, the three parameters ( p , p, T ) vary according to the following laws. For a n altitude h up to 11,000m P
= ~ ~ ( T lh + id% ~ h )
P
=
T
= To+Thh=To
P Th RT=Po(l+T,h)
(4.87) (4.88) (4.89)
For an altitude h less than 11, OOOm, the pressure p, the density p and the temperature T decrease when the altitude h increases. For a n altitude h between 11,000 m and 20,000 rn P
7'11
*
( h - 11,000)
= Pile P = plleR-9 (h- 11,000) ~ i i = RT = 216.66K
(4.90)
Fur a n altitude h between 20,000 m and 32,000 m
(4.91)
4.3 External eflorts
89
___
Th
=
1 10-3K/m
T20 = 216.66 K p20 = 5,474.72 Pa
p20 = 0.08803 kg/m3
For an altitude h between 32,000m and 47,000m
--9--] I2 T h
(4.92)
Th = 2.8 10-3K/m T32 = 228.66 K
4.3.3
p32 = 867.98 Pa
p32 = 0.01322 kg/m3
Aerodynamic efforts
Aerodynamic efforts, force and moment, have the following general configuration Force
=
1
-~SV:CF 2
(4.93) (4.94)
The centre of gravity of the aircraft G is taken as the centre of reduction of the moment. The symbol S is the surface of reference, in general, the surface of the wing. The symbol C is the length of the reference, in general, the mean aerodynamic chord of the wing. The notation Vz is the modulus squared of the aerodynamic velocity Va
(4.95) REMARK 4.14 It is common American practice to use for the length of reference .t the wing span .t = b for roll and yaw aerodynamic coefficients and the mean aerodynamic chord .t = C for pitch aerodynamic coefficients. In Europe, usually the length of reference l is the mean aerodynamic chord l = C.
The coefficient vectors CF and CM, whose components are dimensionless, are defined by their projection in the different frames. In the aerodynamic frame F a , they have as expressions
CX"
-CD (4.96) Dynamics of Flight: Equations
90
4
)I::( Cl”
- Equations
(4.97)
CM” =
The coefficients CO, Cy, C L are frequently used. The signs “minus” historically exist. They maintain the notion of the force of drag ~PSV’CDand of lift i p S V 2 C ~ which are positive in the majority of cases of flight; for drag, it is always true until proven otherwise! In the body frame Fb, the aerodynamic coefficients are expressed as
(4.98)
(4.99) The symbol CA is called the coefficient of the axial force and C N ,the coefficient of the normal force. The coefficients CZ, Cm, Cn are frequently used. REMARK 4.15 A superscript sign “A” is recommended by the standard for aerodynamic coefficients to show their aerodynamic characteristics. For the standard, the coefficients without this superscript sign are dedicated to the sum or the resultant, of the aerodynamic forces and propulsion. However the notation without it for aerodynamic coefficients, is accepted as long as there is no risk of confusion. This is the case in this document because the propulsion efforts are indicated in another way. REMARK 4.16 Just as the suffix “b” was introduced for the body frame Fb, it is logical to give this symbol as a superscript to Cx, CY,Cz. This is not a standard practice, since the body frame does not have a subscript in the standard proposal. REMARK 4.17 The coefficients Cl, C m , Cn are commonly used for moment. These coefficients are, by definition, the coefficients of moment with respect to the aircraft center of mass G (Hypothesis 20). REMARK 4.18 There might sometimes be some confusion between C y = C c and C Y . With the lateral equations of the movement, CY could be used as the equations that are often projected into the frame Fb.
The relationship that exists between the coefficients of the force expressed in the body frame Fb and these coefficients expressed in the aerodynamic frame F a , is obtained by
CFb = TbaCF”
(4.100)
thus
-CO COS
COS
PO - C y sin POCOS a, + C Lsin
-COsin /3a + C y cos /3a
-CO COS pa sin a a
- C y sin
sin O a - C Lcos a,
(4.101)
4.3 External egorts
91
General characteristics of aerodynamic coefficients These aerodynamic coefficients depend essentially on aerodynamic angle of attack, that is t o say the orientation of the aerodynamic velocity Va with respect t o the aircraft, such as the angle of attack a a and the sideslip angle pa (Section 2.2.7, p. 3 2 ) , for the general orientation, and the rotation velocities p t , q:, rk for the local angles of attack (Equation 3.55, p. 57). The coefficients also depend on the Mach number M when the compressibility effects become perceptible (high subsonic). The Mach number (Equation 6.20, p. 162) itself depends on the velocity V, and the velocity of sound a, which is itself dependent on the altitude. Finally, the coefficients can be dependent on the Reynolds number which in turn is a function of the velocity V a , of the air density p, and of the temperature T , for any given aircraft. It must be remembered that the knowledge of the altitude h, the velocity and "the angles of attack", can lead to the evaluation of the aerodynamic efforts. All these parameters being kinematic, they clearly appear in the flight equations. Obviously these coefficients are dependent on the geometry of the aircraft and in particular on the position of the aerodynamic control surfaces. The aerodynamic coefficients are the object of more or less sophisticated models. For an almost static approach, a simple solution is given by the linearization with respect to the influential parameters for general usage. For example
C L = CLa(aa - a a o ) + CLSm 6m + CLM M
9 2 + CLq V
with CL^ gradient of lift with respect to the angle of attack
CLa =
dCL da
-
and CLq gradient of lift with respect to the normalized pitch velocity
q,etc. Then
dCL v CLQ = -89
e
This formulation is extended for the roll and yaw velocity. REMARK 4.19 For the normalized angular velocity as q l / V , it is common American practice to use for the length of reference t the wing semi-span .t = b / 2 for roll and yaw derivatives and the mean aerodynamic semi-chord l = E/2 for pitch derivatives. In Europe, usually the length of reference t is the mean aerodynamic chord .t = E .
Nevertheless, the classical exception to the linearization is the coefficient of the drag CO which does not depend on the angle of attack but on the squared angle of attack
CO =
CDO+ICCL~
This type of model is especially useful for an analytical study of the aircraft dynamics after linearization. For a study by numerical means, the non-linear models of aerodynamic coefficients are generally used. Some frequency models can prove t o be necessary for the study of certain dynamics. Works on aerodynamics define these coefficients in detail starting with the external geometry of the aircraft. Dynamics of Flight: Equations
4.3.4
Propulsion efforts
The purpose here is not t o suggest a sophisticated model of propulsion forces but rather, from a simple model, to give the expression of the modulus of this force. This would also include its point of application and its orientation as well as establishing the expression of the propulsion efforts in the general equations. A simple model generally used t o define the modulus of thrust is
F
= k,pV,Xd~
(4.102)
The symbol X depends on the type of engine; the symbol p is the air density. The modulus of the aerodynamic velocity is represented by V, with Ic, as a constant and 62 representing the position of the throttle, between 0 and 1 inclusive. The following values characterize approximately the type of engine for propeller propulsion and high bypass ratio turbofan X x -1 for turbojet engine with no fan X x0 for turbojet engine with after burner Xx1 X x2 for the ram-jet The purpose is to calculate the combination of propulsion forces and moments with respect to the aircraft center of mass G, the origin of the body frame Fb, and expressed in Fb. Take an engine which delivers a thrust F whose point of application M , has for coordinates (zL,&, ,&) in the body frame Fb. The symbol F is the modulus of the vector F . This engine is oriented with respect to the aircraft at angles of the pitch setting of the engine a , and yaw setting pm. These angles are defined by analogy with the angle of attack and the sideslip angle (Section 2.2.7, p. 32) by associating F to the velocity V. Obviously, it is assumed that the engine thrusts the aircraft forward. avz > 0 The engine thrusts “downwards” p,, > 0 The engine thrusts “towards the pilot’s right” Under these conditions, the resultant of the efforts in the body frame Fb is written
Fb
=
F
(
cos pm cos a , sinp, cos p, sin a,
) ( ;!) =
F:
(4.103)
Expressed in the aerodynamic frame Fa, the equations are obtained thanks to Fa = TabFb
Fa = F (
+
cos Pacos Pm COS(Q, - a,) sin Pm sin PO COS ,&sin ,Bm - sin ,& COS p m C O S ( Q ~ - am) - cos 0 , sin(a, - a,)
) ( ) =
(4.104)
The moment of the thrust F with respect t o G, in the body frame Fb, is written
MF = GMxF
(4.105) (4.106)
4.3 External eflorts
93
plane parrallel to the symmetrical plane of the aircraft
x ' ,~y'$ , z ' ~ arc parallel to the axis xB' ye zg of the body F~UIK
Rg
Figure 4.3: Orientation of the thrust with respect t o the aircraft frame
(4.107) Thus the combination of the propulsion force and moment is written
CF~ n
resultant force
F =
(4.108)
k l
C M ~ ~ n
resultant moment
MF
=
(4.109)
i= 1
n =
number of engines
Simplifying hypotheses (Hypothesis 21) In general pm = 0, the engine is set with an axis parallel t o the aircraft symmetrical plan, and am is very weak. When there are several engines, it is necessary t o add the combination of the efforts of each engine. Symmetry is most often respected. Even if another engine exists at y h # 0, there will be another symmetrical engine at -y$.
Dynamics of Flight: Equations
4 - Equations
94
This fact will make the terms in y L disappear in the moment expression, except for the case of thrust dissymmetry, encountered when there is an engine failure. In assuming that 6, = 0, the engines are symmetric and the setting a, is low and identical for all of the engines; then it can then be noted that (4.110)
(4.111)
(4.112) With these simplifying hypotheses, only a pitch moment appears.
4.4
Flight Dynamics equations
The purpose of this paragraph is to collect all the results obtained previously (Section 3.4, p. 66)(Section 4, p. 71) in order to write the flight dynamic equations of an aircraft.
In principle, that is the objective of this work. The rest of the book will consist in exploiting the simplified forms of these equations. Before beginning the reassembling of these force equations (Section 4.4.1, p. 95), the moments (Section 4.4.1, p. 95) and kinematics equations (Section 4.4.2, p. 98), the vectorial equations which are the basis of these general equations should be recalled.
Fundamental vectorial equations or equations of efforts The fundamental equations are defined (Section 4.1, p. 71), for the force equations (Equation 4.13, p. 74), (Equation 4.12, p. 74) and for the moment equations (Equation 4.16, p. 74)
dv: G
m AI,G = m- dt
= m g,
1 + -pS 2 V:CF + F
(4.113) (4.114)
The terms on the left are calculated in section (4.1.1), p. 77 and in section (4.2), p. 79 and the terms on the right in section (4.3), p. 82.
Kinematic vectorial equations
95
4.4 Flight Dynamics equations
The kinematic equations calculated in (Section 3.4, p. 66) are associated with the fundamental equations. These equations give the relationship of the following kind: derivative of position equal to velocity. They complete the preceding equations, where acceleration has not been calculated under the form of the second derivative of position but as a derivative of velocities. It is necessary t o have these positions show up explicitly, since the external efforts depend on them. For the kinematic equations associated with the linear velocities this is obtained by expressing the inertial velocity of the aircraft VI,Gin two different ways (Equation 3.114, p. 68) and (Equation 3.115, p. 68) (4.115) The kinematic velocity VI,can thus be written in a particular way, at the origin of the kinematic equations for the velocities of translation (Equation 3.116, p. 68)
+
(4.116) O,EXAG dt The velocity of the aircraft with respect to the normal Earth-fixed frame FE is VI,
VI,
=
(Equation 3.10, p. 46).
The kinematic equations associated with the angular velocities are written through
(Equation 3.121, p. 69)
ObE
= abo+OoE
(4.117)
The terms on the right are expressed as functions of the derivatives of the angles between frames (Equation 3.74, p. 60) and (Equation 3.75, p. 60) (4.118) The angular velocity of the aircraft with respect to the normal Earth-fixed frame FE is ObE (Equation 3.22, p. 49).
4.4.1
Equations of efforts
Force equations The fundamental force equations are expressed (Equation 4.113, p. 94) thanks to the results of the preceding paragraphs. A
= m g, m AI,G = m-m ~ , G
+ z1 p V~:CF + F
dt These equations are projected in the vehicle-carried normal Earth frame Fo
(4.119)
- vz
Dynamics of Flight: Equations
The equivalence of the notations is recalled here (Equation 4.41, p. 78)
(4.121) with the different explicit terms. The “complementary” acceleration linked t o the Earth sphericity (Equation 4.43, P. 78)
AA:
=
+ vg + v;
VNVZ Vg tanLtG VEVZ - VEVN tanLtG
1
(Rt + h )
(4.122)
The “complementary” acceleration linked to the Earth rotation (Equation 4.45, p. 78)
AA;
=
Rt
(
+
+
2VE sin LtG Rt(Rt h ) sin LtG cos LtG 2vz cos LtG - 2vN sin LtG 2vE COS LtG f&(& h ) COS2 LtG
+
+
)
(4.123)
The gravitational force m g,. has been defined in (Section 4.3.1, p. 82). The matrix of transformation of the vehicle-carried normal Earth frame F, t o the body frame Fb (Equation 2.49, p. 29) cos 8 cos 1c, sin 1c, cos 8 - sin 8
sin 8 sin 4 cos 1c, - sin 1c, cos 4 sin 8 sin 4 sin II, cos 1c, cos 4 cos 8 sin 4
+
+
cos 1c, sin 8 cos 4 sin 4 sin 1c, sin 8 cos 4 sin 1c, - sin 4 cos 1c, cos e cos 4
(4.124) The aerodynamic coefficients in the aircraft axis Fb ( - C A , C Y , -CN) are expressed most often as functions of the aerodynamic coefficients in the aerodynamic axis Fa (-CO, Cy, -C L) by the relationship (Equation 4.101, p. 90)
(c) ( =
-CO
COS
-CD COS @a
+
pa - Cy sin pa COS a, C Lsin a , - C ~ s i n p , Cycosp, sin 0, - Cy sin sin pa- C Lcos a, COS
+
(4.125)
The propulsion forces in the aircraft axis Fb are often expressed as functions of the modulus of the thrust and its orientation in these axes (Section 4.3.4, p. 92) (4.126)
with
FP
=
Fi
cos pmz cos am2 cos Prni sin ami
(4.127)
All these equations can be projected in the body frame Fb. With this solution, the aerodynamic and propulsion forces are simply expressed, and on the other hand, the writing of the accelerations and of weight are more complicated.
4.4 Flight Dynamics equations
m Tb,
97
( *) +
ipSV2
( ) + ( $, )
(4.128)
Equations of moments The fundamental moment equations are expressed (Equation 4.114, p. 94) thanks t o the results of the preceding paragraphs.
(4.129) These equations are projected in the body frame Fb. The derivative of the angular momentum H z , G ,relative t o the inertial frame Fz, has been calculated (Equation 4.52, P. 80)
+
+
Ap - E+ - Fq + rq(C - B ) - Epq Frp D ( r 2 - q2) Bq - Fp - D+ + r p ( A - C ) E(p2 - r 2 ) - Frq + Dpq C7:-Ep-Dq+pq(B-A)+Erq+F(q2-p2)-Dpr
+
(4.130) The complementary terms of the angular acceleration
( A f i i , G ) b = ( A I H i , G ) b+ ( A 2 f i i , G ) b
(4.131)
The first complementary term (Equation 4.57, p. 81)
( A I H ~ , G=) ~
(
+
+
+ +
+
A(rqt - qrt) (C - B)(qrt rqt) 2pt(Fr - Eq) 2D(rrt - qqt) B ( P t - r p t ) 4- ( A - C)(prt r p t ) 2qt(Dp - F r ) 2 E ( p p t - f r t ) C(Qrt- P q t ) + (B- A)((rPt + p q t ) + 2rt(Eq - 0 ) 2 q q q t - p p t )
+
+ +
(4.132) The second complementary term (Equation 4.58, p. 81)
(A2fii,G)b =
(
rtqt(C - B ) - Eptqt + Frtpt + D(r,"- 4:) rtpt(A - C ) + E(p,"- r,")- Frtqt + Dptqt ptqt(B - A ) + Ertqt + F(q: - p,")- Dptrt
(4.133)
The Earth angular velocity Slt expressed in the body frame Fb (Equation 4.56, p. 81) sin 0
+
cos sin 8 cos 4 + sin 4 sin $
(4.134) Dynamics of Flight: Equations
4 - Equations
98
The aerodynamic coefficients are defined in section (4.3.3))p. 89. The moment of propulsion forces are defined in section (4.3.4))p. 92. (4.135)
with
4.4.2
Mki
= Fi
zLi
y h i cospmi sin a,i - z t i sinp,i COS&^ cos a,i - xLi COSP,~ sin a m $ sin Pmi - yL, cos pmi cos a,i
xki
Kinematic equations
The kinematic equations of position and of angular positions defined in section (3.4), p. 66 are expounded here.
Kinematics equations of positions (Section 3.4.1, p. 67) The vectorial relationships (Equation 3.116, p. 68) (Equation 4.116, p. 95) give the kinematic velocity VI,
VI, =
dAGo dt
+n o ~ x A G
(4.137)
thus projected in the vehicle-carried normal Earth frame Fo (Equation 3.118, p. 68) (4.138) The latitude and longitude A L t and LgG are defined by (Equation 2.14, p. 22) and (Equation 2.15, p. 22).
Angular kinematic equations (Section 3.4.2, p. 69) The vectorial relationship (Equation 3.124, p. 70) (Equation 4.118, p. 95) projected in the body frame Fb, is written (4.139) thus
(l)
4.4.3
=
(
1 0 0
sin&tanO cos4tanO cos4
%
cos 8
cos 8
L g c sin L t G
(4.140)
Exploitation of the equations
The equations of aircraft flight written in section (4.4.1))p. 95 and in section (4.4.2))
p. 98 enable the temporal simulation of the flight of an aircraft as well as obtaining
4.4 Flight Dynamics equations
99
other information. Without going into the details of the calculation of simulation, these calculations are organized in the following general manner. These equations are first order equations. They contain only the first derivatives of the states. In the first step, they were written under the form of
function with X the vector of the state and Y the output vector which depends on the states. The symbol X is made up of the derivatives of the states of the four vectorial equations, for the forces (VN, VE, Vz), for the moment ( p , q, T ) , for the kinematic equations of position ( A L t , Lgc, h ) , and for the kinematic equations of the angular positions (4, 8, $). The integration of 2 leads to the calculation of these twelve states. The problem is now to verify if the knowledge of these states leads to the calculation of the external efforts and the complementary terms of acceleration. In other words, is it possible to calculate the vector Y in order to reach the function ( X , Y ) and recalculate X ? The parameters of the output vector Y on which depend the external efforts now must be examined. 0
The force of gravitation m g r The force of gravitation only depends on the altitude h which is already part of the state. At a higher level of precision, it would depend on the latitude Ltc and the longitude Lgc, which are also part of the state.
0
The aerodynamic efforts $pS V ~ C and F $pSL V ~ C M The air density p is not part of the state but is deducted from the altitude h through the laws of the standard atmosphere (Equation 4.87, p. 88) t o (Equation 4.89, p. 88). The modulus of the aerodynamic velocity V, as well as its direction in the body frame (a,, pa) are necessary to calculate V2 and the aerodynamic coefficents CFand CM. These parameters are not part of the state. To calculate them, it is possible to use the following method. The kinematic velocity VI, of the aircraft in the Earth axis (V', VN, Vz) leads to the calculations of VI,,71,) X I , by the relationships (Equation 3.108, p. 66). The angular relationships defined in (Section B.2.2, p. 210) lead to the access of a k , PI,, PI, from 71,)X I , , 8, 4, $ which have already been determined. Wind is an external perturbation. It is known data of the simulation and thus a w ,pw, pw are known. From a w ,pw, pw and a h , ,& it is possible to calculate a,, p, by the relationship established in (Section 2.3.1, p. 39). The modulus V, is calculated from Vk and the components of wind (Equation 3.96, p. 63). Thus V,, a,, P, are accessible and the aerodynamic forces can be calculated. It must be noted that the aerodynamic coefficients depend on p , q, r but these parameters are part of the state.
0
The propulsion efforts These efforts depend essentially on V, and on p, which have already been determined for the aerodynamic efforts.
0
The accelerations Dynamics of Flight: Equations
100
4 - Equations
The accelerations A A s and AAR can be calculated. They only depend on the state (Equation 4.122, p. 96) and (Equation 4.123, p. 96). The angular accelerations AHi,Gcan equally be calculated as they, too, only depend on the state (Equation 4.130, p. 97) to (Equation 4.134, p. 97). Thus the problem is looped or closed since all the external efforts and accelerations can be calculated.
Part I1
Simplified equations
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5
Simplified equations The general equations for Flight Dynamics of an aircraft have been established in the previous chapter (Section 4.4, p. 94). These equations are useful for studies requiring a high level of precision, or for the study of high speed aircraft or spaceplane flight. For a number of study cases, this precision is superfluous and simple equations are sufficient. The first simplification consists in assuming that the Earth is pat and fized (Section 5.1, p. 104). This hypothesis leads t o the simple equation form whose validity range is limited t o a Mach number below two, which corresponds t o an acceleration error with respect to the weight of approximately one per cent. The evaluation of the value of the flat and fixed Earth hypothesis is carried out in section (5.1.4), p. 118. This value, expressed in terms of precision, is determined by a detailed examination of the different types of errors carried out under the hypothesis of a flat and fixed Earth. It is done with the support of the quantitative results. These errors are shown t o be dependent on the hypothesis used as a work method or, in the case of experimentation, the type of instrument used for measurement. Four types of error have been identified. This analysis is used not only for the force equations but also for the moment equations and the kinematic equations. The equations being simpler, it is possible to develop several ways of writing the equations, in particular for the force equations. These equation forms are linked t o the choice of derivative velocity, of derivation frames (relative frames) and of the projection frames. The second form of the force equations is attractive because of its simplicity (Section 5.1.1, p. 105). Amongst other things, it allows the wind t o appear not in the midst of the external forces but in terms of acceleration. This second form equation is the result of the derivation of the aerodynamic velocity V,, and the result of using the body frame Fb as the relative frame and the aerodynamic frame Fa as the projection frame. The first form equation is the result of the derivation of the kinematic velocity Vk. Several variants of the force equations under the first form are presented because they are often met in the literature. The number of equation forms is limited for the moment equations because of the difficulty of defining an aerodynamic angular velocity fla. This difficulty is overcome
103
5 - Simplified equations
104
if it is assumed that there is a rotating wind velocity field (Section 5.2, p. 125). If it can be assumed that in the vicinity of the aircraft, the wind velocity can be modeled by a velocity of a given point and an angular velocity, then the wind will be a kind of vortex whose rotation axis and the angular velocity would be known. Another more advanced simplification consists in assuming that there is a uniform wind velocity field (Section 5.3, p. 130). This hypothesis in particular includes the case of flight without wind. The decoupling of equations leads to the study of aircraft flight with a reduced number of equations. The purpose is to separate a certain number of equations from the fundamental twelve first order equations, without jeopardizing the quality of the obtained results with this reduced system. The chapter decoupled equations (Section 5.4, p. 132) deals with the following three cases of decoupling. 0
0
0
5.1
The decoupling of the navigation equations (Section 5.4.1, p. 135), that is to say the geographical positions (z,y) and the heading (+) equations, is perfectly alright within the framework of the flat and fixed Earth hypothesis. Without these hypotheses, strictly speaking, only the longitude could be decoupled. The decoupling of the lateral equations (Section 5.4.2, p. 137) will lead to the conservation of only five decoupled longitudinal equations. This decoupling is allowed on the condition that some hypotheses are often verified. It corresponds to flight within a vertical plane with the wings horizontal. The decoupling of longitudinal equations (Section 5.4.3, p. 145), will lead to the conservation of only four decoupled lateral equations. This decoupling is more difficult to justify than the others and the lateral equations should be handled with caution.
Flat and fixed Earth equations
Two simplifications often used in Flight Dynamics are applied here which consist in assuming that the Earth is flat and fixed (Hypothesis 3) and (Hypothesis 17). The equations of paragraph (Section 4, p. 71) are reused with these hypotheses. Two peculiar angular velocities cancel out each other. The Earth being fixed, there is no longer rotation of the Earth frame FE with respect to the inertial frame FI (Section 2.2.2, p. 21)(Equation 3.62, p. 58)
In the same way, due to the flat Earth hypohesis, the angular velocity between the vehicle-carried normal Earth frame F, and the Earth fixed reference frame FE is zero (Section 2.2.4, p. 26) (Equation 3.74, p. 60). Thus, whatever the geographical position of the aircraft, the direction of the local vertical remains parallel to itself. O0E
= 0
Based on the non-rotation of the Earth, the Earth-fixed reference frame FE becomes Galilean, that is to say an inertial frame. If the fixed Earth hypothesis is added, the
5.1 Flat and fixed Earth equations
.___~.
105
vehicle-carried normal Earth frame F, remains parallel t o the Earth fixed reference frame FE. The vehicle-carried normal Earth frame F, therefore becomes a Galilean frame as well. The consequences of these hypotheses will appear under the terms of acceleration (Equation 4.40, p. 78) (Equation 4.52, p. 80). The two complementary accelerations of translation cancel out each other (Equation 4.43, p. 78) and (Equation 4.45, p. 78)
hthermore, the two complementary angular accelerations equation (4.57), p. 81 and equation (4.58), p. 81 cancel out each other, due t o the fact of the non-rotation of the Earth.
With these hypotheses, there is no distinction between gravitation g, and gravity g, the difference coming from the rotation of the Earth. The kinematic velocity Vk (Section 3.2.1, p. 45), because of equation (3.19), p. 48 becomes the inertial velocity of the aircraft.
Therefore, the inertial acceleration A ~ , G (Equation 4.40, p. 78) is expressed as follows
These last two relationships confirm the Galilean character of the vehicle-carried normal Earth frame Fo.
5.1.1
Force equations
The force equations are reused (Equation 4.120, p. 95) taking into account the hypotheses of the flat and fixed Earth (Equation 5.3, p. 105). A
mAZ,G = m-
dVzTG
dt
= mg,
+ a p s V;CF + F
(5.7)
and with these hypotheses, g, = g (gravitation = gravity) (Section 4.3.1, p. 82) can be written. Just as in the preceding chapter, the force equation (Equation 5.7, p. 105) is projected in the vehicle-carried normal Earth frame F,.
The form above is difficult to manipulate, especially due t o the aerodynamic and propulsion efforts whose projection in the Earth fixed reference frame is rather complicated. The simplifications furnished with the flat and fixed Earth hypotheses, enable other force equations to be written to simplify this expression. These different writings Dynamics of Flight: Equations
are regrouped into two families: the first form is when the kinematic velocity Vk is derived and the second when the aerodynamic velocity V , is derived. For the first form, three variants are suggested as functions of the choice of the relative frames and projections frames. The simplification of the writing of the external forces is done by looking for a projection frame close to the aircraft.
First form, derivation of the kinematic velocity VI, (components) Relative frame: body frame Fb Projection frame: body frame Fb By using the body frame Fb as the relative frame and with the flat and fixed Earth hypotheses, 0 = n b E = Qbo (Equation 3.22, p. 49), therefore (Equation 5.6, p. 105) yields
thus (5.10)
The force of gravity (Section 4.3.1, p. 82) is expressed in the body frame Fb mgb
= mgTb0
( 8) ( =mg
- sin8 cososin+) cos e cos 4
(5.11)
The aerodynamic and propulsion forces are naturally expressed more simply since the projection frame is linked to the aircraft and the complete equations take on the form U;
+ qw,,b - To;
w; + p v ; -qu; rrq
(
- sin8 cos8sin4 cos e cos 4
)+
apSV2
( i? ) + ( 3)
(5.12)
- c N
and the aerodynamic coefficients in the body frame Fb are expressed by (Equation
4.101, p. 90)
(
-CA !:N)
=
(
-CO cos a, cos 0,- Cy sin 0,cos a,
+ C Lsin a,
-CO sin pa + Cy cos pa -CO cos pa sin a, - Cy sin a, sin P, - C Lcos a,
(5.13)
The components of the propulsion force can be expressed under another form thanks to (Equation 4.103, p. 92), and usually F; and F: are equal to zero.
107
5.1 Flat and fixed Earth equations
First form first variant, derivation of the kinematic velocity attack, sideslip angle)
Relative frame: body frame
Fb
Projection frame: body frame
v k
(angle of
Fb
The first variant of the first form is obtained by expressing the components of the kinematic velocity v k and its derivatives not in U ; , U;) wi but as a function of its modulus Vk and by its angular position a k , p k relative to the aircraft. This is thanks to the relationship
The components of the preceding equation (5.12)) p. 106 take on another form. The relative acceleration obtained in equation (5.9)) p. 106 is broken down as follows (5.16) The angular velocity
flkb
was calculated previously (Equation 3.78, p. 60) where
(5.17) thus is projected in the body frame Fb
When the complementary acceleration is added n b o x v k (Equation 5.9, p. l06), the complete force equations first form, first variant, are obtained
-
~~
~
Dynamics of Flight: Equations
108
5 - Simplajied equations ___
-
= mg
(
)+
- sin8 cos8sin4 cos 8 cos 4
______
ips1/,2
( >?) ( 3) -CN
+
(5.19)
First form second variant, derivation of the kinematic velocity VI,(angle of attack, sideslip angle)
Relative f h m e : body frame
Projection frame: kinematic frame Fk
Fb
The second variant of the first form is obtained by changing the projections frame in relationship t o the first variant. Here the kinematic frame Fk is taken as the projection frame. This approach will simplify the terms of acceleration, but in doing so, to the detriment of the terms of aerodynamic and propulsion forces. The relative acceleration of the equation (5.9), p. 106, thanks to the results of equation (5.17), p. 107, take on the following form (5.20) The equation (Equation 5.9, p. 106) projected in the kinematic frame
Fk
gives (5.21)
A!,G
(
=
mgk
Vk
V k (&
cos p k
Vk(&
+ rcoscyk
-psinak)
+ p sin /3k cos ak - q cos PI; + r sin
= mgTko
(8) ( = mg
- sin yk cosyk sinpk cos pk cos yk
a k
sin &)
)
(5.23)
(5.24)
Thus the complete equations are obtained under the first form, second variant Vk
+ r cos cyk - p sin c y k ) COS P k + p sin P k COS a k - q cos p k + r sin a k sin pk ) ~k
Vk (&
= mg
(
- sinTk
cosyksinpk cos pk cos yk
(,&
)+
$ p s v:Tka
( -g ) ( 3 ) -CL
-k I f k b
(5.25)
5.1 Flat and .fixed Earth equations
109
The aerodynamic forces will make the components of the matrix T k a (Equation 2.68, p. 36) appear, that is t o say the “wind angles” a,, p, and p,. The wind bank angle pw can be considered as zero in the majority of cases. The propulsion forces will make the components of the matrix T k b (Equation 2.67, p. 35) appear, that is t o say the “kinematic angles” a k and p k . This equation form is sometimes found in the literature. That is the reason for mentioning it here.
First form third variant, derivation of the kinematic velocity Vk (climb angle, azimuth)
Relative frame: kinematic frame Fb Projection frame: kinematic frame Fk
The third variant of the first form, just as the two preceding variants, can be obtained by keeping the modulus of the kinematic velocity Vk under an explicit form but by changing the relative frame and the projection frame. This form is often used when the equations are decoupled in longitudinal and lateral equations. As regards the decoupled longitudinal equations, this is the form most often used. By taking the kinematic frame Fk as the relative frame instead of the body frame Fb, used in the preceding first form, the inertial acceleration can be written (5.26) The angular velocity vector atohas already been calculated in equation (3.79), p. 61. This expression projected in the kinematic frame Fk =
( )
Thus (5.27) Thus the complete equations m
The expression of the aerodynamic forces and the propulsion forces make the wind p,, pw) and the kinematic angle of attack ( a k , p k ) appear respectively via angles (aw, the transformation matrices T k a and T k b . But this form is simplified in the framework Dynamics of Flight: Equations
110
5 - Simplified equations
of the decoupling hypothesis, leading t o "pure" longitudinal equations (Section 5.4.2, p. 137). The equation (5.28), -p. 109 is written, since
+ C Lsin a , 0 -CLcos aw - COsin aw -CO cos cyw
(5.29) Without wind, a supplementary simplification comes from the annulation of the wind angle of attack a,.
Second form, derivation of the aerodynamic velocity Va Relative frame: body frame Fb Projection frame: aerodynamic frame Fa The essential difference with respect t o the first form consists in working not on the derivative of the kinematic velocity Vk, but on that of the aerodynamic velocity Va. As a consequence, the projection frame will change t o the aerodynamic frame Fa, in which the aerodynamic forces are expressed more easily. Moreover, these aerodynamic forces explicitly depend on the modulus of the aerodynamic velocity Va and the angles of attack 0, which will be able to appear directly in the terms of acceleration, thanks t o the derivative of the aerodynamic velocity V , . The force equations are put in the following form.
Calculation of the accelerations (5.30)
AI,G =
The substantial derivative [l]of the wind velocity has been calculated in section (E.4.1), p. 251, as well as the aircraft angular d o c i t y f i b , with respect to the vehicle-carried normal Earth frame F,, and equal to n = n b E because of the flat Earth hypothesis (Equation 3.22, p. 49)
dt
= &mv;vk
(5.31)
By taking the body frame Fb as the relative frame for the derivation of the aerodynamic velocity Va (Section E.4.2, p. 252) (5.32)
5.1 Flat and fixed Earth equations
111-
~~~~~
~
By breaking down the derivative of the wind velocity V, (Equation 5.31, p. 110) thanks t o VI, = Va V, (Equation 3.17, p. 48)
+
The inertial acceleration AZ,Gobtained is then divided into three terms: the first dVb term -&, the second term AA = a b o x v a GRmv;v,.
+ GRmV0,Va and the third term DVZ, =
dVb First t e r m : the relative acceleration -g
(5.34)
This expression has already been calculated in equation (E.66), p. 252, projected in the aerodynamic frame F, (5.35)
+ G.mV;,V,.
Second t e r n : the complementary acceleration AA = nbo xV, term AA can also be written AA
=
(&bo+&mV$)V,
This (5.36)
It must be remembered that the wind is defined in the vehicle-carried normal Earth frame F, by
By projecting this in the body frame Fb, the following expressioin is obtained 1
(5.37) wz;
The calculation of (Ghp;o~Vz)~ has been performed in section (D.6), p. 238. Moreover =
(0 -Q
-0'
(5.38)
;p)
P
is projected in the aerodynamic frame Fa
~~
Dynamics of Flight: Equations
+
Only the first column of the matrix (& bo &mVE)a is calculated to be able to be multiplied by Vz which has only one component on the xa axis. This term A A is broken down into four elements
= AAtN
AA"
+ A A t R + AA:L + AA:
(5.40)
with
The term AA,N is cancelled out if the wind is modeled by a vortex. The term A A w ~ corresponds to the vortex components of the wind. The term AA,L corresponds to the translation gradients of the wind and is null if the wind is modeled by a vortex. Finally, the term AAI, is linked to the aircraft's kinematic angular velocity. When all the calculations are done
(5.44) AAGL
= V a CosPO COS ,Oa
cos2 a a
pa
- cos2 a, sin - COS sin a,
sin pu tan pa sin 0, 0
COS sin2 -sin2a,sinpu COS 0, sin
) ( 2) ) )(
(5.45)
AA;
= Va
0 - sin a, sin pU COS a,
0 0
- COS @a
0 COS%
sin a a sin
(5.46)
Third t e r m : the complementary acceleration due to wind DV, = @mVGV,. This term depends only on the wind and it will not be zero only if the wind and the gradient of wind are not null. It can be noted that DVO, = ( @ m V ~ V , ) o
(5.47)
5.1 Flat and fixed Earth equations
113
when projected in the aerodynamic frame Fa (5.49)
Calculation of external forces The force of gravity is expressed in the aerodynamic frame Fa mga
= mgTao
(H ) ( = mg
- sin Y a
COSYa
sinpa
COS Y a COS p a
)
(5.50)
To keep the states of the general system of equations, that is to say to be able to replace the climb angle Y a by the angle of attack and p a y mga can be written in the following form
mga = TabTbomg
mga
= mg(
( H)
+ cos 8 sin 4 sin pa + cos 8 cos 4 cos Pa sin sin 8 sin a a + cos 8 cos 4 cos aa
- sin 8 cos aacos pa
sin 8 sin @a
COS a a
+ cos 8 sin 4 cos pa - cos 8 cos 4 sin a, sin
@a
(5.51)
The aerodynamic and propulsion forces naturally can be written in the aerodynamic frame Fa (Equation 4.97, p. 90) (Equation 4.104, p. 92).
Recapitulation of the force equation - second form The preceding calculated terms, in this paragraph, enable the writing of the force equations under a second form. These equations are projected in the aerodynamic frame Fa.
m
i- m A A " + m D V ; =
A variant of the expression of gravity is available (Equation 5.51, p. 113). The complementary acceleration A A can be derived as the sum of four terms. ~~~
~
Dynamics of Flight: Equations
114
5
AA;
=
Va
-p sin
- Simplified equations
0
+ r COS a, + r sin a, sin p,
p sin P a COS a, - q COS p a
The complementary acceleration peculiar to wind is
This form is attractive. The acceleration appears under the derivative form of the parameters, having a direct influence on the external forces. The external forces are simply expressed without having the wind term intervene. The components of wind appear under the form of accelerations and must be calculated in the body frame Fb (Section D.6, p. 238).
The first forms of the force equations have been presented here because they are often found in works on Flight Dynamics. But they also point out a difficulty when it is necessary t o evaluate the aerodynamic angle of attack and the sideslip angle (a,) p a ) which does not result from the integration of the derivative of the state. It is then necessary t o use the equations with complicated trigonometrical formulas. It is clear that this difficulty only appears in the presence of wind. Without any wind, the two forms are equivalent.
5.1.2 Moment equations Derivation of the kinematic angular velocity
flk
5.1 Flat and fixed Earth equations
Relative frurne: body frame
Fb
115
Projection frurne: body frame
Fb
The moment equations are recovered from section (4.4.1), p. 95 while taking into account the hypothesis of the flat and fixed Earth (Hypothesis 3) and (Hypothesis 17). This leads t o the elimination of AlH;,, and A2H;,, (Section 4.2, p. 79). The equation (4.130), p. 97 is expressed in the form (5.53) The derivative of the angular momentum with respect to the body frame written
Fb
is
(5.54) The expression of the kinematic moment at the center of mass G with respect t o the vehicle-carried normal Earth frame Fo is written Ho,G
= IGOk
(5.55)
These equations are projected in the body frame Fb
+
Ap - 3'4 - E+ rq(C - B ) - Epq + Frp + D ( r 2 - q 2 ) -Fp + Bq - Di. + r p ( A - C ) + E(p2 - r 2 ) - Frq + Dpq - E p - Dq + Ci. + p q ( B - A ) + Erq + F(q2 - p 2 ) - Dpr (5.56)
With the expression of the propulsion moment
with b yM i COS Pmi sin ami - z t i sin Pmi z M COSP,~ C O S C Y~~ COS^,^ sinam* b x t i sin Pmi - yLi cos pmicos a,*
zL,
MFP = Fi (
(5.57)
At this point, a second form is not possible since the aerodynamic angular velocity 0 , has not been defined. It must be remembered that the aerodynamic coefficients of moment depend on the following aerodynamic angular velocities p : , q:, ry:, rx: (Equation 3.54, p. 56) and (Section 4.3.3, p. 89) Pa
b
=
P-PY,
Qa b b
=
Q-
b
(5.58)
QXw b
b
b
and rxa = r - rxw ry! = r - ry, The local wind angular velocity in the vehicle-carried normal Earth frame F, (DO,, etc) are known data of the problem. Their projection in the body frame Fb (py&, etc) calls for the calculations done in (Section D.6, p. 238). Dynamics of Flight: Equations
116
5.1.3
5
- Simplafied equations
Kinematic equations
Kinematic equations of position The kinematic equation of position is linked to the kinematic velocity v
k
(Equation
3.116, p. 68) (Equation 4.116, p. 95)
=
v k
dAG" dt
+~ , E x A G
This equation will be written taking into consideration that AG = A 0
dAG" -
dAO" dt
-
dt
+-dOG" dt
(5.59)
+ OG and (5.60)
With the flat and fixed Earth hypothesis (Hypothesis 3) and (Hypothesis 17), the point 0 is fixed and the normal Earth-fixed frame FE merge in the vehicle-carried normal Earth frame F,, then noE
The kinematic velocity
v k
v k
dAO" dt
=o
(5.61)
is finally written v k
The kinematic velocity
=
=
dOG" dt
(5.62)
is expressed in the vehicle-carried normal Earth frame
(Equation 3.107, p. 66)
(5.63) By definition, the position of center of mass in the vehicle-carried normal Earth frame F, is written
OG"
=
( $)
(5.64)
with z& = -h, where h is the altitude. From equation (5.62), p. 116 the kinematic equations are linked to the kinematic velocity vk projected in the vehicle-carried normal Earth frame F, (5.65) Thanks to the relationship equation (3.17), p. 48 between the three kinematic, aerodynamic and wind velocities
5.1 Flat and fixed Earth equations
117
these equations can be written under another form. Projected in the vehicle-carried normal Earth frame F,
(5.66)
(5.67)
(5.68) It must be remembered that the components of the wind velocity V, in the vehiclecarried normal Earth frame F, , that is to say U:, U:, 20: are considered as the known data of the problem. In order t o keep the states of certain forms of the general system of equations, V,O can also be written in the following manner
(5.69) Thus the kinematic equations give
The first two equations are not expressed, as the writing is rather complicated, and usually they are used less than the last equation because of the decoupling process (Section 5.4, p. 132).
Angular kinematic equations The angular kinematic equation (Equation 3.121, p. 69) linked to the kinematic angular velocity of the aircraft nk is written
Dynamics of Flight: Equations
5 - Simplified equations
118 with the flat and fixed Earth hypothesis (Hypothesis 3)
The equation (3.123), p. 70 projected in the body frame Fb, therefore takes on the form
() ( =
1 0 0 cos+ 0 -sin+
- sin8 cos8sin4 cosOcos+
) (l )
(5.72)
or even after inversion 1 s i n + t a n 8 cos+tanO -sin4
( f ) (: =
5.1.4
c??!
CO8
dl
)(
-
(5.73)
The consequences of flat and fixed Earth hypotheses
The appendix which corresponds to this paragraph is (Section E.6, p. 257).
The flat and fixed Earth hypothesis, obviously, leads to some inaccuracies with the simplified model. These inaccuracies are the consequences of these hypotheses, and the purpose of this section is to evaluate this imprecision. Previously, two terrestrial frames were defined, the normal Earth-fixed frame FE (Section 2.1.2, p. 15) and the vehicle-carried normal Earth frame Fo (Section 2.1.3, p. 16). These two frames are linked to the Earth, the first one is fixed relative to the Earth and the second one follows the vertical direction as seen by the aircraft. Thus, in the framework of the flat and fixed Earth hypotheses, these two frames merge and become an inertial frame. Before these hypotheses, these two frames are relative frames and can generate four relative accelerations A , , G by the two successive derivations of the position and velocity. Therefore four complementary accelerations AAG appeared AZ,G
=
A r , G -k
AAG
(5.74)
REMARK 5 . 1 There exists the same type of development with the derivative of the angular momentum H, but the quantitative results are really negligible. That is the reason why this development was not accomplished in this document. Th: order of magnitude of the complementary angular acceleration p , q, 7: links to A H , is lO-' rad/s2 in straight flight and 10-' r a d / s 2 in turning flight. On a transport airplane, this last value corresponds to an acceleration given by a variation of an angle of 10-7 rad on the pitch control surface. That is to say, absolutely negligible.
One of these four relative accelerations already has been evaluated for the calculation of the inertial acceleration AZ,Gin (Section 4.1.1, p. 77). When the flat and fixed Earth hypotheses are made, the relative acceleration A r , G becomes the inertial ~~
'For the acceleration two terrestrial frames are taken into account. Although it is difficult or fruitless to imagine other frames, it is theoretically not infeasible.
119
5.1 Flat and fixed Earth equations
acceleration AI,G.Therefore the error made is the complementary acceleration AAG. Then there are four possible errors depending on the choice of the relative frames. The problem now lies in the understanding of what kind of experimental situations give these errors. These four errors will be analysed and quantified from a parametric point of view. The first step is to obtain the two relative velocities. With the vehicle-carried normal Earth frame F, as relative frame, the relative velocity v k p or the flat kinematic velocity, is achieved V k p
=
dOG" dt
(5.75)
With the normal Earth-fixed frame FE as relative frame, the relative velocity v k is generated. This velocity already has been calculated in (Section 3.2.1, p. 45) and is named kinematic velocity. v k
=
dOGE dt
(5.76)
The velocity v k p should be equal to the velocity VI, if the Earth was supposed flat, since the two frames F, and FE merged. There is also a relation between VI, and V k p (Equation E.92, p. 258)
The second step is to obtain the four relative accelerations. The four relative accelerations will be generated by the derivation of the kinematic velocity v k and flat kinematic velocity v k p relative to the normal Earth fixed frame FE and the vehiclecarried normal Earth frame F,.
