AE345NotesJan25 - AIRCRAFT FLIGHT DYNAMICS AND CONTROL...

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AIRCRAFT FLIGHT DYNAMICS AND CONTROL January 25, 2008 Dennis S. Bernstein Department of Aerospace Engineering The University of Michigan Ann Arbor, MI 48109-2140 [email protected] C o p y r i g h t 2 0 0 8
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Contents Chapter 1. Review of Kinematics 3 1.1 Points, Particles, and Bodies 3 1.2 Physical and Mathematical Vectors 3 1.3 Dot Product 6 1.4 Angle Vector and Cross Product 6 1.5 Frames 8 1.6 Signed Angles 11 1.7 Particle-Fixed Frames 11 1.8 Physical Matrices 12 1.9 Orientation Matrices 18 1.10 Euler Angles 21 1.11 Frame Derivatives 24 1.12 Angular Velocity Vector 25 1.13 Euler Angle Derivatives 28 1.14 Transport Theorem 29 1.15 Double Transport Theorem 31 1.16 Solid and Open Frame Dots 31 Chapter 2. Aircraft Kinematics 35 2.1 Frames Used in Aircraft Dynamics 35 2.2 Velocity Vector 38 2.3 Rotational Kinematics 42 2.4 Vector Derivatives in Rotating Frames 44 2.5 Cross Product 45 2.6 Problems 46 Chapter 3. Review of Dynamics 49 3.1 Newton’s First Law for Particles 49 3.2 Newton’s Second Law for Particles 50 3.3 Forces and Moments 50 3.4 Change in Angular Momentum 52 3.5 Continuum Bodies 57 3.6 Angular Momentum Relative to Center of Mass 58 3.7 The Inertia Matrix 61
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iv CONTENTS 3.8 Problems 62 Chapter 4. Aircraft Dynamics 65 4.1 Flight Dynamics and Control 65 4.2 History of Aircraft Stability and Control 65 4.3 Aerodynamic Forces 65 4.4 Translational Momentum Equations 66 4.5 Rotational Momentum Equations 71 4.6 Aircraft Equations of Motion Resolved in F AC 73 4.7 Problems 74 Chapter 5. Linearization 77 5.1 Taylor series 77 5.2 Linearization Procedure 78 5.3 Trigonometric Functions 78 5.4 Steady Flight 79 5.5 Linearization in F AC 80 5.6 Linearization in F S f 82 5.7 Problems 83 Chapter 6. Static Stability and Stability Derivatives 85 6.1 Control Surface Deflections 85 6.2 Aerodynamic Coefficients 85 6.3 Lift and Drag Polar 86 6.4 Linearization of Forces in F S f 88 6.5 Moment Stability Derivatives 95 6.6 C m α 0 95 6.7 C l β 0 99 6.8 C n β 0 107 6.9 Effect of Adverse Control Derivatives 108 6.10 Linearization of Moments in F S f 110 6.11 Problems 111 Chapter 7. Linearized Equations of Motion 113 7.1 Longitudinal Equations of Motion 113 7.2 Lateral Equations of Motion 114 7.3 Linearized Longitudinal Equations and Transfer Functions 116 7.4 Linearized Lateral Equations and Transfer Functions 118 7.5 Problems 120 Chapter 8. Analysis of Flight Modes 121 8.1 Eigenflight 121 8.2 Phugoid Mode 122 8.3 Short Period Mode 125 8.4 Phugoid Approximation 126
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CONTENTS v 8.5 Short Period Approximation 127 8.6 Problems 128 Chapter 9. Control Concepts 131 9.1 Problems 131 Chapter 10. Control of Aircraft 135 10.1 Problems 135 Appendix A. Mathematical Background 137 A.1 Vectors and Matrices 137 A.2 Complex Numbers, Vectors and Matrices 139 A.3 Eigenvalues and Eigenvectors 142 A.4 Single-Degree-of-Freedom Systems 144 A.5 Matrix Differential Equations 146 A.6 Eigensolutions 148 A.7 State Space Form 149 A.8 Linear Systems with Forcing 150 A.9 Standard Input Signals 151 A.10 Laplace Transforms 152 A.11 Solution of ODE’s 156 A.12 Initial Value Theorem 157 A.13 Final Value Theorem 158 A.14 Laplace Transforms of State Space Models 159 A.15 Pole Locations and Response 161 A.16 Stability 164 A.17 Routh Stability Criterion 165 A.18 Problems 166 Appendix B. Frequency Response 175 B.1
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