Series Preface xiii Glossary xv 1 Introduction 1 1.1 Background 1 1.2 Overview 2 1.3 Customs and Conventions 6 2 Coordinate Systems 7 2.1 Background 7 2.2 The Coordinate Systems 7 2.2.1 The inertial reference frame, FI 7 2.2.2 The earth-centered reference frame, FEC 8 2.2.3 The earth-fixed reference frame, FE 8 2.2.4 The local-horizontal reference frame, FH 8 2.2.5 Body-fixed reference frames, FB 10 2.2.6 Wind-axis system, FW 12 2.2.7 Atmospheric reference frame 12 2.3 Vector Notation 13 2.4 Customs and Conventions 14 2.4.1 Latitude and longitude 14 2.4.2 Body axes 14 2.4.3 'The' body-axis system 14 2.4.4 Aerodynamic angles 15 3 Coordinate System Transformations 17 3.1 Problem Statement 17 3.2 Transformations 18 3.2.1 Definitions 18 3.2.2 Direction cosines 18 3.2.3 Euler angles 21 3.2.4 Euler parameters 25 3.3 Transformations of Systems of Equations 26 3.4 Customs and Conventions 27 3.4.1 Names of Euler angles 27 3.4.2 Principal values of Euler angles 27 4 Rotating Coordinate Systems 31 4.1 General 31 4.2 Direction Cosines 34 4.3 Euler Angles 34 4.4 Euler Parameters 36 4.5 Customs and Conventions 38 4.5.1 Angular velocity components 38 5 Inertial Accelerations 43 5.1 General 43 5.2 Inertial Acceleration of a Point 43 5.2.1 Arbitrary moving reference frame 43 5.2.2 Earth-centered moving reference frame 46 5.2.3 Earth-fixed moving reference frame 46 5.3 Inertial Acceleration of a Mass 47 5.3.1 Linear acceleration 48 5.3.2 Rotational acceleration 49 5.4 States 53 5.5 Customs and Conventions 53 5.5.1 Linear velocity components 53 5.5.2 Angular velocity components 54 5.5.3 Forces 54 5.5.4 Moments 56 5.5.5 Groupings 56 6 Forces and Moments 59 6.1 General 59 6.1.1 Assumptions 59 6.1.2 State variables 60 6.1.3 State rates 60 6.1.4 Flight controls 60 6.1.5 Independent variables 62 6.2 Non-Dimensionalization 62 6.3 Non-Dimensional Coefficient Dependencies 63 6.3.1 General 63 6.3.2 Altitude dependencies 64 6.3.3 Velocity dependencies 64 6.3.4 Angle-of-attack dependencies 64 6.3.5 Sideslip dependencies 66 6.3.6 Angular velocity dependencies 68 6.3.7 Control dependencies 69 6.3.8 Summary of dependencies 70 6.4 The Linear Assumption 71 6.5 Tabular Data 71 6.6 Customs and Conventions 72 7 Equations of Motion 75 7.1 General 75 7.2 Body-Axis Equations 75 7.2.1 Body-axis force equations 75 7.2.2 Body-axis moment equations 76 7.2.3 Body-axis orientation equations (kinematic equations) 77 7.2.4 Body-axis navigation equations 77 7.3 Wind-Axis Equations 78 7.3.1 Wind-axis force equations 78 7.3.2 Wind-axis orientation equations (kinematic equations) 80 7.3.3 Wind-axis navigation equations 81 7.4 Steady-State Solutions 81 7.4.1 General 81 7.4.2 Special cases 83 7.4.3 The trim problem 88 8 Linearization 93 8.1 General 93 8.2 Taylor Series 94 8.3 Nonlinear Ordinary Differential Equations 95 8.4 Systems of Equations 95 8.5 Examples 97 8.5.1 General 97 8.5.2 A kinematic equation 99 8.5.3 A moment equation 100 8.5.4 A force equation 103 8.6 Customs and Conventions 105 8.6.1 Omission of Δ 105 8.6.2 Dimensional derivatives 105 8.6.3 Added mass 105 8.7 The Linear Equations 106 8.7.1 Linear equations 106 8.7.2 Matrix forms of the linear equations 108 9 Solutions to the Linear Equations 113 9.1 Scalar Equations 113 9.2 Matrix Equations 114 9.3 Initial Condition Response 115 9.3.1 Modal analysis 115 9.4 Mode Sensitivity and Approximations 120 9.4.1 Mode sensitivity 120 9.4.2 Approximations 123 9.5 Forced Response 124 9.5.1 Transfer functions 124 9.5.2 Steady-state response 125 10 Aircraft Flight Dynamics 127 10.1 Example: Longitudinal Dynamics 127 10.1.1 System matrices 127 10.1.2 State transition matrix and eigenvalues 127 10.1.3 Eigenvector analysis 129 10.1.4 Longitudinal mode sensitivity and approximations 132 10.1.5 Forced response 137 10.2 Example: Lateral–Directional