[ ]
E
&,EE
dVf d dOGE = - - dt = dt dt
&,E0
=
(5.79)
&,,E
=
(5.80)
Af,OO
=
[
d dOG" -- dt - d t dt
]
(5.78)
(5.81)
Thus the inertial acceleration can be expressed under four alternative representat ions
(5.82) (5.83) (5.84) (5.85) Dynamics of Flight: Equations
120
5
- Simplified equations
Practical situations The analytical development of these complementary accelerations AA will be made below, but from a practicable point of view, what do they mean? Take two examples to illustrate these four errors AA when the flat and fixed Earth hypotheses are done. Imagine a pilot who controls the kinematic velocity VI; relative to the normal Earth fixed frame FE,that is to say relative to a fixed reference on the Earth. Suppose he maintains the vector Vk at a constant in modulus and direction, with regards to the Earth. The relative acceleration A,.,EEwill be equal to zero (Equation 5.78, p. 119) and the pilot thinks he is in a steady state flight. Actually, the states are not constant since the inertial acceleration AI,Gis not equal t o zero, but is equal to the complementary acceleration AAEE which depends on velocity. Thus, this is not exactly a steady state flight and the resultant of the external efforts is not equal t o zero. The aircraft is submitted to a non-zero acceleration although with flat and fixed Earth hypotheses, it should be with zero acceleration. With this first representation, the trajectory of the aircraft is a straight line with a constant speed, and the altitude will increase. To find this situation, the velocity has to be evaluated relative to the Earth, that is to say the Vk:velocity, and its components have to be displayed in fixed Earth axes V f ; so, it depends on the measurement devices. For the second representation (Equation 5.83, p. 119) the difference with the first representation comes from the display of the components of the kinematic velocity VI;which should be given in the local reference as seen by the aircraft Vg. In this situation, the airplane trajectory will follow the Earth curvature at constant speed. For these two situations, neither the trajectories nor the inertial acceleration are the same although the pilot could assume to be within the same circumstances. For the two other representations equation (5.84), p. 119 and equation (5.85)) p. 119, the measured velocity is the flat kinematic velocity V k p or the velocity relative to the vehicle-carried normal Earth frame F,, that is to say the velocity in the local reference as seen by the aircraft. The second example is the process of the measurement of a flight test. If the measurement devices for the position and velocity have a Earth fixed reference frame, the results will drift in accordance with the first representation (Equation 5.82, p. 119). Generally speaking, it is the case with the cinetheodolite and classical telemetry devices. On the other hand, if the flight test uses onboard measurement devices with a local reference, the fourth representation (Equation 5.85, p. 119) will work. It could be the case with onboard radar devices. It should be mentionned that the acceleration measurements generally give the inertial acceleration when the velocity measurement give local velocities. With a numerical simulation within the flat and fixed Earth hypotheses framework, the error made is linked t o the “belief’ the operator has in the parameter he processes. Actually, it depends on the definition of the velocities the operator uses. Analytical development of the complementary acceleration AA The complementary acceleration AA has been developed in section (E.6)) p. 257. The results are given below. For the first representation with the terrestrial relative acceleration A?.EE (5.86)
121
5.1 Flat and fixed Earth equations and expressed in the normal Earth frame Fo
The complementary acceleration AAR depends on the Rotation (subscript R ) of the Earth through the Earth's angular velocity Rt. For the second representation with the pseudo-terrestrial relative acceleration A r . ~ ,
AAE, with A A s
= AA~+AAR = O,EXV~
(5.88) (5.89)
and expressed in the normal Earth frame F,
AA;
=
Rt
+h
VNVZ
+ Vl tanLtG
(5.90)
Relative t o the first representation, a new complementary acceleration A A s appears. The subscript S is given because this complementary acceleration depends on the Spherical Earth hypothesis through the factor &. For the third representation with the pseudo-vehicle relative acceleration A r , , ~
AA,E with
=
AAA
+ A A D + AAs + AAR
A A A + AAD =
daZE
dt
xOG
(5.91) (5.92)
and expressed in the normal Earth frame F,
Relative t o the second representation, two new complementary accelerations A A A and AAD appear. The first one AAA is linked to the Acceleration (subscript A ) components due t o the derivative of VE, VN and Vz. In the second one AAD, the Distance (subscript 0)of the aircraft relative t o the normal Earth frame F,, appears with the components X G , Y G and ZG. For the fourth representation with the vehicle relative acceleration AT,,,
AA,,
=
AAA
+ AAD + AA'D + 2AAs + AAR
(5.95)
Dynamics of Flight: Equations
5 - Simplified equations
122
___.
with
AA'D = - ~ ~ , E X ( ~ ~ ~ E X O G )
(5.96)
and expressed in the normal Earth frame F,
(5.97) Relative t o the third representation, a new complementary acceleration AA'D appears. This term is similar t o the term AAD where the distance of the aircraft relative to the normal Earth frame F,, appears with the components X G , YG and ZG. Numerical evaluation of the errors From the first representation (Equation 5.82, p. 119) to the fourth (Equation 5.85, p. 119), the number of the complementary acceleration AA increases. There are five terms AAR,AAs, AAA,AAD and AA'D. The two last AAD and AA'D depend on the distance of the aircraft from the origin of the Earth reference frame. The values of these complementary accelerations are always negligible. This is due to the factor 1/(& I z ) ~ . For an example, at a distance of 300 km from the origin, the order of magnitude of the errors is 10-4 g (g is the value of gravity, i.e. about 9.8 r n / s 2 ) . The value of the first term AAR is given in figure 5.1 and figure 5.2 for two azimuths, in the first figure towards North and in the second one towards East. In these figures, the three components of AAR expressed in the normal Earth frame Fo are presented, the northern component, the eastern component and the vertical component. The low-right figure gives the modulus of AAR. The value of the acceleration is given per g, in order to have a non dimensional unit. When the airplane flies towards the North in the northern hemisphere, the northern component of the complementary acceleration is equal to zero on equator and on the poles, and reaches a maximum value for a 45" latitude with a value of 0.17 % of g whatever the speed might be. The eastern component is always negative and its modulus increases with the speed and latitude to reach 1 % of g a t A4 = 2 a t the northern pole. This component can be considered as a positive inertial force and it tends to make the aircraft turn towards East. The vertical component is independent of the speed and attains a maximum value of 0.34 % of g on the equator. When the airplane flies towards the East in the northern hemisphere, the northern component of the complementary acceleration is always positive and increases with the speed and latitude to reach 1 % of g at M = 2 at the northern pole. This component tends to make the aircraft turn towards the South. The eastern component is equal to zero. The vertical component increases with the speed and attains a maximum value of 1.35 % of g at the equator for a Mach number equal to 2. Whatever the azimuth is in the northern hemisphere, this complementary acceleration AAR linked t o Earth angular velocity, tends to make the aircraft turn in a clockwise manner. It is the well known Coriolis effect. Its value is in the range of f 1 . 3 5 % of g for M < 2 and in the range of k0.85 % of g for a subsonic airplane with M < 1. The value of the second term AAs is given in figure 5.3 and figure 5.4 for two azimuths, the first azimuth towards North and the second one towards East.
+
5.1 Flat and .fixed Earth equations
123
Figure 5.1: The complementary acceleration AAR links t o the Earth Rotation flight towards North
Figure 5.2: The complementary acceleration AAR links to the Earth Rotation flight towards East When the airplane flies towards the North in the northern hemisphere, the northern component and the eastern component of the complementary acceleration are equal t o zero. Only the vertical component increases as a parabolic function of the Mach number and is independent of the latitude. The value of this component is equal to 0.18 % of g at M = 1 and 0.74 % of g at M = 2. On the other hand, if the airplane flies towards the East, the northern component increases with latitude. For a latitude of 45", its value is 0.1 % of g at M = 0.75 and 0.75 % of g at M = 2. There is a singularity at the poles, coming from the notion of constant azimuth flight which is, Dynamics of Flight: Equations
124
5 - Simpli-fied equations
at the pole, a turning flight for eastern or western flight. If the latitude is less than 60°, the value of the complementary acceleration AAs linked to Earth sphericity, is in the range of f 1 . 5 % of g for M < 2 and in the range of h0.7 % of g for a subsonic airplane with M < 1. It must be remembered that this term is multiplied by two when the fourth representation is used.
Figure 5.3: The complementary acceleration AAs links to the Earth sphericity flight towards North
Figure 5.4: The complementary acceleration AAs links to the Earth sphericity flight towards East The value of the third term AAA is given in figure 5.5 and figure 5.6 for two azimuths, the first azimuth towards the North and the second one towards the East. The origin of the Earth frame is located at a latitude of 45", and the complementary
5.2 R o t a t i n g w i n d velocity field equations
125
acceleration is calculated as a function of the distance and the horizontal acceleration VN, VE from 0 g to 1g . The distance increases from 100 m t o 300 km on a logarithmic scale. This acceleration A A A is independent of the vertical acceleration Vz. When the airplane flies towards the North in the northern hemisphere, the only non-zero component is the vertical component which increases with distance and acceleration. As an example, with 1g of acceleration at 100 km from origin, the value is 1.5 % of g . If the airplane flies towards the East, only the eastern component increases with distance and acceleration. With 1g of acceleration at 32 k m from the frame origin, the value is 0.5 % of g , and at 100 krn from its origin, the value is 1.6 % of g .
Figure 5.5: The complementary acceleration A A A links to the acceleration flight towards North
5.2
Rotating wind velocity field equations
If the hypothesis is made that the wind is modeled around the aircraft by a local wind angular velocity aW, the notion of aerodynamic angular velocity aa can be defined (Equation 3.55, p. 5 7 ) . That situation corresponds t o the wind vortex hypothesis (Hypothesis 10) (Equation 3.50, p. 5 5 ) . Thus, there exists a frame linked to the atmospheric particles in which the field of particles velocity is uniform. For an area in the vicinity of the aircraft, it can be written (Equation 3.23, p. 50)
The atmospheric angular velocity with respect t o the Earth vehicle-carried normal Earth frame F,, gives
aw, projected in the (5.99)
Dynamics of Flight: Equations
126
5 - Simplified equations
Figure 5.6: The complementary acceleration AAA links to the acceleration flight towards East Thus the aircraft has a angular velocity with respect to the atmosphere Oa = Obatm,and as 0 k = 0 b the ~ result obtained is
na
=
fik -
n,
(5.100)
If, for example, this is projected in the vehicle-carried normal Earth frame PO,then (5.101)
The Jacobian matrix of V,, &mV:, then takes a particular form as it becomes a skew-symmetric matrix (Section 3.2.1, p. 45)(Equation 3.38, p. 53) (Equation 3.50, p. 55)
0
-rk
qg
(5.102)
because of the fact that ux;
=
vy;
= wz; = 0
rx;
=
ry;
=r;
qx;
=
qz; = q;
PY;
= Pz; = P ;
(5.103)
It must be noted especially that the substantial derivative of V, takes on the following form dV; dt
= GRmv;v,
=
n2,xvI,
(5.104)
5.2 Rotating wind velocity field equations
127
It is possible to understand that in another frame, for example the body frame F b , the components of & m V w will have the same characteristics as before, that is t o say
(5.105)
5.2.1
Force equations
- second form
The force equations in the first form are not directly affected by this hypothesis of the rotating wind velocity field because the terms of wind do not explicitly appear there. However, the force equations in the second form (Section 5.1.1, p. 105) can be written by reformulating the expression of inertial acceleration (Equation 5.33, p. 111)
These different terms can be expressed because of the calculations performed in section (5.1.1), p. 105 and are thus projected in the aerodynamic frame Fa
with
AA:,,
= 2Vu
sin pa (p:
r;
0
COS cta
COS cta
+
- p: sin cyU r: sin a a ) -
COS
pa (5.110)
Dynamics of Flight: Equations
5 - Simplified equations
128
5.2.2
Moment equations
- second form
Just as for the force equations, the moment equations in the first form are not explicitly affected by the hypothesis of the wind angular velocity field. However, with this hypothesis, it becomes possible to exploit a second form for the moment equations which is equivalent to that of the force equations. Indeed, the aerodynamic angular velocity 52a appears and it can be used as V a was for the force equations. Derivation of the aerodynamic angular velocity
52a
Relative frame: body frame Fb Projection frame: bodg frame Fb The angular momentum H,,G is expressed as a function of the aerodynamic angular velocity 52, and the local wind angular velocity 52,
From this, the derivative of H,,G with respect to the vehicle-carried normal Earth frame F,, can be found by using the body frame Fb as a relative frame. This derivative can be expressed thanks to the relationship (Equation 3.5, p. 44)
The aircraft is assumed to be solid (Hypothesis 15) so (5.113)
The equation (5.112), p. 128 therefore is written (5.114)
By deriving the wind angular velocity 0, with respect to the vehicle-carried normal Earth frame F,, the following expression is obtained (5.115)
In the vehicle-carried normal Earth frame F,, the local wind angular velocity is assumed temporarily and spatially (Section 3.2.1, p. 45) constant (Hypothesis s), since the wind velocity field is frozen and varies linearly as a function of the distance (Hypothesis will therefore be zero 9). The derivative
d52;dt
an; at
+ GRmnt0,vk:= 0
The equation (5.115), p. 128 evolves into the following form
(5.116)
129
5.2 Rotating wind velocity field equations
Thus again, since the cross product of
n,
by itself is zero, the result is (5.117)
By developing the term obtained
nk X I G n k
with
n k
= na+ 0, the following expression is
Projected in the body frame Fb, the different terms are written
The term (Q,,xIGn,)b is expressed, and the terms naxlIGna, f I a x I ~ Q uand C~,XIIGQ, are deduced from ( S t , x l I ~ n , ) ~by the modification of the subscript of p , Q , r*
(5.121)
In this second form of the moment equation, with respect to the preceding form
(Equation 5.56, p. 115), only the expression of the derivative of the angular momentum
changes d % t G . This explicitly makes the components of the aerodynamic angular velocity 52: appear in the body frame Fb (p:, q:, T : ) . These components of a:, in general, directly influences the aerodynamic efforts (Equation 3.54, p. 56) (Section 4.3.3, p. 89). In the case where the aerodynamic coefficients would depend on (pg, qz, r,O) instead of (p:, q:, T : ) , it is sufficient t o derive Sza not in the body frame Fb but in the aerodynamic frame Fa and to project these equations in Fa (5.122)
5.2.3
Kinematic equations
Only the kinematic angular equations (Equation 5.73, p. 118) can use another form by introducing the wind angular velocity nw.These equations are written with the Dynamics of Flight: Equations
5 - Simplified equations
130 vectorial relationship (Equation 5.100, p. 126) frame Fb
thus
($) ( =
1 sin$tan@ cos$tan@ 0 cos$ -sin@ 9 cos e
cos e
nk
=
Qa
+ 52,
) [( $ )
projected in the body
+Tbo($,@y'$')
( )] (5.123)
5.3 Uniform wind velocity field equations It has been demonstrated in (Section 3.2.1, p. 45) that it can be assumed that the field of atmospheric velocity is frozen (Hypothesis 8), that is to say, independant of time, in the framework of Flight Dynamics of the aircraft. The field of atmospheric velocity in the vicinity of the aircraft, therefore, can be modeled by (Equation 3.51, p. 56)
V , , N = V w , ~ + & ~ V MN o,
(5.124)
The symbols M and N represent positions of two particles in space. So that the velocity of an atmospheric particle M is known, it is possible to calculate the velocity of the particle N by the relationship above. Moreover, if it can be assumed that the wind velocity field is uniform (Hypothesis l l ) , the velocities of all the atmospheric particles are identical.
V w , ~ = V w ,=~V ,
whatever M, N and t are
(5.125)
Thus in this case
GRmv;
= 0
(5.126)
This last result, corresponding t o the hypothesis of a constant wind velocity whatever the aircraft position is and whatever the time is, will be exploited in order to evaluate its influence on the general equations. The equations which result from this calculation are the simplest possible. Nethertheless, they are of a huge practical consequence and are used for numerous studies.
5.3.1
Force equations
When the accelerations are expressed by the derivatives of the kinematic velocity v k , the constant wind hypotheses do not explicitly appear. They indirectly appear during the transformation from v k to Va for calculating the external forces. By using the second form (Section 5.1.1, p. 105), the constant wind hypothesis clearly appears (Equation 5.126), and is translated by the following inertial acceleration expression (Equation 5.32, p. 110)
5.3 Uniform wind velocity field equations
131
has already been calculated in The derivative of the aerodynamic velocity equation (5.35)) p. 111 as nkxVu in equation (5.42)) p. 112 and equation (5.46), p. 112. By using the relationship (Equation 5.52, p. 113), the force equations are written
i.h
m
( mg
h a Va COS pa
(
V, (p: sin a, - r: COS a,) Va[sin pa(p: COS aa + r: sin a, ) - 9:
b a V, -
+
- sin y , cosy, sinp, cos Y, cos p,
COS
I&,
) + ( -g ) + ( 3) ipsv:
1
Tab
-
(5.128)
-CL
The wind velocity field being uniform (Hypothesis 11)) there is no atmosphere angular velocity with respect to the Earth fixed reference frame FE and f l w = n a t r= n0~. Thus (Equation 5.100, p. 126), equality between the aerodynamic and kinematic angular velocities is established Thus
( 2t ) ($)=(;) r
=
(5.130)
r
5.3.2
Moment equations
Just as for the force equations, the second form of moment equations (Section 5.2.2, p. 128) explicitly gives the result of the constant wind hypothesis. The relationship (Equation 5.117, p. 129) gives the derivative of the kinematic moment, with f l w = = 0 and a, = (Equation 5.130, p. 131)
natrn~
nk
(5.131) Thus with the moments of external forces calculated in equation (4.130))p. 97, the moment equations are found projected in the body frame Fb
+
+
Ap - Fq - E?: rq(C - B ) - Epq Frp + D(r2 - q 2 ) -Fp + Bq - D?:+ r p ( A - C ) E(p2 - r 2 ) - Frq + Dpq -Ep-Dq+C?:+pq(B-A)+Erq+F(q2-p2)-Dpr
+
(5.132) The results of the constant wind hypotheses are recalled (Equation 5.130, p. 131)) and are useful for the exploitation of these equations P
b
= P,
b
=
9,
r =
b
9
(5.133)
r,
Dynamics of Flight: Equations
5 - Simplified equations
132
This demonstrates that with a uniform wind velocity field, the two forms of moment equations are equivalent because the relationships of equation (5.56), p. 115 have been found again.
5.3.3
Kinematic equations
The kinematic equations of position (Equation 5.69, p. 117) linked to the kinematic velocity Vk are not modified with respect to those given by the flat and fixed Earth hypothesis, except that the components of wind in the Earth fixed reference frame U:, U:, w : are constant (5.134)
(5.135)
The kinematic angular equations (Equation 5.72, p. 118) and (Equation 5.73, p. 118) linked to the kinematic angular velocity of the aircraft ak have not been modified, but it is possible to exploit the preceding results (Equation 5.130, p. 131)
(n) ( I ) = ( $ ) b
=
() ( =
5.4
1 0 0 cos+ 0 -sin+
- sin8 cos@sin+ cos8cos#
)(
(5.136)
(5.137)
Decoupled equations
The interest in decoupling a system of equations is obviously to simplify the study by reducing the number of relationships. This is particularly useful for the analytical study of the equilibrium and the dynamics of the aircraft. The decoupling is much less interesting when the study follows a numerical procedure. The idea of using the decoupling of an equation system, naturally comes from the study of certain phases of the aircraft’s movements. To analyse the performance of takeoff, it does not seem senseless to modelize only the dynamic equation carried by the horizontal axis x,. In fact, practically speaking, it can be acertained that when an aircraft goes down a runway in its acceleration phase, its trajectory remains rectilinear (it follows the axis of the runway while staying on that runway). Thus it seems useless to set up lateral force equations on the yo axis, or lift equations on (2, = zb) axis, as well as the moment equations as the pitching velocity is zero.
5.4 Decoupled equations
133
The same goes for the rectilinear climb without turbulence. Experiments have shown that the aircraft is piloted only by the forward or backward movement of the stick (pitching moment ) and the “throttles” (thrust), and the aircraft remains in a vertical plan. It seems therefore natural to work on a two-dimensional problem in the vertical plan and only use the two force equations of lift and drag and the equation of the pitching moment. In the framework of the flat and fixed Earth hypotheses, it does not come to mind t o use either the kinematic equations of geographical position ( i ~GG) , or the heading t o study the performances of the aircraft. These performances naturally seem t o be independent of the geographical positions. It does not matter whether the aircraft is at New York or Paris to be able to evaluate the climb rate, just as it is unimportant t o be heading North or South. In this paragraph, this notion of decoupling will be formalized, identifying the different types of decoupling and establishing practical examples.
(4)
Decoupling of a system of equations This paragraph is t o show that the phases of movement of the aircraft, under certain conditions, can be analysed with a reduced number of equations without impoverishing the quality of the analytical model used. The system of equations of Flight Dynamics can be put in the state form, that is t o say
X
X U W
= F(X,U,W)
(5.138)
the state vector the control vector the wind vector or vector of atmospheric perturbations
In the vector F,each component is a function of the components of the vectors X, U and W. This system can be partitioned. Thus
X
=
( ci )
with
Three types of decoupling, summarized in the sketch above, can be distinquished.
The first type of decoupling is encountered when F1 does not depend on either
x2, or u2, w2
(5.141) The first system X1 is not influenced by the second system X2, and can be studied independently. The second system X2 is “controlled” by the first system via XI and Dynamics of Flight: Equations
134
5
0
- Simplified equations
X
X=
I
navigation
Figure 5.7: Decoupling sketch eventually via U1 and W1. This second system cannot be studied independently from the first, but its dynamics are independent. This type of decoupling is encountered with kinematic equations of navigation.
The second type of decovpling is encountered when the parameters of the second system X2, U2, W2 on which the first system depends, are constant. For example, if F1 depends on X2, but with X2 constant. However, the first system can be studied independently from the second. In general, the systems are decoupled on the controls and the wind. The first system F1 does not depend on, or only slightly depends on, U2 and W2. The condition X2 constant, can be encountered when the system X2 is in equilibrium (Section 7.1.1, p. 180), stable and is not perturbed. It is represented as the following form
The perturbations of the system X2 can come from the system X I , the command U, or the wind states We. Either these perturbations are zero, for example U, = We = 0, F2 = F2(X2e,U, = 0, We = 0), or the conditions of equilibrium (that is to say the value of Xze) cancels the transmission of these perturbations. Thus for X2 = X2e, F2 takes the form F2 = F2(X2,). This type of decoupling is encountered when
the lateral equations are decoupled to obtain the decoupled longitudinal equations.
The third type of decovpling is encountered when X2 is a function of XI. This case can be met on several occasions: 0
The system X2 is stable and perturbed but with dynamics sufficiently high with respect to the first system to assume that the second system X2 will find its equilibrium rapidly. The equilibrium equations can be solved (U2 = 0)
and Xae can be expressed as a function of X I , U1, W1. In the first system, X2 can be eliminated by this dependence and the first system X1 is found to be independent of X2 and therefore decoupled. A variant can be brought in by piloting the system X2. This allows a value to be given to n parameters ( n corresponds to the number of controls) of X2. This piloting can be done either by man or automatically. This particular case does not correspond to the study
5.4 Decouvled eauations
135
of the “natural” aircraft, and the validity of the decoupling will be linked t o a particular form of utilization of the aircraft. 0
0
On the contrary, the parameters are submitted to slow dynamics and, with a perturbation at a high frequency or zero average, these parameters could be considered as constant. Finally, t o simplify the matter, the working hypothesis is used where all the states of X2 are constant. In that case, it is necessary t o return to the framework of the second type of decoupling but this time as a working hypothesis and not as a characteristic of the system studied. It is routine to make this last hypothesis in order t o decouple the longitudinal equations and obtain the lateral decoupled equations.
This third type of decoupling is more difficult t o justify than the preceding ones. The examination of respective dynamics is more fastidious. As a function of the aircraft, in the case of flight and the type of perturbation, it is a matter of connecting the dynamics of the aircraft to each state of X2. An experimental verification could later justify the hypotheses which will be done. For example, for the evaluation of the equilibrium or of the dynamics of the system, the numerical calculation on the system not decoupled will allow the validation of the hypotheses of decoupling. The hypotheses of decoupling can depend on the studied problem. In the case of a study of lateral equilibrium, the preceding hypotheses corresponding t o the longitudinal equilibrium are very well adapted. In the case of a study on the lateral dynamics, these hypotheses could be adapted if the longitudinal dynamics is high. In the opposite case, however, other hypotheses might be better adapted. The simplest hypothesis: the constant longitudinal parameters, could possibly work. It is a matter of adapting the hypotheses t o each case study.
5.4.1
The decoupling of navigational equations
Navigational equations are the kinematic equations that define the position and the heading of the aircraft (26,96,$). By regrouping the force equations XF, the moment XM and kinematics X c together, the following system is recovered. If it is assumed that the wind W does not depend on the heading $ XF
= FF(XF,XM,Xcp,U, W
XM = FM(XF,XM,Xcp,U, W
O )
forces moments
O )
(5.144)
XcP = Fcp(Xcp,XF, X M ,WO) kinematics XcC = Fcc(Xcc,XcP,XF, X M ,W
or by putting
x
=
(:)
O )
navigation
(5.145)
XCP Dynamics of Flight: Equations
136
5 - Simplified equations
the system is written in the following form X
= F(X,U,Wo)
X c c = Fcc(Xcc,X,W")
(5.146)
with
A decoupling of the first type thus appears. What this means is that the first system X does not depend on the second X c c . It can therefore be studied independently. On the other hand, to exploit the navigational equations X c c , it is necessary to work conjointly with the two systems. If the wind depends on the heading @, X must be increased in the heading state @ and XcCis obviously reduced from the same state @. These navigational equations are decoupled from the system of force and moment equations for two reasons: 0
0
They correspond to a state of position which cannot intervene in the expression of inertial acceleration, where only the velocities intervene. The navigational positions do not have any influence on the external forces because of the horizontal plane isotropism of the atmosphere. Only the altitude h has an influence on the external forces, by the intervention of the air density p.
REMARK 5.2 If the kinematic equations of the inclination angle and the bank angle (0, 4) are not decoupled, it is because they define the vertical thus the direction of
the weight, one of the external forces.
REMARK 5 . 3 If the flat and fixed Earth hypothesis is not made, the navigational equations are no longer decoupled since the geographical position has an influence on the complementary acceleration terms and on the gravitational force through the medium of the heading and the latitude.
The notion of wind independant of the heading merits being stated more precisely. This hypothesis appears when the components of wind velocity V, and the wind gradient &mV, need to be expressed in the body frame Fb from the known data of wind in the vehicle-carried normal Earth frame F,. The passage of the body frame Fb to the vehicle-carried normal Earth frame F, is a function of three angles 4, 8, @. The bank angle 4 and the longitudinal inclination angle 8 make up part of the states of the system of force equations because of the presence of weight. The heading @ only appears for the representation of the wind as an external perturbation. The wind is defined in the vehicle-carried normal Earth frame F, and explicitly appears in the equations by its components in the body frame Fb. These last components are thus a function of the heading $. For them, not to be dependent of the heading $, which would correspond to the framework of the hypothesis (Hypothesis 29) "the wind does not depend on the heading", one of the following configurations has to be stated:
137
5.4 Decoupled equations 0
0
No wind. In this case the problem of the influence of the heading is obviously avoided. A wind velocity V, and a wind gradient CEWmV, with uniquely vertical components, respectively 20; and This is brought t o light in the formulation of wb= TbOw".
YE.
0
0
A trajectory of the aircraft at a constant heading. A theoretical definition of the wind in an intermediate frame deducted from the vehicle-carried normal Earth frame F, by the heading $.
5.4.2
Decoupled longitudinal equations
At the beginning of this chapter, an experiment approach of decoupling has been mentioned which suggests the existence of a flight in which the trajectory is contained in the plan called longitudinal ( x , z). If this observation is used as a guide, conditions can be looked for that are sufficient for this type of flight. The flight in this plan ( x , z) will thus be regulated by two force equations, one projected on x , and the other on z, a moment equation on y , two kinematic equations of position on x and z and, finally, an angular kinematic equation on y . By using the force equations in a second form (Equation 5.52, p. 113), the moment equation (5.56), p. 115 and the kinematic equation (5.73), p. 118, the longitudinal states associated with the equations will be
ZG
force
Va(X)
aa(4
moment kinematic position angular kinematics
q(Y> ZG (x)
h(z)
8(y)
It has been previously shown (Section 5.4.1, p. 135) that the kinematic equation on is decoupled from the others. Thus the states that are left are
Naturally the complementary system of this longitudinal system will constitute the complementary states of the complete system. This will be true after having taken out the state of the decoupled navigational equations (ZG,YG, $). (5.148) This last system will be called the lateral system. Now the sufficient conditions for decoupling can be looked for, by searching for the second type of decoupling (Section 5.4, p. 132), that is t o say X2 = X P =~constant. This will be obtained with an equilibrium (X2 = 0) from the second system Xz undisturbed by XI,the controls U or the wind W . Clearly speaking, the question Dynamics of Flight: Equations
is to look for the conditions under which, each term of the equations of the second lateral system
become zero, independently of the values of the states of the first longitudinal system
and of the controls of the first system
and of the wind =
W O t
(U;,
U;,
w;,(6wmv;)
(5.150)
The conditions found will be those of an equilibrium of a lateral system, undisturbed by the longitudinal system. The analysis of the conditions cancelling the different terms of the lateral equations of force, moment and kinematic is found in section ( F . l ) , p. 265. These conditions are described below. The values of lateral states are equal to zero. These conditions come mainly from the aerodynamic lateral force equal to zero.
The values of lateral controls are equal to zero, coming from the aerodynamic lateral force and moments
6, = 6,
(5.152)
= 0
In consequence, due to the kinematic equation, a constant heading is obtained $
= constant
(5.153)
For the wind, the following conditions are obtained in the body frame Fb pyb, TY,
b
vyL = O
= pzb, = 0
=0
-
-
b rxu
or
U& =
o
(5.154)
which is translated in the vehicle-carried normal Earth frame Fo by either for uy: = 0 ry:
= -rxL
uxo, +vy: py:cos$ +qx:sin?I, pzzcos$+qzO,sin$
= uxC, cos$ sin $ = uxc, = 0 =
0
(5.155)
5.4 Decoupled equations
or for
U :
=
139
o - u ~ s i n $ + u ~ c o s @= 0
(5.156)
For the first case (Equation 5.155), UX; can be considered as a constant. It means that the wind gradient is oriented towards the aircraft heading. For the second case (Equation 5.156), the condition means the horizontal wind has only a component oriented towards the aircraft heading. Mass and geometric symmetry of the aircraft: two products of inertia equal zero
D=F
= 0
(5.157)
The lateral aerodynamic coefficients C Y , C1, Cn independent of the longitudinal parameters a a ,q, Va,h, ,S S,. The moments and lateral propulsive forces equal to zero and independent of the longitudinal parameters (5.158)
In practice, this condition is verified if the propulsion is symmetrical. This is not the case with an engine failure on a multi-engine, or if there is an influence of the downwash of the propeller or the gyroscopic torque of the engine’s rotating parts.
These conditions lead to a flight with horizontal wings and n o sideslip, an the vertical plane which is called pure longitudinal flight. The heading of the aircraft is constant and the vertical plan coincides with the aircraft’s symmetrical plan.
A physical approach to this decoupling can be justified later by the following arguments. The longitudinal flight is a flight in which the trajectory is situated in a plane. So that this trajectory stays in this plane, it is necessary not to have any external forces perpendicular to this plane and to have the moments of the external forces perpendicular t o this plane. By choosing one of these characteristic planes, here the plane of the aircraft symmetry, the weight in this plane imposes a zero bank angle (4 = 0) since the weight vector stays vertical by definition. The zero lateral aerodynamic force imposes zero sideslip (p = 0). The symmetrical plane is therefore mixed with the vertical plane. The vertical plane could have been chosen in the beginning to arrive at the same result but this time by the lift instead of the weight. As the moments of external forces should not be in this plane, the lateral controls are zero (6l = 6, = 0). To avoid disturbing this situation, the roll and yaw velocities ( p , r ) must be zero. To all intents and purposes, they represent the derivatives of the bank and sideslip angles The wind, in order not to disturb the longitudinal flight, must not “evolve laterally”. The wind angular velocity flwmust be perpendicular to the symmetrical plane.
(4)8).
The decoupled longitudinal equations of flight
These equations come from the preceding hypotheses (Equation 5.151, p. 138) to (Equation 5.158, p. 139). It ensues from these hypotheses, some preliminary results
achieve below.
Dynamics of Flight: Equations
140
5
-
- Simplified equations ____
Angular relations The inclination angle of the aircraft 8 is expressed by a simple form due to 0, = 4 = 0 (Equation 2.85, p. 40)) the inclination angle being the sum of the angle of attack a, and the aerodynamic climb y,
e
= a,+?,
(5.159)
Moreover, if 0, = 0 then 4 = 0 leads to (Equation 2.86, p. 40) an aerodynamic bank angle pa equal to zero Pa
(5.160)
= 0
and with p a = pa = 0 (Equation 2.87, p. 40) the aircraft azimuth aerodynamic azimuth xa
+
It can be equally noted that with the right wing axis Yb.
+ is equal to the
(5.161)
= Xa
0,= 4 = 0, the aerodynamic axis y,
coincides with
Expression of wind The wind V, and @mV$ is expressed by its components in the vehicle-carried normal Earth frame aircraft azimuth oriented F,, deduced from the vehicle-carried normal Earth frame F,, by a rotation of the heading $ of the aircraft. The frame F,, represents the vehicle-carried normal Earth frame Fo whose axis x, is aligned with the heading of the aircraft. The plane (x,, z,) coincides with the vertical plane containing the trajectory of the aircraft. Thus the transformation matrices are obtained (Section A.3.1, p. 198)
T,,
cos+
= Tbo(+,O= 4 = 0 ) =
sin$
0
0
1
(5.162)
These two matrices are obtained with the hypotheses of the longitudinal flight, d, = 0 for the first one and p, = 0 for the second one. cos0 0 -sin8
T~,= Tbo(e,+= 4 = 0) =
L a
= T3a(Xa =
+=
O,%,Pa = 0) =
cosy,
0 siny,
-siny,
0 cosy,
(5.163)
(5.164)
The heading can now be considered constant, as the wind is defined in the vehiclecarried normal Earth frame aircraft azimuth oriented F,. In this frame F,, the components of the wind velocity V, are expressed by
v;
=
(z)
= T , , V ~=
WL
(
U:
--U;
cos+ sin
+ vz sin$
++ w;
cos 1c)
(5.165)
5.4 Decoupled equations In the body frame
~
~
_
141
Fb
(5.166) In the aerodynamic frame F, U: COS Y, U:
sin
- w& sin
+ w;
cos
(5.167)
If the wind velocity Vw can be any value whatsoever inspite of the decoupling operation of longitudinal equations, this is not the case for the wind gradient ChmV; which takes on a particular form (Equation F.29, p. 269). Projected in the vehiclecarried normal Earth frame aircraft azimuth oriented F,
(amv:),
=
(
uxc, 0 -qxc,
0 qz& 0 0 0 wzc,
)
(5.168)
The components of this matrix are the known data of the wind, the gradients in the vertical plan of the aircraft's trajectory. Projected in the aerodynamic frame Fa with (Equation 5.164, p. 140), the gradient takes on the form
(CGwmv:)"
=
T,,(CGwmV~)CT,, uxo, 0 qz;
(5.169) (5.170)
The expression of acceleration
The different terms of acceleration of the second form (Equation 5.33, p. 111)
will be expressed in the following simplified forms thanks t o the decoupling hypotheses. The first term (Equation 5.35, p. 111) with pa = 0
(5.173)
Dynamics of Flight: Equations
The second term equation (5.46)) p. 112 with equation (5.151))p. 138 (5.174) The third term thanks t o the expressions of the wind previously defined (Equation 5.170, p. 141)
( & ~ v ~ ) " =v ~VQ
(
uXC,
2 + W ~ C sin , ya + COS ya sin yQ(qx&- q z L ) 0 sin2 ya - qxC, cos2 ya + cosyo.siny,(~xC,- W ~ L )
cos2 ya
-qzL
1
(5.175)
The fourth term
DVZ = (cGwmVzV,w)" = T,C(&ADV~V,)"
=
( ) du; dug dwt
(5.176)
with (GmV;)' already calculated (Equation 5.168, p. 141)) and TQccomes from equation (5.164), p. 140 dut
=
dv:
= 0
dwz
=
+ q z k t & ) + siny,(qxC,uC, - wzC,wL)
COS~,(UXC,UC,
COS yQ(wzkw;
- qxkuk) + sin yQ(uxLuk+ qzLwL)
(5.177)
Expression of the external forces Gravitational force equation (5.50)) p. 113 and equation (5.160), p. 140 mg"
= mg(
- sin y,
0 cos YQ
)
(5.178)
Aerodynamic force equation (5.52)) p. 113 and equation (5.151)) p. 138
+psv,2c;
=
$pSV,2
( ) -CD
(5.179)
OL
Propulsion force equation (5.52)) p. 113 and equation (5.151)) p. 138
F" = TQb(aQ)PQ = 0) Fb =
+ sin aQ~ , b 0 - sin aQF,b + cos aQF,b cos a, F i
(5.180)
With equation (4.104)) p. 92 and a symmetric propulsion, the following form is obtained
F" = F
cos P m Cos(aQ- a,) 0 - COS& sin(a, - a,,)
) ) ( =
Ft"
(5.181)
5.4 Decoupked __________ equations
___.______
143
______
Moment equation The moment equation of pitching (Equation 5.56, p. 115) with the decoupling hypotheses (Equation 5.151, p. 138) and (Equation 5.157, p. 139) and
p=r=O
F=D=O
(5.182)
thus
Bq
=
3pSlV; C m + M k v
(5.183)
with the decoupling hypotheses equation (5.158))p. 139 and equation (5.164))p. 140
Mky = MgY
(5.184)
Kinematic equations In the vehicle-carried normal Earth frame aircraft azimuth oriented F,, the kinematic equations linked to the kinematic velocity Vk projected in F, (Equation 5.69, p. 117) are written
V i = T,aV,"+V&
(5.185)
thus = V,cosy, +uc, y& = 0 i& = - A = -V,siny,+wL
X&
(5.186)
The kinematic equations linked t o the kinematic angular velocity
f2k
are written
9 , = 0 o
=
q
q = o
(5.187)
Decoupled longitudinal flight equations Regrouped here are the previously obtained results. This longitudinal flight takes place in a vertical plane coinciding with the aircraft's symmetrical plane. It is controlled by two force equations (propulsion and sustentation), a moment equation of pitching and two kinematic equations (vertical velocity and pitching velocity). There exists a kinematic navigation equation but decoupled from the preceding equations. The propulsion equation comes from equation (5.172), p. 141 and equation (5.178), p. 142 t o equation (5.181))p. 142, expressed in the aerodynamic frame Fa
Dynamics of Flight: Equations
5 - Simplified equations
144
_____
The sustentation equation comes from equation (5.172), p. 141 and equation (5.178), p. 142 to equation (5.181), p. 142
Thanks to equation (5.197) the term &a - q could be replaced by -?,. Pitch moment equation (5.183), p. 143
Bq = ~ p S I V ~ C r n + M ~ v
(5.190)
It must be recalled that the expression of the aerodynamic pitch velocity qa on which depend the aerodynamic coefficients are written in the body frame Fb (Equation 5.59, p. 115) q,b
= 9 - 9% b
(5.191)
From equation (F.34), p. 270, it can be written
As the second form of equation is not available on the moment equation, the wind effect appears indirectly in external efforts. The pitch moment equation (5.190), p. 144 is an example. The wind, known in frame F,, comes from qx: and indirectly influences the aerodynamic pitch moment coefficient through q i . Kinematic equations of vertical and pitch velocity (Equation 5.186, p. 143) and (Equation 5.187, p. 143) are written (5.193) (5.194) Kinematic equations of distance decoupled (Equation 5.186, p. 143) are written X&
= V ~ C O S+ U~:,
(5.195)
The angular relationship must be denoted (Equation 5.159, p. 140) 8
= ff,+y,
(5.196)
thanks to which, it is possible to eliminate the variable 9 by replacing the kinematic equation of pitch velocity in equation (5.194), by
This practice is common as it simplifies the exploitation of the equations since the inclination angle 9 does not explicitly intervene in the expression of the external efforts, contrary t o the angle of attack a.
5.4 Decoupled equations
145
Decoupled longitudinal flight equations and uniform wind velocity field In section (5.3), p. 130, it has been shown that a uniform wind velocity field can be translated by the relationhip equation (5.126), p. 130
thus hereafter (Equation 5.168, p. 141) becoming
The previous equations (Equation 5.188, p. 143) to (Equation 5.194) take on the form
with (Equation 5.57, p. 115)
as well as (Equation 5.59, p. 115) Qa b
= (2
(5.199)
The longitudinal equations (Equation 5.199, p. 145) constitute the simplest obtainable form and thus represent the final result of the preceding developments. The last simplification will be done t o show equilibrium (Equation 7.24, p. 189).
5.4.3
Decoupled lateral equations
In section (5.4.2), p. 137 the system of aircraft equations has been divided into a longitudinal system
and a complementary lateral system
It has been shown that the longitudinal system can be decoupled from the lateral system under the conditions of the second type (Section 5.4, p. 132) by the equilibrium of the lateral system. Thus, X 2 = 0 whatever XI,is obtained. This equilibrium is special, since Xae = 0, with as a consequence, each of the lateral equation states equal to zero. Dynamics of Flight: Equations
5 - Simplified equations
146
The possibility of the decoupling of this type for the lateral system is now going to be examined. The question is whether XI = 0 whatever Xz,can be obtained. By using the same approach as in the longitudinal system, a condition sufficient for decoupling is looked for by trying to cancel each term of effort, or complementary acceleration, of the longitudinal equations with whatever values the lateral states may be. The sustentation equation ( 5 . 5 2 ) , p. 113 is taken to obtain ci, = 0. First, what must be done is to cancel the component of weight (Equation 5.51, p. 113)
+ cos 8 cos 4 cos a,)
m g (sin 8 sin a,
This term will be zero independently of 7T
8= 2
4, if
and a0 = O
or if IT
a, = -
2
and 8 = 0
Besides the fact that this represents the case of unusual flight, these conditions do not permit the cancellation of the component of weight on the propulsion equation (5.51), p. 113
+
m g( - sin 8 cos a, cos ,& cos 8 sin # sin ,k10 + cos 8 cos # cos p, sin a,) With 8 = and a, = 0, the component of weight -mgcosp, a, = :, 8 = 0, the component of weight mg(sin 4 sin P0
+ cos 4 cos PO)
is obtained. With
= mg cos(4 - PO)
is obtained. Thus, it is impossible to find the conditions on the longitudinal parameters (VO, a,, q, h, 8) which will lead to the cancellation of the weight components, on both the sustentation and the propulsion equations no matter what the lateral parameter values are, especially the sideslip angle Pa and the bank angle 4. From this statement, it appears that the type of decoupling obtained for the longitudinal flight can not be reconducted on the lateral flight. It is impossible to cancel each component of external forces of the longitudinal equations whatever the values of the lateral states are. A less demanding approach consists in looking for a situation of longitudinal equilibrium that is independent of the lateral states in order to obtain longitudinal states constant. If it is admitted that the aerodynamic forces of lift and drag are independent of the lateral states (and especially the sideslip angle P), the question is to show that the components of weight, and the complementary accelerations, are equally independent of the lateral states. For the sustentation equation, the component of weight is independent of the lateral states in the first order if the bank angle # is around zero. Thus cos4 x 1 can be admitted. By assuming that the wind is zero (AA, = 0 ) and the sideslip angle Pa is around zero, the complementary acceleration (AAk) is equal to /3, cos a,p - q
+ Pa sin a O r
5.4 Decoupled equations
147
In addition if is around zero at the first order approximation, then the complementary acceleration is equal t o
This term will be independent of the lateral states P a and p , if the sideslip angle and the roll angular velocity p are small enough so that the product is of second order with respect to the pitch velocity q. This special kind of situation will be found at the time of linearization (Section 6, p. 157). For the propulsion equation, with the same hypotheses as for the sustentation equation, at the first order approximation, the component of weight is equal to Pa
mg(- sin 8 + cos 8 sin a,) Thus this expression is independent of the lateral states and 4. The complementary acceleration (AAk) is zero without any special hypotheses. This analysis allows the demonstrated result t o appear farther along than at the time of the linearization operation (Section 6.4, p. 170). Around this case of rectilinear horizontal and symmetrical flight, without wind ( P a = 4 = 0)) the lateral equations are decoupled from the longitudinal equations if the sideslip angle ( P a ) and the bank angle (4) remain small. If not, the decoupling cannot be of the second type (Section 5.4, p. 132) because of the fact that it is always necessary to compensate a component of weight as a function of the sideslip angle (pa) and the bank angle (4)) with more or less lift and drag t o obtain an equilibrium. In the most general cases, only the third type of decoupling is possible as a solution (Section 5.4, p. 132). The question here is t o define the longitudinal states with an appropriate model for the case treated. The longitudinal state is worth remembering: Va, a a , q, h, 6. If an equilibrium is studied, the longitudinal equations to the equilibrium will furnish this model. In a general manner of speaking, the influence of the altitude on the external efforts is ignored. This is a useful simplification that is hardly restrictive. p
= constant
The kinematic equation of the inclination angle (Equation 5.73, p. 118) is balanced. Thus 9 = 0 leads t o the relationship
q =
T
tan4
(5.200)
The kinematic equation of altitude (Equation 5.69, p. 117) shows that sin?,
=
-w; +Va
h Va
(5.201)
The other states V a , a a ) 8 can be expressed as a function of the lateral states, making it necessary to solve the equilibriated longitudinal equations. If the studied problem is a case of non-steady flight, the question is t o find the best adapted state for each longitudinal state as a function of its own dynamics and perturbations provoked by the lateral movement. Thus, for the simplest solution, Dynamics of Flight: Equations
5 - Simplified equations
148
either the state is assumed constant or the state is the solution of an equilibriated longitudinal equation. It can be admitted that the longitudinal states are piloted (thus constant) if that corresponds to a certain reality. The most “comfortable’) analytical solution consists in using the invariables of all the longitudinal states as a working hypothesis. This solution gives good results in most situations. Decoupled lateral equations The lateral force equation (Equation 5.52, p. 113) is written
md,
Va
-m V, +m V, +m
V, =
+
+mV,(-psina,
+ r cosa,)
sin a , COS a, sin pa cos p, (92,b - qx,)b (- (p: cos2P, + py& sin2P,) sin a ,
+ (ry:
sin2 pa + r& cos2
cos a,)
p, sin Pa (-uxL cos2 a, + VYL- W Z ; sin2 a,) + m d v i rng(sin 8 cos a , sin Pa + cos 8 sin 4 cos p, - cos 8 sin a , cos 4 sin PO) COS
4pSV:Cy
- F,bsinp, COS a,
+ Fy”COS&
- F,bsin a, sinp,
(5.202)
The component of the force of propulsion in (Equation 5.202)
-F,b sin ,8,
+
COS a a
Fy”COSpa - F ’ sin a, sin p,
can also be expressed in the form (Equation 4.104, p. 92)
The calculation (Section D.6, p. 238) of the components of (G~rnv~)~ can obtain the expressions of pz:, pyb,, r y b , , r&, U & , vy,,b WZ:. The calculation of DVZ = (CGwmV;V,)” (Equation 5.49, p . 113) can obtain the expression of dvk dvz
=
+ +
+
cos 8 cos II, cos pa(sin 8 sin 4 cos II,- sin II,cos 4) - sin sin fla (cos II,sin 8 cos 4 sin #sin II,))duG (- sin PaCOS sin II,cos 6 cos Pa(sin 8 sin 4 sin II, cos II,cos 4) - sin a a sin @,(sin 8 cos 4 sin II,- sin 4 cos II,))dv:
(- sin Pacos
+
+
(sin /?a COS aa sin 8
+ COS
/?a COS
+
8 sin 4 - sin a a sin @a
COS 8 COS 4)dwL
(5.203)
with
(2;) dw;
=
(
U
X
~
rx0,uz -qx;u;
-U ry0,vg ~
+ qZ;w;
+ p.;v;
+ wz;w;
+ vy0,vG - p.0,~;
(5.204)
The moment equations (Equation 5.56, p. 115) assuming that the aircraft is symmetric (Hypothesis 4) D = F = 0 can be written as follows. The roll moment equation is
Ap - E?:+ rq(C - B)- Epq
=
$pSt!V:CZ
+ MF,b
(5.205)
5.4 Decoupled equations
149
The yaw moment equation is
C?:- Efi + p q ( B - A ) + Erq = $pStV:Cn
+ MF,b
(5.206)
The kinematic equation of the bank angle and heading (Equation 5.73, p. 118) is
6
= p+tane(qsin4+rcos4) =p+IIsinO
(5.207) (5.208)
It must be remembered that the aerodynamic moment coefficients depend on the aerodynamic angular velocity p:, q t , r y : and r x : , (Equation 3.54, p. 56) (Section 4.3.3, P. 89)
(5.209) The atmospheric perturbations mb,, q x b , , etc, are introduced in the moment equations in this way (Section D.6, p. 238).
Decoupled lateral equations with uniform wind velocity
In (Section 5.3, p. 130), it has been shown that a uniform wind velocity field can be translated by the relationship (Equation 5.126, p. 130) thus all the terms p&, py%,
ryb,,
rxL, U X : , vy,,b
WZ;
and dv; are equal to zero. Then
the lateral force equation (5.202), p. 148 is written
mb,Va
+
mV,(-psina,
+
$pSV:Cy
+ r coscy,) + cos 8 sin 4 cos Oa - cos 8 sin eta cos 4 sin Oa) - F,bsinOacosa, + F ~ ~ c o -s F:sina,sin& ~, (5.210)
= mg (sin 8 cos cya sin /?a
The moment equations and the kinematic equation do not change but the aerodynamic angular velocity is equal to the kinematic angular velocity, then
(5.211)
5.4.4
The consequence of lateral and longitudinal decoupling
It has been shown the decoupled longitudinal equations is a second type of decoupling (Section 5.4, p. 132). Then, the pure longitudinal flight (Section 5.4.2, p. 137) gives no errors within the framework of decoupling hypotheses '. In practice, these hypotheses ~~~~~~~
2The errors are relative to the non-decoupled equations.
Dynamics of Flight: Equations
5 - Simplified equations
150
could be well verified and in that case, the longitudinal flight is perfectly modeled with the decoupled equations. On the other hand, the lateral flight is associated with the third type of decoupling, and the more the flight is lateral, the more the errors are important. The decoupling is only rigorous around the level horizontal flight without wind and with weak sideslip and bank angle. The purpose of this paragraph is to evaluate these errors. Therefore, three methods will address this issue and the numerical evaluation will be made around a flight of a large commercial transport airplane. This flight is classical cruise flight a t an altitude of 30,000 f t and a Mach number of 0.8. The first method deals with the modal approach and gives the errors on the characteristic of the modes. That is to say the errors on the eigenvalues (frequency, damping ratio and time-constant). As with the first method, the second one is linked with the modal approach, but from a magnitude viewpoint. The variation of the magnitude contribution of each mode on the aircraft response is examined through the eigenvectors. The third method requires the gramian approach to throw light on the difference of energy of the signals between the coupled and decoupled model.
Mode
- Eigenvalue
The characteristics of the longitudinal and lateral modes are examined around different equilibriums. These equilibriums correspond to different values of sideslip angle and yaw rate r. Three yaw rates are taken into account: the null value for the straight flight, and two values of 0.57 O / s and 1.15 "/s which correspond to a turning flight with a bank angle of 13 O and 26 O. The results of relative differences between coupled and decoupled model are given in figure (5.8). Generally speaking, the rapid modes like the short period and rolling convergence are not affected by the decoupling. The dutch roll is only slightly influenced (from 1 % to 2 %) by the sideslip angle but not by the yaw rate. The slow modes are more affected. Then, the phugoid and spiral modes can change 50 % from the coupled model, depending of the sideslip angle and yaw rate. It should be noted that, whatever the yaw rate is, with zero sideslip angle, the decoupling has little influence on the modes.
Magnitude contribution of mode
- Eigenvector
The previous method gives information on the frequency, damping-ratio or timeconstant. However we can imagine, for example, that the frequency does not change although the magnitude of the response of the aircraft to a perturbation changes. This magnitude can be evaluated through the components of the eigenvector which give the contribution of one mode on the response of one state under the influence of one perturbation. To simplify the calculation, the perturbations are taken as initial conditions on the states x:. The influence of the initial condition x: of the state k on the state xi, through the mode j is evaluated thanks to the product mfj of the . symbol * right eigenvector component vij by the left eigenvector component u ; ~ The denotes the transpose and conjugate of the vector. (5.212)
5.4 Decoupled equations
151
Short Period frequency
Short Period damping ratio
I
-2'5
"0
4
8
12
-5 0
I
(b) 4
8p12
::Fi Phugoid frequency
0 - -
-25
-50 0
Altitude Convergence mode
50
4
8,312
Dutch Roll frequency
oh I -2:h I
25
251
I
-
-25
-50 0
4
8 p 1 2
Dutch Roll damping ratio
-50 0
4
8 s 1 2
Rolling convergence mode
5m I I
1
2.5
-2.5Oi-i---5
0 Yaw rate r:
4
Spiral mode
8 p 1 2
-r=O.O"/sec - - - -
r=0.57 "/sec
---
r = ] . ] 5 "/sec
Figure 5.8: Decoupling influence on the modal characteristics Then, the temporal response of the state x, t o an initial condition x f , is
j=1
The superscripts and are for the coupled system and the decoupled system. Therefore, a criterion ci about the magnitude for the state xi, can be defined as
(5.213)
The numerator is a length and the denominator is the maximum modal magnitude. Then, the range of the criterion is (0, 1). The decoupling does not have the same influence on each state. So, in order to have the consequence on the whole system, Dynamics of Flight: Equations
152
5
- Simplified equations
a global criterion C g l o b a l is built with the sum of the criterion ci weighted with the coefficient pi associated with each state xi. n
E Pi i= 1
(5.214)
With
(5.215)
The rangepf variation Si of the state xi gives the maximum variation whatever the perturbation is. The different ranges of variation Si are linked together with a time scale and an altitude scale. The time scale, based on rapid modes, links the angular velocity with the angle. The rapid modes are the short period mode and the dutch roll mode, and the given time scale is 0.5 s. The altitude scale, based on the total altitude ht, links the relative velocity to the altitude. The relative velocity belongs to the angle family. The total altitude is derived from the kinetic energy theorem, so that (5.216)
With these scales, only one range of variation Si is needed to define the others. The figure (5.9) shows the variation of the global criterion C g l o b a l applied on the whole states, that is to say longitudinal and lateral. This variation depends on the sideslip angle p and the yaw rate T . Three yaw rates are considered like for the mode analysis. The four presented curves correspond to four initial angle perturbations, one on the relative velocity, one on the angle of attack, one on the sideslip angle and the last one on the bank angle. Noted the classical result: with no sideslip and no yaw rate the decoupling is perfect. Globally, the relative change of magnitude between the decoupled system and coupled system is around 20 %, except for a bank angle perturbation, for which the magnitude is higher. Another way to analyse the decoupling is to consider the longitudinal system on one side and the lateral system on the other side. For example in the figure (5.10) the two above curves correspond to the longitudinal system alone. That is to say, only the variation on the longitudinal states are considered, submitted to longitudinal perturbations. The same the two lower curves, but for the lateral system. In this case, the relative change of magnitude between the decoupled system and coupled system is lower, around 10 %,
-
Energy contribution Gramian
The energy of the response on each state can be analysed thank t o the Gramian method. This approach is a kind of mixing of the two previous methods, because the
5.4 Decoupled equations
153
Relative longitudinal velocity 100
,
2
0
4
6
8
,
7 10,
12
10
a =0.01 rad
Angle of attack
UN=0.01
,
0
2
4
5
8
1
0
P(d&)
100
-
Sideslip angle ---IT-
p =0.01 rad
--
1
2
P(deg)
1 0 0 - ---~
0
2
Bank angle ----
4
4 =0.01 rad I
6
8
10
12
Figure 5.9: Decoupling influence on the magnitude of the states, for the whole system (longitudinal and lateral)
energy of the signal is sensitive to its magnitude and frequency. A physical understanding is associated with the grey surfaces seen in the figure (5.11). The energetic length of the state z ( t ) is denoted Ilz(t)lI. This length is defined through its square value
This value is evaluated thanks to the observability Gramian. The figure (5.12) shows the variation of the energetic length applied on the whole states, that is to say longitudinal and lateral. The figure (5.13) shows the variation of the energetic length applied on the longitudinal states and on the lateral states.
Dynamics of Flight: Equations
Relative longitudinal velocity
UN=0.01
Angle of attack a =0.01 rad 100 1 --7
(4
.-
20
0
r = 1.15"/s 0
2
r = 0.57"/s r = 0. I~
4
6
8
10
12
P(deg)
Sideslip angle
100
-___
p =0.01 rad
I ' - 100
Bank angle $I =0.01 rad ~
Figure 5.10: Decoupling influence on the magnitude of the states, for the separate system (longitudinal for the two above curves and lateral for the two lower curves)
Figure 5.11: Energy contribution on the coupled and decoupled system
et-6
0
5-
m
a"
m
P
0
0
0
N
0
0
P
Lateral
o
m
o
0
m
0 0
Global criterion
o
~
Lateral
Global criterion
0
0 0
m
o
o
o
1
o
w
1
~
o
m
o
Longitudinal o
r
e,
v
n
Global criterion
\
Longitudinal
Global criterion
n
o
o
m
E -
no
P
N
0
o
o
O
I
o
C
o
O
O
O 0
m
o
W
o
--1
E
g
I QII
l
v
L
n
5
Longitudinal and lateral
\;
P
o
Global criterion
o
N
Longitudinal and lateral
W
o
Global criterion
n
a
7
C 0
Longitudinal and lateral
P
Global criterion
o
N
Longitudinal and lateral
Global criterion
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6
Linearized equations
The more or less simplified equations previously studied were all non-linear equations. These equations are exploited without any difficulty by numerical methods especially to simulate the aircraft’s movement. However, when the question is to analyse the dynamics of a system and to synthesize a control law, the majority of tools offered by the automatic control scientist can only be put into use in linear systems. This is why it is necessary to linearize the equations of the aircraft’s system in order to have a model adapted to the study of its dynamics. The Zinearined equations (Section 6, p. 157), are part of the simplified equations but their major importance in the analysis of flying qualities justifies their development in a separate chapter. These linear equations of the aircraft system provide a simplified model representing an aircraft in a certain validity range, around the initial conditions of linearization. This validity range depends on the initial conditions and the type of linearized equations. Thus the longitudinal equations usually have a larger validity range than lateral equations. The linearixation method (Section 6.1, p. 158) is first expounded, then a numerical and analytical process is proposed to exploit this method. Numerical Zinearixation (Section 6.2, p. 160) can be implemented for every flight condition with non-decoupled equations and non-analytical external effort models. This numerical linearization has a large domain of application but it does not give an inside view of the phenomena as the analytical linearization does. The analytical linearixation carried out on the decoupled longitudinal (Section 6.3, p. 161) and lateral (Section 6.4, p. 170) equations leads to a more limited implementation, due to its heavy calculations. However it has the advantage of being an explicit parametric study of dynamics and thus favorable to a physical approach to phenomena. At this point, it can be shown that linearization around a steady state flight with wind modifies the state matrix with respect to steady state flight without wind. This signifies that the modes and thus the flying qualities can be a function of wind.
157
158
6.1
6 - Linearized equations
Linearization method
The system is put in a state form (Section 5.4.1, p. 135) (6.1)
= F(X,U,W")
X
The differentiation of these equations leads to a linear system. This differentiation is made about the initial conditions defined by the equations = F(Xi,Ui,WY)
Xi
(6.2)
This differentiation is representative of a development of the Taylor series in the first order. The differentiation gives dX
+
= GRAJDFX~ dX GRADFU~ dU
+ G;r~rnFw~ dW"
(6.3)
with
and the matrix GRmFxi is the Jacobian matrix of F with respect t o the vector X (Section 3.2.1, p. 45). The rows of GRmFxi are made up of the partial derivatives of one of the components of F with respect t o each of the X components. These partial derivatives are calculated about the initial conditions Xi, Ui, WO. Thus (GWmFxiis a matrix with constant coefficient.
\ with
F n x i i Fnx2; .. . Fnxni
Fmxki =
J
dFm
d x k (Xi,Ui,WO)
with m and Ic varying between 1 and n. The matrix (GWmFu,is the Jacobian matrix of F with respect to the control vector U, calculated about the initial conditions Xi,
ui, w:.
The matrix @mFwi is the Jacobian matrix of F with respect to the wind vector calculated about the initial conditions Xi, Ui, Wy. The initial conditions Xi, Xi, Ui, WO can be any value whatsoever. This is the case for the proposed method later on, in order to look for a state of equilibrium (Section 7.2, p. 186). In general, linearization is made around a state of equilibrium ( e ) such as Xi = X, = 0 (Section 7.1, p. 180). It has been shown [6] that since the Zinearixation is made around a state of equilibrium, the linear systems from two different state bases represent the same system. However for the general condition, the linear system is not intrinsically linked W O ,
6.1 Linearazation method
159
t o a non-linear system. The linearization around a state representing a trajectory of an aircraft could have this same property, found in the linearization around a state of equilibrium. In this way, let the system (Equation 6.1) without wind for simplification X
= F(X,U,O)
A state base transformation Y = T(X) is applied t o it. Let see how the linearized system is obtained, about any point (Xi,U;),so that Xi = F(Xi,Ui,O)
In order t o linearize the system, the transformation equation is linearized
dY
= &mTidX
In this new state base the linearized system is written Y
= &mTidX
However if the state base transformation is first applied on the non-linear system and the linearized process is made later, the result differs. Thus
Y
= GRAKDTX
Thus after linear ization Y
= ~(GRADT)X~+GADT~X
The two approaches give the same result only if Xi is equal t o zero, that is t o say when the point (Xi,Ui) is an equilibrium point. If the linearization around any initial state can be made mathematically, its properties are not clearly defined. All the Xi = F ( X i , U i )are not linked t o trajectories. These trajectories are defined as the system answers to a realizable control U(t). The equations of equilibrium are written (Section 7.1.1, p. 180)
If the linearization is made around the equilibrium state, X, = Xi and U, = Ui. Most of the time, equilibrium is defined around the conditions of zero wind, WO = W,O = 0, then 0
= F(Xe,Ue,O)
(6.8)
The aerodynamic coefficients can depend on the derivatives of the state X. For example, the lift coefficient C L and the pitch coefficient C m are a function of the temporal derivative of the aerodynamic angle of attack 6,. Thus the general form of the system is the following X
= F(X,X,U,W')
(6.9)
where the effort equations have several components of X in the terms of acceleration, as is the case for the lateral equations. A practical difficulty then comes up when Dynamics of Flight: Equations
6 - Lanearazed eauatzons
160
looking for whatever initial conditions with Xi # 0. In fact, in the preceding case, it was sufficient to give Xi, Ui and Wy in order to find Xi via the equations given by F. Now, a iterative step is necessary to find Xi such as Xi = F ( X i , X i , U i , W F )
(6.10)
The linearization of (Equation 6.9) makes a new term appear dX
= cGwmFxi dX
+ ~G~ADFX; d X + ~G~ADFu; dU + GRADFw~ dW"
(6.11)
or again dX
=
- GRADFX~)-' (GRmFxidX + @mFui dU
(11
+ @mFwi
dw")
(6.12)
This linearized system is frequently shown with the following notations dX
= AxdX+BudU+BwdW"
(6.13)
If no ambiguity is possible, the notation d X , dX, du,dw" which represents the increment relative to the initial value, is often dropped and replaced by X, X, U, W" (Hypothesis 25). The matrices Ax,Bu, Bw are
6.2
AX
=
(1, - @ADFX~)-~GRADFX~
BU
=
(11 - G R A D F X ~ ) - ~ @ A D F U ~
BW
= (II~ &~FX~)-~GII~ADFW~
(6.14)
Numerical linearization
In the framework of a numerical simulation of aircraft flight, the non-linear equations X
= F(X,X,U,W")
(6.15)
are put in the calculation program. It is therefore entirely possible to construct a linearization procedure from these equations. In the following paragraphs (Section 6.3, p. 161) and (Section 6.4, p. 170), the matrices Ax, Bu, BW will be analytically constructed from the linear system. This method shows a number of interesting points but it is rather complicated to put into practice. Its application is limited to special cases. However, the numerical linear method can be put into practice whatever the case of flight and modeling of external forces might be, including a non-analytical modeling case. An example would be when the aerodynamic coefficients are given in the form of graphs. The working out of this numerical linearization method is illustrated by the linear program found in section (G.l), p. 275. The matrices Ax and B are built by columns. For example, for the matrix Ax, thanks to a numerical variation of one of the components of X , A x j , the variation induced on all of the components of X, AX(Axj) can be calculated by some terms of the acceleration. The j t h column of Ax will therefore be equal to A X ( A z , ) . In
161
6.3 Longitudinal linearized equations
this manner of provoking a variation of the angle of attack Axj = Aaa, the induced variations on the components of acceleration A X ( Aaa) will be calculated. A practical difficulty can appear in the definition of the values of Axj. If these values are too high, the difference cannot represent the local slope around the initial conditions since the terms over the first order in the Taylor series can no longer be neglected in comparison to the first order term. This first order term is the linear term represented by the matrix Ax. If these values are too low, it is possible to reach computer precision and make calculation errors. For example, a numerical linearization made on a working station for a commercial aircraft in its flight envelope, gives this range for the increment Axj in order to stay within a good accuracy. For the increment on the components of velocity U , U , w, its value should be between 10-3 m / s and 10-1 m / s . For the increment on the components of angular velocity p , Q, r , its value should be between 10-3rad/sand lO-lrad/s for p and r , and for q between 10-4rad/s and 10-3 radls. For the increment on the altitude h, its value should be between 1m and 10 m. For the increment on the inclination angle 8, its value should be between 10-2 rad and 10-1 rad. For the increment on the bank angle 4, its value should be between 10-4rad and 10-2rad. It seems useful to determine the miminum and maximum values of each increment for the case which is examined. This numerical linearization is used, in particular, for the general research of equilibrium (Section 7.2, p. 186).
6.3 Longitudinal linearized equations The analytical linearization performed in this section presents the advantage of an explicit parametric study of the dynamics of an aircraft which clearly explains the physical interpretation of phenomena. However, it is more complicated to implement than the numerical linearization. As a consequence, it is only usable for special simplified cases. This linearization is performed on the decoupled longitudinal flight equations (Section 5.4.2, p. 137). Before attacking the linearization of longitudinal equations as such (Section 6.3.2, p. 165), a certain number of preliminary linearizations must be made (Section 6.3.1, p. 161).
6.3.1
Preliminary linearizations
Reduced velocity The reduction of the velocity to a nondimensional value simplifies the writing of the linearized equations. The reduced velocity will be denoted Va and it is equal to the ratio between the velocity Va and the initial velocity Vai. The quantities AV, represent the change from the initial values (most of the time the equilibrium values) and from a physical point of view is equivalent to the quantities dVa defined in equation (6.5), p. 158
AV,
= Va - Vai etc
(6.16) Dynamics of Flight: Equations
6 - Linearired equations
162
(6.17) wind velocity (6.18) thus (6.19) etc. Etkin [5] suggests a system of equations where all of the variables are reduced t o nondimensional variables. For example, ci, p , q, T , are nondimensionalized by the the mass by the factor inertia by the factor the air density by the factor time by the factor If the reduction of the velocity simplifies the writing factor of the coefficients of linearization, the reductions of the other parameters do not have the same advantage.
6,
k,
F.
&,
&,
Linearization with respect to the Mach number The Mach number M is not part of the state of the aircraft for classical forms of equations. It is therefore necessary to show the variation of Mach AM around the initial conditions in function of the states, here AV, and Ah, for the linearization. The Mach number is defined by
M = - -Va a
(6.20)
a2 = yRT
(6.21)
with the speed of sound a
Here, y represents the adiabatic constant of the air and is equal to 1.4 under normal conditions. This y is only a local utilization and must not be confused with the climb angle y of the trajectory inclination. The differentiation of equation (6.20), p. 162 gives
dM -
M
The differentiation of equation (6.21), p. 162 gives
da 2a
=
dT - 1 dT - -dh
T
T dh
dV, Va
1 dT --dh 2T d h
Thus
dA4 A4
-
--
(6.22)
6.3 Longitudinal linearized equations
163
The temperature gradient Th (Equation 4.1, p. 88) as a reduced form and will be denoted Th
dT Th= dh
Th Th= Ti
and
(6.23)
from which the change of M from the initial condition AM = M - Mi is
(6.24) This relationship could be written
Linearization with respect to the altitude The altitude h intervenes in the expression of the external forces through the air density p. For the linearization with respect to the altitude, the air density gradient Ph must be evaluated (6.25) By definition
Laws of the standard atmosphere (Equation 4.84, p. 87) (Equation 4.85, p. 87) give
(6.26) p
=
pRT
(6.27)
Thus, after the differentiation of equation (6.27)
(6.28) from which the air density gradient
Ph
is
(6.29) its reduced form ph
= ph
-Th
Linearization of pitch velocity in pure longitudinal flight
(6.30)
6 - Linearized equations
164
The aerodynamic coefficients depend on the aerodynamic pitch velocity q t (Equa-
tion 3.54, p. 56) (Section 4.3.3, p. 89) with the expression of q:
The state of the aircraft contains the kinematic pitch velocity qk usually denoted as q. The question here is to linearize q t with respect to q and the components of the wind. pitch velocity q&. The expression of qxb, in pure longitudinal flight has been calculated in (Equation F.34, p. 270) =
qxk
- sin 8 cos e(uxL - wzL)
+ qZ; sin2 8 + qx& cos28
(6.32)
where, after linearization
AqxL = -
+
(COS 28, (wZ&- u X l i ) sin 2Oi(qZLi - qxLi)) A8 sin 8; cos O,(AuxL- Awz&) + sin2 8,AqZC, cos2 8,Aq.L
+
(6.33)
The relationship equation (5.159), p. 140 valid for pure longitudinal flight is (6.34)
Finally, by differentiating equation (6.31), the result obtained is
Aqt
= Aq-
(6.35)
Linearization of thrust With the thrust model (Equation 4.102, p. 92)
The constant k , is a characteristic of the engine. The linearization with respect to the altitude h, the aerodynamic velocity Va and the position of the throttles 6, is accomplished in this manner
dF
= km-Va dP x Sxdh
dh
+ Xk,pV,X-lG,dV + k,pV,Xd(Sx)
(6.37)
Translated into a change from the initial conditions, with the preceding equation (6.17), p. 162 and equation (6.30), p. 163 (6.38)
This relationship could be denoted
1.
6.3 Longitudinal linearixed equations
6.3.2
165
Linearization of longitudinal equations
This linearization is performed in section (G.3), p. 295 on the non-linear, decoupled, longitudinal flight equations (Section 5.4.2, p. 137) which are represented by the following form. It must be remembered that the components of XL, UL,WLL,WGLand WRL,represent the change from the initial value (Hypothesis 25), that is to say the increment relative to this initial value, symbol i .
XL = F(XL,XL,UL,W')
(6.39)
with the state vector
XLt = [Va,& a , Q , h, %I
(6.40)
ULt = [6,,6,]
(6.41)
with the control vector
and, to simplify the writing, WO is divided into three elements (6.42)
with the wind linear velocities ( L )
WLL~
(6.43)
the gradients of the wind linear velocities ( G ) (6.44)
the angular or rotational wind velocities ( R ) (6.45)
This wind vector has a restricted size in the case of longitudinal flight with respect to general cases such as lateral. After linearixation around the initial conditions, symbol i , the system takes on the form
(11 - @mFXLi)AX~= GRADFXL~AXL + GRADFuL~AUL + GAIDFWGL~AWGL +@ADFWLL~AWLL +UhmFw R L A ~ W RL (6.46) and the result is
~ ~ - @ A D F X=L ~
[1
-a& 0 0 0 1+aiy& 0 0 -;q& 1 0 0 1 0 -ay& 0 0
0 0 0 0 1
1
(6.47)
Dynamics of Flight: Equations
6 - Linecrrized equations
166
The matrix GRADFxL~, which is a part of the state matrix of the longitudinal system, takes on the form
(6.48)
The state matrix
AX, (Equation 6.13, p.
AX
160) and (Equation 6.15, p. 160)) is equal t o
= (1, - GRADFXL,)-'GRADFXL,
(6.49)
The matrix GRADFXL, is due to the effect on aerodynamic force of the angle of attack derivative A, (Equation 6.112, p. 298),(Equation G.136, p. 302) and (Equation G.162, p. 304). If this effect is null, the state matrix AX is equal t o GRADFxL,. The first row is made up of the coefficients that came from the linearization of the propulsion equation (5.188)) p. 143, the second row from the kinematic angular equation (5.197))p. 144, the third row from the moment equation (5.190))p. 144, the fourth from the kinematic altitude translation equation (5.194)) p. 144 and the last from the sustentation equation (5.189))p. 144.
REMARK6.1 The matrix GRADFXL~ expresses the longitudinal dynamics of the aircraft
and depends on the wind gradients. Thus the aircraft, when it crosses an area of established wind gradient, can see its modes and thus its dynamics modified.
These coefficients have the following general form. The first row axu results in the linearization of the propulsion equation (5.188))p. 143 made in equation (G.107)) p. 298
6.3 Longitudinal linearized equations
167
The second row axa results in the linearization of the kinematic angular equation (5.197), p. 144. All the axa are the opposite sign of axy given in (Equation 6.54, p. 168) except for
axaz = -axy, for z = (U,a, h, g ) axaq = 1 - aXyq
(6.51)
The third row axq results in the linearization of the moment equation (5.190), p. 144 made in equation (G.157), p. 304
(6.52) The fourth row axh results in the linearization of the kinematic translation of altitude equation (5.194), p. 144 made in equation (G.167), p. 305 axh, hh, The other axh
= Vai shy,, Vai COS^^, are equal to zero
=
(6.53)
The fifth row axy results in the linearization of the sustentation equation (5.189), p. 144 made in equation (G.131), p. 301
Dynamics of Flight: Equations
6 - Linearized equations
168
[
The matrix of controls &mFuLi takes on the form
GRADFUL~ =
-buy, bup
buvx -buyx b?
buym
buyx
buvm
)
(6.55)
with from equation (G.108), p. 298
(6.56)
from equation (G.132), p. 301
(6.57)
from equation (G.158), p. 304
(6.58)
The matrices interpreting the wind perturbation take on the following forms. The matrix of perturbation associated with the wind translation velocities is
(6.59)
With from equation (G.109), p. 298
bwvu = bwvw =
- sin Tai qxLi - COS yaiEx& - cos Y~~
@Li + sin yaizLLi
(6.60)
from equation (G.133), p. 301 (6.61)
6.3 Longitudinal linearized equations
169 -
from equation (G.168), p. 305
1
(6.62)
The matrix of perturbation associated with the wind translation of velocity gradients is
(GW~FWGL~ =
[
bwvux -bw?ux bw";
bwvwz -bw?wz bw?,
b ~ u x bwrwz
(6.63)
With from equation (G.llO), p. 298
(6.64) from equation (G.134), p. 301
(6.65) from equation (G.160), p. 304 bwqux =
e2p ''i Cmq sin 28, 2SV&
(6.66) The matrix of perturbation associated with the wind rotations is
(6.67)
With from equation ( G . l l l ) , p. 298 bwvqx = - COS Toi sin ?ai -
sin Y~~ + Sqpie COS2 o i c D q mV2 (6.68) Dynamics of Flight: Equations
170
6 - Linearized equations
from equation (G.135), p. 302
(6.69) from equation (G.161), p. 304
(6.70)
6.4
Lateral linearized equations
In this section, the results of the preliminary linearizations are used (Section 6.3.1, p. 161). With regard to the linearization of the longitudinal equations, two problems appear when dealing with the linearization of the lateral equations. 0
0
The longitudinal states will appear in the lateral equations under a form that is a function of the flight situation studied. Depending on the hypotheses, several results of linearization will be obtained. Generally speaking, there is no such thing as a perfect decoupling between lateral and longitudinal equations. The wind does not have the simple form it had with the longitudinal flight and the most general case should be treated. The wind is defined in the vehiclecarried normal Earth frame F,. These components need to be expressed in the body frame Fb under a linearized form (Section G.2, p. 283). This operation makes the angle of heading $, the bank angle 4 and the inclination angle 8 appear. Two of these angles create a coupling: the heading with the navigational equations and the inclination angle with the longitudinal equations.
The linearization of lateral equations is made in section (G.4), p. 305. The lateral non-linear equations (Section 5.4.3, p. 145) are represented by the following form. It must be remembered that the components of XL, Xi, Ui, WLI,WGIand WRI,represent the change from the initial value (Hypothesis 25), that is t o say the increment relative to this initial value, symbol i. XI
= F(XL,X I ,XI,U,
WO)
(6.71)
with the lateral states vector
XIt = [ P a , P , r , $ , $ ]
(6.72)
and the longitudinal states vector
XLt = [Va,%,Q,eI
(6.73)
6.4 Lateral linearized equations
171
The altitude h does not appear because the influence of its eventual variation has been neglected (Hypothesis 28). The control vector
Ult = [6&]
(6.74)
and, to simplify the writing, the wind vector WO is divided into three elements
WO
=
(E)
(6.75)
with the wind linear velocities ( L )
(6.76) the wind linear velocities gradients (G)
(6.77)
wGit
The angular or rotational wind velocities ( R )which represent the wind velocity gradients that are perpendicular to the radial axis (Section 3.2, p. 45) are
(6.78) After linearization around the initial conditions, symbol i , the system takes on the form (11 - (r;rnP~i~FXii)AXl =
cGwmFxiiAX1+ (6WmFUiiAU1+ ~ G ~ ~ F x L ~ ~ A X L +@ADFWLI~AWLI + GRADFWGI~AWGI (6.79) +(r;rnPrnFwmiA WRI
These matrices are calculated in section (G.4), p. 305 for the most general cases. The reader can refer to this if, in particular, he wishes to analyse the turning flight. To simplify things, the results of linearization around a case of rectilinear steady state flight with horizontal wing, is presented in this section. This already has great practical interest. The state matrix Ax is equal to (11 - cGwrnFX~~)-'(r;rnPmFx~~, (Equation 6.13, p. 160) and (Equation 6.15, p. 160). The matrix cGwmFx~~is due to the effect of the state derivatives, if they exist, and of components of acceleration generally depending on the product of inertia E (Equation 6.80, p. 172). If this product of inertia E is null the state matrix AX is equal to GRADFXI~.
Linearization around a rectilinear steady state flight with horizontal wing The conditions of steady state flight with horizontal wing lead to the following hypotheses: 0 0
The linearization is performed around the steady state flight,
Pai = 0.
The initial conditions (Hypothesis 26) in the sideslip angle and azimuth are zero, = pi = 0. Dynamics of Flight: Equations
6 - Linearized eauations
172 0
0
0
With the conditions of zero sideslip angle and propulsion symmetry Pm = 0, the aircraft flies with its wings horizontal, q5i = 0. And equilibrium is a case of longitudinal flight, ei - cyai = Y ~ ~ . The angural velocities of roll and yaw of equilibrium are zero, pi = ri = 0
.
A hypothesis of symmetry is made, that is non-restrictive in practice. The aircraft is geometrically symmetric (Hypothesis 4) and as pi = pi = ri = 0 then c y , = Cl, = Cn, = 0.
On the other hand, it can be assumed that the wind is known in the vehicle-carried normal Earth frame aircraft fuselage oriented F f , which is oriented towards the initial inclination angle of linearization 8,. The terms duz, dvc, dw;, are calculated in equation (5.48), p. 113. The terms uxLi, vy;,,, w&,, qx”,, q&, are calculated in equation (D.75), p. 240 to equation (D.77), p. 240. The result is
0 II1
- GRAIUJFXI~ =
0
0 0
0
1 0 0 1
(6.80) 0 0
0 0
The hypothesis has been made that the aerodynamic lateral forces do not depend on the derivative of the aerodynamic sideslip angle ,& and this leads to obtaining a “1” on the first row associated with the lateral force equation. If this hypothesis is not must be recalculated. made, the first term of (11 - GRADFXI~) The inversion of (II1 - (GWmFxii)gives
(6.81)
or
(111 - cl;mP~~FXi~)-l =
1 0
l-m
0 0
0 1
50 0
0
0
0
z o o
1 0 o 0 1 0 0 0 1
j
(6.82)
It can be noted that for the inertial product E = 0, this matrix is equal to the identity matrix 11. Usually, the right term of equation (6.81) which follows the identity matrix II1 is almost equal to zero. The multiplication of the terms on the right of equation (6.79), p. 171 by this matrix only affects the two equations of yaw and roll moment.
6.4 Lateral linearized eauations
173
The matrix GRmFxl,,which is almost the state matrix of the lateral system, takes on the form
(6.83)
The first row is made up of the coefficients that came from the linearization of the lateral force equation (5.202),p. 148. The second row is obtained from the roll moment equation (5.205), p. 148 and the third came from the yaw moment equation (5.206), p. 149. The fourth and fifth rows came from kinematic angular inclination angle equation (5.207), p. 149 and azimuth equation (5.208), p. 149. The matrix (GWrnFx~l~ represents the influence of the longitudinal states on the lateral equations. They take on the following form
(6.84)
The coefficients of the matrix (Cr;wmFxl,and the matrix (GWAIDFXLI~ have the following general form. The first row axp (Equation G.184, p. 312) results from the linearization of the lateral force equation (5.202), p. 148 hPP
=
sin 2aai 2 (qGi - qx;,) sin a,, -~ (-dwEi vai
+
cos2 a a i
+
~z;,
sin2 a,, - 2ry0,,
c o d i - du& sine,) + -(du;, cos Oi - dw& sin ei) Va i COS
Dynamics of Flight: Equations
174 ____________
6 - Linearized equations --__
The first row a;xp (Equation G.184, p. 312) of the matrix GRADFXLI~ relative t o the longitudinal states, is
(6.86) The second row axp (Equation G.196, p. 315) results from the equation of the roll moment (Equation 5.205, p. 148)
The second row axp (Equation G.197, p. 315) relative t o the longitudinal states, is axpv
c1. = 2+piStVai2
axp,,
= 0
axP,
=
-
=1
-
*Pe '
.
A
F, + A-(9,VaiA
COS^,^ sina, - zm sin&)
ri(C - B ) - Ep, A p. se2 2
A Vai ( -ryLi Clp + PY;; Clr,
+ pz;,
CZr,)
(6.88)
The third row axr (Equation G.209, p. 318) results from the linearization of yaw moment equation (Equation 5.206, p. 149)
pise2
axr,
=
1.-
axr4 '
. =1
-
axr+
=
-
2 c
B-A VaiCnp - ~ i - C
p. se2
2 c
p. se2
Vai [Cnp(wzzi - v y Z i ) - qzO,,Cnr, - q x ~ , ~ ~ n r , ]
21 c1 Vai
[qxLiCnp
+ (ux:~ - vyLi)(Cnr,
-
Cnr,)]
(6.89)
6.4 Lateral linearized equations
175
The third row itxr (Equation G.210, p. 318) of the matrix (GWADFXLI~ relative t o the longitudinal states, is
p ( B - A ) - Eri axrq = C (6.90) The fourth row ax4 (Equation G.217, p. 319) results from the linearization of the kinematic bank angle equation (5.207)) p. 149 about initial conditions with a zero bank angle 4i = 0 ax+p
= 0
ax+p
=
ax+,
1 = tanOi
ax++ = qi t a n & ax++ = 0
(6.91)
The fourth row ax4 (Equation G.217, p. 319) of the matrix G . A J D F X relative L ~ ~ t o the longitudinal states, is
(6.92)
The fifth row ax$ (Equation G.220, p. 320) results from the linearization of the kinematic azimuth equation (5.208), p. 149 about initial conditions with a zero bank angle +i = 0
ax$+
=
0
(6.93)
The fifth row ax$ (Equation G.220, p. 320) of the matrix GRADFXL~~ relative to the longitudinal states, is
(6.94) ~
Dynamzcs of Flaght: Equatzons
6 - Linearized equations
176
_______-
[
The matrix of controls @mFuli takes on the following form
GRADFUI~ =
buPl
b r bun
bUPn
b i n ] bupn
(6.95)
With issue from equation (G.176), p. 309
bupl
=
S 4 pi Vai Cy61
bup,
=
$pi-VaiCy6n m
rn
S
(6.96)
from equation (G.192), p. 314
(6.97) from equation (G.205), p. 317
(6.98)
[i i i)
The matrices expressing the atmospheric perturbation have the following form. The wind velocities of translation bwPu
0
(GBPADFwLI~ =
bwpv 0
bwPw
(6.99)
with afterwards equation (G.181), p. 311
(6.100) The wind translation velocities gradients
(6.101)
6.4 Lateral linearized equations
177
with afterwards (Equation G.182, p. 311)
= 0
bwPwz
(6.102)
with afterwards equation (G.199), p. 316
(6.103) with afterwards equation (G.213), p. 319 bwrus = 0 bwrvy = 0 bwrwz
= 0
(6.104)
The wind angular velocities
with afterwards equation (G.183), p. 311
+
S
= - +pi- l ( C y p cos 8, Cyr, sin 8,) m 1 5 bwPPz = -w;, sin aaicos 8, - cos sin 8, - +pi-[Cyr, sin 8, Vai rn bwpq, = 0 bwPqz = 0 1 5 bw& = -- u;, - cos a,,cos Oi - sin sin 8, - i p i-lCyr, cos Oi Vai m
bwpp,
+
bw&
S = - +pi-l(Cyr, cos Oi - Cyp sin Oi) m
(6.106)
with afterwards equation (G.200), p. 316 bwpp,
=
-3-
bWPpz
=
-1-
pise2
A pise2
'
A
Vai(ClpCOS Oi + CZr, sin 8,) VaiCZr, sin 8,
Dynamics of Flight: Equations
6 - Linearized eauations
178
bwp,,
=
-1-‘
bwpry
=
-1-
pise2
A Vai Clr, cos Oi
pise2 Vai (Glr, cos Oi ‘ A
-
CZp sin Oi)
(6.107)
with afterwards equation (G.214), p. 319 bwrpy =
-$L p. se2 Vai (Cnpcos Oi
bwrpz =
--1.-
C
bwrqa: = 0 bwrqz = 0 bwr,.,
=
bwrry =
pise2
‘ C
-$-1- -
pise2
C pise2
‘ C
+ Cnr, sin Oi)
VaiCnr, sin Oi
V a iCnr,
cos Oi
Vai (Cnr, cos Oi
- Cnp sin Oi)
(6.108)
Linearization about a steady state flight without wind If the preceding case is taken, but with the supposition that the initial state corresponds t o a state without wind, for example an equilibrium without wind, then all the wind terms of the type ux&, qx& etc, are zero. As a consequence, the coefficients a X & , axp+ and axr+ cancel each other out and the kinematic azimuth equation is decoupled. The linearized lateral system goes from the fifth order t o the fourth order because the azimuth angle no longer has any influence on the external efforts. Moreover the matrix G R A D F XisL ~ cancelled ~ out, which means that the longitudinal states no longer influence the lateral equations. This last result leads to the conclusion that there is a true decoupling between the longitudinal and lateral equations. The result of section (5.4.3),p. 145 is discovered again. This result is very important because it shows that for a linearization around a steady state flight without wind and with a zero bank angle 4 = 0, the true decoupled lateral equations exist without a special hypothesis on the longitudinal parameters. Among other things, it is not necessary to “pilot” the longitudinal state. Finally, it can be remarked that the coefficients axpd and axrd equally cancel out each other and that the coefficient ax& is reduced to cos Oi. The wind perturbation matrices & m F w u i (Equation 6.99, p. 176) and G R ~ F W G I ~ (Equation 6.101, p. 176) cancel out each other and the aircraft is no longer sensitive t o the wind translation velocities and their gradients.