Dynamics 140 10.2.1 System matrices 140 10.2.2 State transition matrix and eigenvalues 140 10.2.3 Eigenvector analysis 142 10.2.4 Lateral–directional mode sensitivity and approximations 144 10.2.5 Forced response 148 11 Flying Qualities 151 11.1 General 151 11.1.1 Method 152 11.1.2 Specifications and standards 155 11.2 MIL-F-8785C Requirements 156 11.2.1 General 156 11.2.2 Longitudinal flying qualities 157 11.2.3 Lateral–directional flying qualitities 158 12 Automatic Flight Control 169 12.1 Simple Feedback Systems 170 12.1.1 First-order systems 170 12.1.2 Second-order systems 172 12.1.3 A general representation 177 12.2 Example Feedback Control Applications 178 12.2.1 Roll mode 178 12.2.2 Short-period mode 184 12.2.3 Phugoid 188 12.2.4 Coupled roll–spiral oscillation 198 13 Trends in Automatic Flight Control 209 13.1 Overview 209 13.2 Dynamic Inversion 210 13.2.1 The controlled equations 212 13.2.2 The kinematic equations 215 13.2.3 The complementary equations 221 13.3 Control Allocation 224 13.3.1 Background 224 13.3.2 Problem statement 225 13.3.3 Optimality 231 13.3.4 Sub-optimal solutions 232 13.3.5 Optimal solutions 235 13.3.6 Near-optimal solutions 241 Problems 243 References 244 A Example Aircraft 247 Reference 253 B Linearization 255 B.1 Derivation of Frequently Used Derivatives 255 B.2 Non-dimensionalization of the Rolling Moment Equation 257 B.3 Body Axis Z-Force and Thrust Derivatives 258 B.4 Non-dimensionalization of the Z-Force Equation 260 C Derivation of Euler Parameters 263 D Fedeeva's Algorithm 269 Reference 272 E MATLAB Commands Used in the Text 273 E.1 Using MATLAB 273 E.2 Eigenvalues and Eigenvectors 274 E.3 State-Space Representation 274 E.4 Transfer Function Representation 275 E.5 Root Locus 277 E.6 MATLAB® Functions (m-files) 277 E.6.1 Example aircraft 278 E.6.2 Mode sensitivity matrix 278 E.6.3 Cut-and-try root locus gains 278 E.7 Miscellaneous Applications and Notes 280 E.7.1 Matrices 280 E.7.2 Commands used to create Figures 10.2 and 10.3 281 Index 283

Aircraft Flight Dynamics and Control addresses airplane fl ight dynamics and control in a largely classical manner, but with references to modern treatment throughout. Classical feedback control methods are illustrated with relevant examples, and current trends in control are presented by introductions to dynamic inversion and control allocation. This book covers the physical and mathematical fundamentals of aircraft flight dynamics as well as more advanced theory enabling a better insight into nonlinear dynamics. This leads to a useful introduction to automatic flight control and stability augmentation systems with discussion of the theory behind their design, and the limitations of the systems. The author provides a rigorous development of theory and derivations and illustrates the equations of motion in both scalar and matrix notation. Key features: Classical development and modern treatment of flight dynamics and controlDetailed and rigorous exposition and examples, with illustrationsPresentation of important trends in modern fl ight control systemsAccessible introduction to control allocation based on the author’s seminal work in the fieldDevelopment of sensitivity analysis to determine the infl uential states in an airplane’s response modesEnd of chapter problems with solutions available on an accompanying website Written by an author with experience as an engineering test pilot as well as a university professor, Aircraft Flight Dynamics and Control provides the reader with a systematic development of the insights and tools necessary for further work in related fields of flight dynamics and control. It is an ideal course textbook and is also a valuable reference for many of the necessary basic formulations of the math and science underlying flight dynamics and control.