+
7
Equations for equilibrium The last case of the simplifications of equations begun in Simplified equations (Section 5 , p. 103)) will be developed here. The equilibrium (Section 7, p. 179) is a special case of the general equations of dynamics. These equations of equilibrium generally speaking correspond to the study of the performance of the aircraft. At first, the notions of equilibrium (Section 7.1, p. 180) or pseudo-equilibrium (Section 7.1.2, p. 182) are defined. The equilibrium definition that has been chosen, is the one linked t o the state representation. Thus, the aircraft will be in equilibrium when the derivative of the state vector of the principal system X is equal to zero. Physically, this means that there is an equilibrium when all of the states that have an influence on the external efforts, or the complementary terms of acceleration, are constant. All equilibrium flights correspond to a spiral trajectory such as the turning flight. The rectilinear flight could be considered as a particular spiral trajectory with an infinite radius. The principal pseudo-equilibrium is the climbing flight. In order t o assure that the conditions for the resolution of equilibrium of a linear system are present (Section 7.1.3, p. 182)) it is necessary t o complete the equation system with as many independent equations as there are controls. In general, these conditions can be practically extended to a non-linear system that represents the aircraft. To avoid difficult numerical resolutions, it is desirable to keep in mind the decoupling phenomenon of the longitudinal and lateral movements when choosing the supplementary equations. The equilibrium conditions having been defined, a method for the numerical research of equilibrium (Section 7.2, p. 186) is suggested, based on the linearization of the equation system around any flight situation. This numerical method implemented in Fortran (Section H, p. 321), allows for the research of any kind of equilibrium or pseudo-equilibrium without any special initialisation with a free choice of the supplementary conditions of the equilibrium definition. It equally detects a poor formulation of these equilibrium conditions, for example when the conditions are not independent or when they ignore the decoupling effects. General equilibrium (Section 7.3, p. 188) is evoked when the flat and fixed Earth hypotheses are not made. Within the decoupling frame, longitudinal equilibrium
179
180
7 - Equations for equilibrium
(Section 7.4, p. 188) and lateml equilibrium (Section 7.5, p. 190) are given. These are the simplest equations of the document but they are rich with multiple practical information for the analysis of aircraft flight. However, the exploitation of these equations is not one of the aims of this book. A choice had to be made in organizing the order of the chapters equilibrium and the chapter linearixed equations. As the linearized equations is a simplified system of equations but for the analysis of the dynamics of the aircraft which includes the equilibrium, the equilibrium equations appear as more simplified than the linearized ones. That is the reason for the choice made. The inverse choice should have been made because in general, the equations are linearized around a steady state flight given by equilibrium. This problem does not change anything in the formal writing of the linearized equations, since the differentiation is made around the initial conditions which can be those of equilibrium or others.
7.1
Equilibrium notions
The physical notion of equilibrium is rather intuitive. It corresponds to a stabilized situation where “the elements” do not evolve. The transformation of this idea to a rigorous analytical definition can sometimes lead to some difficulty. The sum of the external efforts equal to zero corresponds to the equilibrium definition usually used. Pure longitudinal flight becomes part of this definition frame when the wind is zero, but the steady state turning flight is not included in this formulation. In fact, in this last case, there exists an acceleration not equal to zero which is “equilibrated” by an aerodynamic force. This case can be treated all the same by placing it in the relative frame and by examining the “relative equilibrium” case seen in this frame. Howerver this type of equilibrium will depend on the choice of the relative frame. The definitions of equilibrium in the literature are numerous. The notion of equilibrium from the state representation given by automatic control scientists will be retained.
7.1.1
Definition of equilibrium
For the system put in the state form (Equation 5.138, p. 133), the following notion of equilibrium’ will be adopted For a system in the state form X = F(X,U) There is equilibrium if X = 0 whatever time t with U = constant This equilibrium is associated with a point of equilibrium, or a singular point, defined by the state vector X, and a control vector U, such as F(Xe,Ue) = 0. For the aircraft, the role that this definition plays will be examined from a practical point of view. With the example of the aircraft, it is clear that if the three kinematic navigational equations are integrated on the representation of the state, the notion of equilibrium will be reduced to the situation of an aircraft in a fixed position with respect to the Earth2. What that means is that the aircraft is on the ground! This ‘Some authors join a stability notion to this definition. It is not the case here. 2The derivative of the geographical position 5 and y has to be zero, as well as the azimuth derivative 21 = 0.
181
7.1 Eauilibrium notions
situation holds no practical interest and experience in flight proves the existence of equilibrated flight situations, for example when the aircraft is cruising. In fact, as has been shown before, the navigational equations have been decoupled (Section 5.4.1, p. 135). Therefore, if the representative state equations used are those of effort and kinematic equations without the navigational equations, a decoupled system is generated. If this decoupled system is equilibrated, very useful kinds of equilibrium are obtained. All equilibrium cases belong to the equilibrium class of steady state level turning flight, with a particular case, the rectilinear steady state flight, which is only a turning flight with an infinite radius! While turning, the azimuth changes # 0, and it is shown more precisely that the derivative of the azimuth is = constant. This is the confirmation of the non-equilibrated navigational azimuth equation. To obtain this equilibrium, the system of equations must be decoupled. The choice of the “dominating” system that needs to be equilibrated, does not cause a problem in the case of the navigational equations, if the physical sense is refered to, when decoupling. It can be remarked that the external efforts do not depend on the navigational states. Equilibrium therefore corresponds to a situation where all the states, that have an influence on the external eforts or the acceleration terms, are constant.
4
4
REMARK 7.1 In the framework where the hypotheses of a flat and fixed Earth are not made, the decoupling of navigational equations can no longer be completely made and the equilibrium will be obtained at constant latitude, that is to say for an East or West azimuth.
The decoupling can be continued. The decoupling between the longitudinal and lateral equation (Section 5.4,p. 132) has been examined. It appears that there could be a lateral equilibrium X i a t = 0 with whatever longitudinal movement, but the opposite is not possible (Section 5.4.2,p. 137). Therefore there are only two possibilities of equilibrium: a particular equilibrium in lateral, or a general longitudinal and lateral equilibrium. A decoupling could be imagined that could be obtained by changing the base of the state. This would allow an association of the equilibrium with each sub-system. As there is no chance that the new states obtained, by these base changes, do have any physical significance, what will become of these new equilibriums? Nevertheless, nothing indicates that a representation of a new physical state cannot be found in association with a new equilibrium. For example, if there is a thrust that is independent of the velocity, a change in the variable between the velocity module and the equivalent velocity Veqsuch as poVA = pV2, would certainly allow the steady state climb to be admitted into the class of equilibrium. In the representations of classical states, this non-zero climb angle flight is not an equilibrated flight but a pseudo-equilibrated flight which will be developed in the following section. These remarks show that there is some difficulty in defining the equilibrium rigorously. This difficulty is, in part, linked to the notion of the decoupling of the system which might be dependent on the base of the state representation which is not unique. An extension of the notion of equilibrium could be made by admitting the orbits into the class of equilibrium, that is to say the periodical trajectories of period T such as X(T + t ) = X ( t ) with a constant control U = constant. Dynamics of Flight: Equations
7 - Equations for equilibrium
182
7.1.2
Pseudo-equilibrium
The notion of “pseudo-quilibrium” or relative equilibrium, is used practically speaking as it corresponds t o a partial equilibrium. What is meant here is a partition of X equal t o zero. The most common example is equilibrium at nonzero climb angle. To obtain this pseudo-equilibrium, the kinematic altitude equation h = V sin y is substracted from the equation system. Thus, it is no longer necessary to force h = 0 and the altitude is not obliged t o stay constant. This approach will accept the case of an aircraft climbing in a nearly constant climb angle y as being in a state of equilibrium. It is not possible to admit just any kind of pseudo-equilibrium since these simplifying hypotheses must be justified either experimentally or theorically. The relinquishment of one or several equations of the system for the resolution of equilibrium produces results close to reality inspite of the reduction of the validity of the model. In the case of constant climb, this is justified by the very slow variation of the air density p in function of the altitude which concludes that p is a local constant. This justification is confirmed by the large time constant of the exponential altitude convergence mode, associated with the kinematic altitude equation. Another case of pseudo-equilibrium deals with the acceleration phase on the ground during takeoff. The moment equation and lift are supposed to be equilibrated. The propulsion equation is “dynamic” with a derivative of the velocity module not equal t o zero. All these equilibriums or pseudo-equilibriums are linked t o the notion of the aircraft’s performance, just as the study of dynamics is associated with the notion of the flying qualities.
7.1.3
The conditions of equilibrium
Here, the question is how to define an equilibrium or an pseudo-equilibrium and the practical consequences that proceed from this definition. In order to do this, the conditions of the resolution of an equilibrated system must be examined. In most cases, the system is strongly non-linear and only a numerical resolution is viable. However some useful information is furnished by the resolution of a linear system. These results could be extended in general to the cases of equilibrium of non-linear systems. Thus a linear system is X
= AxX+BUU+BWW
(74
There is the special case where the wind W and the components of controls U are known around the equilibrium We, U,. The state X in equilibrium is determined by writing the conditions of equilibrium
Xe
=
o
Thus There is only one solution to equilibrium for a value of U, and We. This signifies that when the wind conditions are given, there is only one state of equilibrium for
7.1 Equilibrium notions
183
the position of the controls. Otherwise, for a stable aircraft having a linear system behaviour, it is enough to position the control surfaces to attain the only position of equilibrium, for example, defined by the velocity, the altitude, etc. This equilibrium will be achieved through the modes of the aircraft of which certain are very long and others badly damped. This can thus constitute only a rudimentary means of piloting. However this result still remains fundamental and of great practical importance for the understanding of the behaviour of the aircraft. Then the aircraft, without the pilot, recognizes by itself the vertical position through the spiral mode and the altitude through the altitude convergence mode. It even recognizes the latitude through the navigational modes but with a dynamic so low that it has no pratical sense. In the general cases, it is necessary to find the vector
z =
(E)
(7.3)
such as
x = o The aircraft system is then written
AZAZ = 0 with
AZA = [ A x B u B w ] the dimension of the square matrix Ax the dimension of Bu the dimension of Bw the dimension of AZA is therefore
is is is
nx n nxm, n x m,
nx ( n + m,
+ m,)
There will be a non-trivial solution to the equilibrium if it can be written
AZZ,
=
Z,
with the squared matrix Az non-singular of order (n to zero. The equilibrium solution is given by
(7.6)
+ mu + m,)
and 2, not equal
Therefore, in order to define the equilibrium of the aircraft, the question is to complete the system of aircraft equations AZA,by using the independent equations specifying the values of the state, the controls or the wind in order to obtain Az. The number of these independent equations must be equal to the number of controls increased by the number of wind states. ~
~
~
~~~~~
Dynamics of Flight: Equations
7 - Equations for equilibrium
184
These independant supplementary equations are given by
Thus
In the particular case seen previously (Equation 7.2, p. 182), these m, mentary equations were
+ m,
supple-
In most cases, it is possible to define the equilibrium by the values of the aircraft state. For example, in the longitudinal flight, it is possible to fix the altitude h and the velocity V. In the case of the pseudo-equilibrium, it is taken into account by fixing the first values of Zo to the values not equal to zero. z o
=
(Ze)
t 7.9)
The n first values of Z, and Xe correspond to the derivative of the aircraft state. In the case of equilibrium Xe = 0, for a pseudo-equilibrium, certain of these values can be not equal to zero. For example, in the case of pseudo-equilibrium in a non-zero climb angle y # 0 the kinematic altitude equation is written /Le
= Vesinye
(7.10)
To process this particular case, it is sufficient to fix the value of h e in X e , that is to say to intervene on the first values of Z,. Another met hod of processing the pseudo-equilibrium consists in eliminating certain equations of the system. In the preceding example, it is necessary to eliminate the kinematic altitude equation (7.10) as has been shown in the beginning of this chapter. This elimination of the equation does not modify the number of supplementary equations to add to the system in order to solve the equilibrium. Generally there are four supplementary equations, the number of controls. Thanks to this example, the consequences on the results of equilibrium due to the definition of pseudo-equilibrium will be examined. Take the eliminated equation in the system
The pseudo-equilibrium will free the relationship of equilibrium fi(X) = 0, here Ve sin ye = 0. Thus in this particular case, the climb angle Y e does not have to stay at zero and the climb or the glide of the aircraft can be considered as equilibrated or rather pseudo-equilibrated. For the other equations, the freed constraints are of the type such as the angle of attack/pitch control for the moment equation of pitch, angle of attack/velocity for the equation of sustentation, etc. In the flight situation of pseudo-equilibrium, xi can be a varying state, as in the above example the altitude h
7.1 Eauilabriurn notions
185
is varying. However to define the pseudo-equilibrium, in general, a initial value must be given t o zi. This pseudo-equilibrium will thus be defined for a value of xi, which is no more a state but a parameter of the system of equations. Here the climbing flight will be defined by a value of the climb angle y, the result of the resolution of the equilibrium, however it will be around the initial value of the altitude hi defined previously as a parameter. The altitude h is no longer a state variable but a parameter. With the numerical research of equilibrium (Section 7.2, p. 186) it is shown than only an element of Z (Equation 7.16, p. 187) is a result of equilibrium and as ki = fi(X) is no longer an equation of the system, zi is no longer a state and is no part of Z and not a result of equilibrium resolution. It has been shown that the aircraft system can be decoupled into a longitudinal and lateral system, by means of several hypotheses (Section 5.4, p. 132). When these two systems are rigorously decoupled, the search for equilibrium must take into account an independent lateral equilibrium and longitudinal equilibrium. This remark must not be neglected even in the situation where the two systems are not rigorously decoupled. This decoupling corresponds, nevertheless, to a more or less marked physical reality. If the numerical difficulty of the resolution of equilibrium is t o be avoided, the consequences of decoupling must be taken into account. With the numerical research method of equilibrium (Section 7.2, p. 186), the case of equilibrium with three longitudinal conditions and one lateral, by playing with the coupling, have been nevertheless resolved. Practically speaking, this signifies that two supplementary independent equations will be taken with the longitudinal parameters (for example h = h e , V = ),(I and with the lateral parameters (for example p = p e , 4 = 4 e ) . It must be remembered that the two longitudinal parameters must be defined in order to define the two independent supplementary equations because there are two longitudinal controls, the pitch control and the throttle. There is the same problem for the lateral as there is a control for roll and a control for yaw. If there were a supplementary control, another parameter would have to be defined. Thus for a triplane aircraft with a canard and a horizontal back tail, it is possible to define a supplementary condition for equilibrated flight. Thus while cruising in a classical aircraft, for a given altitude, if the velocity is defined as a supplementary equation of equilibrium, the angle of attack is imposed by the equilibrium. However, for a triplane, it is possible t o define the velocity and the angle of attack independently. This is the supplementary degree of freedom. The resolution of equilibrium is performed in several steps. 0
The choice of what system to equilibrate. the general equations the general equations without the navigational equations the longitudinal or lateral equations
0
The definition of the level of equilibrium.
A true equilibrium by taking X e = 0 in Z,, or a pseudo-equilibrium by taking certain components of X e not equal to zero or by eliminating certain equations from the system. 0
The characteristics of equilibrium. Dynamics of Flight: Equations
7_- - Equations for equilibrium -
_186 ____
The wind being most of the time considered as a known quantity, it is necessary t o determine the m, values of the components of the wind vector We t o equilibrium. The simplest case is equilibrium without wind, with We = 0. The m, supplementary equations which characterize equilibrium. In general, for longitudinal equations, there is a pitch control 6, and thrust control 6, and lateral control, roll 61 and yaw 6,. Therefore it is necessary t o define four independent supplementary equations. To take into account the preceding remark, it is preferable t o take two equations associated with the longitudinal states and two equations associated with the lateral states. For the longitudinal equations, it is possible to fix the altitude h and the velocity V (or the angle of attack a).However it is ill-advised to fix the pitch velocity q because this supplementary equation in the case of pure longitudinal flight is not independent, since the kinematic pitch velocity equation (& + i, = q ) gives q = 0 t o the equilibrium. However it is possible t o define a pseudo-equilibrium as a resource, for example t o a constant angle of attack c i = 0 thus i/ = qe = qe. It is possible t o fix the velocity V and the angle of attack Q as an altitude h in order that the relationship between V and Q through the lift equation will be satisfied. However if a climb or glide pseudoequilibrium flight is looked for by eliminating the altitude kinematic equation, for a given altitude the lift equation cannot be satisfied for any pair of V, a. In this case, the supplementary equations are not independent. For the lateral equations, the sideslip angle ,O and the bank angle 4 or even the yaw velocity T can be fixed. However fixing the roll velocity p must be avoided; it is practically zero in equilibrium and constitutes a weak independent supplementary equation.
7.2
Numerical research of equilibrium
In section (7.1.3), p. 182 the conditions of equilibrium of a linear system have been examined. Here a numerical research method to resolve equilibrium of the non-linear system is proposed based on the results of section (7.1.3), p. 182. The method is based on the linearization of the non-linear system around whatever known initial state denoted “i’’. (7.11) The linearization (Section 6.1, p. 158) is written
AX = A x A X + B U A U + B W A W
(7.12)
with
AX=X-Xi AU=U-Ui
AX=X-Xi AW=W-Wi
The upplementary equations are written Zce
=
CXX+DUU+DWW
(7.13)
187
7.2 Numerical research of equilibrium Linearized, they take the form of
(7.14)
0 = CXAX+DUAU+DWAW
This linearization is used for the research for the solution of equilibrium on the nonlinear system linearization for which
For true equilibrium Xe = 0 for the pseudo-equilibrium a part of Xe can be not equal t o zero, where
AX = Xe-Xi By placing as before (Equation 7.3, p. 183)
(7.16)
Z The equation is obtained
AzAZ
=
(7.17)
AZ,
with
(7.18)
(7.19) and
AZ
(7.20)
= Az-' AZ,
The difference AZ thus foresees the conditions of equilibrium conditions (i )
(e)
from the initial
(7.21) where Ze
=
AZ+Zi
(7.22)
The initial state Zi is a known quantity of the problem. The difference between the estimated state of equilibrium Ze and the initial state Zi, AZ is calculated (Equation 7.20, p. 187) thanks t o the difference between the derivative of the state vector expected Xe and its value at this step Xi. Equilibrium can thus be estimated by equation Dynamics of Flight: Equations
7 - Equations for equilibrium
188
(7.22), p. 187. This process of calculation will be done again until the convergence of the solution, by reinitializing each step Zi by Z e . An example of computer code is available in section (H), p. 321, showing how this method can be numerically implemented. The case treated assumes We = Wi = 0, with the hypotheses of a flat and fixed Earth. In the case of a transport aircraft, this method converges very quickly, whatever the case of equilibrium might be. The convergence is even assured with three supplementary longitudinal equations and only one lateral one; for example, h, a,8 and 0.The solution of the equilibrium depends on the longitudinal and lateral coupling.
7.3
General equilibrium
Here, starting with the general equations (Section 4.4, p. 94), the question is to comment the conditions of the equilibrium of the aircraft with the hypothesis of a spherical, rotating Earth. Equilibrium is defined by the derivative of the state X equal to zero, as in the case for general equations
V ' = V' = Vz . . . p =q =r . h =. ALt. 4 =$ =8
= 0 for the forces equations = 0 for the moments equations = 0 for the kinematic equations of position = 0
for the kinematic angular equations
The kinematic navigational equation (4.139), p. 98 L g G = . . . is not mentioned, since it has been decoupled from the others; LgG does not intervene either in the expression of external forces or in the expression of acceleration terms. The practical and immediate consequences of these conditions of equilibrium define flight at a constant altitude ( h = 0) and a constant latitude (A& = 0). The aircraft will fly in a circle centered on the world axis North-South, in a plan parallel to the equatorial plan and the kinematic equations of position will find V ' = Vz = 0. The values of the other parameters need a longer analysis that is not the purpose of this document. With the flat and fixed Earth hypotheses and gravity independent of the latitude, the kinematic equation is freed from latitude and a generalized equilibrium will correspond to a level turning flight.
7.4
Longitudinal equilibrium
In the framework of decoupling hypotheses (Section 5.4.2, p. 137), and relative to the second form equations (Equation 5.188, p. 143) to (Equation 5.196, p. 144), the longitudinal flight in equilibrium is translated by
This is flight with a constant altitude ( h = 0) and a zero pitch velocity ( q = dr, +qa = 0). Thus this is a rectilinear level steady state flight. The equation (5.188), p. 143 to equation (5.196), p. 144 give the following relationships to equilibrium by integrating
7.4 Longitudinal equilibrium
189
the results of equation (7.23), p. 188. With a field of uniform wind velocity (Equation 5.199, p. 145) the result is
F C O SCOS(CY, ~, - a,) - + ~ S V , ~ C=D m gsiny, F C O S ~ , sin(a, - a,) + ~ ~ S V ~=C m L gCOSya M i v ++pStV;Cm = 0
q = o
(7.24)
If the vertical wind is zero (w: = 0) then the aerodynamic climb angle is zero (?a = 0). Nevertheless, in the frame of a pseudo-equilibrium, it is possible to keep the first three equations propulsion, sustentation and moment, with 7, different than zero. Three hypotheses are often used: 0
0
0
Moderate aerodynamic climb angle such as sin x and cos ?a x 1. This hypothesis is justified for most transport aircraft and allows the decoupling of the propulsion and sustentation equations with respect to the aerodynamic climb angle. Thrust parallel to the aerodynamic velocity, which comes back to imposing a, = a,. This hypothesis justified in cruise flight, allows the decoupling of the propulsion and sustentation equations with respect to the thrust. Thrust moment with respect to the center of mass G zero A4; = 0. It is assumed here that the thrust vector goes through the center of mass. ?his hypothesis has really been verified by most combat aircraft and it is acceptable for transport aircraft.
It can also be noted that on the majority of aircraft, p, hypotheses, the equations are written
x 0. With all these
F - ~ ~ S V Z C= D mgya ~ ~ S V : C L= mg Cm = 0
(7.25)
Clearly stated Thrust minus Drag = Climb angle . Weight Lift = Weight Coefficient of aerodynamic moment = 0 It can be shown that the moment equation C m = 0 is the strongest because it is independent of climb angle ya, mass m , altitude h and velocity Va. This equation gives a relationship that is somewhat linear between the position of the pitch control 6, and the angle of attack a a . I n equilibrium, the stick pilots the angle of attack. The sustentation equation shows that the lift is constant for moderate climb. The C L being linked to the angle of attack for a given altitude p and a given mass, the angle of attack pilots the velocity in equilibrium. Finally, the propulsion equation shows that to have a positive climb angle Y,, the aircraft needs a positive propulsion bilan, thus thrust superior to drag. Dynamics of Flight: Equations
190
?
________
7.5
- Equations for equilibrium
Lateral equilibrium
In the framework of decoupling hypotheses, equilibrated lateral flight (Section 5.4.3, p. 145) is translated by
4
The kinematic angular equation (5.208), p. 149 = . . . can be decoupled if the field of wind velocity is zero; see end of section (6.4), p. 170 and section (5.4), p. 132. In this case, equilibrium is defined by the four first zero derivatives. The equation (5.202), p. 148 to equation (5.208), p. 149 give the relationships t o equilibrium by integrating the results of equation (7.26)) p. 190. With a field of zero wind velocity, the following is obtained mV,(-psina,
rq(C - B) pq(B - A)
+ +
rcosa,) = mg [sin 8 cos a, sin p, cos B(sin 4 cos p, - sin a, cos 4 sin p,)] - +psv,2cy F [cosp, sin p, - sin@, COS P, cos(aa - a,)] -
+
+
+
Epq = +pSl?V:CZ+ Mba
+
Erq = +pSl?V:Cn Mkz p = - tan 8(q sin 6 + r cos 4 )
(7.27)
It must be remembered that equilibrated longitudinal flight (Equation 5.73, p. 118) gives 0 = 0, thus 4 = r tan 4. After integration of this result in equation (7.27), p. 190 it yields
mV,r cos a,( 1
+
tan 8 cos$ t a n 4 =
mg sin 8 cos aa sin O ,, mg cos B(sin 4 cos Pa - sin a, cos 4 sin p,) +psv;cy
F [cospa sin Pm - sin pa cos ,&,cos(a, - a,)]
C-B+E-
cos 4
-
P =
-r-
tan 8 cos 4
(7.28)
It is frequent to adopt the following simplifying hypotheses: 0
The inclination angle of the aircraft is weak 8 M 0, this leads to p x 0 (last part of equation (7.27), p. 190).
0
The angle of attack is weak a, x 0.
0
The engine angle of attack is weak amM 0.
7.5 Lateral equilibrium 0
0
0
191
The engines do not create a roll moment Mka = 0. This hypothesis is well verified. The engines do not create a yaw moment Mk, = 0. This hypothesis is well verified, except for the case of engine failure on a multi-engine aircraft. The sideslipe angles p, and sinp, = p,, sin p, = pm.
pm
being weak, this gives:
COS&
= cospm = 1,
After integration of these hypotheses, the simplified equations are written in the following form
In addition, the fuselage axis xb is usually the principal axis of inertia, then E = 0 and the equation of yaw moment is reduced t o C n = 0.
Dynamics of Flight: Equations
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Part I11
Appendices
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Appendix A
Transformation matrices between frames
A.1
Transformation matrices from frames FI to FE and from FI to F,
The inertial frame FI (Section 2.1.1, p. 14), which is a Galilean frame, is a geocentric inertial axis system. The origin of the frame A being the center of the Earth, the axis “south-north” ZI is carried by the axis of the Earth’s rotation, axis XI and yz keeping a fixed direction in space. The normal Earth-fixed frame FE (Section 2.1.2, p. 15), is linked to the Earth. The origin 0 is a fixed point relative to the Earth and the axis ZE is oriented following the descending direction of gravitational attraction g,. located (Section 4.3.1, p. 82) on 0. This frame is therefore fixed relative to the Earth (Figure 2.7, p. 22). The axis zo of the vehicle-carried normal Earth frame Fo (Section 2.1.3, p. 16) is oriented towards the descending direction of the local gravitational attraction g, in G, the center of gravity of a aircraft. The axis z, is therefore the direction of the gravitation as viewed by the aircraft (Figure 2.7, p. 22). A general transformation matrix is defined for the two cases, the tranformation from FZ to FE,and from Fz t o Fo (Section 2.2.2, p. 21) and (Section 2.2.3, p. 26). lrstrotation -wt about the axis ZI 2nd rotation -Lt about the axis y k = yi
2
X’ cos(-wt)
- sin(-wt)
cos(-wt) 0
195
O 0 1
) ( =
coswt -sinwt 0
sinwt 0 coswt 0 0 1
A - Transformation matrices between frames
196
cos(-;
TLt
=
=(
0 - sin(-; -sinLt 0 cosLt
0 sin(-; - Lt) 1 0 - Lt) 0 cos(-f - Lt) 0 -cosLt 1 0 -sinLt
- Lt)
T i E = Tut T L t =
- sin Lt cos wt
sin Lt sinwt cos Lt
sin wt coswt 0
- cos Lt cos wt
cos Lt sinwt - sin Lt
(A.3)
Transformation from FZ to FE: The latitude is Lt = Lto and the stellar time is wt = wto, then the transformation matrix TiE is TIE. Transformation from FI to Fo: The latitude is Lt = LtG and the stellar time is ,; is Tz, wt = WtG, then the transformation matrix a
A.2
Transformation matrix from frames FE to F,
As defined in equation (2.14), p. 22 and equation (2.15), p. 22 the longitude of G and the latitude of G with respect to 0 are
Thus, the transformation matrix T E from ~ the normal Earth-fixed frame FE (Section 2.1.2, p. 15), to the vehicle-carried normal Earth frame Fo (Section 2.1.3, p. 16) is obtained thanks to the two transformation matrices obtained previously (Section A.1, p. 195)
Tzo =
TEZ
all
=
- sin LtG cos UtG
sin WtG - cos LtG cos WtG sin LtG sin wtG cos WtG cos LtG sin wtG 0 - sin LtG cos LtG - sin Lto cos wto sin Lto sin wto cos Lto 0 sin wto cos wto - cos Lto cos wto cos Lto sin wto - sin Lto
= sin Lto cos wto sin LtG CoswtG
+ sin wtG sin LtG sin Lto sin wto
A.2 Transformation matrix from frames FE to F,
+
a12 a13
197
cos LtG cos Lto = - sin w t sin ~ Lto cos w t o COS w t sin ~ Lto sin w t o = COS L ~ COS G w t sin ~ Lto cos w t o sin Lto sin wto cos LtG sin w t ~ - sin LtG cos Lto
+
+
+
+
a23
= - sin wto sin LtG COS w t ~ cos w t o sin LtG sin w t ~ = sin oto sin w t ~ COS w t cos ~ wto = - sin oto cos L ~ COS G w t ~ cos wto cos L ~ sin G w t ~
a31
=
a32
=
a21
a22
+
COS Lto COS wto
+
sin L ~ cos G W t G + cos Lto sin w t o sin L ~ sin G W
~ G
cos LtG - sin w t COS ~ Lto cos oto + cos w t cos ~ Lto sin w t o = COS Lto COS w t o cos LtG COS w t ~ cos Lto sin wto cos L ~ sin G wtc + sin Lto sin LtG - sin Lto
a33
+
+
Then
+
+
a13
= sin Lto sin L ~ G ( CwOtSo COS w t ~ sin w t sin ~ w t o ) cos LtG cos Lto = sin Lto (cos w t sin ~ w t o - sin w t sin ~ wto) = COS L ~ sin G Lto(cos w t cos ~ oto sin w t o sin w t ~ -) sin LtG cos Lto
a21
= sin LtG(coswt0 sinwtG - sinwto coswto)
a22
=
C O S ( ~ ~ G
a23
=
COS L
a31
=
COS
a32
=
COS Lto
a33
=
COS
all a12
+
--do)
~ sin(wtG G - wto)
Lto sin L ~ G ( Cwto O S COS w t +~ sin w t o sin w t ~ -) sin L t c cos LtG sin(wt0 - W
~ G )
Lto COS L ~ (COS G w t o COS w t +~ sin oto sin w t ~+) sin Lto COS L t c
Finally with COS LgG
sin LgG
+
= C O S ( ~ ~-Gw t o ) = coswto C O S W ~ G sinwtG sinwto = sin(wtG - w t o ) = coswto sinwtG - sinwto cosoto
The transformation matrix TE, is obtained
First row all a12 a13
sin Lto sin LtG cos LgG + cos LtG cos Lto = -sinLtosinLg~ = cos L ~ sin G Lto cos LgG - sin L ~ cos G Lto =
Dynamics of Flight: Equations
A - Transformation matrices between frames
198
Second row
Third row a31
a32 a33
A.3 A.3.1
= cos L to sin LtG cos L ~ G sin L to cos LtG
-cosLtosinLg~ = cos L to cos LtG cos LgG =
+ sin Lto sin LtG
Transformation matrix from frames F, to Fb First angular system
The rotation which allows the transformation of the vehicle-carried normal Earth frame F, to the body frame Fb corresponds to the transformation of the frame determining the orientation of one solid to another. Three angles are necessary (Section 2.2.5, p. 27) lst rotation .1c, azimuth about axis zo 2nd rotation 8 inclination angle about axis yc 3'd rotation 4 bank angle about axis xb
These three transformations are associated with two intermediate frames F, and F f .
Figure A . l : Intermediate frames The frame Fc is deduced from the vehicle-carried normal Earth frame Fo by a rotation of the azimut @ of the aircraft. The frame F,, represents the vehicle-carried normal Earth frame whose axis xc is aligned with the heading of the aircraft. The subscript ('," stands for the course or heading oriented frame. The frame F f is deduced from the course oriented frame F, by a rotation of the inclination angle 8. The subscript ((f" stands for the fuselage oriented frame.
X" = T + X C
xc= TeXf cos$
To
= Tcf =
(
C T e
-sin8
Xf = T#Xb -sin$
0
0
1
;
si;8)
0 cos8
A.3 Transformation matrix from frames F, to Fb
TOT4 = gcb
199
sin 8 sin 4 sin 8 cos 4 cos 4 - sin 4 - sin 8 cos 8 sin 4 cos 8 cos 4 cos 8
Tob = T.$TO T4 = cos 8 cos II, sin 8 sin 4 cos II,- sin II,cos 4 sin II,cos 8 sin 6 sin 4 sin II, cos II,cos 4 - sin 8 cos 8 sin 4
+
A.3.2
cos II,sin 6 cos 4 + sin 4 sin II, sin 8 cos 4 sin 11, - sin 4 cos II,
Second angular system
There exists another system of rotation which is sometimes used lStrotation $ transversal azimuth about the axis z, Znd rotation 4' lateral inclination about the axis xc 3rd rotation 8' pitch angle about the axis Yb
0 COS^' 0 sin$' cos$ sin$
cos 8' cos II,- sin II,sin 8' sin 4' sin II,cos 8' cos II,sin 8' sin 4' - cos 4' sin 8'
+
-sin# cos#
-sin$ cos$ 0
0 1
+
- cos 4' sin II, sin 8' cos II, sin II,sin 4' cos 8' cos 4' cos II, sin 8' sin II,- cos II,sin 4' cos 8'
sin 4'
cos 6' cos +'
Dynamics of Flight: Equations
A - fiansformation matrices between frames
200
A.4 A.4.1
Transformation matrix from frames F, to Fa and from F, to Fh Transformation matrix fkom frames F, to F a
The transformation of the vehicle-carried normal Earth frame Fo to the aerodynamic frame Fa is defined by three angles (Section 2.2.6, p. 31) lst rotation Xa aerodynamic azimuth angle about the axis zo 2nd rotation ?a aerodynamic climb angle about the axis yoia 3'd rotation pa aerodynamic bank angle about the axis X a
X0
COS X a COS Ta
Toa
=
sin x a COS ?a
- sin
A.4.2
COS X a
sin Ta sin p a
- sin x a COS F a
sin X a sin
+
sin pa
+ sin
Xa
sin xa sin Ta - COS X a sin p a
COS p a
COS X a COS pa
COS
sin Ta sin pa
COS p a COS X a
sin
Transformation matrix from F, to
COS pa COS r a
Fk
I
By analogy with the previous process, Tok is obtained with a substitution, in To,, of Yk, pa for pk and Xa for X k .
ya for
A.5 A.5.1
Transformation matrix from frames Fb to Fa and from Fb to Fk Transformation matrix from Fb to Fa
The transformation of the body frame Fb to the aerodynamic frame Fa is, in reality, the transformation of one vector to another, from the fuselage axis vector xb, to the
201
A.6 l+ansformataon matrix from frames Fk to Fa
aerodynamic velocity vector xa. The axis x, is carried by the aerodynamic velocity Thus only two rotations will be necessary lStrotation -cYa about the right wing axis Yb 2nd rotation Pa about the axis z , = zi
Va.
cosp,
T a a=
A.5.2
1
sincu,
0
COSCY~
- sin @a COS a,
- sin cua
COS P a
- sin
sin
Transformation matrix from Fb to
By analogy with the previous process, for cYk and P a for @ k .
A.6
0
0 -sincy,
sin @a COS ,& sin
=
0
coscy,
COS cya COS P a
Tba
-sinp,
Tkb
COS
Fk
is obtained with a substitution, in
Tab,
of
Transformation matrix from frames F' to Fa
The transformation of the kinematic frame Fk to the aerodynamic frame Fa will allow the kinematic velocity Vk to be connected to the aerodynamic velocity V , . These two velocities axe made up with the wind velocity V,. Therefore, it is not surprising to see the angles for "wind" indication appear. The transformation from frames Fk to Fa is accomplished with three rotations (Section 2.2.9, p. 35) lst rotation -a, wind angle of attack about the axis Yk 2nd rotation P,, wind sideslip angle about the axis zkio 3'd rotation p, wind bank angle about the axis Xa
TPw
=
1 0 0 cosp, 0 sinp,
-sinp, cosp,
Dynamics of Flight: Equations
A - Transformation matrices between frames
202
cosa, 0 sina,
Y a w=
cos a , cos p,
Tka
- cos a , sin 0 , cos p, - sin a,,,sin p,
cos a , sin p, sin p, - sin a , cos p,
cos p, cos p,
- sin p, cos 0 ,
- sin a , sin pWcos p, cos a , sin p,,,
sin a , sin pw sin p, cos a , cos p w
sin pW
=
sin awcos p,
0 -sins, 1 0 cosa,
+
+
The transformation of Fk to Fa is therefore defined by a , and pw, wind angle of attack and wind sideslip angle. These angles could have been defined by the inverse transformation Fa to Fk, or by the inversion of the order of rotations ( a , and pW).It is a question of convention and as for example, for an inversion on the two rotations a , and pw, this gives
cos a , cos p,
- sin p, cos p, - sin p, sin a , cos p,
cos a , sin p, sin a ,
A.7
sin p, sin p,
- cos p, sin a , cos p,
- sin p, sin a , sin p,
cos p w cos p w
- sin p, cos 0 , - cos p, sin awsin p,
sin p,,, cos crw
cos a , sin p,
Probe angle of attack and sideslip angle
h n s f o r m a t i o n matrix from the bodp frame Fb to the probe frame Fa The probe for the measurement of angle of attack and sideslip angle is usually mounted with a rotation axis parallel to the body axis zb (Section C.5, p. 220). This leads to the following transformation matrix
Xb
=
Tp,T-,;, xa= Tba xa
COSPL, - sinpas
Tb,
=
(
0 cosa;,
O
sina;,
0
0
0 -shahs; 0
rotation about
zb
rotation about Y a
COSCY~,
cos a:, cos pLs - sin cosa;, sinp;, cospL, 0 sin a;,
@As
- sin ahs cos pis
- sina;, sin@:, cos a;,
A.7 Probe angle of attack and sideslip angle
203
Due to the particular rotation axis zb and particular transformation from Fb to Fa, the angle of attack and sideslip angle measured by the probe ahs, are not exactly conventional as defined in section (A.5), p. 200. This particular rotation axis leads to an inversion in the order of rotation between cy and p. The relationships between these two sets of angles are calculated in section (C.5), p. 220.
@LS
Dynamics of Flight: Equations
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Appendix B
Angular relationships
B.l
Relations between angles of attack and sideslip angles
The matrices of transformation between the aircraft body frame Fb and the aerodynamic and kinematic frames F, and Fk, are linked by the following relationship
The calculus of recalled
is not completely developed here but the Tba structure is
'II'bk'II'k,
cosa,cosp, sin@, cosp,sina,
=
Tba
-sina, 0 cosaa
X cosp, X
The element on the first column and second row yields
for a, = 0, sinp, = sin@
+p,) and as -f 5 p 5 f
The element on the first column and third row yields COS
Pa sin a,
= sin arc cos pk cos awcos Pw - sin p, sin a k sin P k
for a, = 0 then cosp, sina, = sinak cos(pk sina, = sinak; then
a,
=
ak
for
a, = O 205
+ p,) if
+ cos a k sin a , cos pw (B.3)
and thanks to equation (B.2) lr
lr
--
B - Angular relationships
206 If sideslip angles
P,,
(Pk,
PO)
are assumed equal t o zero this relation yields = sin a k cos a,
sins,
+ cos ak sin a,
and sina,
= sin(ak +a,)
for
Pk
= P, = Pa = O
The element on the third column and first row yields -
sin a,
= cos a k cos P k (cos a, sin p, sin P, - sin a, cos p, ) cos a k sin pk sin pw cos P, - sin a?k (sin a?, sin p, sin P, cos a?, cos p,)
+
+
sin aa = cos p, (cos a k sin a,
+ sin a k cos a,)
The element on the third column and third row yields = sin a
COS
sin p, sin p, - sin a, cos p, ) sin p, cos P, + cos ak sin a , sin p, sin P,
k COS P k (COS a,
+ sin a k sin P k
+ cos a, cos p, (B-7)
if
Pk
= P, = 0 then
cos a, = cos p, (cos a k cos a, - sin a k sin a,) cos aa = cos p, COS(Cuk a,)
+
The element on the first column and first row yields COS a a COS
Pa
=
COS ~k COS P k COS CY,
- sin a k
with the sideslip angles PO,,& and
COS
sin a , cos p,
P, - sin P,
COS a k
sin P k
P, equal t o zero then
COSaa
= cos a k cos a, - sin a k sin a,
cosa,
= cos(ak +a,)
for
Pk
= P, = Pa = 0
(B.lO)
The element on the second column and second row yields COS
Pa
ifa,=O
=
- sin ,&(cos a,
sin P,
COS pw
+ sin a, sin p,) + cos P k cos P,
cos p,
( B. l l )
B.2 Relationship between the angles of attack, inclination, and azimuth
207
then
Particular case: For a, = 0 the relation equation (B.2) yields pa = P k equation (B.12) yields cos& = cosp, cos(pk + p,). Then pw
= 0
when
aw
=o
(B.13)
By analogy with the previous process, for the sideslip angles zero the equation (B.8) and equation (B.lO) yield p,
= 0
For the sideslip angles (B.lO) yield
+ PW, and the relation PO, P k and pw equal to
when the sideslip angles ,8 are equal to zero
(B.14)
pa, pk and p, equal to zero the equation (B.4) and equation
General case: The aerodynamic angle of attack a, and sideslip angle P O can be obtained as a , ak,p k , function of wind and kinematic angles of attack and sideslip angles a W p,, independently of p,. The sideslip angle pa is determined by equation ( B . l ) and the angle of attack a, is determined by equation (B.3) and equation (B.lO).
B.2 Relationship between the angles of attack, inclination, climb, bank, sideslip and azimuth The matrices of transformation between the Normal Earth-fixed frame F, and the body frame Fb and aerodynamic frame Fa, are linked by the following relationship
which gives the relations between the angles. However, two others relations are also available.
Depending on the angles known and the angles to be determined, one of these three relations will be chosen. The first one seems simpler in most cases, but an example with the others will be given. All the relations found in this section, are transposable to the kinematic angles by the exchange of the subscript “a” to the subscript “k”. First the expression of the transformation matrices is recalled cos 8 cos $ sin 8 sin 4 cos $ - sin $ cos 4 sin$cos8 sinOsinq5sin$ +cos$cos4 - sin 8 cos 8 sin 4
+
cos $ sin 8 cos 4 sin 4 sin $ sinOcos#sin$ - sin+cos$ cos e cos 4 Dynamics of Flight: Equations
B - Angular relationships
208 sin
COS a, COS
- COS a a sin - sina,
Tab =
COS xa COS 70
p,
0
COS xa
sin 70 sin p a
- sin xa COS
- sin
B.2.1
COS
sin a, COS pa sin pa cos a,
- sin a a
COS pa COS X a
sin 70
+ sin X a sin pa
COS ^/a sin
COS /.La COS
Determination of inclination, bank and azimuth angles
The first relation Tub = ToaTab is used. The calculus of To,T,b is not developed here. Only the elements of the third row and first column of Tab, are calculated.
Determination of inclination angle 8 The element on the first column and third row yields
(B.16) The element on the second column and third row yields (B.17) (B.18) for p, = 0 then cos8sin4 therefore if pa = 0 then
=
COS 7 ,
sin pa
(B.19)
4 = 0, and reciprocally
if Pa = 0
(p, = 0
then
4 = 0)
(B.20)
The element on the third column and third row yields COS 6 COS
4
=
- sin 7 a COS
+
sin a a - COS 7,sin p, sin a a sin COS a, COS pa COS 7, (B.21)
for
0,= 0 then COS 6 COS 4
+
= - sin 7, sin a a COS COS 7, cos pa COSO = COS(Qa 7), for pa = p, = O
+
(B.22)
B.2 Relationship between the angles of attack, inclination, and azimuth
209
Finally these three relations (Equation B.16))(Equation B.20) and (Equation B.22)) yield for
= Q, = 0 if pa = 0
then then
8 = a, + (pa = O
qj = 0)
(B.23)
These relations are transposable to the kinematic angle of attack and climb angle by the exchange of the subscript (‘U” to the subscript ‘(k”,thus for ,& = 4 = 0 if P k = 0
+
then then
8 = ak ~k (pk = 0 w Q, = 0)
(B.24)
+,
Determination of azimuth angles Xa By calculating the two first elements of the first column of Tab, the following relations are obtained. The element on the first column and first row yields COS
COS
+
=
COS
COS
+ sin pa COS
- sin
- sin
COSX a COS
COS a, COS X a
sin ya sin pa
sin xa COS pa - sin a, COS pa cos xa sin ya sin xa sin pa
This relation shows that for
(B.25)
=0
Therefore with pa = 0 and pa = 0 the following relation is obtained
Then
The element on the first column and second row yields sin
+
COS 8
=
COS a a COS
sin x a COS
- sin
- sin pa COS COS x a COS pa sin sin pa cos xa
+
This relation shows that for sin@cosO =
COS
sin X a COS
sin X a sin ya sin pa
COS
- sin a a COS pa sin x a
(B.29)
=0 - sin
COS/.i.la
sin ya
sin X a sin ya
+ sin
(B 30)
sin pa cos xa
Dynamics of Flight: Equations
B - Angular relationships
210 With
/3a
= 0 and pa = 0 the following relation is obtained
sin2()cos8 = sin X a COS a a COS 7, - sin a a sin Ya = sin X a Cos(aa + Ya) = sinXacos8
(B.31)
Then sin$
= sinx,
(B.32)
Finally these two relations (Equation B.28) and (Equation B.32) yield (B.33)
Determination of bank angle pa The aerodynamic bank angle pa can be determined through the equation (B.17), then cos 8 sinp, = sin 4 tan Ya tan (B.34) COS
for
COS /3a
+
= 0 then
(B.35) for Ya = 0 then (B.36) For small values of the aerodynamic climb angle and sideslip angle 3/, between the aerodynamic bank angle pa and the bank angle 4 yields sinpa =
B.2.2
cos 8 COS Ya COS /3a
sin 4 x cos 8 sin 4
the relation (B.37)
Determination of angle of attack say sideslip pa and bank pa angles
Thanks to the second relation between the transformation matrices, another kind of relationship between angles can be found. The angles pay a a ) /3a are determined as a function of 8, 4, $, Ya, X a thanks to the relation The matrix T0b is fully known by 4, 8, $. For Tea, X a and The matrix Tea can be rewritten with the following notations
are assumed known.
B.2 Relationship between the angles of attack, inclination, and azimuth
21 1
(B.40) (B.41)
(B.42)
Tsao
0
a12
a22
a32
4 1 2
4 2 2
=
Tcao =
) )
0
(
0
0 b12
0 b22
0 0
a12
a22
a33
(B.43)
(B.44)
The basic relation is used (B.45) This matrix has the following structure (B.46)
(B.47)
(B.48)
Method for solving First step: Determination of a,, and 0, Dynamics of Flight: Equations
B - Angular relationships
212 On the first row of
TaoTob @a
= T a b it can be obtained that sin@, = ai2, so
= arcsinai2
7r
--
2
5p <-
therefore /3, is determined and also cospa, then
7r
a - 2
(B.49)
can be achieved with (B.50)
Second step: Determination of pa With the third row, first and second element, the following system can be solved (B.51) which gives
(B.52) These last relations about the bank angle are more complicated than the equation (B.34), p. 210. This is an example where the method used is important in order to simplify the final result.
Particular case, relations between sideslip and azimuth angles For 8 = 0 the second column of is equal to (B.53) and for 70 = 0 the first row of l l ' ~ ~ ,is, equal to (B.54) From the equation (B.49)
If in addition, it is assumed that q5 = 0
B.2 Relationship between the angles of attack, inclination, and azimuth
B.2.3
213
Third determination of the bank angle
Another way for the determination of pa can be used through the third relation Tao = The two last terms of the third column of Tao are calculated. Then
Tab.Tbo.
sin 8 cos
+ cos pa cos 8 sin 4 - sin a a sin pa cos 8 cos 4
sin
+ COS
sin 8 sin
=
COS ?a
cos 8 cos 4 = cos ?a
sin pa cos pa (B.57)
This yields sinpa COSpa
= =
sin 8 cos a, sin pa sin 8 sin a a
With the hypothesis if cospa > 0
pa = arctan
+ cos pa cos 8 sin 4 - sin aasin pa cos 8 cos 4 (B.58) COS ?a
+ cos Pa cos 8 cos 4
(B.59)
cos 70 -T
< pa < T and the result achieved for
tan 4
+ tan
: '; ; ;
(tan e 1 tan 8 c o
+
-
~ ~ ~ ; s
(Section C.2, p. 216)
+ sgn(sinpa)
T
(B.60)
Dynamics of Flight: Equations
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Appendix C
Relationships between angles and velocities
C.1
Velocity components of Va,
Vk,
Vw
The fundamental relationship between the Earth velocities (Equation 3.17, p. 48) gives
v,
=
vI,-v,
The wind velocity V, is known in the vehicle-carried normal Earth frame F,
v;
=
(2) w:
Components expressed in the body frame F ,
vg
= vL-TboV;
then
w; - w;
with U:
= u::cosecos++vt',sin$cose
U;
= u;(sinOsin+cos$
w,b
=
+
+w; cos 8 sin u:(cos$sinOcos+ +wt', cos e cos
+
- wEsin8
- sin$cos+)
+ uz(sinesin+sin$ + c o s ~ c o s + )
+ sin+sin$) + ut',(sinOcos+sin$
-
sin+cos$) (C.2)
215
C - Relationships between angles and velocities
216
and in another form
Components expressed in the kinematic frame
vt
= v,k-l[lkoV$
then
(q
with
= Tko(
WW
The components of Tk, are functions of
C.2
~ k x,k
;)
WW
and pk (Section A.4, p. 200).
Aerodynamic angle of attack a a and sideslip angle Pa
From the expression Vg = TbaV,",the relations between two forms of components of Vg, otherwise U:, U:, w: and Va, a a , p,, are search for. Basic definitions give and
V:=
(i)
However, in any case the modulus are always the same, and by definition Va va
-4
=
Then with Tba (Section A.5, p. 200), it is obtained
so
> 0 so (C.6)
217
C.3 Aerodynamic climb and azimuth angles ?a and Xa
with
-5 < Pa < $ according to convention, then Pa
From the knowledge of
= arcsin
($)
the angle of attack a. can be determined COScYa
=
sin&,
=
-K <
with
4
Va COS Pa
:
(C.10)
Va COS D o
< K tana ,
=
w:
(C.11)
U:
depends on As the function arctan is defined between - $ and $, the expression of the sign of cos ao. determined a a in quadrants 1 and 4. > 0 the expression = arctan If COS is in quadrants 2 or 3. If COS^, < 0 then Therefore
3,
(C.12)
As a conclusion according to convention
-n <
aa
if
U:
<0
Cta
=
:
w b arctan- + s g n ( w , ) ~ U:
(C.14)
C.3 Aerodynamic climb and azimuth angles T a and The angles ?a and Xa are defined between the aerodynamic frame F a and the vehiclecarried normal Earth frame Fo+ The relationships looked for will thus be obtained from the vectorial relationship
Expressions between the components of VE, U:, U:, wz and Va, ?a, Xa, will be determined. The process is similar to those of section (C.2). Therefore the velocity modulus expressed in the two frames yields
(C.15) ~~
-~
Dynamics of Flight: Equations
C - Relationships between angles and velocities
218 Then
Vz = ToaV:
( :i )
then
=
w:
Therefore siny, =
Va
(
COS Xa COS ’)’a
sinXacosya - sin y,
)
(C.16)
-2 according to the convention -:5 ya 5 $; then -WZ
= arcsin -
Ya
(C.17)
va
and
(C.18) With the convention C.2, p. 216)
-7r
<
xa
< 7r
and the previous result achieved for
cta (Section
(C.19)
C.4
“Wind” angle of attack aw and sideslip angle
Pw
The“wind” angle of attack and sideslip angle are angles that connect the kinematic frame Fk and the aerodynamic frame Fa. The relationships looked for will thus be obtained from the vectorial relationship
Vk
= TkaVz
Expressions between the components of V: , U:, U:, w t and V a , a w ,pw will be determined. The process is similar t o those of section (C.2), p. 216. Therefore the velocity modulus expressed in the two frame yields
Then
With Tka (Section A.6, p. 201), it yields cos a, cos pw sin ctw cos pw
(C.21)
pw
C.4 “Wind” angle of attack a, and sideslip angle
Therefore sin@, =
219
$ and according to the convention -4 5 0, 5 4 pW
From the determination of convention -7r 5 a, <_ 7r
k
= arcsin-V Q
(C.22)
VQ
p, the expression of a, can be written according to the
(C.23)
With the results of section (C.l), p. 215, the expression of the aerodynamic velocity can be found, depending on the kinematic and wind velocity, then
( 3) ( =
v k - U;
) =v, (
cos a , cos p, sin@, sin a, cos Pw
)
(C.24)
where (C.25)
sinp,
=
k
-vw cos a, cos pw vk
- U$
(C.26)
then tanp,
=
-VW
v,-
k U$
cos a, (C.27)
With another convention for the definition of the “wind” angle of attack and sideslip angle (Section A.6, p. 201), due to the order on the angle of rotation succession a,, pw the following relations are obtained. These relations result from a recalculation of equation (c.21), p. 218 but with T ’ k a (Section A.6, p. 201) instead of T k Q .
( :!) vQ ( =
WQ
sinp, =
k VQ
vacos a,
cos a , cos p, cosof,sinp, sin a ,
) k
tanp, = 21,
4
Dynamics of Flight: Equations
C - Relationships between angles and velocities
220
C.5
Measurement of angle of attack and sideslip angle with an aerodynamic probe
An aerodynamic probe, or sonde, installed on the aircraft measures a local angle of attack and sideslip angle ( ( 2 0 8 , p c 8 ) . The problem here is to find the relation between these local angles and the aircraft angles, that is to say the angle of attack and sideslip angle referred to the velocity at the center of mass (ac, @ a ) . To define these relations, the first step is to calculate the velocity referred to the probe V;,s. The results of section (D.5), p. 236 will be used.
Kinematic velocity at the probe station This calculation is made in section (D.5), p. 236. The probe position S is expressed in the body frame Fb relative to the center of mass position G GSb=
(q )
(C.28)
ZS
The equation (D.49), p. 237 yields
REMARKC.l As mentioned in the nomenclature, the abbreviated notation U; instead of u ; , ~are , used for the components of the center of mass velocity.
C.5.1
Aerodynamic velocity of the probe
The relation (Equation D.56, p. 238) yields
The wind is known in the normal Earth-fixed frame F,, and the angles therefore intervene.
4, 6, $. will
Angle of attack and sideslip angle at probe station The aerodynamic probe frame Fc,s, whose axis xc8carried the probe velocity V a , s , allows the definition of the angle of attack and sideslip angle at probe station (ass, Pas),with respect to the body frame Fb. The elaboration of the matrix Tba,S is made on the same basis as those of the elaboration of the matrix Tba (Section A.5, p. 200). The only change is the value of the angle; with Tbc,s it is the local angle of attack and sideslip angle (etas, pas).
C.5 Measurement of angle of attack and sideslip angle with an aerodynamic probe221
and
(C.33) then from equation (C.31), p. 220 and equation ( C . 3 0 ) ,p. 220
or
The probe velocity Va,s has t o be calculated from its components in the body frame Fb, in particular the angular velocities p , q, r intervene. Relationship between probe angle of attack and sideslip angle a,,, Pas, and aircraft angle of attack and sideslip angle pa The usual relations for the aircraft (Equation C.8, p. 216) and (Equation C.10, p. 217) b
sinPa = 21, Va
sina, =
w: Va C O S P ~
(C.37)
with
(C.38) U b, b
va w;
=
U; - U ;
= U; -v; = w; - w ;
The probe measures a a s and pas and possibly Va,s; therefore from the previous equations (Equation C.30) and (Equation C.31), (Equation C.38), the following relations can be written b
U, b b Wa
+
= Va,s COS^,^ COS Pas - (-uX,ZS b b (q - qz;>z; - ( r - ~y;)yg) b b = Va,s sin pas - ( - ~ y , ~ s (T - ~x;)& - ( p - pz;)~;) = sin&,, COS Pas - (-WZ:Z~ ( p - p~;)& - ( q - q x L ) x i )
va,s
+
+
(c.39) (C.40) (C.41)
Dynamics of Flight: Equations
222
C - Relationships between angles and velocities
The aerodynamic velocity of the aircraft Va can be calculated from U:, U:, 20: and then the expression (Equation C.37, p. 221) of sinpa and sins,, can be obtained
(C.43)
Probe angle of attack and sideslip angle a,,, Pas, with a sideslip probe linked to the fuselage Usually the sideslip probe is linked to the fuselage. Thus its sideslip rotation axis is parallel to the fuselage axis zb and not to the aerodynamic axis Za. The aerodynamic frame of the probe whose axis xas carried the aerodynamic velocity V,,S a t the probe station, defines the probe angle of attack and sideslip angle, which are not exactly the conventional ones defined in (Section A.5, p. 200). These new angles are denoted a:, and pi,. (C.44) This transformation matrix is defined in section (A.7), p. 202. T’ba
cos ahscos p;, cos ahssin Pb, sin a:,
=
(C.45)
o x
with a rotation Ph, about the axis zb, and a rotation a:, about the axis equation (C.36), p. 221 can be written Sinai,
=
tanp;,
=
w; - 20,b - wzb,z; U;
b
- U,
- VYL&
+ ( p - pY”,>y; +
Va,s
(T
-TxL)z;
Ya.
The
- (q - qxL)Z$ b b - ( p - p,)zs
(C.46)
Relations between a:,, Ph8 and ass, pas These relations cannot be obtained by the expression T’ba = Tb,. The two transformations from the aerodynamic frame to the body frame are not strictly equivalent. A third rotation is missing! However, it is legitimate to write the equality of the
C.5 Measurement of angle of attack and sideslip angle with a n aerodynamic probe223 aerodynamic velocity components expressed in the body frame Fb. This amounts to writing the equality of the first column of T'ba with those of Tba, then cos ahscos pkS = cos a,, cos pas cos a:, sin ,8Ls = sin pas sin a;, = cos passin a,, with sin a:,
=
sin a,, cos pas
(C.47)
tan pkS = tan pascos aaS
or tan aas = tan ahscos pLS
(C.48)
sin pas = sin PAScos a:,
Dynamics of Flight: Equations
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Appendix D
Kinematic relationships D. 1 Fundamental kinematic relation Temporal derivative of a vector This relationship will allow a vector to be derived, with respect to time, in any kind of frame and will be essential during the procedures leading to the equations of Flight Dynamics.
Establishment of the fundamental wlation
dXodt
-
-
dxl + n , , x x dt
5, 5,
The derivative with respect to the time of a vector X in the frame Fo, is equal to to the derivative of this vector X with respect to the time t in the frame Fl, which must be added the cross product of the angular velocity of the frame F1 relative to frame Fo, n10,by the vector X. Pmof; The vector expressed in the frame Fo, Xo, can be represented as a function of its expression in the frame F1, X1, thanks to the transformation matrix from the frame F' to F1, To1
The projection of the vector X in the frames FO and F1 yields to a relation which represents three analytical relations among components of X. This equation (D.2)) p. 225 will be derived with respect to time t. So, each component of X is derived in the frame Fo or F1. The formal notations of this operation are dt dXO and These three new components represent, as for example the
5.
components of the vector derived from Xo, that is to say 225
s,
5,
expressed in the frame
D - Kinematic relationships
226
Fo since this frame Fo is stayed in. Therefore the derivation of X o and X1 is correctly formulated by the notation the equation (D.2), p. 225
(s)'(s) . 1
and
From this the derivation of
dX1
1
With pre-multiplication by Tlo this yields
or 1
dX'
(D.3)
dt The m ,trix Tol has the property of transformation matrices from handed frame to another frame (Equation 2.3, p. 19), then
ort hogon 1 right-
Thus leading to the derivation
mt
-T+ dt
dT Tt-- = 0 dt
or
so T t g is a skew-symmetric matrix noted
&=
(
0
Rz
-Ry
-Rz
Rx
With the matrix & is associated the vector
&X
Ry
-ox)
0
0
at = ( Ra:
Ry Rz ), so that
= nxx
represents the cross product of 0 by X , from which we have equation (D.3), which can be rewritten
(Z) 1
($)l
=
+ni0xx1
with the vector f l h l associated with the skew-symmetric matrix h4i&o
I.
D.l Fundamental kinematic relation
227
The vector Oi0 is the projection in the frame F1 of the angular velocity vector of the frame F1 relative to the frame Fo. The elements of the skew-symmetric matrix T&* are the components of these angular velocity vector expressed in the frame F1. The equation (D.5) is a vectorial relation projected in the frame F1, under its intrinsic form, this relation is enunciated
+
dXo - dX1 OlOXX dt dt This is the fundamental relation of kinematics (Equation D.l) which is essential to obtain the numerous equations used in mechanics. For the particular case where the vector X is a product of a matrix II by a vector Y, there is some relevance to particularize the derivation of the matrix II from those of vector Y . This situation is encountered with the angular momentum, where H = IIO, otherwise X = IIY. First it can be written dX1 This leads t o the intrinsic relation
dlY1 dII1 - -Y dt dt and now, the derivation of the matrix 1, with
dY1 +I-dt
so that
dt with a multiplication by
"10
on the left, and
'IT01 on
the right
Then with the use of equation (D.6) a01 t d'ITo1 n odt = T O l T = %lo
To1 = at OITol = -'ITo1-t d'ITo1 - -*lo dt dt dt The previous expression is written m 1 0
l)$(
=
*loII1
- II1%lo
+ (?tl)
All these terms are projected in the same frame F1, so the intrinsic relation is obtained
Dynamics of Flight: Equations
228
D - Kinematic relationships
The equation (D.8) and equation (D.9) allow to write
d X o- - -=-Y+I-d ( I o Y o ) do dt dt dt
d Y o dOYO
dt
dlI1
dYo
=~loIIY-II~loY+-Y+IIdt dt
dt
in another way
+
+
d(IoYo) - a,, x (IIY)- (Ialo) X Y -Y dlI1 I-d Y o (D.lO) dt dt dt By carrying on with the calculation, the result of the fundamental relation equation (D.7) is reached - -
+
dOYO - = n,,x(IY) - ( I a 1 O ) x Y ) dt d1 dY1 dOYO - -Y + I+ a10 x (IIY) dt dt dt d0Y0 dlY1 dX1 dXo - - - -=+ a , , x ( I I Y ) = -+a1oxx dt dt dt dt Thus the fundamental relation of the kinematics applied to the vector X = IY is recovered. T w o characteristics of the angular velocity vector From equation (D.6)
projected in the frame
n
F1
From equation (D.4) and equation (D.6) *ol
=
t a 0 1 - T 0 1 r
= -*lo
and thus the relation =
001
-a10
(D.ll)
With a third frame F2, it can be written To2
= TOlTl2
F'rom equation (D.6)
so
and thus the relation
= *lo
+ WL a20
=
a21
+010
(D.12)
I
D.2 Inertial linear velocitg
229
D.2 Inertial linear velocity The inertial velocity of the aircraft center of mass G, V Z , Gis, the velocity relative to the inertial frame F I , and is defined by
dAG' dt
VI,G =
The purpose of this appendix is to examine the various expression of this velocity. With the normal Earth-fixed frame FE as relative frame and in order to make the kinematic velocity VI, appear, this relation is obtained
+-
dAG' -dOG' - dA0' dt dt dt dAOE ~EZXAO doGE + $ 2 ~xOG ' VI,G = dt dt VZ,G =
+
+
The point 0 is fixed on the Earth, A 0 is fixed in the frame FE, so
VZ,G =
= 0 then
dOGE
+ ~ E I X A+O~ E I X O G dt
VZ,G =
dOGE dt
+~ E I X A G
(D.13)
and according to definition (Equation 3.10, p. 46)
dOG -E dt
-
v k
then
Moreover, it is feasible to directly express the inertial velocity V I , Gwith the vehiclecarried normal Earth frame F, as the relative frame
V I , G = dAGo dt
+ 0,' x AG
(D.15)
For these two expressions equation (D.13) and equation (D.15), the projection frame which gives the simplest form is the vehicle-carried normal Earth frame Foe If the detailed expression of V I , Gis not looked for, a faster demonstration of the kinematic relation can be made by noting that
+ (no'- ~
dOGE dAG" VI, = - dt dt
E I xAG )
dAGo dt
+
=- f l o x A~G
(D.16)
Calculation of equation D.13 Dynamics of Flight: Equations
D - K i n e m a t i c relationships
230 With (Equation D.29)
According to definitions (Figure 2.7, p. 22)
(
AGO =
-(Rt
+ h)
)
and from equation (3.11), p. 46
so
(D.17)
Calculation of equation D.15
0 (
Y
)
O
j)
=
Here the (Hypothesis 2) spherical Earth hypothesis intervenes. If the fact were taken into account that the Earth is a ellipsoid, f i t will not be equal to zero. From equation (D.30), p. 233 =
(
-&tG cos LtG -J!JtG
wtG sin LtG
with (Equation D.31, p. 233) and (Equation D.32, p. 233)
&tG = ALg-Qt L t G = ALt
)
D.3 Angular velocities
231
then
(D.18)
Kinematic relation By equaling the two expressions of VZ,G(Equation D.17, p. 230) and (Equation D.18, p. 231), the following kinematic relation is obtained (D.19)
D.3 Angular velocities D.3.1
Determination of the Earth angular velocity
i2EI
From equation (3.7), p. 44 it can be written that
GEI=
TIE. =
-=( m Z E
dt
E dTzET =
G
(
0
-nzo
nzo -QYo
- sin Lto cos wto sin wto sin Lto sin wto cos wto cos Lto 0
nyo
0 nxo
4 x 0 ) 0
- cos Lto cos wto
cos Lto sin wto - sin Lto
+ sin Lto sin wtocjto cos LtoLto sin wto + sin Lto cos wtoLjto
- cos LtoLto cos wto
- sin LtoLto
- sin Lto cos wto sin Lto sin wto sin wto cos wto - cos Lto cos wto cos Lto sin wto
-S2xo
=
- cos wtowto sin Lto cos wto
(D.20)
cos wtoLjto
- sin wtoLjto
0
x
X (D.21)
cos Lto 0 - sin Lto
- sin wtowto sin Lto sin wto
-wtosinLto - cos wtocjto cos Lto cos wto - sin wtocjto cos Lto sin wto Qxo = -WtocosLto = =
-ay,
(
= - COS Lto cos wto - cos LtoLto cos wto
+ sin Lto sin wtoLjto 1 Dynamics of Flight: Equations
D - Kinematic relationships
232
=
+ cos Lto sin wto (cos LtoLto sin wto + sin Lto cos wtoLjto ) + sin2 L t o t t o t t o (cos2Lto cos2o t o + cos2 Lto sin2 wto) +&to (- cos Lto cos wto sin Lto sin o t o + cos Lto sin wto sin Lto cos wto)
+ t t o sin2 Lto = Lto
-&to cos Lto
+&to sin Lto
(D.22)
Therefore the time of the normal Earth-fixed frame FE is the reference time wto, the time of point 0. F’rom equation (2.16), p. 22 the following equation is obtained
wto = -Rt
(D.23)
Otherwise, the point 0 is fixed with respect to the Earth, its latitude is constant and
Lto = 0
(D.24)
Another way to find this result is to admit that only the angular velocity of the normal Earth-fixed frame FE relative to the inertial frame F I , is the Earth angular velocity Rt, and (D.25)
Moreover the following kinematic expression of
QkI is achieved (D.26)
from which
Lto = O
and
wto
=-Rt
(D.27)
Thus finally the below expression of the Earth angular velocity f 2 ~ 1is retained cos Lto
- sin Lto
(D.28)
The projection of f 2 ~ 1in the vehicle-carried normal Earth frame F, is f2~1 =
T , I ~ L , = fit
(
(D.29)
D.3 Angular velocities
233
D.3.2 Determination of the angular velocities Calculation of no^ By analogy with the equation (D.22), p. 232 of obtained
~ E I the ,
&,E
and RoI
following expression is
(D.30) The kinematic relation is achieved with equation (2.14), p. 22
from which after a temporal derivation and with equation (D.27), p. 232 L t o = 0 and
wto = -Rt, then
(D.31) (D.32)
Calculation of f l , ~ From equation (3.9), p. 45
with a projection in the vehicle-carried normal Earth frame F, and from equation (D.30), p. 233 and equation (D.29), p. 232
thus
(D.33)
D.3.3 Determination of
C2bo
The expression of f 2 b o is a function of ~,8, 4, and can be obtained from equation (3.7), p. 44 by @bo = Ir&% as the previous calculation of ~ E I With . a more “physical” Dynamics of Flight: Equations
D - Kinematic relationships
234
approach it can be written with reference t o section (2.2.5), p. 27 and section (A.3), p. 198
y;
=
(
T$y;= '
cos$
sin$ 0
-sin$ cos$ 0
0 -sin$
TO=(
= T,jTex! =
(
then
=
cos8 0
1 0 0 cos4 0 -sin4
D.3.4 Determination of
f&,b
and
- sin$
=
CO;$
)
(D.34)
- sin 4
01 sin8 0 ) 0 cos8
x x x x
coseco~~ cos8sinq -sin8 X
(
) ( :) (
cos4
-sin8
xg
0 0 1
X
) (H ) (
- sin8 cosesin4 cos8cos4
=
cos 8 cos $
(l )
)
(D.35)
f&b
The expressions of n a b and n k b are functions of a and p, and can be obtained from = &ab as the previous calculation of 0 ~ 1 With . equation (3.7), p. 44 by a more "physical" approach, the following relation can be written with reference to section (2.2.79, p. 32 and section (A.5), p. 200.
D.3 Angular velocities
235
(D.37) By analogy with the previous process, the following expression is obtained
(D.38)
D.3.5
Determination of f l k o
The expressions of the components of n k o are functions of "k, p k , X k and can be as the previous calculation obtained from equation (3.7), p. 44 by T L k Y = %ko . a more "physical" approach, the following relation can be written with of 0 ~ 1 With reference to section (2.2.10), p. 36 and section (A.4), p. 200.
projected in the kinematic frame
Fk
(D.39) projected in the vehicle-carried normal Earth frame Fo
Dynamics of Flight: Equations
D
236
- Kinematic relationships
D.4 Geographic position relationship The geographic positions x, y, z are linked to the latitude Lt and longitude Lg, since there are two different ways to give the position of the aircraft relative to the Earth. In the vehicle-carried normal Earth frame F,, the components of the different position vectors are from section (D.2)) p. 229
AGO =
(
by definition
OGO =
-(Rt
+ h)
)
(D.41)
(z)
(D.42)
XG
AGO
= AO"+OG"
from figure (2.7)) p. 22
AOE =
AOo = T 0 ~ A O E = -Rt
(D.43)
( -it)
cos Lto sin LtG cos Lgc - sin Lto cos LtG - cos L t o sin LgG COS Lto COS LtG cos Lgc sin Lto sin LtG
+
(D .44)
(D.45)
thus
x$ = &(cos Lto sin LtG cos Lgc
- sin Lto cos LtG) y z = -Rt cos Lto sin L ~ G z& = - (& 4- h) Rt (cos Lto cos L ~ cos G L ~ G sin Lto sin LtG)
+
D.5
+
(D.46)
Velocity field of the aircraft
The purpose of this appendix is to find the relations between the local velocity v k , S , V , , S a t any point of the fuselage and the aircraft center of mass velocity v k , V , ,
s
D.5 Velocityfield of the aircraft
237
the angular velocity components p , q, r and the wind velocity V w .These relations will be useful, in particular for the measurement of angle of attack and sideslip angle with a probe placed anywhere on the fuselage.
Kinematic relation The point S is linked to the aircraft. The kinematic velocity of S, V k , S , that is to say the velocity relative to the Earth, according to definition, is equal to vk,S
dOSE dt
=
and the relation between the velocities of a rigid system issue from equation (3.1), p. 43, with G the center of mass of the aircraft yields Vk,S
=
Vk,G
+SGxabo
(D.47)
The components of the position of the point S expressed in the body frame Fb, centered at G, are
GSb =
(q)
(D.48)
ZS
Thus, the kinematic velocity projected in the body frame
Fb
The classical notation is recalled, but abbreviated, with U: for u : , ~ , for v : , ~ , w: for w : , ~and v k for V k , G . This is a situation where confusion is conceivable between the components of the velocity of the point S and point G.
Wind velocity It is a matter of determination of the velocity of the air particle W placed at time t at the point S. Relative to the Earth and according to definition (Equation 3.15, p. 47) dOWg dt The equation (3.51), p. 56 allowed us to express this velocity as a function of wind velocity at G, V w , ~ . v w , s
=
so projected in the body frame Fb
See nomenclature
Dynamics of Flight: Equations
D - Kinematic relationships
238 with the projection of (EwmVE in components U & , T&, etc.
Fb,
calculated in section (D.6), p. 238 for the
Aerodynamic velocity The aerodynamic velocity of the point S, VO,s,according to definition is equal to
(D.52) and as
(D.53) (D.54) thus with equation (D.49), p. 237, equation (D.51), p. 237 and equation (D.54), p. 238
(D.55)
U;
- U; - ux;x;
U; - U; - uy:y;
w; - 20,6
- wz:z;
+ ( q - qz;,.; + - rx;,x; (T
+ ( p - py:>y;
- ( r - T&)& - ( p - pz:>z; - ( q - qx:,x;
) ( ) =
4,s U$,,
Wa,s
(D.56)
D.6
Wind velocity field, GRADVL
The problem in question is t o find the expression of @mVg but projected in the body frame Fb; that is t o say, the wind velocity field expressed in the body frame. The matrix &mVZ is known in the vehicle-carried normal Earth frame F, (Equation 3.32, p. 52).
The calculation of &mVOU, projected in the body frame Fb, is rather long. So it will be split up into three parts: one for the azimuth rotation $, one for the inclination angle rotation t9 and the last one for bank angle rotation 4. These three steps correspond t o the successive transformation from the vehicle-carried normal Earth frame F, t o the course oriented frame F,, to the fuselage oriented frame F f and to the body frame Fb (Section A.3.1, p. 198).
then
(D.58) (D.59)
D.6 Wind velocity field, CGWmV;
azimuth rotation
239 -
+ so (D.60) cos+
sin+
cos$
0
0
uxC,
((Gwmvo,)c =
0
.-sin+
(D.61)
1
-ry$
rx:
-gxc,
vy:
-p~$
pyc,
wzc,
(D.62)
with
(D.63)
ry;
+
+
+ 920, sin = q z ; cos $ - p.$ sin II) = r y ; cos2 r x ; sin2
p ~ c= , p z z cos
++
+ ++
pyc, = py; cos$ sin+ gx; = gx; cos - py; sin r x c , = r x ; cos2 r y ; sin2
+
+
+ + (ux;
- vy;) cos II,sin
++
-u
(vy;
x ~cos )
+
+ sin
(D .64)
(D.65)
inclination angle rotation 9 (D.66)
T O = (
cos9 0
-sin9
01 sin8 0 )
(D.67)
0 cos9
(D.68) with = u x cos2 ~ 9 = vy& w Z i = uXc,sin2 9 uxt;
VYt
+ wz; sin29 + (9.: + wZ; cos2 8 + (9.:
- gz):
sin 9 cos 9
- gx:) sin 8 COS 8
(D.69)
Dynamics of Flight: Equations
240
D - Kinematic relationships
pzt= pz& COS e - rX& sin e
+ qxLsin2 e + (uX; - wZ;)sin e + py; sin e
q z i = qz; cos2 e
ryf
= r y z cos e
py; = py& cose - ry&sin8
+ +
qxf = qx; COS^ 8 qz; sin2 8 rx& = rxC, cos e pz: sin e
bank angle rotation
+ (wZ; - u X ; )sine
COS
e
(D.70)
COS
e
(D.71)
+ (D.72)
T$ =
(
1
0
0 cos+ 0 sin4
-sin+ cos4
(D.73)
(D.74) with
(D.75)
(D.76)
(D.77)
Appendix E
Accelerations E.l
Inertial acceleration of the center of mass G
The inertial acceleration AZ,Gof the center of mass of a solid is obtained by the derivative with respect to the inertial frame FI of the inertial velocity VZ,Gof the center of mass, (Equation 3.18, p. 48). This inertial velocity is the temporal derivative of the G position with respect to the inertial frame FI. Thus
AZ,G
CPAGI -
= --dt2
dt
through equation (D.13), p. 229 and equation (3.10), p. 46
The relative frame used for the derivation is the vehicle-carried normal Earth frame FO
The first term can be written
+- dt +nEI
XAG
xdAGo dt
term B
term^
The second term ~ , I x V Z , G or ,term D, of equation (E.2), can be directly calculated with a projection in the vehicle-carried normal Earth frame F,. The angular velocity 241
242
E
- Accelerations
and inertial velocity V7,c have been already calculated (Equation 3.73, p. 59) and (Equation D.17, p. 230). As regards the angular velocity, they have been calculated in
flzI
section (D.3))p. 231.
So the term D O,IXVI,G
the term A
the term B (Equation D.29, p. 232) (Equation D.16, p. 229)
dnOE'xAG
(t
)O
= Rtttc
(
- sin L t c
0
- cos L t c
)( X
-(&
+ h)
)
the angular velocity of the Earth is assumed to be constant C4t = constant (Hypothesis 22) and the equation (D.19)) p. 231 is used
the term C
dAG"
0
cos LtG
- sin LtG
-h
E.l Inertial acceleration of the center of mass G
243
and from equation (D.19), p. 231
h =
vz
Thus the final result
with
The inertial acceleration (Equation E.2, p. 241) can be presented in another form in order t o give prominence t o the flat and fixed Earth hypothesis which will be used later on
Therefore the inertial acceleration AI,Gcan be modelized by
AI,G = A ' , + A A ~ + A A R
(E.11)
As the hypothesis of a spherical Earth is made, the angular velocity no^ (Equation 3.74, p . 60) is not equal t o zero and the acceleration AAs is also not equal t o zero. The increment of acceleration AAs is the complementary acceleration due to the terrestrial sphericity (subscript S ) . As the hypothesis of a rotating Earth is made, the angular velocity ~ E isInot equal to zero (0,# 0) and the acceleration A A R is also not equal Dynamics of Flight: Equations
244
E - Accelerations
to zero. The increment of acceleration AAR is the complementary acceleration due to the terrestrial rotation (subscript R ) . Then AA;
=
(
2 Rt + h
VE -VN -VE tan LtG
)(
x )z: -
(E.14)
This term AAR can be achieved from the two previous results AZ,G(Equation E.7) and A A s (Equation E.15) through complementarity, and it corresponds to the elements which contain the Earth angular velocity Q t . From this finally AZ,G = A t , + A A s + A A ~
(E.16)
with (E.17)
(E.18)
E.2 Two forms for the derivative of the kinematic velocity Vj From the derivation of Vk (Equation 3.11, p. 46) with respect to the vehicle-carried normal Earth frame F,
(T)"(8) =
(E.20)
- Vz
the derivation of Vk with respect to kinematic frame Fk is obtained thanks to equation (3.1), p. 43 (E.21)
245
E.3 Inertial angular momentum deriuatiue
as
(2) dvo = Tok (2) O
k
(E.24)
thus, this relation is obtained from equation (E.20)) p. 244 and equation (E.22)) p. 245 (E.25)
E.3
Inertial angular momentum derivative
The complement of the inertial acceleration for the moment equation is the inertial angular momentum derivative. The purpose of this appendix is to achieve this essential result.
The angular momentum The angular momentum of the aircraft with respect to G (center of mass of the aircraft ) and with respect to the inertial frame FI (Equation 4.48, p. 79) is equal to
(E.26)
The inertial angular momentum derivative This derivation is expressed vectorially with respect to a relative frame, for HE,G the body frame F b , and for AHG the normal Earth-fixed frame FE.
~~
Dynamics of Flight: Equations
E - Accelerations
246
dHi,G dt
- dH'?~ dt
term A
+n~z x HE,G
+-dAH; dt
term B
term C
+nb,qXAHG term D + ~ E I X A H Gterm E The notation A I H ; , ~for the sum of terms B, C and D will be used
(E.28) This term will be equal to zero if the fixed Earth hypothesis is made ~ E =I 0 ) . The notation A2H;,G for the term E will be used
AZH;,,
(nt = 0 or also
= QEZXAHG
(E.29)
so (E.30)
Calculation of term A mZ,G TermA= dt
As the expression of HE,Gis known in the body frame Fb, the relative derivation is done in this same frame. The inertia matrix IIG is defined by (Equation 4.29, p. 76) and the angular velocity nbE is defined by (Equation 3.22, p. 49) and also denoted nk
H;,~ =
fi$tE
-E
-D
C
Ap
-F4
Cr
-Fp -Ep
-Er -Dr -Dq (E.31)
the derivative of angular momentum Hk,G (E.32)
247
E.3 Inertial anaular momentum derivative dH
as the aircraft is a rigid system (Hypothesis 13) IL is a constant and 3 = 0. Usually, the plane (xb,yb) of the aircraft is a symmetry plane so that F = D = 0 , and the expression of the angular momentum H is rather simplified.
A p -Fq -Er - F p -Dr Cr - E p -Dq Crq - Epq - Dq2 - Bqr + Dr2 + Frp Arp - E r 2 - Fqr - C r p + Ep2 + Dpq Bpq - Dpr - Fp2 - Apq + Erq + Fq2 rq(C - B ) - Epq + F r p + D ( r 2 - q 2 ) r p ( A - C ) + E ( p 2 - r 2 )- Frq + Dpq pq(B - A ) Erg F(q2 - p 2 ) - Dpr
+
(E.33)
(E.34)
(E.35)
+
Ap - E+ - F q + r q ( C - B ) - Epq+ F r p + D ( r 2 - q 2 ) B q - F p - D+ + r p ( A - C ) + E(p2 - r 2 ) - Frq + Dpq C+ - Ep - Dq pq(B - A ) + Erq F(q2 - p 2 ) - Dpr
+
+
(E.36)
Calculation of term B with
Hk,G =
(
Ap-Fq-Er Bq-Fp-Dr Cr - E p - Dq
)
and
The transformation matrix Tb0 is obtained from equation (2.49), p. 29
cos LtG cos 8 cos q!~- sin LtG sin 8 cos LtG sin 8 sin 4 cos - sin q cos 4) sin LtG cos 8 sin 4 cos LtG cos sin 8 cos 4 sin 4 sin $) sin LtG cos 8 cos 4
+
+
+
+ +
(E.38) sin 8 cos $ sin 8 cos 4
+ sin 4 sin 1c, (E.39) ~
~~
~~~
Dynamics of Flight: Equations
248
E - Accelerations
the angular velocity of the Earth ~
E Iprojected ,
thus
(f ) (
(TermBlb =
x
(TerrnB)b = nb,,xHb,,, =
in the body frame Fb, will be denoted
Ap-Fq-Er Bq - F p - Dr Cr - Er - Dq
)
qt(Cr - Ep - Dq) - rt(Bq - Fp - D r ) rt(Ap - Fq - E r ) - pt(Cr - Ep - Dq) P t ( & - F p - Dr)- d A P - Fq - E r )
(E.40)
(E.41)
Calculation of term C
AH;
=
(E.42)
I$&,
with a derivation relative to the body frame Fb b
(E.43)
As the aircraft is rigid (Hypothesis 13), I& is a constant and
% = 0 thus (E.44)
The first term is equal to zero. It comes from the derivation of equation (3.64), p. 58 and with equation (3.66), p. 59, L t o = 0 so
(%) E
From equation (3.22), p. 49
From equation (E.39), p. 247
= nth0
(
- sin L t o 0 - cos L t o
)
=o
E.3 Inertial angular momentum derivative
249
Finally
(E.45) and projected in the body frame Fb
(TermC)b =
(-%- E
-F -D B
[( _8) (a)l
-E - CD )
x
(E.46)
Calculation of term D
TermD = ~ ~ E x A H G
(E.48)
The Earth angular velocity OEIhas been calculated for the term B (Equation E.39, is defined by (Equation 3.22, p. 49)
p. 247) and Q,E
Apt -Fpt -Ept thus
(TermD)b =
(;) ( x
- Ert - Drt - Dqt + Crt
- Fqt
+ Bqt
Apt - Fqt - Ert -Fpt Bqt - Drt -Ept - Dqt Crt
q(Crt - Ept ( T ~ T ~ D = ) ( ~~ ~ E x A H=G ) r(Apt ~ - Fqt p(Bqt - Fpt
+
+
- Dqt) - r(Bqt - Fpt - D r t ) - E r t ) - p(Crt - Ept - Dqt) - Drt) - q(Apt
-
Fqt
- Ert)
(E.49)
Calculation of term E (E.50) The components of this vectorial expression have been previ term B and C
lculated for the
Dynamics of Flight: Equations
E - Acceleratzons
250
Reassembly of the terms A, B, C, D and E Finally the inertial angular momentum derivative of the aircraft, can be written as
(E.52) with
(E.53)
+-dAHk dt
term C
projected in the body frame Fb
+
Ap - E+ - Fq + rq(C - B ) - Epq Frp + D ( r 2 - q 2 ) Bq - F p - D+ + r p ( A - C ) + E(p2 - r 2 ) - Frq + Dpq C?:- E p - D q + p q ( B - A ) + E r q + F(q2 - p 2 ) - Dpr (E.54)
with
E.4 Derivation of the aerodynamic velocity
251
and sin 8
(E.57)
E.4
Derivation of the aerodynamic velocity
The derivative of the aerodynamic velocity V , yields a derivative of angle of attack tu, and sideslip angle ,& which could have an influence on the aerodynamic coefficients. The difficulty comes from the evaluation of the temporal derivative of the wind velocity V , . The objective is also to find a relation between the derivative of the modulus, the angle of attack, the sideslip angle of the aerodynamic and kinematic velocity, Va, &a, b a and l i k , t u k , j k .
E.4.1
Wind velocity variation V,
Spatial variation of V , The wind velocity V , is defined, independently of the aircraft, in a spatial domain. In the vicinity of the aircraft, the following relation was obtained (Equation 3.32, p. 52)
dVz
= &mVtdXo
(E.58)
applied t o the air particle W which occupies the aircraft center of mass G position, then
Temporal variation of V , It is a matter of finding the increment of wind velocity V , during the time dt and along the aircraft trajectory. The problem of this temporal derivative holds in the necessity to link the temporal variation of wind velocity V , with the aircraft trajectory, as this velocity is defined, independently of the aircraft, in a spatial domain. The spatial variation of V , , d V w u ,is~known , (Equation E.58, p. 251). If the temporal variation of the velocity is examined along the aircraft trajectory, the variation of position d x , d y , dz are linked together via dt through the components of the aircraft kinematic velocity V I , . in the vehicle-carried normal Earth frame F,, it can be written
(E.60) This is the substantial derivative of V , [l],or the Lie derivative. In other words it is the derivative of V , in the v k direction. According t o the Taylor hypothesis, the Dynamics of Flight: Equations
252
E - Accelerutzons
atmosphere is assumed to be frozen (Hypothesis S), that is to say no evolution with respect to time, so
then
E.4.2
Calculation of the aerodynamic velocity derivative
Next to the calculation of wind velocity derivative (Section E.4.1, p. 251), the calculation of the aerodynamic velocity derivative can be achieved relative to the vehicle-carried normal Earth frame Fo. In order to simplify this problem, the calculation will be made within the framework of flat and fixed Earth hypothesis (Hypothesis 3) and (Hypothesis 17) thus the two terrestrial frames FE and Fo will merge. Thus the expression of the aerodynamic velocity derivative is (E.62) with a derivation relative to the body frame Fb (E.63) Then (E.64) and the terms are then calculated. Expression of the aeroda(nanzic velocity derivative
projected in the aerodynamic frame Fa, (Equation 3.14, p. 47) and (Equation 3.77, p. 60)
(E.66)
Expression of the kinematic velocity derivative (E.67)
253
E.5 Probe acceleration - load factor
projected in the kinematic frame Fk, (Equation 3.12, p. 46) and (Equation 3.78, p. 60) (E.68)
Expression of the wind contribution, with equation (3.22), p. 49 and the flat and fixed Earth hypothesis
The matrix T b o is a function of 4, 8, @ (Section 2.2.5, p. 27) and it must be recalled that the wind is usually known in the vehicle-carried normal Earth frame Fo, such as (GRmV;)o =
(
ux;
uyo,
uz;
vx; wx;
vyo, wy;
wz;
vz;
)
This yields to another expression of the aerodynamic velocity derivative (Equation E.66). From equation (E.64), and expressed in the aerodynamic frame Fa
so
another notation can be used, (Equation D.74, p. 240), inside equation (E.70)
E.5
Probe acceleration - load factor
The analytical development proposed in this appendix could be useful for the process of in flight measurement. To clarify the purpose, a practical example is taken. There Dynamics of Flight: Equations
254
E - Accelerations
is an accelerometer onboard positioned at point SA. There is a velocity measurement device positioned at point SV. How could the acceleration and velocity at the aircraft center of mass G, be evaluated? For the velocity, the answer is given in section (D.5), p. 236 where the equation (D.56), p. 238 expresses the aircraft kinematic velocity as a function of the measured aerodynamic velocity, the kinematic angular velocity and the wind velocity and gradient. For the acceleration, the first problem is to examine what the nature of the measurement is, and then, the second point will be the analysis of the relationship between the aircraft acceleration and measurement. In this first point lies the load factor idea. The second point will indicate what kind of measurement is needed to generate the aircraft acceleration. The load factor The measurement of acceleration is equivalent to a weighing of a mass mSA, already described in equation (4.61), p. 83. In a classical weighing the scale is fixed relative to Earth, the acceleromater is an unsteady scale with a given mass to weigh. This given mass mSA can be denoted the subjected mass because this mass is subjected to the reaction of the accelerometer R and to the gravitational attraction gr,SA at the accelerometer sonde (SA) location. The accelerometer device could be called the sonde or probe. The fundamental equation of Mechanics (Equation 4.13, p. 74) leads to ~SAAI,SA = mSAgr,SA 4-R
(E.73)
Physically, the accelerometer experiences the reaction R and this parameter is provided by the measurement. As the subjected mass is known, the measurement obtained offers the value of Accelerometer measurement
= AI,SA- gr,SA
(E.74)
with the vector AI,SAequal to the inertial acceleration at the sonde station SA. From this result, the concept of load factor n can be introduced. The load factor is equal to the ratio of the accelerations to the value of gravitation gr,SA. The gravitational attraction gr,SA can be considered as an acceleration. (E.75) According to the definition, the sign of n is chosen so that the value of the load factor n in horizontal steady state flight, is equal to 1 and not -1. This definition was supported by the accelerometer experiment, but the load factor is a general notion, and for the aircraft the load factor at the center of mass G, is equal to
(E.76) It has to be mentioned that the load factor n is a vector so that it has three components depending on the projection frame. Usually the term load factor is also associated with the z component of n in the aerodynamic frame Fa. Another form for the definition of n is the ratio of the massic forces to the weight value. The massic forces are the weight and the inertial force, that is to say all the terms of the fundamental equation
E.5 Probe acceleration - load factor
255
of Mechanics which explicitly depends on the mass. The inertial force -mAi,G is the opposite of the mass multiplied by the inertial acceleration, so that this inertial force can be inserted in the force class. Then
(E.77) the fundamental equation of Mechanics can be written as (E. 78) or 0
= nGmgr,G -k F a e r o
Fthrust
(E.79)
With this last relation, a new expression of the load factor is found depending on the aerodynamic force F a e r 0 and the propulsive force F t h r u s t . nG
=
Faero
+ Fthrust
mgr,G
(E.80)
In this last form of the load factor n, it can be seen that a change in the aerodynamic force or the thrust create a load factor that people can experience in the aircraft. The load factor is an experienced weight as it renders the massic forces. However passengers in the aircraft are submitted to the aerodynamic force, the thrust and the weight. For an example, if there is no aerodynamic force (and thrust), there is a free fall and passengers will feel a 0 g flight, that is to say a non-gravity or weightlessness flight. During this free fall, the only force applied to the aircraft, is weight. It is quite paradoxical that the gravity or weightiness feeling at 19, is experienced when weight is balanced by the ground reaction, that is to say when people are on the Earth’s surface. Obviously, that is the normal feeling, and when a man is only submitted to weight, it is an unusual situation. However, this Og situation is found when only weight is applied. When the value of load factor is greater than 1 then the weight is overbalanced by external force, as for example by lift. Probe acceleration By probe acceleration, what is meant is the acceleration a t any point on the aircraft, as for example at the probe or sonde station SA. Let SA be the measurement point, so that GSAis the distance between the center of mass of the aircraft and the probe. ‘It should be noted that the used of the word force is a trick in order to have a elegant writing of the fundamental equation of Mechanics. The inertial force is not a force and does not at all have the characteristics of a force. A force has a concrete existence from a physical point of view. The forces generate the acceleration which conveys the behavior of the system due to the application of the forces. The acceleration says “if there is a force, there is a velocity variation”. Although nobody can apply or create an acceleration on a system, forces can be created whose consequences in terms of motion are expressed by acceleration. When a pilot makes a pull-up maneuver, passengers are crushed back in their seat because of the lift increase and not because of the centrifugal inertial force. However, the centrifugal inertial force says that the trajectory will change due to this lift and the airplane will climb. Dynamics of Flight: Equations
E - Accelerations
256 This vector is expressed in the body frame GSb, =
(q:)
Fb
ZSA
the acceleration in S A is defined thanks to the fundamental relationship of kinematics (Equation 3.1, p. 43)
(E.81) The local kinematic velocity V k , S A has already been calculated in equation (D.49), p. 237. The acceleration A S Aexpressed is an acceleration relative to the vehiclecarried normal Earth frame Fo. If this frame is supposed to be Galilean or inertial, this acceleration will therefore become inertial. The angular velocity Slbo comes from equation (3.22), p. 49. Projected in the body frame Fb this relation can be written AiA
=
(
+
U: qziA G;+fxiA-ljz;A
w:
+h ; A
-qXiA
) + (i) ( wi + 6 uk
x
+ qZ;A
-rYiA r x i A -pz;A
+PYiA -qxiA
Flom this relation, the aircraft acceleration at the center of mass G, can be extracted. It is the components in the body frame Fb ( U k , ri)k, u j k ) which can be expressed as a function of the kinematic velocity of G (uk,V k , wk), the kinematic angular velocity of the aircraft (p, q , T ) , the angular acceleration ( l j , q, +), the probe position ( Z S A , Y S A , ' S A ) and the probe acceleration A S A .The angular velocity can be obtained from gyrometer measurement and the angular acceleration from the derivative of this measurement. The kinematic velocity of G can be obtained from a velocity measurement at another station S V on the aircraft (Equation D.49, p. 237). The probe acceleration comes from the accelerometer measurement through the equation (E.74), p. 254. If the accelerometer measurement components in the body frame Fb are denoted U:, U: and a!, then the equation (E.82), p. 256 can be rewritten. (E.83) and from equation (4.71), p. 85
~~~~~~~~~
~
2Caution must be taken because this accelerometer measurement conveys the acceleration ASA and not the load factor whose conventional sign is the opposite of ASA.
E.6 Relative accelerations - consequences of flat and fixed Earth hypotheses
257
then
If the accelerometric probe S A , is not placed at the same station as the velocity probe S V , it needs to calculate the velocity V k , S A at the S A probe position, via the center of mass velocity V k , G , in order t o use the previous equation (D.49)) p. 237. The velocity V ~ , SatVthe velocity probe position is expressed as a function of probe positions x i v , y i v , z g V , and the expression of the acceleration coming from equation (E.82)) p. 256 can be rewritten. If the distance between the velocity probe S V and the acceleration probe S A is denoted (E.85) then
+
+ +
,sv - rvrC,sv sv - PWk,SV b ,SV - quk,SV
+ +
-(q2 r 2 ) A x i qpAy: r p A z $ + p q A x i - (p2 r 2 ) A y : r q A z i + p r A x i q r A y i - (p2 + q 2 ) A z i
+
(E.86)
With this formulation, the velocity components at the velocity probe station U k , S V , V k , S V , w k , S V intervene directly and there is no need t o calculate the velocity components at the center of mass G, uk, V k , wk in order to achieve the acceleration of the aircraft center of mass u k , z j k , w k . So, with equation (E.84), p. 257 and equation (E.86), p. 257, and through the various measurements described above, the acceleration of the aircraft center of mass u k , i ) k , 2i)k can be evaluated.
E.6
Relative accelerations - consequences of flat and fixed Earth hypotheses
Four relative accelerations will be calculated by successive derivations of position and velocity relative t o two terrestrial frames, the normal Earth fixed frame FE and the vehicle-carried normal Earth frame F,.
Relative velocities
v k p and v k The inertial velocity is defined by (Equation 3.18, p. 48)
VZ,G
=
+-
dAG' - dAO' dOG' dt dt dt
(E.87)
Dynamics of Flight: Equations
258
E - Accelerations
The first term is expressed through a relative derivation with respect to the normal Earth fixed frame FE and the second term through a relative derivation with respect to the vehicle-carried normal Earth frame F,
VI,G =
dAOE dt
dOG" + ~ E I X A+O+n 0 i x O G dt
(E.88)
The point 0 is fixed relative to the Earth, then
d A O-E dt
(E.89)
- 0
according to definition, the flat kinematic velocity v k p
=
dOG" dt
(E.90)
-
then
Taking into account the expression of the kinematic velocity v
the relationship between the kinematic velocity V obtained:
k
k
(Equation D.13, p. 229)
and flat kinematic velocity v
k p
is
(E.92) Relative accelerat ions The four relative accelerations will be generated by the derivation of the kinematic velocity v k and flat kinematic velocity v k p relative to the normal Earth fixed frame FE and the vehicle-carried normal Earth frame F,.
Ar,EE
= Ar,oE
=
Ar,oo
-
d
] [ ]"
dOGE = T - d t [ dt E d v k - -
E
dvz = d dOGE dt dt dt
[ [
dvfp d dOG" dt = d t dt dvgp d d O - - - G" dt = dt dt
] " ]
E
(E.93) (E.94) (E.95) (E.96)
E.6 Relative accelerations - consequences of fiat and fixed Earth hypotheses
259
Thus the inertial acceleration can be expressed under four alternative forms (E.97) (E.98) (E.99) (E.100) The problem now is to find the expression of the four complementary accelerations AA.
First representation with the terrestrial relative acceleration A r , E E The inertial acceleration AZ,G is elaborated through the derivative of the particular expression equation (D.13), p. 229 of the inertial velocity VZ,G.The relative frame is the normal Earth fixed frame FE. (E.lO1) therefore with equation (D.13)) p. 229 AZ,G
=
(E.102) The first term is the relative acceleration Ar,EE. The second term is equal to zero because the Earth angular velocity f2t and the latitude Lto of the origin of FE,are constants (Equation D.28, p. 232). The third term is equal to ~ E Z X since V ~ AG ) = A 0 O G . The vector A 0 is a constant vector in the frame FE and the derivative of O G relative to FE is the definition of the kinematic velocity v k . Therefore, expanding the fourth term, finally the expression of the inertial acceleration AZ,Gyields
+
and the complementary acceleration is denoted A A R
The subscript R is given because this complementary acceleration depends on the Rotation of the Earth through ~ E Z The . first term is the Coriolis acceleration and the second one is the centripetal acceleration. Projected in the normal Earth frame Fo, the following relation is obtained thanks to equation (D.29), p. 232 and section (D.2), p. 229. (E.105)
Dynamics of Flight: Equations
E - Accelerations
260
AGO
v;
=
=
(
-(Rt
+ h)
( 2vz )
)
(E.106)
(E.107)
-
Therefore
This term already has been calculated in section (E.l), p. 241 in another way, for the general expression of the inertial acceleration. Second representation with the pseudo-terrestrial relative acceleration A r , ~ o The process is similar to those developed for the previous first representation, but the relative frame will be the vehicle-carried normal Earth frame F,.
(E.109) thus
AZ,G =
m;+-dn2"Ezx AG + S ~ E xZ dAG" dt
dt
dt
+ nozx (Vk + ~
E xAG) Z
(E.110)
This approach will give the expected result after a lot of calculation. In order to take advantage of the previous calculations, the relation obtained previously from equation (E.103), p. 259 and equation (E.104), p. 259 can be considered (E.111) and with the fundamental relationship of kinematics equation (3.1), p. 43, it is possible to derive the kinematic velocity Vk relative to the vehicle-carried normal Earth frame F O*
The first term is the expected relative acceleration A r , ~and o the second term is a new complementary acceleration named AAs.
AZ,G = A r , ~+oAAs
+ AAR
(E.113)
E.6 Relative accelerutions - consequences of flat and fixed Earth hypotheses
261
The subscript S is given because this complementary acceleration depends on the Spherical Earth hypothesis through &. Projected in the normal Earth frame F,, the following relation is obtained thanks to equation (3.74), p. 60 and equation (D.2), p. 229
(E.115)
v;
=
( 2vz )
(E.116)
-
This second representation is the chosen form for the general inertial acceleration used in this document. It is the reason why the complementary acceleration AAs has already been calculated in section (E.l), p. 241.
Third representation with the pseudo-vehicle relative acceleration A r , , ~
The kinematic velocity v k is no longer used. The concerned velocity is now the flat kinematic velocity v k p . In accordance with the process used for the second representation, the relation obtained previously from equation (E.113), p. 260 has to be considered
AI,G
&,E,
dv: + AAs + AAR + AAs + AAR = dt
(E.118)
With the fundamental relationship of kinematics equation (3.1))p. 43 and equation (E.92)) p. 258, it is possible to derive the flat kinematic velocity Vkprelative t o the normal Earth fixed frame FE (E.119)
(E.120)
9
According to the definition, the derivative is equal t o the flat kinematic velocity thus the second term is the opposite of the fourth term, and
Vkp,
AI,G =
dv'p
dt
+-dnzE x O G + AAs + AAR dt
(E.121)
Dynamics of Flight: Equations
262
E - Accelerations
The relative acceleration A r , o ~is equal to the first term and the complementary acceleration is composed of the last three terms
The two terms A A s and AAR already have been calculated for the first two representations. Therefore the second term of equation (E.121), p. 261 has to be evaluated. The angular velocity 52zE is known from equation (3.74), p. 60 and equation (E.115), p. 261. The position of the aircraft center of mass G relative to the vehicle-carried normal Earth frame F, is simply defined by
OG" =
(3)
(E.122)
The derivative of 52& gives
(Rt?h)?
(
-VE N' - tan LtG
) (E.123)
After the cross product with OG and thanks to the kinematics equation (3.118), p. 68 and equation (3.72), p. 59
h LtG
= vz
VN = ALt = -
Rt
+h
this result is achieved d52zExOG = A A A + A A D dt with (E. 124) and
E.6 Relative accelerations - consequences of flat and fixed Earth hypotheses
263
The first term is the complementary acceleration AAA linked t o Acceleration (subscript A ) components due to the derivative of V E ,VN and Vz. The second term is the complementary acceleration AAD in which the Distance (subscript 0 ) appears with the components X G , YG and ZG. Therefore, the last version of the equation (E.121), p. 261 is obtained
AI,G =
dt
+ AAA + AAD + AAs + AAR
(E.126)
Fourth representation with the vehicle relative acceleration A,,,, In accordance with the process used for the third representation, the relation obtained previously from equation (E.126), p. 263 has to be considered
AI,G =
dvk"p dt
+ AAA + AAD + AAs + AAR
With the fundamental relationship of kinematics equation (3.1), p. 43 and equation (E.92), p. 258, it is possible to derive the flat kinematic velocity v k p relative t o the normal Earth frame F,
The term n & X V k is equal to AAs (Equation E.114, p. 260); the problem is to calculate the third term thanks to equation (E.115), p. 261 and equation (E.122), p. 262. This term will be denoted A A D ~
AAD~ = - ~ ~ , E x ( ~ ~ , E x O G )
(E.128)
This term is similar to the term AAD. Therefore the expression of the inertial acceleration is obtained
AI,G =
n i p
dt
+ AAA + AAD + A A D ~+ 2AAs + AAR
(E.130)
Dynamics of Flight: Equations
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Appendix F
State representation and decoupling
F.1
Decoupling conditions for the longitudinal equat ions
The problem is to find the conditions so that each term of the lateral equations is equal to zero independent of the value of the longitudinal states (Section 5.4.2,p. 137).
F.l.l
Lateral force equation
From the second form equations of forces (Equation 5.52,p. 113), the second component is taken in account. Aerodynamic lateral force
This term will be equal to zero, independent of the longitudinal state Va if C y = 0. The coefficient of lateral force Cy essentially depend on /3a, r: and p t . Thus the condition must be
as
265
F - S t a t e representation and decoupling
266
Then this condition (Equation F.2) on angular velocity components, could be expressed as the new condition p=r b
py,
= r y L
= r X ,
b
= 0
=
o
(F.3)
In equation (F.2) p a = 0 because the aircraft is supposed to have a geometrical plane of symmetry. This coefficient C y also depends on the yaw lateral control 6,. Therefore
S, = 0
(F.4)
This last condition is also associated with the geometrical plane of symmetry hypothesis. Otherwise, the longitudinal states must have no influence on the coefficient C y when
then
C y independent of
a,, q i , V a , h
(F-5)
This condition is usually well verified for “reasonable” angles of attack, that is to say before stall. From an equivalent point of view, the longitudinal controls ,S S,, etc must not have any influence on Cy.
Cy
independent of longitudinal controls
(F.6)
Lateral force due to propulsion The condition is Fy” = 0, and as
Fy” = -F,b when
@a
COS a a
sin&
+ F i COS@,
- F,b sina, sinp,
= 0 (Equation F.2, p. 265), then
If the propulsive forces are symmetric, which is the usual situation, then
Fi =0
(F.9)
Or with equation (4.103), p. 92, this condition is equivalent to Pm = 0. In the body frame Fb the propulsive force must not have any component on the axis Yb, whatever the values of the longitudinal parameters aa, q:, V a , h, ,S 6,. In other words no lateral propulsive force.
F. 1 Decoupling conditions for the longitudinal equations
267
Weight mg
rn g
+ cos 9 sin 4 cos p,
(sin 8 sin p, cos a,
- cos 8 cos 4 sin a, sin p,)
when P, = 0 (Equation F.2, p. 265), then this term is reduced t o mg cos 8 sin 4. In order for this term be equal t o zero independent of the longitudinal state 8, this condition has t o be satisfied
4 = 0
(F.10)
So the aircraft flies with horizontal wings. Azimuth $ The azimuth kinematic equation (5.73), p. 118 with gives
4=0
(Equation F.lO, p. 267)
r
$=cos e
(F.ll)
the angular velocity r = 0 is a previous condition (Equation F.3, p. 266), so azimuth is constant
11) = 0 the
$ = constant
(F.12)
Then, with 4 = 0 and $ constant, the aircraft will fly with horizontal wings within a vertical plane merged with its geometrical symmetric plane. First term of acceleration AA:
V,
(F.13)
( r cos a, - p sin a,)
this term will be equal t o zero independent of V, and a, if p=r
= 0
(F.14)
This condition was already found for the aerodynamic lateral force (Equation F.3, p. 266).
Second term of acceleration (wind and aircraft state participation ) This term is composed of three elements
First element AAZN -
V, cosp, sina, cos a, sin/3,(qzwb
-
q x bw )
(F.15)
this element will be equal t o zero for p, = 0; this condition was already found for the aerodynamic lateral force (Equation F.3, p. 266) Dynamics of Flight: Equations
268
F - State representation and decoupling
Second element AAtR V, cos c u , ( r y ; sin2 p,
+rx;
cos2 p,) - V, sin cu,(p& cos2p,
+ py;
sin2
(F.16) with
p,
= 0 the next condition is
= 0
rx; =py;
(F.17)
on r x : this condition was already found for the aerodynamic lateral force (Equation F.3, p. 266).
Third element AA:, V, cos p, ( - U X ;
+ vy;
cos2a , sin p,
sin p, - w&, sin2 a, sin P,)
(F.18)
this term is equal to zero for p, = 0, which was a previous condition.
Third term of acceleration (the only wind participation) It is a matter of making the second row of
( Q ~ v L )=~ Tab(cGWmVL)bV; v~
(F.19)
equal t o zero equation (5.47), p. 112 and equation (5.52), p. 113. Tab(@,
= 0)
=
cosa,
0 sincu,
-sincu,
0 cosa,
(F.20)
Taking in account the previous conditions obtained equation (F.3), p. 266 and equation (F.17), p. 268 r x bw
(cswmv;)bv;
= r y b , = p& = py,
=
(
ux:
0
-Q x w
0 vy:
0
= o qz:
0
WZt
) (3)
(F.21)
then
(F.22) With
p,
= 0, this third term will be equal to zero if
either
vy;
=0
or
=0
(F.23)
F . l Decoupling conditions for the longitudinal equations
269
Wind conditions The problem is to clearly give an equivalent to the conditions imposed on the wind parameters in the body frame F b with other conditions in the normal Earth-fixed frame F,. These conditions here, are described in the body frame F b and come from equation (F.3), p. 266, equation (F.17), p. 268 and equation (F.23), p. 268 (F.24) in order to “translate” them in the vehicle-carried normal Earth frame F,, it has to be noted that in the particular vertical plane where the aircraft flies, the wind is expressed in a simpler form. The course oriented frame Fc associated with this plane is obtained from the vehicle-carried normal Earth frame F, by an azimuth rotation $, whose transformation matrix (Section A.3.1, p. 198) is equal to
K
O
cos$ -sin$
=
sin$ cos$ 0
0 0 1
(F.25)
and the fuselage oriented frame F f merge with the body frame F.10, p. 267), then
Fb
as 4 = 0 (Equation
(F.26) The matrix (6wm7 will be expressed in this frame Fc, from t,,e previous results with the condition vyL = 0
(&mV”,b =
(
ux;
Ob -qxw
0 0 0
q&
0 WZL
)
(F.27)
otherwise
(F.28) We obtained
(&mv;)c
=
(
uxc,
oc
-qxw
0 0 0
qzc,
0 WZL
)
(F.29)
with uX; W~C,
qz; qx;
+ W & sin2e sin e cos qqX; q.,) b uX&sin2 e + w,& cos2 e + sin e cos e(qX%- q zbw ) - sin e cos e ( U x ; - wZ,b) + qz; cos2 e + qx; sin2 e sin e COS e(ux; - w&) + qzw b sin2 e + qx; cos2e
= uX; cos2 e
= = =
-
-
(F.30) (F.31)
Dynamics of Flight: Equations
270____
F - State representation and decoupling
-
The terms p", T& and VYC, are equal to zero. This means there cannot exist a wind gradient outside the vertical plane which contains the aircraft trajectory. By analogy with the previous process the useful inverse relations are obtained (Section D.6, p. 238)
ux% =
w&
'
-
Qxw
=
Qzw
6
+ w z sin2 ~ e + sin e cos e(qX& - q z L ) + wZ; cos2e - sin e cos e(qx& - qz;) sin e cos - w z ~+) qzL cos2 e + qxLsin2 e - sin 8 cos e(ux; - wZL)+ qz; sin2 e + qxL cos2e u Xcos2 ~ e
= uX&sin2 e
(F.33)
e(Ux;
(F.34)
and in the vehicle-carried normal Earth frame F,
so
( G h m V ~ ) "=
uxk cos2 $ uxC,cos $ sin ?b, -qx;
-uXk cos $ sin $ ux: sin2 $ - q x L sin $
cos $
qz:
cos
qz: sin $ wzc,
This last relation can be compared to the usual writing of (@mV:)O (Equation 3.35, p. 53) and the conditions imposed on the wind parameters in the normal Earth-fixed frame F, are found either for v& = 0 ry;
= -rX;
ux; +vy;
pyO,cos$+qxO,sin$ pzO,cos$+qzO,sin$
= uxC,cos$ sin$ = uxc,
= 0
(F.37)
= 0
With the condition v& = 0 (Equation F.24, p. 269), the problem is to write
so the relation
-uO,sin$+v:cos$
= 0
(F.38)
For the first case equation (F.37) UXC, can be considered as a constant. It means that the wind gradient is oriented towards the aircraft heading. For the second case equation (F.38) the condition means the horizontal wind has only a component oriented towards the aircraft heading. Conclusion on force equations
As a conclusion on force equations, the sufficient conditions for the decoupling of the longitudinal system with respect to the lateral system are found. The lateral aerodynamic force will be equal to zero whatever the values of the longitudinal parameter
F. 1 Decoupling conditions for the longitudinal equations
271
if the coefficient C y is equal to zero. That is to say, if the lateral states are equal to zero and the lateral controls to
(F.39)
Cy
q:, V a ,h,,S
independent of
,S,
(F.40)
The condition @a = S, = 0 is associated with the hypothsesis of a symmetric plane for the aircraft. The lateral component of the propulsive force is equal to zero independently of the longitudinal parameters
F; the condition
ry:
= 0
whatever longitudinal parameters
(F.41)
= rxi = p: = S, = 0 imposes = 0
p=r b
py,
=ry%
=rX,
b
=
o
(F.42)
The lateral component of the weight is equal to zero, so
+ = o
(F.43)
and, as a consequence of these conditions, the azimuth is constant $ = constant
(F.44)
The acceleration terms equal to zero impose supplementary conditions pzb,
= 0
(F.45)
and vyb,
=o
v; = 0
or
(F.46)
All of these conditions relative to the wind are expressed in the vehicle-carried normal Earth frame F, for the two cases. First case, for vy: = 0 ry;
= -rx;
ux; +vy; py;cos$+qxO,sin$ pz;cos$ qzzsin$
+
=
-uxL sin$cos$
= uxc,
= O = O
(F.47)
and second case, with the condition v% = 0, then -u;sin$+vzcos$
= 0
(F.48) Dynamics of Flight: Equations
272
F.1.2
F - State representation and decoupling
Yaw and roll moment equations
The problem is to find the conditions such that all the terms of the yaw and roll moment equation (5.56), p. 115 are equal to zero, in order to obtain a lateral equilibrium, that is to say p = 7: = 0. These equations are expressed on the axis xb and z b of the body frame Fb.
1~s
First term of acceleration The equation on the fuselage axis xb
The equation on the yaw axis zb
To ensure that this equation will be decoupled from the longitudinal equation, the product of inertia F and D must be equal to zero, in order to avoid that 4, issue from the longitudinal motion, disturbing the yaw and roll moment equations = 0
F=D
(F.49)
The result is that the aircraft has a massic plane of symmetry which has to be identical to the geometrical plane. Second term of acceleration Ok XI[G 01, With the previous hypothesis equation (F.49), p. 272 F = D = 0, this yields on the fuselage axis xb
In order that these equations are equal to zero, whatever the value of the pitch angular velocity q, it is necessary to have p = r
= 0
(F.50)
This condition has already been obtained for the forces equations (Equation F.42, p. 271).
Aerodynamic roll and yaw moment ;pSt'V:Cl
and
+pSlV;Cn
These terms will be equal to zero independently of Va if C1 = Cn = 0. This is a similar problem of those encountered for the C y coefficient (Equation F.l, p. 265). The
F . l Decoupling conditions for the longitudinal equations
273
conclusion will be the same. However, the roll control 61 which has little influence on C y has obviously a great influence on Cl and sometimes on Cn. So
61 = 0
(F.51)
Yaw and roll propulsive moment It is a question of achieving Mka = M F, b = 0 whatever the value of the longitudinal parameters a,, q, V,, h, 6,, 6,. This is obtained if the propulsion is symmetric, which is usually the case except in the case of engine failure on a multi-engine aircraft. If a high precision is searched for, the non-symmetry of the propeller flow which creates roll and yaw moment depending, among other factors, on the thrust control 6,, must be mentioned. As the rotating parts of the engines constitute gyroscopic elements whose angular momentum is not equal to zero, if the engines rotate in the same direction, following a pitch motion, a yaw moment will appear. Conclusion about the moment equations The necessary hypothesis for the decoupling and associated with the moment equations is the same as for the force equations. The only supplementary hypotheses deal with the massic symmetry, the roll control and the propulsive symmetry are
D=F = 0 61 = 0 Mka = M Fb, =
(F.52) (F.53) (F.54)
Kinematic equations The concerned kinematic equations are the lateral equations but coupled with the longitudinal system, so the roll equation with 4 is equation (5.73), p. 118
the derivative of bank angle
(J) will be equal to zero independently of q and 8 if 4=0
and
p=r=O
(F.55)
These conditions have already been obtained.
Dynamics of Flight: Equations
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Appendix G
Linearized equations
G.1
Numerical linearization
This routine written in fortran is an example of numerical linearization.
C****************************************************
subroutine linearis C****************************************************
c c c c
This subroutine is for the linearization of the aircraft model about a flight situation given by the initial state * The linearized model is under the following form:
C
c c
Xdot=Alin*X+Blin*U+Adot*Xdot Y=Clin*X+Dlin*U
C C****************************************************
c* Common declarations
*
C****************************************************
include '/edika/libincl' c= Input, output and state variables double precision X(dimstate),Xdot(dimstate) double precision U(diminput) double precision Y(dimoutput1 c= idem for the initial conditions double precision XO(dimstate) ,XOdot(dimstate) double precision U0 (diminput double precision YO (dimoutput) c= increments for the calculus of the gradients double precision dX(dimstate1 double precision dU(diminput)
275
276
G - Linearized equations
G.l Numerical linearization
277
c read(89,*) dX,dU,dXdot c close(89) dX(1)=O .05 dX(2)=0.01 dX(3)=0.01 dX (4)=O 001 dx(5)=0.0001 dx(6)=0.01 dx(7)=0.5 dX ( 8 )=O .003 dx (9)=o .001 dX(10)=0.001
.
dX(ll)=O.Ol dU( 1) =O .001 dU(2) =O .001 dU(3)=0.001 dU(4)=0.001 dxdot (1) =O .5 dxdot (2) =O .001 dXdot (3)=O 1 dXdot (4) =O 001 dXdot (5)=0.001 dXdot (6)=O .001 dXdot (7)=0.001 dXdot (8)=0.001 dXdot (9)=O 001 dXdot(l0)=0.001 dXdot(ll)=0.001
. .
.
Dynamics of Flight: Equations
278
c
c c
c c
c
G - Linearized equations
usolGg=X(l) vsolGg=X(2) wsolGg=X(3) p=x (4) q=x (5) r=X( 6 ) a1titude=X( 7 ) teta=X(8) phi=X (9) supplementary states psi=X(10) DLt=X( 11) b- the variable useful for the calculation of the efforts and for the output are calculated call environ-terrestrial call calcul-state-associated deltal=UO(l) deltam=U0(2) deltan=U0(3) deltax=UO(4) c- the efforts are calculated call efforts d- the accelerations are calculated call accelerations call calcul-state-associated-dot e- new transformation from explicit states to the state vector if (ii.eq.1) then call equivalence(Xdot,Y) else call equivalence(XOdot,YO) endif enddo
C
c the column i of the state matrix Alin is fulfilled do j=l,dimstate Alin(j,i)=(Xdot(j)-XOdot(j))/(2.*dX(i)) enddo c the column i of the output matrix Clin is fulfilled do j=l,dimoutput Clin(j,i)=(Y(j)-YO(j))/(2.*dX(i)) enddo c the initial state is restored X (i)=XO(i) enddo c the last state is restored DLt=X(dimstate) c the controls are restored deltal=UO( 1) deltam=UO(2 delt==U0 (3 deltax=UO(4)
G.l Numerical linearization
279
do i=l,diminput is=-1 ics=-1 do ii=1,2 is=ics*is c an increment is given to the variable U(i) U( i)=U0(i) +dU(i) c the associated derivative vector Xdot and output vector Y c a- transformation from the control vector U c to the explicit variables (deltal,... ) de1tal=U( 1) deltam=U(21 deltan=U(3) deltax=U(4) c b- the variable useful for the calculation of the efforts c and for the output are calculated call environ-terrestrial call calcul-state-associated c c- the efforts are calculated call efforts c d- the accelerations are calculated call accelerations call calcul-state-associated-dot c e- new transformation from explicit states to the state vector if (ii.eq.1) then call equivalence(Xdot,Y) else call equivalence(XOdot,YO) endif enddo c the column i of the control state matrix Blin is fulfilled do j=l,dimstate Blin( j,i)=(Xdot (j -XOdot(j )/(2*dU(i)) enddo c the column i of the control output matrix Dlin is fulfilled do j=l,dimoutput Dlin( j,i)=(Y (j)-YO( j))/(2*dU(i)) enddo c the initial states for the controls are restored U(i 1=U0(i) enddo Dynamics of Flight: Equations
G - Linearized equations
280 c the last control is restored deltax=U (4)
....................................................
c"""""""'-"'""""""""""-------------
c= calculation of the matrix Adot c=========================r====='==========================
call call call call
efforts accelerations calcul-state-associated-dot equivalence(XOdot,YO)
C
do i=l,dimstate c an increment is given to the variable Xdot(i) Xdot (i)=XOdot (i)+dXdot (i) c the associated derivative vector Xdot and output vector Y c transformation from the state vector X c to the explicit variables (p,q,. . . I usolCgdot=Xdot(l) vsolGgdot=Xdot(2) wsolCgdot=Xdot (3) pdot=Xdot(4) qdot=Xdot ( 5 rdot=Xdot (6 altitudedot-Xdot(7) tetadot=Xdot(8) phido t=Xdot (9) c supplementary states psidot=Xdot(lO) DLtdot=Xdot(ll) c b- the variable useful for the calculation of the efforts and for the output are calculated c call calcul-state-associated call calcul-state-associated-dot c c- the efforts are calculated call efforts c d- the accelerations a r e calculated call accelerations c e- new transformation from explicit states to the state vector call equivalence (Xdot ,Y> c the column i of the state derivative matrix Adot is fulfilled do j=l,dimstate Adot(j,i)=(Xdot(j)-XOdot(j))/dXdot(i) enddo c the initial state is restored Xdot (i)=XOdot(i) enddo c the last state is restored DLtdot=Xdot(dimstate) C==O=P=O=D=PI======'============'='='==============
c the last calculation before going out C=='=IP'='=P======================'=r='==============
call calcul-state-associated
G . l Numerical linearization
281
c======================t====l=============================
c c c c
Saving Saving Saving Saving
of of of of
Alin Blin Clin Dlin
in in in in
the the the the
file file file file
fileAlin fileBlin fileclin fileDlin
...........................................................
open(l83,file='/edika/files/fileAlin',status='old'~ open(l84,file='/edika/files/fileBlin',status='old'~ open(l85,file='/edika/files/fileClin',status='old') open(l86,file='/edika/files/fileDlin',status='old'~ do j=l,dimstate do i=l,dimstate write(l83,1203) Alin(i, j) enddo enddo do j=l,diminput do i=l,dimstate write(l84,1203) Blin(i, j) enddo enddo do j=l,dimstate do i=l,dimoutput write(l85,1203) Clin(i, j) enddo enddo do j=l,diminput do i=l,dimoutput write (186,1203) Dlin(i, j enddo enddo 1203 format(8(1x,el7.11)) close (183) close (184) close (185) close (186)
Dynamics of Flight: Equations
G - Linearized equations
282 enddo enddo do i=l,dims,r do j=l,dime-r Clin-r(i,j)=clin(i,j) enddo enddo do i=l,dims-r do j=l,diminput Dlin-r(i,j)=Dlin(i,j) enddo enddo endif return end C****************************************************
subroutine equivalence(Xdot,Y) C****************************************************
c this subroutine is for the transformation from c the explicit variables (pdot, qdot, etc c to the state and output vector Xdot and Y C****************************************************
c* Declarations
*
C****************************************************
include '/edika/libincl' double precision Xdot (dimstate),Y(dimoutput) c===================================pP=================
Xdot(l)=usolGgdot Xdot(2)=vsolGgdot Xdot(3)=wsolGgdot Xdot (4) =pdot Xdot (5)=qdot Xdot (6)=rdot Xdot(7)=altitudedot Xdot(8)=tetadot Xdot (9)=phidot c supplementary states Xdot (10)=psidot Xdot(ll)=DLtdot Y(l)=altitude Y (2) =mach Y(3)=alpha Y (4)=beta Y (5) =p Y (6)=q Y (7)=r Y (8)=psi Y (9)=teta
G.2 Wind velocitg field linearization
G.2
283
Wind velocity field linearization
Wind is defined in the vehicle-carried normal Earth frame F,; the problem is t o express the linearized form of the wind in the body frame Fb. The expression of the wind velocity field (&mVL) is given in section (D.6), p. 238.
Linearization with respect to the azimuth $ The linearization with respect t o the azimuth $ in a turning flight situation presents a validity domain that is obviously limited. Linearization of equation (D.63), p. 239
Linearization of equation (D.64), p. 239
Linearization of equation (D.65), p. 239
Wind defined at the initial azimuth Dynamics of Flight: Equations
284
G - Linearized equations
The generality of the problem is not affected if the wind is supposed to be known in the vehicle- carried normal Earth frame F,, oriented by the initial azimuth of the aircraft $, (Section A.3.1, p. 198). It comes down in the previous equations to take $, = 0, so
AUX; = Aux; + (rx;, - ry”,)All, AVY: = Avy; + (ry& - rxEi)All, A W Z ~= AwzL AP: A& Ary; Am: Aqx; ArxL
= A&,
+ qz;,All,
= Aqz; - pzziAll, = Ary; (uxO,, - vy;,)A$
+ + &,,,All,
= Am: = Aqx; - m & A $ = Arx; (vy& - ux”,)All,
+
(G.10) (G.ll) (G.12) (G. 13) (G.14) (G.15) (G. 16) (G.17) (G.18)
and also (G.19)
Linearization with respect to the inclination angle 8 Linearization of equation (D.69), p. 239 taking into account the results of equation (G.10), p. 284 to equation (G.18), p. 284
Auxf, = AUX; cos28, + ( r X & - r y & ) cos2 &A$ + AwZ; sin2 Oi + (Aqx; - Aqz; - (py& - p.Li)A$) sin 8, COS 8, + (sin 2Oi(wZEi- uX;,) - COS 28,(qz;, - qxLi))A8 Avyf, = AUY; + (ry& - rX;,)A$ AwzL = Aux; sin2 8, + (rXO,, - r y & ) sin’ 8,A$ + AwzL cos’ 8, + (Aqz; - Aqx; - (pZ& - m;,)A$) sin 8, COS 8,
+ (- sin28,(w~;, - uXzi)+ c 0 s 2 8 , ( q ~ ;-~ qx”,)) A8
(G.20) (G.21)
(G.22)
Linearization of equation (D.70), p. 240
AqzL = ( AqZ; - pZiA$) cos2 8, + (Aqx; - m;, A$) sin2 8, + (AuxG - AWZ; ( r X & - ry;,)A$) sin@,cos@, - (COS 28i(wz;i - uX;,) + sin ae,(qz;i - &,,)) A8
+
(G.24)
285
G.2 Wind velocity field linearixation Aryf
+
= Ary; cos 8, (uXii- uy;,)A11, cos 8, +(Am; q~;, A?)) sin 8, ( -rY&sin 8,
+
+
+ mzi cos &)A8
(G.25)
Linearization of equation (D.71), p. 240
Amf
= (Am; -
Aqxi
Arxf
=
=
+ q~;, A$) cos 8, - (Ary; + (ux;,
(m:, sin 8, + ry;,
- uyzi)A?))sin 8,
COS &)a8
(G.26)
+
(Aqx; - m;, A@)cos2 8, (Aqz; - pZziA$) sin2 8, (Awz; - nux; - ( r X O , , - ryO,,)A11,)sine, c o d ,
+ + (cos 28,(wZii- uXEi)+ sin 28, (qZ& - qXLi))A8
+ (uY& - U X " , ) ) ~ ?cos8, )) +(Am; + &,,A$) sine, + (-rX;, (Arx;
Linearization with respect to the bank angle Linearization of equation (D.75), p. 240
(G.27)
sine,
+ p.;, cos8,)AB
sin 4,
+ r y L i cos &)A@ (G.33)
(G.28)
4
Linearization of equation (D.76), p. 240
AqZL Ary;
= Aqzi cos 4,
+ Aryf sin 4, + (-q&
= A r y i cos $i - Aqzf sin 4, - ( r y f , sin 4,
+ 4.1,
cos 4,)A+
(G.34)
Linearization of equation (D.77), p. 240
Aqx;
= AqxL cos 4,
+ Arxf sin 4, + ( -qxfi
sin 4,
+ r x f , cos $,)A+
(G.36)
Dynamics of Flight: Equations
G - Lineam'zed equations
286
Arx;
=
Arxf, cos 4, - Aqxf, sin 4, - ( r x f , sin 4,
+ qxf,, cos +,)A4
(G.37)
In order t o simplify the writing of the equations, the following notations are introduced
13; = PY; g)= qz;
- qx;
?: = rx;
- ry;
and and and and and and
- P.;
ii; = UX; - uy; 6; = uy; - wz; 6; = wz0, - ux;
AC; = Am;
- Apz;
A@; = Aqz; A?: = Arx; Aii; = Aux; A$; = Auy; ACE = Awz;
- Aqx; - Ary; - nuy; - Awz; -
Aux;
(G.38) (G.39) (G .40) (G.41) (G.42) (G.43)
Wind velocity Aeld linearization The expression of the linearized components of the wind gradient (GwmV, expressed in the body frame Fb as a function of the components expressed in the vehiclecarried normal Earth frame F,, oriented towards the initial azimuth, is finally calculated from the equation (G.29), p. 285 t o equation (G.37), p. 286 in which the equation (G.20), p. 284 t o equation (G.28), p. 285 are taken in account.
+ +
AuXL = AuX; cos2 8, Aw.; sin2 8, - AGE sine, cos6, +aux4,A+ aux+,All, auxe,A€J
+
(G.44)
with
Auy;
aux4,
= 0
aux+, auxe,
=
-&,
sin 8, COS 8,
+,?;
cos2 8,
= 6:,sin28, - @Zi~ 0 ~ 2 8 ,
+ Auy0, cos2 4, + Awz; cos2 8, sin2 4, +A& cos 8, sin 4, cos 4, + A@),sin2 4, cos 8, sin 8, + A?: sin 8, sin 4, cos 4, (G.45) +avyd, A 4 + avy+, A$ + avye, A8
= AUX; sin2 4, sin2 8,
with avYd,
=
avy+,
=
avye,
=
+
(sin 24,(wZO,,cos2 ei - uY& uX;, sin2 ei COS 24, @ ( ,; COS 8, ?Zi sin 8,))
+
+
+
sin 8,COS e,)
?Gi (- cos2 4, + sin2 8, sin2 4,) + fi;, sin 8, cos e, sin2 4, +Ei cos 8, sin 4, cos 4, - 2iiLi sin 8, sin 4, cos 4, sin 4, (sin 4, ( -G& sin 28, + @Eicos 28,) + cos 4, (-jj;, sin 8, + ?Ei COS e,))
AWZL = Aux; sin2 8, sin2 4, + AuyL sin2 4, + AwZ; cos2 8, cos2 4,
-A& cos 8, sin 4, COS 4, +A@),sin 8, cos 8, cos2 4, - A?: sin 8, sin 4, COS 4, +awzd,A+
+ awz+,AlC, + awze,AO
(G.46)
G.2 Wind velocitv field linearization
287
with = sin 24,(vyO,, - wz;,cos2 8, - uXzisin2 8, - tj;, sin 8, cos 8,)
awZ4,
+,?; sin 8 ; ) sin 8, COS 8, cos2 4, + q;, COS 8, sin 4, COS 4,
- cos 24, (fi& cos 8,
fi;,
awZ+,
=
awze,
= cos 4, (sin 4, (fi& sin 8, - F;, cos 8,)
+?Ji(sin2 8, cos2 4, - sin2 4,)
Apz,b
+
=
COS
4,( -G;, sin 28, +, @;
+ 2ii;,
sin 8, sin 4, cos 4,
COS 28,))
sin 24, (-nux: sin2 8, AvYL - AwZO,cos2 8,) 2 cos Oi(cos2diApZL sin2 +,Am;)
+ +
+
A G sin 28, sin 24, - sin 8,( ArxL cos2 4, + AryL sin2 4,) - +apz4,A+
+ apz+,A$ + apze,A8
4
(G.47)
with
=
apz+w
%, -sin 28, sin 24, + cos 8,(qXzisin2 4, + qzLi cos2 4,) 4
+ sin2 8,) + ii;, sin 8, cos 24, - cos 8, (rX& cos 24, + ry;, sin 24,) sin 8,(pZ& cos 24, + py;, -?& sin 4, cos 4,(1
apZew =
- sin 4, cos 4, (
-*:,
sin 28,
+ Q;,
-
COS 28,)
sin 24,)
Aqz& = - AG; sin 8, cos 8, cos 4, + Am; sin 8, sin 4, cos 4,( AqzL cos2 8, Aqx: sin2 8,) + Ary; cos 8, sin 4, +aqz#,A$ aqz@,All, aqze,A8
+
+
+
+
(G.48)
with
aqz4,
=
aqzrow
=
sin 4, (qz;, cos2 8, + qX;, sin2 8, - t~;, sin 8, cos 8,) + cos 4,(r&, cos 8, + p&, sin 8,) - cos 4, (mLisin2 ei + pzzi cos2 8,) + qxzi sin ei sin 4, -
+?,;
aqze,
sin 8, cos 8, cos 4,
= sin 4,( -ry;,
sin 8,
+ ii;,
COS 8, sin
4,
+ py;, cos 0,) - cos 4i(GJ,cos 28, +
sin 28,)
@Ji
sin 28, Ary& = AGJ- 2 sin 4,
+ Am: sin 8, cos 4, - sin 4, (AqzL cos28, + Aqx; sin2 8,) + Aryg cos 8, cos 4,
+ary4,A@
+ ary+,A$ + arye,A8
(G.49) Dynamics of Flight: Equations
G - Linearized equations
288 with
+ py;, sin 8,) - COS 4, (qz& cos28, + qX;, sin2 ei - G ; ~sin 8, COS 8,) sin 4,(p.Eicos2 8, + pyci sin2 8,) + qX;, sin 8, cos 4, -?& sin 8, COS 8, sin 4, + ii;, COS 8, COS 4, cos q5i ( -ry;, sin 8, + py;, cos 8,) + sin di(+E, sin 28, + GZi cos 28,)
ary4,
= - sin 4,(ry& cos 8,
arY+,
=
arye,
=
-
Am:
sin (Aux; sin2 8, - AvY; Aw,; cos2 8,) 2 cos Oi (Am: cos2 $i Apz; sin2 4,) A 0 +sin 28, sin 24, - sin 8,(Ary; cos2 4, + Ayx: sin2 4,) 4 (G.50) + a p ~ + , A 4 apy+,All, apye,A8
+
+
+
+
+
with
+
apy4,
= - sin 24,@ ( ,; cos 8, FE, sin 8,) COS 24, (uX;, sin2 8, wZ;, cos2 8,
apyll,
=
+
+
+ ,+; sin 8, COS 8, - vY;,) @o,, -sin 24, sin 28, + cos qqX;, cos2 4, + qz;, sin2 4,) 4 +.?;2 (1 + sin2 8,) sin 24, - ii;, sin 8,cos 24, - cos O,(ry$, cos 24, + rxEi sin 24,) - sin 8,(&,, cos 24, + p.& + sin 4, cos 4,( -G;, sin 28, + &, 28,) AG; sin 8, cos 8, cos 4, + Am: sin 8, sin 4, + cos 4,(Aqx; cos2 8, + AqzL sin2 8,) + Arx: COS 8, sin 4, 2
apye,
=
sin 24,)
COS
Aq&
=
+aqx4,A$
+ aqX+,A$ + aqxe,A8
(G.51)
with
+ + G;, sin 8, cos 8, + + cos 4, (m;, cos2 8, + p:, sin2 8,) + qZLisin 8, sin 4,
aqx4,
=
- sin 4, (qx;,
aqx+,
=
-
aqxe,
-?,; sin 8, cos 8, COS $, - ii;, COS 8, sin 4, = sin 4, ( - r X O , , sin 8, &, cos 8,) COS 4, (G:, sin 28, tij;, COS 28,)
Arx;
cos2 8, qZ:, sin2 8, cos 4,(rXLicos 8, py;, sine,))
+ +
+
=
-AGE sin 8, cos 8, sin 4, + ApyL sin Oi cos 4i - sin 4,(Aqx; cos2 8, Aqz; sin2 8,) Arx; cos 8, cos 4, +arxd,A+ arx+,All, + arxe,A8
+
+
+
(G.52)
G.2 Wind velocity field linearization
289
with
+ + cos2 ei +
- COS 4,(qX:,
+ tij:,
aw,
=
arw,
= sin 4, (py& p Z z i sin2 8,) qZ:, sin e, cos 4, +?$ sin 8, COS 8, sin 4, - ii; COS 8; COS 4, = COS 4; ( -rx& sin 8, p:, COS 8,) - sin 4, (gii sin 28, tij;, COS 28,)
arxe,
cos28, qZ:, sin2 8, - sin 4; ( r X & COS 8, pz& sin 8,)
sin 8, COS 8,)
+
+ +
Wind velocity field linearization relative to an initial bank angle equal to zero 4, = 0
Back t o the equation (G.44), p. 286 t o equation (G,52), p. 288 and setting zero
4, to
AUX; = nux; cos2 8, + Aw,; sin2 8, - Ag; sin 8; COS Oi
+ auX+,A$ + auxe,A8
+aux4,A+
(G.53)
with = 0
aux4, auxllr, auxe,
+
-fiti
= sin 8, COS 8, F;, cos28, = 6;sin 28, - GL COS 28,
AvYL = AvyL
+ avy4, A 4 +- avyq, A$
+- avyewAB
(G.54)
with =
awllr,
= -F& = 0
avye, AwZ;
fit,cos Oi + ?Ei sin 8,
avy4,
= AwZLcos2 8,
+ AGE sine, cos 8, + awz#,A+ + awZ+,A$ + awze,A8 (G.55)
with awz4,
=
- COS e,fi& - sin Of&
+ ?zi
awZ+, = f i sin ~ e,~ cos 8, sin2 ei awzew = -tij& sin 28, QZ,COS 28;
+
ApzL
= ApzL cos Oi - Arx; sin 8,
+ apz4,A4 + apZQwA$ + apze,,A8 (G.56) Dynamics of Flight: Equations
290
G - Linearized equations
with apz4, = VY;, - uXO,, sin2 8, - wZ;,cos2 8, - ,j;, sin 8, cos Oi 0 qzWi c o d i iiki sine, aPZ+W apze, = -rX;, cos 8, - pZLisin 8,
+
Aq&
+ Aqz; cos2 ei + Aqx; + aqz+,A$ + aqze,A8
-AGg sin 8, cos Oi
=
+aqzd,A+
sin2 Oi (G.57)
with
Ary;
+ py;,
=
ry;,
aqz+, aqze,
=
-py;, sin2 8, - p z ; , cos2 8,
=
-G;,
= A p ~ sin t 8,
cos Oi
sin 8,
aqz4,
COS 28, -
+ Ary;
&, sin 28,
+ F ; ~ sin 8, COS 8,
+ aryd,A+ + ary+,A$ + arye,A8
cos 8;
(G.58) with -qZ;, cos2 8, - qX;, sin2
ary4, ary+,
= qXZisine,
arye,
=
=
+ ii;, cos& + p~;, cos 8,
ei + G;, sin 8, cos 8,
-rY&sin 8,
= Apy; cos 4 - ArY; sin 8,
+ apy#,A+ + apy$,A$ + apys,A8(G.59)
with apyd, spy+,
+
= uX;, sin2 8, wz;, cos2 8, = qxwi O COS^, - ii;, sinei
aPY@, = AqxL
-ry&
+ @E,sin 8, COS ei - v ~ o , ,
cos8, - m;, sine,
+
+
= AG; sin Bi cos Oi Aqx; cos2 Oi A& sin2 8, +aqx$,A+ aqx+,A$ aqxe,AB
+
+
(G.60)
with aqx4,
= rXZicos 8,
+ py;
sin 8,
aqx+w = -py;, cos2 ei - pZLisin2 8, - FE, sin ei COS ei aqxe, = &i sin28, tijEi cos28,
+
Arxk
= Apy; sin 8,
+ Arx;
cos 8,
+ arx4,A+ + arx+,A$ + arxe,A8
(G.61)
G.2 Wind velocity field linearixation
291
with -qxLi cos2 8; - qZGisin2 = qzLi sin 8, - ii; cos Bi = - r X L i sin 8, pZLicos Bi
arX4, arX+, arxe,
=
ei - zijki sin ei cos 8,
+
Wind velocity Aeld linearization relative to an initial inclination angle equal to zero Oi = 0 Back t o the equation (G.44)) p. 286 to equation (G.52)) p. 288 and setting 8, t o zero = Aux;
Aux:
+ aux4,A+ + aux+wA$ + aux6,Ae
(G .62)
with
=
Avyb,
aux4,
= 0
aux+, aux6,
= ,? : = -qwi -0
+ +
+
Avy; cos2 4, AWZ; sin2 qhi AfiZ sin 4i cos 4, +avy$,A+ avy+,All) avye,A8
+
(G.63)
with
(G.64) with
Apzb,
sin 24, = 2 (Avy; - Awz;)
+ Apz: cos2 4; + Am; +apz@, A 4 + apz+, A$ + apze, A8
with aPZ4w
=
apz+w apze,
-
=
fiz,sin 244 + ij;, q X G i sin
2
+i
sin2 4, (G.65)
cos 24,
+ qzGi cos2 4i -,?:
sin 4, cos 4,
-rxLi cos 24, - r y L i sin 24, -, @: sin 4, cos 4, cos 28, Dynamics of Flight: Equations
292
G - Linearized eauations
with
aqz4, aqz$, aqze, AryL
=
- sin 4,Aqz;
= =
sin 4, -pzLi cos 4i -qz&
+
+
cos #+ GO,,sin 4,
TYO,,
= pyzi sin 4i - t5Li cos 4i
+ AryO, cos 4, + ary+,A+ + ary+,A$ + arye,AO (G .67)
with
(G.68) with apy4,
=
apyrl,
=
+ ij;, qx& cos2 4, + qz;,
apyew
=
-ryzi
-Pzi
sin 2&
cos 24i
?Gi +2 sin24, sin 24, + tjz,i sin cos q!+
sin2 4,
cos 2 4 - T X &
$i
with aqx4,
=
-qx& sin 4i
+ rXzicos 4,
aqxqw = -py& COS^, - ii& sin#, aqxe, = pzisin +i t5zi cos 4,
+
Ar& with
=
- A & , sin +i
(G.70)
+ Arxz cos 4, + arx$,A4 + arx$,A$ + arxeWAqG.7l)
G.2 W i n d velocity field linearization
293
Wind velocity field linearization relative to an initial bank angle and inclination angle equal to zero 4i = 8, = 0 Back to the equation (G.44)) p. 286 t o equation (G.52)) p. 288 and setting the initial bank angle 4, and inclination angle 8, to zero, the expressions will be simpler. These relations can be considered as the simplest relation of the wind linearization and can be used in the situation of a steady state rectilinear flight (G.72) (G.73) (G.74) (G.75) (G.76) (G.77) (G.78) (G.79) (G.80)
Linearization of dv; The term dv; is the second component of DVZ = (C~;;WADV~V,)~ (Equation 5.49, p. 113). This term appears in the lateral force equation (5.203), p. 148 and is made of three terms associated with the three components of (C~;;WADV~V,)~ (Equation 5.48, p. 113).
Advi Adv&
=
Adv&
+ Adv& + Advzw
(G.81)
-
AduO,
+du&
+du&
(- sin Pai COS a,, COS ei COS +i + cos Pai(sin Oi sin 4i cos $, - sin $J~ COS 4i) - sin a,,sin Psi (cos qisin 8, cos 4, + sin 4, sin $ i ) )
(- cos PO,cos a a i cos ei cos ll,, - sin PO,(sin Bi sin 4, cos $J,- sin qi COS 4i) - sin a,, cos Pai(cos ll,,sin 8, cos 4, + sin 4, sin ll,,)) A@, (sin Pai cos a,, cos 8, sin $, - cos Pai(sin Oi sin 4, sin I), + COS qi COS 4i) - sin a,; sin P,, (- sin ll,i sin 8, cos 4, + sin 4, cos $+)) All,
+
+dugi
(cos Pai(sin 8, cos 4, cos qi sin t)i sin 4i) - sin a,, sin PO,(- cos ll,i sin Oi sin 4, cos 4, sin t,bi)> A 4
+du&
(sin Pai cos aaisin 8; COS qi cos Pai cos Oi sin 4, cos ll,i - sin aaisin Pai COS ll,i cos Oi COS + i ) A8
+du&
+
+
(sin Paisin a,, cos 8, COS $i - cos aaisin
Pai(cos
Qi
sin Oi cos 4,
+ sin 4, sin ll,,)) ACW,
(G.82)
Dynamics of Flight: Equations
294
G
Advz (- sin Paicos a,, sin $, cos ei + cos (sin 8, sin +, sin $i cos $i cos 4,) - sin a,,sin PO,(sin 8, cos 4, sin $, - sin +i COS$1)
=
A&:"
+
(-
+du&
COS
Paicos a a i sin $! cos 8,
- sin Fa,(sin 8, sin 4, sin $,
+ cos $, cos 4,)
- sin a,, cos Pai (sin 8, cos +i sin $! - sin
+, cos $,)) A@,
(- sin Pai cos a,, cos $, cos 8,
+dug,
f cos psi(sin 8, sin 4, cos $, - sin $, cos + i ) - sin a,, sin Pai (sin 0, cos
+, cos qi + sin 4, sin $,)) A$
cos c$i sin $, - cos $, sin 4,)
+dv&
(COS Pai (sin vOi
+du&
(sin PO,cos a,, sin $, sin 9, COS$^,, cos@,sin sin $, - sin a,i sin Pai cos 8, cos (sin P,, sin a,, sin $i cos 8,
+ sin a,, sin Pai(sin 8, sin +, sin $, t cos 4, cos $,)) A+
+,
+
+dv&
- cos a,, sin Pai(sin 8, cos 4, sin $, - sin
-
Adv&
- Lineam'zed equations
Adwz
(sin Pai cos a,, sin Oi
cai
(G.83)
sin Pai cos 8, cos 4,)
- sin a,; cos paicos 0, COS c$,)AP,
+
+dw& +dw& +dw&
+, cos $,)) Aa,
+ cos Paicos 8, sin 4, - sin
(cos Pai cos a,, sin 8, - sin Paicos 8, sin +,
+dw&
+, sin $,) A8
(cos Pai cos 8, cos 4, sin a,,sin PO,cos 8, sin +,) 4 4 sin 8, sin 4i (sin Pai cos a,, cos 8, - cos sin a,, sin sin 8,cos di)A8 (- sin Pal sin a,, sin Oi - cos a,, sin PO,cos 8, cos 4i)Aaa
+ ~
(G.84)
Linearization of d v t relative to an initial sideslip angle and azirr-9th angle equal to zero pi = $, = 0 The simplification of the expressim du; is relevant when the initial sideslip angle Pi and azimuth angle $, are equal t o zero. Eventually, these conditions are usually achievable. So Adu&
+ sin a,, sin 8, cos -A$ du& cos +, + A 4 du;, sin 8; cos 4, + A0 du& cos 8; sin +, Adut COS 4, + AD, du& sin aOisin 4i + A$ du& sin Oi sin 4,
= Aduz sin 8, sin 4i - A@, du;, (cos a,,cos 0,
A~u& =
+i)
(G.85j
-A+ du& sin 4; (G.86) Advzw = Adwz cos 8, sin 4, A@, dwzi (cos a,,sin 8, - sin aai cos 8, cos 4,) +A@dw& cos 8, cos 4, - A0 dw& sin 8, sin 4, (G.87)
+
Linearization of dut- relative to an initial sideslip angle, azimuth angle and bank angle equal to zero Pi = $, = = 0
+,
G.3 Linearization of the longitudinal equations
295
The previous equation (G.85) to equation (G.87) with the bank angle equal to zero
+, = 0 , yield
Furthermore, if the the initial inclination angle is equal to zero, Oi = 0, then COS a a i Apa - duo,,All, Adv& = Adv& = A d v i Advtw = -dw$i sin aaiApa + dw& A+
Finally the expression of dv;
(G.91) (G.92)
(G .93)
equation (G.81) appears as
(G.94) The linearization of Adu;,
Adwg, Adw;
equation (5.48), p. 113 yields
(G .95) (G.96) (G.97)
G .3
Linearization of the longitudinal equations
G.3.1
Linearization of the propulsion equation
The propulsion equation (5.188), p. 143 below, expressed in the aerodynamic frame Fa, is linearized
From equation (4.104), p. 92 the term cosa,F,b F cos(aa - a,) COS Pm.
+ s i n a a F i can be written as
Linearization of the acceleration terms
Dynamics of Flight: Equations
296
G - Linearized equations
and the reduced velocities (Equation 6.16, p. 161) to (Equation 6.19, p. 162) V a , etc, are introduced by dividing each term by the initial velocity Vai, then
Linearization of the external forces - mgcosyai Ay, - !jphiSVzCDiAh - piSvaiCDiAVa
(CDb,Mi(AVae iThiAh) + CDCYaACYa + CDqAqt-3piSV: CDduaAdr- + C ~ b m A 8 m ) ( Va -F, sin(aai - a,) cos/3,Aao -$piSV:
-
i
+F,
COS(cYa;
- a,)
3/,
COS
(PhiAh + AAVa +
(G.lO1)
with equation (6.33), p. 164
Aqi = Aq
- Aqx,b
(G.102)
The drag coefficient CD is often modelized by
CO = COO+ICCL~ Then the angle of attack
(G.103)
derivative of CO
CDCY, = ~ ~ C L C L C U ~
(G. 104)
G .3 Linearization of the longitudinal equations
297
The linearization of the CD proposed here is a classic one; if the reader has another modeling of the CO, he can adapt the linearization to his particular case. With the dynamic pressure, it can be noted that qpi = $piV2 the previous expressions are simplified and the linearized equations of external forces are written
sqp e Va i
+-cD,
(- sin Oi cos Oi (Aux; - A W Z + ~ )sin2 BiAqZk + cos2 OiAqxk) (G.105)
The previous results are gathered and divided by mVai in order to obtain the linearized propulsion equation
(G.106) with
Dynamics of Flight: Equations
298
G - Linearized equations
____.__
(G. 108)
(G.109)
(G. 110)
(G.112)
If the system is linearized relative to a steady state flight with an aerodynamic climb angle equal t o zero 7, = 0, the coefficients ax and bw will be simplified. Particularly the term axv, and the term axv7. The terms bw are equal to (G.113) (G .114)
(G. 116) (G.117) (G.118)
G.3 Linearization of the longitudinal equations
299
Furthermore, if the system is linearized relative to a steady state flight without wind, all wind terms for the initial conditions (subscript i ) are equal to zero. So, the terms axv,, axvy,axv,, are simplified. The terms bw are reduced to (G. 119) (G.120) (G.121)
bwvq,
=
- cos yai sin yai
+ mV2 cos2 eicoq
(G.122) (G. 123)
G.3.2
Linearization of the sustentation equation
The sustentation equation (5.189), p. 144 below, expressed in the aerodynamic frame Fa, is linearized
+
From equation (4.104)) p. 92 the term - sina,F,b cosa,F,b can be written as -Fsin(cu, - cu,)cos~,, otherwise from equation (5.197)) p. 144 &a - q = -?,. With a process similar to those applied t o the propulsion equation (Section G.3.1, p. 295)) the sustentation equation is linearized. Linearization of the acceleration terms
then with the rearranged terms
-mV,, ATa
+
mV,, AV, (-qz:,
c
2
sin2yui - qxwicos yui) Dynamics of Flight: Equations
G - Linearized equations
300
Linearization of the external forces
- F, sin(aai - a,)
C O S ~ ,
PhiAh
+ AAVa + (G.127)
with equation (6.31), p. 164 and equation (6.33), p. 164 Aqt
= A q - Aqx;
then
1;1, - SgpiCLSmA6m - sin(aai - a,) cos&ASx SX;
(G.128)
G.3 Linearization of the longitudinal equations
301
The previous results are gathered and divided by mVai in order to obtain the linearized sustentation equation
with
(G .132)
(G.133)
Dynamics of Flight: Equations
(G.136) If the system is linearized relative to a steady state flight with an aerodynamic climb angle equal to zero y, = 0, the coefficients a x and bw will be simplified. In particular, the term axy,, and the term axyr. The terms bw are equal to (G .137) (G.138) (G.139) (G.140) (G.141) (G.142) Furthermore, if the system is linearized relative to a steady state flight without wind, all the wind terms for initial conditions (subscript i ) are equal to zero. The terms axyv, il"y7, and axya, are concerned. The element within CLq disappears. The terms bw are reduced to (G.143) (G.144) bwyux = cosyai siny,, bWYW% =
+"'pi sin 2oicLq 2mV2
- COSY,, sin yai - e s q ~ i sin 2 o i c L q
2mV2
~
(G.145) (G.146) (G.147) (G.148)
G.3.3
Linearization of the moment equation
The moment equation (5.190), p. 144 below, expressed in the body frame Fb, is linearized
Bq
=
tpSeV:Cm+M~6,
(G.149)
G.3 Linearixation of the lon.qitudina1 equations
303
with equation (5.57), p. 115 M F ~=
~ c o s ~ , ( z ~ ~-x&sina,) o s a ~
(G.150)
denoted
M F i = FZ r b
(G.151)
with
z t bM
=
b COSP~(ZM COSCY, - x&
sincu,)
(G.152)
The linearization of the moment equation gives
+
+
e
$piSIV; (CmqAq; V ai z'kF,(phiAh
+
e + CmbAb, + CmGmAGm) Va i
1 AAVa + -A6m) 62;
(G.153)
with equation (6.31), p. 164 and equation (6.33), p. 164 (G.154) Thus, with the rearranged terms
+
? !E !
B
+ -B
(
e
C m a a - -Cmq Va i
(cos 28, (waLi - zlx:,)
+ sin 28,(qz& - qx;,))
&F, A6x Cm6mAGm + 6XiB
(G.155) Thus the linearized moment equation is obtained
Dynamics of Flight: Equations
304
G - Linearized equations
(G.156)
--2S'pi
BVa;
Cmq [cos 28,(wZ:, - u x ~ + i sin 20,(qzLi- qxLi )I
(G.157)
(G.158)
(G.159)
(G. 160)
(G.161)
(G.162)
G.4 Linearization of the lateral equations
G.3.4
305
Linearization of the kinematic equations
The kinematic equations of vertical and pitch velocities (Equation 5.194, p. 144) below, are linearized (G .163) (G.164) Thanks t o equation (5.197), p. 144, this last relation (Equation G.164, p. 305) can be written &a+?,
(G .165)
= q
The linearization of the kinematic equations gives
so axh, axhr
= Vai sin yai = VaiCOSY,,
(G .167)
The others axh, bwh, buh are equal t o zero, and (G .168)
For the kinematic equation of pitch velocity (Equation G.165, p. 305), the coefficient are
amq axax busy bwaz
1 -axyq -hyX for = -buy, for = -bqz for = =
x = ( V , a ,h , y ) y = (m, z)
z = ( U , w , uz, w z , qz, q z )
(G.169)
G.4
Linearization of the lateral equations
G.4.1
Linearization of the lateral force equation
Divided by mVa the lateral force equation (5.202), p. 148 i s linearized ~
b
+ j a i -AVav, - Ap sin a,, + Ar COS aai- (risin aai+ p; COS aai)Aaa
a
- sin a,, - sin
i
( (cos2Pai - sin2 Psi )ijLi AP,
COS aai
COS Pai COS 2aai
+ sin Pai COS p,, A&)
&,, Act, - sin a a i sin 2Pai&, A@,
Dynamics of Flight: Equations
G - Linearized equations
306
+
?ki
- sin a,, (cos2 PaiAp& sin2 PaiAm;) - cos a,, sin 2PQi A@, COS a,,(cos2 ,8, ATX; + sin2 P,, Ayy;) - ((p&, cos2 P,, py;, sin2 P,,) cos a,, (TY;, sin2 P,, + TX;, cos2 PO,) sin a,,) Aa,
+ + + +
+
cos 2PQi( - - u X w 6i cos2 a,,
-
f sin 2Pai( -AuxL cos2 a,, - Am;
-
sin PQiCOS P,, sin 2aaiI.$,, Act,
+
Advz
-
vai
9
-(sin 8, cos a,, cos P,,
+ + -
+ vyLi)AD, sin2 a,, + Avy;)
wz,,6 sin2 a,,
-
vai
cos 8, sin 4, sin PO, - cos 8, sin a,, cos 4, cos ,8, )A&
+ cos 8, sin a,, sin 4, sin P,, ) A 4 9 (cos 8, cos a,, sin Pai sin 8, sin 4, cos Pai + sin 8; sin a,, cos 4, sin P,, )A8 vai 9 (sin 8, sin a,, sin P,, + cos 8, cos a,, cos 4, sin Pai)Aao 9
-(cos 8, cos 6,cos Pai vai
-
-
vai
+ + + + + +
E (- sin Paisin Pm
mv,,
E
(sin/?,,
COS
- cos Pai cos Pm cos(a,, - a,))
Pmsin(a,,
mv,, AE (cosP,, sinpm - sin& mv,,
- a,))
AD,
Aa,
COSP,cos(a,,
-
a,))
(G.170)
For this linearization, the altitude h and the air density p are assumed constant; if not, complementary terms will appear in the expressions of the aerodynamic and propulsion lateral forces.
Particular initial conditions
Some hypothesis can be made without significant consequences on the majority of flight situations. 0 0
0
The linearization is made with respect t o a steady state flight case, so
Dai = 0.
The initial conditions (Hypothesis 26) on the azimuth and sideslip angle are equal t o zero, so $,I = Pi = 0. The aircraft as a geometrical plane of symmetry (Hypothesis 4) so Cyi = 0.
G.4 Linearixation of the lateral equations 0
in
307
The aircraft as a symmetry on the propulsion forces Pm = 0.
The linearized equation of the lateral force equation (G.170), p. 306 is simplified
+
b Advz (-ux,, cos2 a,, - WZ;, sin2 aai + vyLi)APa+ Vai
9
-(sin Oi cos a,, - cos Oi sin a,, COS +,)ABa
+ + +
vai
9
-(cos Oi cos 4,A4 - sin 8, sin +i
Vai
s'm ,,
+pi-
(cyPAPa
A8)
I + -(CypApa + CyrAr, + CybAb,) VQi
+
+ p i s(CySlAS1 CySnASn)
rn
With Advz calculated in equation (G.85), p. 294 to equation (G.87), p. 294
Advt
=
+
+ + +
+
Aduz sin 8, sin 4, Adv; cos 4i + Adwz COS 8, sin +i (du;, (- COS aai COS 8, - sin a,, sin 8, cos 4,) + dv& sin a,, sin 4, +dw& (cos aai sin 8, - sin a,, cos Oi cos $+)) AB,
+
(-du& cos 4, dv& sin 8, sin 4,) Azl, (duzisin 8, cos 4i - dv;, sin 4, + dw;, COS 8, COS 4,) A+ (du&cos 8, sin 4, - dw& sin 8, sin 4,) At9
(G 172)
The components of the linearized wind gradient are calculated in section (G.2), p. 283. The aerodynamic angular velocities (Equation 3.54, p. 56) are
and
Gathering the previous results for the particular initial conditions, and with the expressions of Ap&, Ar&, &, pz& and r x & written as functions of the components of Dynamics of Flight: Equations
G - Linearized equations
308
the wind expressed in the vehicle-carried normal Earth frame F, (Equation G . l , p. 283) to (Equation G.37, p. 286) and (Equation G.44, p. 286) to (Equation G.52, p. 288), we obtain
(G.174)
+ -
(duo,, COS@, - dwGi sine,)
+
8, cos
COS (xai
*P4
=
+
&Pe
=
Va i 9 -(sin Va i
- cos Oi sin aaicos 4i)
1
Vai (dv& sin 4, - cos +i (du& sin ei + dwzi cos e,)) + Va i cos 8, cos 4i apz4, sin a,, - arx4, cos aai 9
1 9 - dwz,sin 8, sin 4i) - -sin 8, sin +i Va i Va i +apze, sin a a i - arxe, cos aai - - (duzi cos 8, sin 4,
G.4 Linearization of the lateral equations
S m
-+pi -t(arxe,Cyr,
309
+ arye,Cyr, + apyewCyp)
(G.175)
The terms apzt,, apytw, aryt,, arxt, with t = [e, 6, $3 are respectively calculated in equation (G.47), p. 287, equation (G.50), p. 288, equation (G.49), p. 287 and equation (G.52), p. 288. The terms duO,, dug, dwg,are calculated in equation (5.48), p. 113. The terms U X ~ , v, y b , , W Z & , q&, q z & , are calculated in equation (D.63), p. 239 t o equation (D.77), p. 240 as functions of the components of the wind expressed in the vehicle-carried normal Earth frame F,. The components of the control matrix
(G.176) The components of the wind perturbation matrix: the first for the linear wind velocity
(G .177) The second for the gradient of the linear wind velocity bwPuz
=
bwPvy
=
bwPwz
sin 4, -sinOiuLi va,
cos q5i Va i sin -- 4,
--
=
vai
sin 24, +2
S
- sin aai- ;pi -tcyp) m
sin 24, +(sina,, + 2 coSeiw;,
+
sin 244 2
m
S
- sin a,, - +pi-mt ~ y p )
(G.178) Finally, the third for the wind angular velocity bwPpy
=
sin $i
--
i
S -e m
- +pi bWPpz
=
cos ($i Va i
+3pi
cos
+ sin aaicos 8, sin2 q!+
(Cyp COS 8, cos2 4,
+ ~ y r sin , 8,
COS
4i)
+ sin a,;cos 6;cos2 c$~ - cos a a i sin Bi cos 4i S
t (Cyp cos ei sin2 +i
+ Cyr,
sin ei cos + i ) Dynamics of Flight: Equations
310
G
bwPqx
=
sin 4, cos eiuLi Va i +;pi
bWPqz
=
S
-e m
+ + sin
aai
sin 28, sin 24,
(Cyr, sin2 8, sin 4,
--sin 4, sin eiw& - a sin
+ cos
cos2 8, sin 4,
+ cos a,, sin2 8, sin 4,
S + + pi .t (cyr, sin 4, cos2 8, - + cyp sin 28, sin 24, + cyr, m bwPrx
=
cos 4, U;, - COS a,, Va i
--
S
- +pi -e
m
bwPry
=
equations
+ i ~ y p s i 28, n sin 24, + Cyr, sin 4, cos2 8,) sin 28, sin 24,
Va i
Qai
- Linearized
COS 8, COS
sin 4, sin2 8,)
4, - sin a a i cos2 4, sin 8,
(Cyr, cos 8, cos 4, - Cyp sin2 4, sin 8,)
sin 4, sin 8,v& - sin a a i sin2 4, sin 8, Va i
S
-+pi-[m ( ~ y r ~ c o s e ~-c ~oysp~c ~ os~4,sine,)
(G.179)
Initial conditions with horizontal wing, so a zero bank angle 4, = 0 With the hypothesis 6, = 0, the coefficients axp and bwp are simplified
+ ux,,b sin +-(cos8,dw& Va i
*P4
*Pv
=
=
1 -(du& sin Oi Va i
dv&
vi
+ wzb,, sin2aai- vy,,b COS + sinB,duO,,)+ (cos 8,duLi - dw& sin 8,) Vai
cos2 a,,
+ dw;,
COS 8,)
9 +COS 8, Va i
G.4 Linearixation of the lateral equations - $pi-t(arxe,Cyr, S
m
311
+ arye,Cyr, + apye,Cyp) (G. 180) arxt, with = [@,$,+I are respectively calculated in
The terms apzt,, apyt,, aryt,, equation (G.56), p. 289, equation (G.59), p. 290 equation (G.58), p. 290 and equation (G.61), p. 290. The terms duc, du;, dw;, are calculated in (Equation 5.48, p. 113). The terms UX;,, v y L i , wzLi, qx",, qz",, are calculated in equation (D.69), p. 239 to equation (D.71), p. 240 as functions of the components of the wind expressed in the vehicle-carried normal Earth frame Fo . The components of the control matrix do not change
The components of the wind perturbation matrix: the first for the linear wind velocity
(G.181) The second for the gradient of the linear wind velocity
bwpvy = bdwz
1
-vaiU &
(G. 182)
= 0
Finally, the third for the wind angular velocity
S
+
bwPpy = - ;pi - t ( C y p COS 8; Cyr, sin Oi) m 1 S bwPpz = -wGi sin a a i cos Oi - cos aaisin 8, - $pi-t Cyr, sin Oi m Va i bwPqz = 0
+
bwPqz = 0
1 Va i
=
-- U&
bwPry
=
- i p i - l (Cyr, cosOi - CypsinOi) m
S
-
S
cos a,,cos Oi - sin &ai sin Oi - ;pi -l Cyr, cos Oi m
bWPrx
(G.183)
Wind known at a zero inclination angle If it is assumed that the wind is known in a normal Earth-fixed frame oriented by a zero initial inclination angle Oi, the components of the wind in coefficients axp Dynamics of Flight: Equations
G - Linearized equations
312
(Equation G.180, p. 311) are simplified. It is found thanks t o the equation (G.72), p. 293 to equation (G.80), p. 293 and equation (D.69), p. 239 to equation (D.71), p. 240; this last results with Bi = 0 axpp
a&#)
sin 2aai = ( q z & - qx&) uxLi cos2 2 sin -- Vai (-dw& cosOi - du& sine,)
+
=
1 -- (du& sin 8, Va i
+ w z ~sin2 , ~ aai- u y ~ ~ COS +Va (du& cos Oi - dwEi sin Oi) i
9 + dw& cos ei)+ cos Oi + (vyLi - wzLi)sin Va i
(G.184)
G.4.2
Linearization of the roll moment equation
The linearized equation of roll moment (Equation 5.205, p. 148) appears in the following form
The roll moment of the thrust force M F ; , is assumed to be a constant with respect to the lateral states. As for the lateral force equation (3.54), p. 56
G.4 Linearixation of the lateral equations
313
then and
SO
ClrAr, ClrAr,
+
= Clr,Ary: Clr,Arx! = (Clr, Clr,)Ar - ClryAryb, - Clr,ArxL
+
(G.187)
Finally this equation is obtained AAp
-
+ -
+
+ -
-
E A + = $piSlV:Cl/3A@a (+piS12VaiClp+E$) A p + (+piS12Va,Clr- qi(C - B)) A r ( r i ( C- B) - Epi) Aq F, ( 2 $ p i S K i C l i X-(y, Va i cos& sina, - Z, sin&)
+
b piSlV: (Cl61A61 + Cl 6nA6n) sin 24,
+ AvYO,+ AwZLcos2 8,) A G sin 28, sin 24i + cos 8,(cos2 4,Am: + sin2 4,APT,;) + 4 - sin Oi(cos24,Ary; + sin2 4,Arx;) +apydWA4+ a p ~ + ~ A+$apyo,A8] sin 28, $piS12VaiClr, A c t sin 4, + Am: sin 8, cos 4, [ 2 - sin 4,(Aqzz cos2 8, + AqxL sin2 8,) + Ary; 8, 4, +ar~4,A4 + ary+wA$ + aryo,A8] $piSe2VaiClp
( n u X Lsin2 8,
COS
-
COS
$piS12VaiClr, [-ACE sin 8, cos 8, sin 4, + Am: sin 8, COS 4, AqzL sin2 8,) Arxz COS 8, COS 4,
- sin 4,(AqxL cos2 8,
+arx4, A+
+
+ arX@, A$ + arxowA81
+
(G.188)
The previous results can be gathered into
E Ap--A+
A
=
+ + + + + + + Dynamics of Flight: Equations
314
G - Linearized equations
with
pis e 2
*Pp
=
3-
*Pr
=
5
axpq =
1-
A
pis e 2
A
E vaiczp+ --Qi A
vai(czr,+ CZr,) - qi-C A- B
- ri(C - B ) - E p
A
The terms apyEw, aryiw, arxEw with F = [e, 4, $3 are respectively calculated with equation (G.50), p. 288, equation (G.49), p. 287 and equation (G.52), p. 288. The components of control matrix are
(G.192) The components of the wind perturbation matrix: the first for the linear wind velocity
(G.193) The second for the gradient of the linear wind velocity bwpll,
=
bwPvy
=
bwPwz
=
sin 28, sin2 ei - -sin 4, (GZr, - Clr,) 2 sin 244 -3- pise2 vai czp-
A
(G.194)
2
cos2 8,
A
sin 28, +2 sin + i ( ~ ~-r CZ~,)) ,
Finally, the third for the wind angular velocity bwPpy
=
--1.-
2
pise2
A
vai(CZPCOS ei cos2 + C Z ~sin , ei COS +i
+i)
G.4 Linearization of the lateral equations bwPpz bwPqx
=
-1-
'
A
Vai (Clpcos ei sin2 4i + Clr, sinei COS 4i)
pise2
Vai ( -CZr, sin2 8, sin 4i - f ~ lsin p 28, sin 24, - Clr, sin +i cos2 e,) A piS.f2 Vai( -Clry sin 4i cos2 oi + Clp sin 28, sin 24i - Clr, sin 4i sin2 ei) = -12 A =
-1-
bwPr,
=
-1-
bwprg
=
-1-
bwP,,
piSP
315
2
Va;(Clr, cos ei cos 4i - c l p sin2 4i sin ei)
pis e 2
* 2
A pise2
Vai (Clr, cos ei cos 4, - CZPcos2 4, sin e,)
A
(G.195)
Initial conditions with horizontal wing, so a zero bank angle c$~ = 0 With the hypothesis $i = 0, the coefficients are simplified. Moreover, the wind is
assumed t o be known in the normal Earth-fixed frame oriented by the initial inclination angle 8;. In other terms it is a question of using the wind results equation (G.72), p. 293 to equation (G.80), p. 293 and equation (D.75), p. 240 t o equation (D.77), p. 240, with 4i = ei = o
' axpr =
axPd
=
1 -
p.S12
'
Vai(Clr,
A
piS P -1-
'
A
+ CZr,)
Vai [Clp (wZ&
- ~i
C-B
A
- w;,)
- q z L i CZr,
+ qx;, C Z ~ , ]
Cli F, a x p ~ = 2+piSlV,,- AA VaiA(ym cos Pm sin am - zTnsin Pm) axpa = 0 ri(C- B) - E p axpq = -
+
A
axpe
=
-1-2
pi
se2 A vai (-rY;iclP + PY;pry + pz&Clr,) I
(G.197)
The components of the control matrix do not change bupl
=
bup,
=
)Fv2clal pi se 3A Vz Clan
(G.198)
The components of the wind perturbation matrix: the first for the linear wind velocity which do not change
Dynamics of Flight: Equations
316
G - Linearized eauations
The second for the gradient of the linear wind velocity
(G.199) Finally, the third for the wind angular velocity
G.4.3
bwP,,
pi se2 - - 4 -Vai A
bWPpz
=
-1-
bWPry
=
-1-
pise2
' A
(Clp cos 8;
+ Clr, sin Oi)
Vai Clr, sin 8,
pise2 Vai (Clr, cos Oi - CZp sin ei)
' A
(G.200)
Linearization of the yaw moment equation
The linearized equation of yaw moment (Equation 5.206, p. 149) appears in the following form
By analogy with the previous linearization of the roll moment equation (G.185), p. 312, it can be written
E A+ - -Ap C
= axrpA@,
+ + + + + +
+ axrpAp + axr,Ar + axrdA4 + axr+Aazl, + axrvAVa + axr,Aq + axrgA8
axr,Aaa burlAdl + bur,ASn bwruAuG bwrvAvz bwrwAwz bwruxAuxO, bwrvyAvyz bwr,,AwzOu, bwr,,ApyO, + bwrpzApyL bwrqtAqxO, + bwrqrAqZL bwrrxArxL bwrryAryOu, (G.202)
+
+
+
+
+ +
317
G.4 Linearization of the lateral equations
with
axr,
= 0
axrq =
- pi(B - A ) - Eri
C (G.204)
The terms apzt,, am<,, arytw, arxt, with t = [e,4,+] are respectively calculated in equation (G.47), p. 287, equation (G.SO), p. 288, equation (G.49), p. 287 and equation (G.52), p. 288. The components of control matrix are
hurl = $-VaiCndl Pi= 2 C (G.205)
The components of the wind perturbation matrix: the first for the linear wind velocity bwrU = 0 bwrv = 0 bwrw = 0
(G,206)
The second for the gradient of the linear wind velocity
C
bwruz =
sin 24,
2
sl"n28, sin2ei - 2 sin di(cnr, - Cnr,)) (G.207)
sin 24,
C
sin 28, cos2 8, + 2 sin 4i(Cnr,
- ~ n r1),
Finally, the third for the wind angular velocity bwrpy = - 4
se2Vai (Cnp cos 8, cos2 c$i + Cnrosin eiCOS 4i) 7 Pi
~~~
Dynamics of Flight: Equations
G - Linearized eauattons
318
bwrpl
= -
Pi se2 + -Vai C
bwrqz = - 4
(Cnpcos 8, sin2 4i + Cnrs sin 8, cos 4i)
Pi se2 7 Vai (-Cnry sin 4i cos2 8 + +Cnpsin 28, sin 24i - Cnr, sin 4i sin2 ei)
Pi se2 bwrrx = - 4 -Vai (Cnr, cos 8, cos 4i - Cnp sin2 4i sin e,) C
bwrry
=
Pi se2
- 4 -VaCi
(Cnr, COS ei COS +i - Cnp cos2 4isin ei)
(G.208)
Initial conditions with horizontal wing, so a zero bank angle 4i = 0 With the hypothesis 4i = 0, the coefficients are simplier. Moreover the wind is assumed to be known in the normal Earth-fixed frame oriented by the initial inclination angle ei. In other terms it is a question of using the wind results equation (G.72), p. 293 to equation (G.80), p. 293 and equation (D.75), p. 240 to equation (D.77), p. 240, with 4i = e, = o
axrg
=
axrp = Pi se2
+-VaiCnr
C
axr4
E
-c qi
=
axrQ = axrc,
= 0
(G.210)
The components of the control matrix do not change
(G.211)
The components of the wind perturbation matrix: the first for the linear wind velocity which do not change
G.4 Linearization of the lateral equations
319
(G.2 12)
The second for the gradient of the linear wind velocity
(G.213)
Finally, the third for the wind angular velocity
bwrry =
G.4.4
Pi se2 -4 -Vai C
(Cnr, cos& - Cnpsin8,)
(G.2 14)
Linearization of the bank kinematic equation
The bank kinematic equation (5.207), p. 149 is linearized Ad
= Ap+tanOi(qicos4, - r i s i n ~ i ) A 4
+
tan@,cos4iAr
+ tan@,sin4,Aq + (1 + tan2 e,)(qisin#, - ri cos#,)AO (G.215)
Then
All the bu4 and bw4 are equal to zero 0 1
tan e, cos 4, tan 8,( ~COS i 4i - ri sin 4i)
0 tan 8, sin 4, (1 + tan2 ei)(qisin 4i ri cos 4i) 0 0
+
(G.217)
Dynamics of Flight: Equations
320
G.4.5
G
- Linearized equations
Linearization of heading kinematic equation
The heading kinematic equation (5.208), p. 149 is linearized
(G.218)
All the bU+ and bw$ are equal to zero
*$+ aX&
*$e
= 0 sin 4, = cos 8, tan 8, = cos 8, (q, sin 4,
-
+ r , cos 4,) (G.220)
Appendix H
Software for the calculation of the equilibrium
H.1
Software for the calculation of the equilibrium
These routines are written in fortran C************************************
subroutine equilibrium C*
c* This subroutine is to find the equilibrium c* state of the aircraft obtained whatever the c* initial position is. C************************************
c cornon declarations include ’/edika/libincl’ CIIPIIIPlllllltrlt=’=~=~~x%%~~~=~aa=a~~=xx~xx=x~~=%x===
c
character*80 text
C’
c usually di1mnax,v=dimstate+di1ninput~11+4=15 integer i integer state-fixe(diminput) avar(diminput) data name,state,equi/’altitude’,’latitude’, % ’II1(LCh’,’alphCL’D’bOta’D % ’p),)q’,’r’D k ’phi’,’teta’,’psi’, % ’deltal’ ’doltam’,’deltan’ ’deltax’/ CP113111=00=D1011111===~==~=========~a==E=~~===~=~Ex==
c system dimensions c----------------------------------------------------
n-stateadimstate n-inputldiminput
321
322
write ( 0 6 , *) write(06,*) vrite (06,* ) write (06, ) write(06,*) write (06,*) vrite(06,*) write (06,* 1 uri to (06 ,*) Write (06,*) write(Ob,*) write (06 ,*) write(06,*) vr ite (06 ,* write(06,*) write(06,*) urite(Ob,*) write (06,* 1 c
H - Software for the calculation of the equilibrium
H.1 Software for the calculation of the equilibrium
323
C************************************
subroutine calc,equilibre(state,fixe) C************************************
c* this subroutine is for the calculation of c* aircraft equilibrium position C************************************
include '/edika/libincl' ~
I
I
I
I
~
~
~
~
P
I
I
~
I
I
~
~
'
~
I
~
~
~
~
~
integer state,fixe(diminput) integer i,j,k,niter,visu,idv,~x double precision vdoti(dimstate) double precision v,min(dirnanax,v),v,cen(dinnnax,v),v-~x(di~x-v)
double precision dX(dimstate) double precision dU(diminput) integer i,dich,i,dich,tot
Dynamics of Flight: Equations
~
~
~
H - Software for the calculation of the equilibrium
324
c= internal variables link to the subroutine call environ-terrestrial call call call call
init-accelerations init-integration init-integ-reg init-integ-serv
i,di ch=O i,dich,tot=O ideter=O imscore=O niter=O pivot,decoupling=l.OdOl C minimum value of error on the deravative of the state vector Xdot c this value validate the convergence test conv,err=l.Od-l2 c minimun gap between the errors for the dichotomy search of dv=dX conv,delta,err=conv,err/lOO C l a t l ~ = l 0 5 = ~ P l t = = t l = = = l f P = I = l t P P S l t x = ~ ~ = ~ = ~ ~ = = ~ = l ~ x =
c= initialisations C = P I ' P I l l l l t l = D ' = l l r = x = = = = ~ a a = ~ ~ ~ x = ~ ~ ~ ~ = = = ~ = a = = = ~ x x ~ ~ = ~ = ~
c display of the equilibrium search write ( 0 6 , *) Display of the equilibrium search: yes=l" read(OS,*)visu C= initialisation to zero of the gradient matrix do i=l,n,v do j-1,n-v grad ( i,j1=O . enddo enddo C= management of the constant states do i=l,n,input vect,err(i+n,state)=O. do j=l,n,v grad(i+n,state,j)=O. I'
enddo
enddo do i=l,n,input grad(n,state+i,state,fixe(i))=pivot-decoupling enddo
H.l Software for the calculation of the equilibrium
325
dX(2)=0.01 dX(3)=0.01 dX (4) =O .001 dX (5) =O .0001 dX(6)=0.01 dX (7 =O . 5 dX(8)=0.003 dX(9)=0.001 dX(10)=0.001 dX(ll)=O.Ol dU(l)=O.OOl dU(2)=0.001 dU(3)=0.001 dU(4)=0.001
c= initialisation of the derivative state vector error Xdot call calc-error(v,vdot,error,vdot,real) if (overflow.eq.1) then write(06,*) 'overflow of the initial variables' call save-state(vsave,-1)
Dynamics of Flight: Equations
H - Software for the calculation of the equilibrium
326 201
v(i)=vi(i)+dv,grad call calc,error(v,vdot,error,vdot-real) if (overflow.eq.1) then v(i)=vi(i) dv_grad=dv,grad/2. got0 201 endif
c
do k=l,n-state grad(k,i)=(vdot(k)-vdotO(k))/dv-grad enddo restoring the state after the increment dv v(i)=vi(i) enddo
c==============================================
c decoupling of latitude LtC and azimut psi c if the Earth is flat and fixed c===============================I=================
if(Earth-spheric.eq.O.or.Earth-rotate.eq.0) then do i=l,n,v grad(i,n-psi)=O. grad(i,n,DLt)=O. enddo do j=l,n,v grad(n-psidot ,j) =O . grad(n,DLtdot,j)=O. enddo grad(n-psidot,n-psi)=pivot-decoupling grad(n-DLtdot,n-DLt)=pivot-decoupling endif
c definition of decoupling is made in def-pseudo-equilibre c imax,decoupled= number of decoupled equations c state-decoupled,dot(i) and state-decoupled(i) are the indices of c the same decoupled state, the first one for the vector Xdot=DZ=dvdot c and the second one for the vector X=DZo=dv do k=l,imax-decoupled c vect,err(state-decoupled-dot (k) )=O. do i=l,n-v grad(i,state,decoupled(k))=O. enddo do j=l,n-v grad(state-decoupled-dot (k) ,j>=O. enddo
H . l Software for the calculation of the equilibrium
327
enddo c====================================P=========
c= calculation of dv=dX=dZO increment of the state c====================================P=========
c c c c c c
Checking if the gradient matrix is non singular with the calculation of deter=determinant(grad) At this point a subroutine is needed for the calculation of determinant of grad and for the inverse matrix of grad calculation of the inverse of grad matrix and of its determinant deter=determinant(grad) inv-grad=inverse(grad) if (deter.eq.0) then write(06 ,*) deter,"the matrix is singular" write(06,*) "redefine the equilibrium conditions" call trace-equilibre(1) call trace-equilibre(2) return endif if(ideter.eq.0)then ideter=l call trace-equilibre(1) endif
c.............Calculation of dv c= calculation of dv through vect,dv=grad\vect-err=inverse(grad)*vect-err c or also DZo = inverse(AZ) DZ C C=
calculation of vect-dv=inverse(grad)*vect,err do i=l,n-v vect-dv(i)=O. do j=l,n,v vect-dv (i)= vect-dv (1)+inv-grad (i,j) *vect-err (j) enddo enddo
c cancellation of dv for the decoupling states c in principle calculation naturally cancels them if(Earth-spheric.eq.0.or.Earth-rotate.eq.0) then vect-dv(n,psi)=O. vect,dv(n-DLt)=O. endif do k=l,imax,decoupled vect,dv(state,decoupled(k))=O. enddo
Dynamics of Flight: Equations
H - Software for the calculation of the equilibrium
328
c............Calculation of the new state do i=l,n,v dv (i =vect-dv (i dvi ( i) =dv (i v (i) =vi(i)+coef-relax*dv (i) enddo C
c calculation of dv(i) maxi and of its indice c for the relaxation coefficient idv-max=1 dv,max-rel=dabs(dvi(l)/dvO(l)) dv,max=dabs(dvi(l)) do i=l,n-state-1 if(dv,max-rel.lt.dabs(dvi(i+l)/dvO(i+l)))then
dv-max=dabs(dv i( i+ 1 ) idv,max=i+l endif enddo 'm \ax dv sur if(visu.eq.l)write(6,*) ',name,state-equi(idv-max) c= calculuation of the error call calc-error(v,vdot,errl,vdot-real) do i=l,n,state vdoti (i)=vdot (i) enddo c display of the equilibrium search if(visu.eq.1)then call trace-equilibre(2) endif c= in case of overflow the relaxation is done 320 if (overflow.eq.1) then if(visu.eq.l)write(06,*) 'overflow of the states' do i=l,n-v dv(i)=dv(i)/2. v (i) =vi(i) +dv (i) enddo call calc-error(v ,vdot,error,vdot-real) do i=l,n,state vdoti(i)=vdot (i) enddo goto 320 endif c..............................................
c the increment dv which gives the minimum error c is search through dichotomy process ...............................................
err,min=errO
H . l Software for the calculation of the equilibm'um
329
err,max=errl do i=l,n-v dv (i) =dv (i ) /2. v-min( i =vi(i) v-max ( i =v ( i v-cen (1)=v,min (i) +dv (i) enddo call calc,error(v-cen,vdot,err,cen,vdot,real)
330 i,dich=i,dich+l i,dich,tot=i,dich,tot+l if(err-max.lt.err,min) then c taking away from the origin err,min=err-cen do i=l,n,v dv(i)=dv(i)/2. v-min (i) =v,cen (i) v,cen(i)=v-min(i)+dv(i) v,max(i)=v-cen(i)+dv(i) enddo call calc,error(v-cen,vdot,err-cen,vdot-real) call calc-error(v,max,vdot,err-max,vdot,real) else c return to the origin err,max=err,cen do i=l,n,v du (i1=dv ( i) / 2 . v-max (i =v-cen(i) v,cen(i) =v-min(i) +dv (i) v-min (i ) =v, cen ( i) -dv( i) enddo call calc-error(v,cen,vdot,err,cen,vdot-real) call calc,error(v,min,vdot,err-min,vdot,real) endif if(dabs(err,max-err,min).gt.conv,delta-err)goto
330
c If the error is not reduce, the calculation of the gradient c is improved if (err-min.ge .O.9999*err0)then if(visu.eq.l)write(06,*)" no reduction of the error" coef~relax~grad=0.5*coef~relax,grad
endif C
if(visu.eq.l)write(6,*)
"number of dichotomy : ",i-dich ~~
Dynamics of Flight: Equations
330
H - Software for the calculation of the equilibrium
i-dich=O c Vectors a r e re-initialized after the dichotomy error=err,cen do i=l,n,v v (i)=v,cen(i) enddo do i=l,n,state vdoti(1)=vdot(i) enddo c The relaxation coefficient is a posteriori evaluated on dv coef,relax,dlO=dabs(v(idv-max)-vi(idv-max))/dv-max c A part of this coefficient is taken into account for the next c calculation with the constraint that it is between 0.1 et 1.2 coef-re1 \ax = dmaxl(O.l,1.2 * coef-relax-dl0) coef,re1 \ax = dminl(1.2, coef,relax) if(visu.eq.l)then write(06,*)" error , coef-relax-dlO , idv-m \ax , coef-re1ax 'I urite(06,5000)error,coef,relax,dv,idv,idv~max,coef~relax else write(06,5000)error endif 5000 format(d9.2,5x,d9.2,7x,i2,7x,d9.2) C
c End of the dichotomy
c= end of the loop condition, check on the error if (error.le.conv,err) goto 400 c= output after 40 loop niter=niter+l if (niter.ge.40) then write(06,*) "too much iterations, stop 'I goto 400 endif C=====P=====DI===IP====rP===DL==I='==I=r=============
c= preparation of the next iteration .....................................................
errO=err-cen do i=l,n-state vdotO (i =vdot(i) vect-err (i) =-vdot(i) enddo c re-initialisation of vi(i) do i=l,n,v vi (i) =v (i) enddo c End of the equilibrium search loop goto 200
H . l Software for the calculation of the equilibrium
331
...................................................
c= final state 400 continue if(visu.eq.l)write(6,*) "total number of dichotomy:",i-dich-tot i-dich-tot=O call trace-equilibre(3) call save-state(v,-l) reynolds=O alphadot=O C*******************************************
c angle limits psi, teta, phi, LtG et LgG C=
c= C= C=
C= C=
aims -piCpsi
C= C*******************************************
call anglimit(psi,teta,phi,LtG,LgG) if(Earth-spheric.eq.1)then call geo-xyz-D endif ...............................................
return end
C************************************
subroutine calc,error(v,l,vdot,l,error,l,vdot,real,l) C************************************
c this subroutine evaluates the aircraft acceleration c vector vdot from its state vector v C************************************
include '/edika/libincl' double precision v-1(dimmax-v) ,vdot-l(dimstate) , vdot-real-l(dimstate),error-1 ........................................
c= transformation towards explicit states call save-state(v-1,-1) reynolds=O alphadot=O .........................................
c atmosphere rhoair, vson, pstat call environ-terrestrial ......................................................
c= calculation of the states vita pcin u,v,w etc call calcul-state ......................................................
c= External efforts seen by the aircraft Dynamics of Flight: Equations
H - Software for the calculation of the equilibrium
332
c these efforts are necessary for the acceleration calculations c-----------------------------------------------------
c= overflow is equal to I if a state is out of limit overflow=O call efforts C==='"===========""======"'========"===============
c= Calculation of the aircraft accelerations call accelerations c these accelerations give the components of the vector vdot C="==l=I'======l==============="=I======================
c= calculation of the error on the state derivative vector vdot(i)
vdot,l(l)=pdOt-pdot,I ~d0t-l(2)=qd0t-qd0t-I vdot-l(3)=rdot-rdot,I v~o~-~(~)=uso~G~~o~-uso~G~~o~-I
v~o~-~(~)=vso~G~~o~-vso~G~~o~-I v~o~-~(~)=wso~G~~o~-wso~G~~o~-I vdot-l(7)=tetadot-tetadot-I vdot,l(8)=phidot-phidot-I vdot-l(9)=altitudedot-altitudedot-I vdot-l(iO)=psidot-psidot-I vdot-l(ll)=DLtdot-DLtdot-I
c
~d0t-l(l2)=DLgd0t-DLgd0t-I
C
c memorisation of vdot before the cancellation of c vdot components associated to the decoupled states do i=l,n-state+l vdot,real-l(i) =vdot-1 (1) enddo C
if(Earth-spheric.eq.O.or.Earth-rotate.eq.0) vdot-l(n,psidot)=O. vdot-l(n-DLtdot)=O. endif do i=l,imax,decoupled vdot-l(state,decoupled-dot(i))=O. enddo
then
c= calculation of the error on the modulus of vdot error-l=O.dO do i=l ,n-state error-l=error,l+dabs (vdot-1(i) enddo c write (06,*) error-1
......................................................
return end C
C C************************************
subroutine save-state (v-local,i-local)
H . l Software for the calculation of the equilibrium
333
C************************************
c This subroutine saves the state in the vector v if i,local=l c and the vector v in the state if i-local=-1 C************************************
include '/edika/libincl' double precision v-local(dimmax,v) integer i-local
c================================P=====================
c= saving the state in the vector v if(i-1ocal.eq.l)then v-local(l)=altitude v_loca1(2)=DLt v-local(3) =mach v_loca1(4)=alpha v,local(5)=beta v_local(6)=p v,local(7)=q v_local(8)=r v_loca1(9)=phi v_local(lO)=teta v-local(ll)=psi v-local(l2)=deltal v,local(l3)=deltam v-local(l4)=deltan v~local(l5)=deltax c v_local(l6)=DLg C
endif .....................................................
c saving the vector v in the state C
c
if(i,local.eq.-1)then altitude=v-local(1) DLt=v,local(2) mach=v,local(3) alpha=v-1ocal(4 1 beta=v_local(5) p=v_local(6) q=v_local(7) r=v_loca1(8) phi=v_local(9) teta=v-local(lO) psi=v,local(ll) deltal=v,local(l2) deltam=v-local(l3) deltan=v~local(l4) deltax=v_loca1(15) DLg=v,local(l6)
C
endif C
Dynamics of Flzght: Equatzons
H - Software for the calculation of the equilibrium
334 return end
C************************************
subroutine init-equilibre C************************************
c This subroutine initializes the parameters useful c for the pseudo-equilibrium calculation C************************************
include '/edika/libincl' c----------------------------------------------------
c indice for ranking in X or n-altitude-1 n-psi=11 n,DLt=2 c indice for ranking in Xdot n-altitudedotP9 n,psidot=lO n,DLtdot=ll c this ranking could be done c for the definition of the
v or DZo
or vect-err or vdot ou DZ
for all the states if necessary pseudo-equilibrium
.....................................................
return end C C**********************************************************
subroutine trace-equilibre(ou) c**********************************************************
c This subroutine keeps the trace of the results obtained c during the equilibrium search c----------------------------------------------------------
include '/edika/libincl' c double precision grad(dimmax,v,dimmax-v) ,vect-dv(dinnnax-v) , c & vect-err (dimmax-v) C=
c double precision vi(dimmax,v) ,v(dimmax,v) ,vO(dimmax-v) c double precision vdotO (dimstate) ,vdot (dimstate) ,vdot-real(dimstate) c double precision error,errO c double precision dv(dimmax,v) ,dvO(dimmax-v) ,dvi(dimmax-v) C
c
common /equilibrium-search/ ,vect,dv(dimmax,v) * c & vect-err(dirmnax-v),vi(dirmnax-v),v(dimmax-v), c & v0 (dimax-v) ,vdotO (dimstate) ,vdot (dimstate) ,vdot,real (dimstate) , c & error,err0 ,dv (dimmax-v) ,dvO (dimmax-v) ,dvi (dimmax-v)
& grad(dimmax-v,dimmax,v)
C
integer ou
H . l Software for the calculation of the equilibrium
335
"
open(lO,file= & '/edika/files/search-equi.res', status='old')
Dynamics of Flight: Equations
336
H - Software for the calculation of the equilibrium
C*********************************************************
subroutine equiv-increment(dX-1 ,dU,1 ,dv,l) C*********************************************************
c This subroutine gives an equivalence between the state c increment and the increment on vector v components include '/edika/libincl' double precision dX,l(dimstate) ,dUJ (diminput) ,dv,l (dimmax-v) C
dv,l(l) = dX-1(7) dv,1(2) = dX,l(ll) dv,l(3) = dX,l(l) / w o n dv,l(4) = dX-1(3) /uaeroCg dv,l(5) = dX,1(2)/vita dv,1(6) = dX-1(4) dv,1(7) = dX,1(5) dv,1(8) = dX,1(6) dv-1(9) = dX,1(9) dv-l(lO) = dX,1(8) dv,l(ll) = dX,1(10) dv-l(l2) = dU-l(l) dvJ(13) = dU,1(2) dvJ(14) = dU-1(3) dv,1(15) = dU,1(4) C
return end
Hypotheses
337
Hypotheses where hypotheses appear, can be found under the term “Hypotheses” of
hYP- 9 hyp. 10 hyp. 11 hyp. 12 hyp. 13 hyp.14 hyp.15 hyp.16 hyp.17 hyp.18 hyp.19 hyp.20 hyp.21 hyp.22 hyp.23 hyp.24 hyp.25 hyp.26 hyp.27 hyp.28 hyp.29
The inertial frame FZ Earth is spherical, Earth radius Rt is assumed constant Flat Earth A symmetrical plane for the aircraft A small Earth flattening Sideslip angles p are zero Angles of attack cy are zero Taylor’s hypothesis, space field of wind velocity is frozen (independant of time) The field of wind velocity is a linear function of the distance The wind is locally modelized by a vortex Wind velocity field is uniform, i.e. constant velocity whatever time and position The aerodynamic climb angle T~ defined with respect to the vertical of the Normal Earth-fixed frame FE The aircraft is assumed to be a rigid solid The mass of the aircraft is assumed to be constant Field of gravity is constant The inertia matrix IIG is assumed to be independant of the center of gravity position Fixed Earth The center of gravity G of the aircraft is the origin of the body frame
Fb
Constant gravitational hypotheses The coefficients of moment defined with respect t o the aircraft center of mass G Simplifying hypotheses related to the propulsion efforts The angular velocity of the Earth 52t is assumed t o be constant The axis XE of the normal Earth-fixed frame, FE, is oriented towards the geographical North The axis, x, of the vehicle-carried normal Earth frame F,, is oriented in the direction of the aircraft’s geographical North Linearized states represents the change from the initial value Linearization: initial conditions are zero Pure longitudinal movement Air density of the atmosphere p is constant The wind does not depend on the azimuth @
Dynamics of Flight: Equations
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References [l] R. L. Bisplinghoff, H. Ashley, and R. L. Halfman. Aeroelasticity. Addison-Wesley,
1957.
[2] B. Etkin. Dynamics of Atmospheric Flight. John Wiley and Sons, 1972. [3] B. Etkin and D.A. Etkin. Critical aspects of trajectory prediction: flight in non uniform wind. In A G 301, volume Vol 1. Agard, 1990.
[4] FAA-RD-74-206, editor. Winds models for flight simulator certification of landing and approach guidance and control systems. FAA, 1974. [5] F.W. Lanchester. Aerodynamics and Aerodonetics. Constable and Co. Ltd., 1908. [6] P. Mouyon, A.J. Fossard, and D. Normand-Cyrot. Nonlinear Systems, volume3 Control of Nonlinear Systems. Chapman and Hall, 1995. [7] J. R. Wertz. Spacecraft attitude determination and control. D. Reidel, 1978. [8] D.Y. Willems. Aircraft dynamics for air traffic control. In AG 301, volume Vol 2. Agard, 1990. The pages where references appear, can be found under the term “References” of the index.
339
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Index side force CY,xxx, 90 tranverse force CY,xxx, 90 yawing moment Cn, xxx, 57, 90 velocity, 33 AI,G,see Nomenclature, Accelerations Aircraft, see Frames, body angular momentum, see Angular momentum angular velocity, see Angular velocity, aircraft center of gravity, 16, 73 link with matrix of inertia, 75 center of mass, 16, 46-48 link with the moment aerodynamic coefficient, 90 constant mass, 17, 74 definition, 5 rigid, 1, 72 symmetrical plane, 16, 31, 77, 80, 81 Aircraft design, 4 Altitude, 24, 26, 69 geometric, 69 geopotential, 69 link t o atmosphere, 87 Angles, xxiii aerodynamic angle of attack, 17, 33, 62 azimuth angle, 31, 63, 200 bank angle, 31, 63, 200 climb angle, 31, 63, 200 climb angle definition, 31 measurement angle of attack, 65 measurement sideslip angle, 65 sideslip angle, 17, 33, 62 azimuth, 27 bank angle, 27 course
A , see Nomenclature, Frames, origins,
see Nomenclature, Kinetics Acceleration, 8 centrifugal acceleration due t o Earth rotation, 83 complementary Earth’s rotation, 78 Earth’s sphericity, 78 fixed Earth, 79 flat Earth, 79 definition, 72 gravitation, 24, 83 constant of gravitation, 83 gravitation = gravity, 85 gravitation is constant, 84 law of gravitation, 83 gravity, 24, 83, 85 function of latitude, 86 inertial acceleration, 6, 72, 77, 241 relative acceleration, 78 relative pseudo-acceler at ion, 78 Aerodynamic angle of attack, 33 sideslip angle, 33 angular velocity, see Angular velocity azimuth angle, 31, 63, 200 bank angle, 31, 63, 200 climb angle, 31, 63, 200 coefficients, xxix, 89 axial force CA,xxx, 90 cross stream Cy, xxix, 90 drag CO, xxix, 90 lateral force Cy, xxix, 90 lift C L , xxix, 90 normal force C N ,xxx, 90 pitching moment Cm, xxx, 90 rolling moment CZ, xxx, 57, 90
341
magnetic course, 37 true course, 37 drift, 37 elevation angle, see inclination angle flight- path azimuth angle, 36 bank angle, 36 climb angle, 36 heading, see Angles, course magnetic heading, 28 true heading, 28 incidence, see angle of attack inclination angle, 27 kinematic angle of attack, 17, 34 azimuth angle, 36, 66 bank angle, 36 climb angle, 36, 66 sideslip angle, 17, 34 lateral inclination, 30 magnetic heading, 28 measurements, 64 pitch angle, see inclination angle, 30 relationships, 39 angles of attack, inclination and climb angles, 40 angles of attack, sideslip angles, 39 azimuth angles, 40 bank angles, 40 relationships with velocities, 61 roll angle, see bank angle table of definition, 42 transversal azimuth, 30 true heading, 28 wind angle of attack, 36, 63, 201 bank angle, 36, 63, 201 sideslip angle, 36, 63, 201 yaw angle, see azimuth angle Angular momentum, 74 aircraft, 80 complementary fixed Earth, 80 definition, 74 derivation, 44, 80, 245
fixed Earth, 81 symmetrical aircraft, 81 inertial, 74 derivation, 80, 245 Angular velocity, xxv aerodynamic definition, 56 aircraft, 80 aircraft angular velocity, 48, 60 components, xxvi Earth angular velocity, 21, 58 components, 81 notation, 81 equations, 70 fixed Earth, 79 flat Earth, 79 fundamental characteristics, 44 inertial, 75, 80 kinematic components, 49, 70 definition, 48, 60 kinematic equations, 70 pitch rate q, 49 roll rate p, 49 yaw rate r , 49 of the aerodynamic frame, 60 of the kinematic frame, 60 wind components, xxvii, 53 hypothesis, 50 modeled by a vortex, 50, 51, 55, 58 modelled by a vortex, xxvii yaw wind angular velocity, 57 A:, see Nomenclature, Accelerations Archimedes’ thrust or buoyancy, see Forces, Archimede at, see Nomenclature, The Earth Atmosphere, 4, 87 law of density, 87 law of pressure, 87 law of temperature, 87 movement, see Turbulence a q o , see Nomenclature, Linearization, state matrix component a q a , see Nomenclature, Linearization, state matrix component
Index Nomenclature, state matrix Ax1, see Nomenclature, state matrix A ~ L see , Nomenclature, state matrix A ~ I see , Nomenclature, state matrix ~ L Isee, Nomenclature, state matrix Azimuth angle, 27
Ax L, see
343 Linearization, Linearization, Linearization, Linearization, Linearization,
B,see Nomenclature, Kinetics Bank angle, 27 Bibliography, see References notation, xix Body frame, 6 b t , see Nomenclature, The Earth b q n , see Nomenclature, Linearization, state matrix component bq,, see Nomenclature, Linearization, state matrix component BUL,see Nomenclature, Linearization, state matrix BUI,see Nomenclature, Linearization, state matrix BWG L, see Nomenclature, Linearization, state matrix BwGl, see Nomenclature, Linearization, state matrix BWLL,see Nomenclature, Linearization, state matrix BWLI,see Nomenclature, Linearization, state matrix BWRL,see Nomenclature, Linearization, state matrix BWR1, see Nomenclature, Linearization, state matrix C , see Nomenclature, Kinetics see Nomenclature, External efforts CA,see Nomenclature, External efforts Cc, see Nomenclature, External efforts CO,see Nomenclature, External efforts Center of gravity, see Gravity, Aircraft
Z,
definition, 73 Center of mass, 16, 46-48, see Aircraft
definition, 73 link with the center of gravity, 73 C L ,see Nomenclature, External efforts CZ, see Nomenclature, External efforts CLQ, see Nomenclature, External efforts CZ", see Nomenclature, External efforts CLq, see Nomenclature, External efforts Cm, see Nomenclature, External efforts Cm", see Nomenclature, External efforts C N ,see Nomenclature, External efforts Cn, see Nomenclature, External efforts Cna, see Nomenclature, External efforts Coefficients, see Aerodynamic Control, 4 Coupling, see Decoupling Cx,see Nomenclature, External efforts C x b , see Nomenclature, External efforts C Y , see Nomenclature, External efforts Cy, see Nomenclature, External efforts CY*,see Nomenclature, External efforts Cz, see Nomenclature, External efforts C z b , see Nomenclature, External efforts
D , see Nomenclature, Kinetics Decoupling, 133, see Equations, decoupled conditions for decoupling longitudinal equations, 265 Density, see Atmosphere Derivation angular momentum, 44 matrix of transformation, 49 positions, 67 substantial derivative, see Gradient of wind vectors, 44 wind velocity field, 283 wind velocity, 50, 51 Drift, 37 du", see Nomenclature, Accelerations, components Dynamics of Flight: Equations
Index
344
DV,, see Nomenclature, Accelerations DV,, see Nomenclature, Wind gradient
duo,, see Nomenclature, Accelerations,
components see Nomenclature, Accelerations, components Dynamic moment, 72 Dynamic resultant, 72 Dynamics, 2 Dynamics and flying qualities, 3 dw;,
E , see Nomenclature, Kinetics Earth, xxii, 3, 4 angular velocity, 21, see Angular velocity atmosphere, 4, 87 center, 14 environment, 4 link with efforts, 67 fixed, 79, 81, 85 flat, 16, 27, 70, 79, 85 flat and fixed influence on navigation equations, 136 geoid, 15, 24 geometry of Earth, 23 gravitation, 16 gravity, 4 mass, 84 radius Rt, 15, 84 spherical, 15, 23 limit of hypothesis, 86 East velocity V', see Velocity, kinematic Efforts, see Forces, Moments aerodynamic, see Aerodynamic, coefficients external efforts, 4 thrust, 2 weight, 2 Equations angles and velocities measurements, 64 angular relationships, 39 decoupled equations, 9, 132 conditions for decoupling, 135, 137, 146
lateral equations, 146, see Equations, lateral equations longitudinal equations, 134, see Equations, longitudinal equations navigation equations, 134, 135 three type of decoupling, 134 equilibrium condition of equilibrium, 182 definition, 180 definition of pseudo-equilibrium, 182 equilibrium and steady state flight, 180 general equilibrium, 188 lateral equilibrium, 190 longitudinal equilibrium, 189 numerical research of equilibrium, 186 resolution of equilibrium, 185 software for numerical research of equilibrium, 321 flat and fixed Earth consequences on acceleration, 118 force, first form, 106 force, first form, first variant, 107 force, first form, second variant, 108 force, first form, third variant, 109 force, general form, 105 force, second form, 9, 110 force, second form, recapitulation, 113 kinematic, angular, 117 kinematic, position, 116 moment, 115 force equations, see Equations, flat and fixed Earth, force, see Equations, lateral equations, force, see Equations, longitudinal equations, force, see Equations, rotating wind, force fundamental kinematic relation, 225 fundamental laws of mechanics, 6, 72 general equations, 94
Index force, 96 kinematic, 98 moment, 97 process of equations, 99 general form, 7 kinematic, 7, 67, see Kinematic, equations definition, 43, 67 fundamental relation, 225 kinematic equations, see Equations, flat and fixed Earth, kinematic, see Equations, rotating wind, kinematic lateral equations, see Equations, linearized.. . conditions for decoupling, 146 equilibrium equations, 190 force equation, 148 kinematic equations, 149 linearization, xxxiv, 10 moment equations, 149 with uniform wind velocity, 149 linearized equations, 9 linearization around equilibrium; 159 Mach number, density, 163 method for linearization, 158 numerical linearization, 160 software for numerical linearization, 275 thrust, 164 velocity, 162 wind linearization, 283 linearized lateral equations conditions of linearization, 171 control matrix, 176 kinematic azimuth angle equation, 175 kinematic bank angle equation, 175 lateral force equation, 173 linearization without wind, 178 roll moment equation, 174 state matrix, 171 wind perturbation matrix, 177 yaw moment equation, 175 linearized longitudinal equations
345 control matrix, 168 kinematic altitude equation, 167 kinematic angular equation, 167 pitch moment equation, 167 pitch velocity, 164 propulsion equation, 167 state matrix, 166 sustentation equation, 168 wind perturbation matrix, 169 load factor, 254 longitudinal equations, 144 angular relations, 139 conditions for decoupling, 137, 265 equilibrium equations, 189 expression of acceleration, 141 expression of forces, 142 expression of wind, 140 force equations, 144 kinematic equations, 143 linearization, xxxii, see Equations, linearized.. . moment equations, 143 second form, 144 simplest equations, 145 with uniform wind velocity, 145 moment equations, see Equations, flat and fixed Earth, moment, see Equations, lateral equations, moment, see Equations, longitudinal equations, moment, see Equations, rotating wind, moment navigation equations, 135 process of equations, 5, 99 pure longitudinal, 40, 58, 139, see Equations, longitudinal equat ions relationships between angles and velocities, 61 rotating wind, 125 force, second form, 127 kinematic, 129 moment, second form, 128 simplest equations of the book, 189, 190 simplified equations, 103 Dynamics of Flight: Equations
346
decoupled equations, 132, see Equations, decoupled equations equilibrium, 180, see Equations, equilibrium flat and fixed Earth, 104, see Equations, flat and fixed Earth linearized equations, see Equations, linearized.. . rotating wind, 125, see Equations, rotating wind uniform wind velocity, 131, see Equations, uniform wind state form, 67 uniform wind velocity force equations, 131 kinematic equations, 132 moment equations, 131 validity range, 4 Equilibrium, 3, 10, see Equations, equilibrium definition, 180 numerical research, 10 numerical research of equilibrium, 186 software for numerical research of equilibrium, 321 et, see Nomenclature, The Earth
F , see Nomenclature, Kinetics
F,see Nomenclature, External efforts,
propulsive F,, see Nomenclature, Frames Fb, see Nomenclature, Frames F,, see Nomenclature, Frames FE, see Nomenclature, Frames Ff , see Nomenclature, Frames Fr , see Nomenclature, Frames Figure notation, xix Fk, see Nomenclature, Frames Flight, 2 Flight dynamics, 5 Flight-path azimuth angle, 36 bank angle, 36 climb angle, 36 Flying qualities, 3
Index
F,, see Nomenclature, Frames Forces, see Efforts, see Equations, forces aerodynamic, 89 Archimede, 4, 83 equations, see Equations, force propulsion, 92 thrust, 92 weight, 83 Frames aerodynamic or air-path F,, 17 angle measurement frame F,,s, 65 angles between, 18 body Fb, 16 course oriented frame F,, 27 fuselage oriented frame F f , 27 Galilean, 7, 73, 105 inertial, 7, 14, 73 kinematic or flight-path Fk, 17 list of frames, xxi normal Earth-fixed FE, 15 orthogonal and right-handed, 13 probe frame F,,s, 65 projection, 6 transformation matrix, 18 unitary vectors, 13 vehicle-carried normal Earth Fo, 16 wind frame, 58 ft, see Nomenclature, The Earth
G , see Nomenclature, Frames, origins
G, see Nomenclature, External efforts g , see Nomenclature, External efforts Galilean, see Frames Geoid, 24 Gliders, 4 Gnopter, 1 g,, see Nomenclature, External efforts CGWmV, see Nomenclature, Wind gradient, components Gradient of wind derivation of wind velocity, 51, 110, 126, 251 expressed in aircraft frame, 238 expressed in body frame Fb, 238 independant of the heading, 136 linearization of wind field velocity, 283
347
Index notations - components, xxvi, 52 pure longitudinal flight, 140 substantial derivative, 110, 126,251 Gravitation, see Acceleration, gravit ation, 24, 82 Gravitational attraction, see Acceleration, gravitation Gravity, see Acceleration, gravity, 24, see Center of ..., Aircraft, 82 Gravity field is constant, 16, 73, 86 Greek symbols a a , see Nomenclature, Angles between frames ass, see Nomenclature, Angles between frames a k , see Nomenclature, Angles between frames a,, see Nomenclature, External efforts, propulsive aW, see Nomenclature, Angles between frames see Nomenclature, Angles between frames pas,see Nomenclature, Angles between frames p k , see Nomenclature, Angles between frames P,, see Nomenclature, External efforts, propulsive pw, see Nomenclature, Angles between frames x,, see Nomenclature, Angles between frames x k , see Nomenclature, Angles between frames AAA, see Nomenclature, Accelerations AAD, see Nomenclature, Accelerations AADI, see Nomenclature, Accelerat ions AAR, see Nomenclature, Accelerations A A s , see Nomenclature, Accelerations A A w ~see , Nomenclature, Accelerations
AA,N, see Nomenclature, Accelerations
A A w ~see , Nomenclature, Accelerations
61, see Nomenclature, External efforts, controls
ALt, see Nomenclature, The Earth ,S see Nomenclature, External efforts, controls
S,, see Nomenclature, External efforts, controls
AV,, see Nomenclature, Linearization, example
S,, see Nomenclature, External ef-
forts, controls see Nomenclature, Angles between frames ~k , see Nomenclature, Angles between frames A, see Nomenclature, External efforts, propulsive pa, see Nomenclature, Angles between frames pk, see Nomenclature, Angles between frames p w , see Nomenclature, Angles between frames n, see Nomenclature, Angular velocit ies flay see Nomenclature, Angular velocities n a b , see Nomenclature, Angular velocities, exemple Ok, see Nomenclature, Angular velocities Of, see Nomenclature, Angular velocities W ~ G see , Nomenclature, The Earth w t o , see Nomenclature, The Earth nW,see Nomenclature, Angular velocities 4, see Nomenclature, Angles between frames $, see Nomenclature, Angles between frames $m,see Nomenclature, Angles between frames ~~~-
__
.
__
-
Dynamacs of Flaght: Equatzons
348
Index t,bV, see Nomenclature, Angles between frames p, see Nomenclature, Standard atmosphere ph, see Nomenclature, Linearization, reduced states ph, see Nomenclature, Standard atmosphere 8, see Nomenclature, Angles between frames
H , see Nomenclature, Altitudes
h, see Nomenclature, Altitudes, see Nomenclature, Standard atmosphere Heading, see Angles h,, see Nomenclature, Altitudes HI,G,see Nomenclature, Kinetics Hypotheses hyp:l -, 14, 73 hyp:lO -, xxvii, 50, 55, 58, 125 hyp:ll -, 130, 131 hyp:12 -,31 hyp:13 -, 17, 72, 77, 80, 247, 248 hyp:14 -, 17, 74 hyp:15 -, 73, 86, 128 hyp:16 -, 77 hyp:17-, 50,68, 70, 79-81,85, 104, 115, 116, 252 hyp:18 -, 16 hyp:19 -, 84, 85 hyp:2 -, 15, 16, 23, 25, 26, 69, 70, 83, 84, 86, 230 hyp:20 -, 90 hyp:21 -,93 hyp:22 -, 242 hyp:23 -, 15, 21, 23 hyp:24 -, 16, 46 hyp:25 -, 160, 165, 170 hyp:26 -, 171, 306 hyp:27 -, 40, 58 hyp:28 -, 171 hyp:29 -, 136 hyp:3 -, 16, 27, 50, 68, 70, 79, 85, 104, 115, 116, 118, 252 hyp:4 -, 17, 77, 80, 81, 148, 172, 306 hyp:5 -, 25
hyp:6 -, 40 hyp:7 -, 39 hyp:8 -, 49, 87, 128, 130, 252 hyp:9 -, 50, 55, 128 Hypothesis, see after Index, list of hypotheses notation, xix 1, see Nomenclature, Kinetics IIG, see Nomenclature, Kinetics Inclination angle, 27 Inertia, see Matrix, inertia Inertial, see Acceleration, Velocity, Frames Inertial acceleration, see Acceleration Inertial frame, 14 Inertial velocity, see Velocity
Jacobian matrix, xxvi, 52 Kinematic, see Equations,..., kinematic angle of attack, 34 azimuth angle, 36, 66 bank angle, 36 climb angle, 36, 66 equations, 7, 67 angular velocity equations, 70 coupling of linear and angular velocity equations, 70 definition, 43 latitude, Earth angular velocity, 59 longitude, Earth angular velocity, 59 navigational relationships, 59 velocity equations, 68 fundamental relation, 225 fundamental relationship, 6, 44 sideslip angle, 34 velocity Vk,34, 46 k,, see Nomenclature, External efforts, propulsive Koenig's theorem, 76, see Matrix, of inertia
l , see Nomenclature, External efforts Lateral movement, see Equations, lateral equations Latitude, 23, 25
Index astronomical, 24 derivation, 59, 68 geocentric, 23 geodesic, 24 geographical, 21, 24, 26 gravitation, 23 Length of reference l , 89 LgG, see Nomenclature, The Earth Linearization, see Equations, linearized equations around equilibrium, 159 lateral equations, xxxiv longitudinal equations, xxxii notations, xxxi numerical linearization, 160 software for numerical linearizat ion, 275 wind linearization, 283 Load factor, 254 Longitude, 21 derivation, 59, 68 Longitudinal movement pure longitudinal, 40, see Equations, longitudinal, 139, 140 U,,, see Nomenclature, The Earth LtG, see Nomenclature, The Earth Lt,,, see Nomenclature, The Earth Ltgd, see Nomenclature, The Earth U,,, see Nomenclature, The Earth L t o , see Nomenclature, The Earth
M , see Nomenclature, Linear velocities Magnetic course, 37 Magnetic heading, 28 Mass, see Aircraft, Gravity, see Earth, mass Matrix identity matrix, xxi Jacobian matrix, xxvi, 52 matrix of inertia constant matrix, 74 definition, 75 Koenig’s theorem, 76 link with the angular momentum, 75 moment of inertia, 75 point of reference, 76
349
product of inertia, 76 projection frame, 76 summation, 76 transformation from frame to frame, 76 operations on matrices, xx rotation associated, xxi, 44 skew-symmetric, 44 transformation matrix, xxii, 19 derivation, 49 Measurement acceleration, load factor, 254 angle of attack, 64, 202, 220 sideslip angle, 64, 202, 220 velocities, 64 weight, 83 Mechanics fundamental laws, 6 MF, see Nomenclature, External efforts, propulsive Missile, 1 MQ, see Nomenclature, Angular velocities Moments, see Efforts, see Equations, moment equations aerodynamic, 89 due to gravity, 86 equations, see Equations, moment moment of external forces, 72 of inertia, see Matrix, of inertia propulsion, 93 thrust, 93 Momentum, see Angular momentum angular momentum, 6 inertial angular momentum, 6, 8 mt, see Nomenclature, The Earth Navigation equations, see Equations, navigation equations Nomenclature, xix Normal Earth-fixed frame, 15 North velocity V N , see Velocity, kinematic Notations, xix, xxxvi 0 , see Nomenclature, Flames, origins Dynamics of Flight: Equations
350 p, see Nomenclature, Angular velocit-
ies, components, see Nomenclature, Standard atmosphere Performance, 3 Pitch, see Angles and Angular velocity p i , see Nomenclature, Angular velocities, components, example Plane of symmetry see, Aircraft, 16 Posit ions components, xxiv derivative, 43, 67 geographic position, 69 longitude, latitude, 69 Pressure, see Atmosphere Probe, 64 acceleration, 254 angles, 202, 220 Product, see Vectors of inertia, see Matrix, of inertia Projection frames, 6 Propulsion efforts, 92 pt , see Nomenclature, Angular velocities, components Pure longitudinal flight, see Equations, longitudinal equations, see Longitudinal movement @E, see Nomenclature, Wind gradient, components pyc, see Nomenclature, Wind gradient, components p:, see Nomenclature, Wind gradient, components q, see Nomenclature, Angular velocit-
ies, components q;, see Nomenclature, Angular velocities, components, example qp, see Nomenclature, External efforts qt , see Nomenclature, Angular velocities, components @E, see Nomenclature, Wind gradient, components q x ; , see Nomenclature, Wind gradient, components a.", see Nomenclature, Wind gradient, components
Index
72, see Nomenclature, Standard atmosphere
r , see Nomenclature, Angular velocit-
ies, components Radius Earth radius Rt, 15 Reference length and surface, 89 notation, xix References [l]-,1, 162 [2]-, 15, 16, 23, 84 p1-7 3, 49 [4]-, 49, 56 [51-, 25 161-9 49 [7]-, 158 [8]-,110, 251 Relationships, see Equations Rigid aircraft, 1 r : , see Nomenclature, Angular velocities, components, example Roll, see Angles and Angular velocity Rt, see Nomenclature, The Earth rt , see Nomenclature, Angular velocities, components F E , see Nomenclature, Wind gradient, components TX;, see Nomenclature, Wind gradient, components r y ; , see Nomenclature, Wind gradient, components
S , see Nomenclature, External efforts
Simplest equations of the book lateral equations, 190 longitudinal equations, 189 Standard, xix Steady state flight, 10 Surface of reference S, 89 Symbols, xix, see Greek symbols, see Nomenclature Symmetrical plane see, Aircraft, 16
T , see Nomenclature, Standard atmosphere
35 1
Index Nomenclature, Transformation matrix T b k , see Nomenclature, Transformation matrix Temperature, see Atmosphere T E ~see , Nomenclature, Transformation matrix Terrestrial model, 4 Th, see Nomenclature, Standard atmosphere T h , see Nomenclature, Linearization, reduced states TZE,see Nomenclature, Transformation matrix Time stellar, 21, 26 Tlo, see Nomenclature, Transformation matrix T k a , see Nomenclature, Transformation matrix Toa, see Nomenclature, Transformation matrix Tab, see Nomenclature, Transformation matrix T o k , see Nomenclature, Transformation matrix True course, 37 True heading, 28 Turbulence, see Gradient of wind, 49 turbulence scale L , 49 T b a , see
U;,
see Nomenclature, Linear velocit-
ies, components, example
UL, see Nomenclature, Linearization,
state vector Ui, see Nomenclature, Linearization, state vector gW, see Nomenclature, Linearization, reduced states ii;, see Nomenclature, Wind gradient, components u x ; , see Nomenclature, Wind gradient, components u y ; , see Nomenclature, Wind gradient, components uz;, see Nomenclature, Wind gradient, components
see Nomenclature, Linear velocities V,, see Nomenclature, Linearization, reduced states V', see Nomenclature, Linear velocities, components Vectors components, xix, xxxvi cross product, xxi derivation, 6, 44 dot product, xxi expressed, xix, 19 notations, xix operations on vectors, xx projection, xix, 6, 19 Velocity, xxiv, 7 aerodynamic, 17, 33, 47, 62, 89 definition V,, 47 air velocity, 47 air-path velocity, 47 angular, see Angular velocity components, xxv definition, 43, 72 flight-path, 46 ground velocity, 46 inertial, 48, 68, 77, 229 kinematic, 34, 68 definition, 46 derivation, 78 East velocity VE,46, 68 flat kinematic velocity v k p , 46 kinematic velocity v k , 17, 46 North velocity V N ,46, 68 Vertical velocity V z , 46, 68 measurements, 64 relationship between linear velocities, 48 relationships with angles, 61 two representations, 61 wind definition, 47 derivation, 50, see Gradient of wind, 251 field of wind velocity, 49, 51, see gradient of wind modeled by a vortex, 50, 58, 126 Va,
Dynamics of Flight: Equations
352
relations hip between velocities, 50, 56 uniform wind velocity, 130 wind velocity, 201 wind velocity V,, 35, 47 Vertical direction, 25, 83 Vertical velocity V z , see Velocity, kinematic VI,see Nomenclature, Linear velocities VI,, see Nomenclature, Linear velocities U;, see Nomenclature, Linear velocities, components, example V k p , see Nomenclature, Linear velocities V N , see Nomenclature, Linear velocities, components Vortex, see Angular velocity, wind V,, see Nomenclature, Linear velocities 6$, see Nomenclature, Wind gradient, components U X ; , see Nomenclature, Wind gradient, components uy;, see Nomenclature, Wind gradient, components V z , see Nomenclature, Linear velocities, c6mponents UZ;, see Nomenclature, Wind gradient, components Weighing, 83 WGL,see Nomenclature, Linearization, state vector WGI,see Nomenclature, Linearizat ion, state vector Wind angles angle of attack, 36, 63, 201 bank angle, 36, 63, 201 sideslip angle, 36, 63, 201 angular velocity, see Angular velocity constant wind velocity, 9, 130 field of wind velocity, 49, 51 derivation, 283
Index gradient, see Gradient of wind notations, xxvi independant of the heading, 136 uniform wind velocity, 130 velocity, see Velocity relationship between velocities, 50, 56 wind velocity, 201 wind velocity V,, 35 vortex, 9, 50, see Angular velocity, 58, 126 wg , see Nomenclature, Linear velocities, components, example WLL,see Nomenclature, Linearizat ion, state vector WLI,see Nomenclature, Linearization, state vector WRL,see Nomenclature, Linearization, state vector WRI,see Nomenclature, Linearization, state vector G;, see Nomenclature, Wind gradient, components WXO,, see Nomenclature, Wind gradient, components wy;, see Nomenclature, Wind gradient, components WZ:, see Nomenclature, Wind, gradient, components
x, see Nomenclature, Positions
XL, see Nomenclature, Linearization, state vector XI,see Nomenclature, Linearization, state vector zb, see Nomenclature, External efforts, propulsive x i , see Nomenclature, Positions, example y, see Nomenclature, Positions Yaw, see Angles and Angular velocity yL, see Nomenclature, External efforts, propulsive &, see Nomenclature, Positions, example
x, see Nomenclature, Positions
Index
353
zL,see Nomenclature, External efforts, propulsive
z;, see Nomenclature, Positions, example
Dynamics of Flight: Equations
