Flight Dynamics, Simulation, and Control

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1 Flight Dynamics, Simulation, and Control For Rigid and Flexible Aircraft Ranjan Vepa CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business

2 Contents List of Acronyms Preface Author xvii xix xxiii 1 Introduction to Flight Vehicles Introduction Components of an Aeroplane Fuselage Wings Tail Surfaces or Empennage Landing Gear Basic Principles of Flight Forces Acting on an Aeroplane Drag and Its Reduction Aerodynamically Conforming Shapes: Streamlining Stability and Balance Flying Control Surfaces: Elevator, Ailerons and Rudder Flaps, High-Lift and Flow Control Devices Introducing Boundary Layers Spoilers Pilot's Controls: The Throttle, the Control Column and Yoke, the Rudder Pedals and the Toe Brakes Modes of Flight Static and In-Flight Stability Margins Power Plant Propeller-Driven Aircraft Jet Propulsion Avionics, Instrumentation and Systems Geometry of Aerofoils and Wings Aerofoil Geometry Chord Line Camber Leading and Trailing Edges Specifying Aerofoils Equations Defining Mean Camber Line Aerofoil Thickness Distributions Wing Geometry 26 Chapter Highlights 30

3 viii Contents Exercises 30 Answers to Selected Exercises 32 References 32 2 Basic Principles Governing Aerodynamic Flows Introduction Continuity Principle Streamlines and Stream Tubes Bernoulli's Principle Laminar Flows and Boundary Layers Turbulent Flows Aerodynamics of Aerofoils and Wings Flow around an Aerofoil Mach Number and Subsonic and Supersonic Flows Properties of Air in the Atmosphere Composition of the Atmosphere: The Troposphere, Stratosphere, Mesosphere, Ionosphere and Exosphere Air Density Temperature : Pressure Effects of Pressure and Temperature Viscosity Bulk Modulus of Elasticity Temperature Variations with Altitude: The Lapse Rate International Standard Atmosphere (from ESDU 77021,1986) Generation of Lift and Drag Aerodynamic Forces and Moments Aerodynamic Coefficients Aerofoil Drag Aircraft Lift Equation and Lift Curve Slope Centre of Pressure Aerodynamic Centre Pitching Moment Equation Elevator Hinge Moment Coefficient 60 Chapter Highlights 61 Exercises 63 Answers to Selected Exercises 65 References 66 3 Mechanics of Equilibrium Flight Introduction Speeds of Equilibrium Flight Basic Aircraft Performance Optimum Flight Speeds Conditions for Minimum Drag 76

4 Contents ix 3.5 Stability in the Vicinity of the Minimum Drag Speed Range and Endurance Estimation Trim Stability of Equilibrium Flight Longitudinal Static Stability Neutral Point (Stick-Fixed) Neutral Point (Stick-Free) Manoeuvrability Pull-Out Manoeuvre Manoeuvre Margin: Stick-Fixed Manoeuvre Margin: Stick-Free 89 Criteria Lateral Stability and Stability 3.12 Experimental Determination of Aircraft Stability Margins Summary of Equilibrium- and Stability-Related Equations 92 Chapter Highlights 95 Exercises 97 Answers to Selected Exercises 101 References Aircraft Non-Linear Dynamics: Equations of Motion Introduction Aircraft Dynamics Aircraft Motion in a 2D Plane Moments of Inertia Euler's Equations and the Dynamics of Rigid Bodies Ill 4.6 Description of the Attitude or Orientation 115 of Motion Aircraft Equations 4.8 Motion-Induced Aerodynamic Forces and Moments Non-Linear Dynamics of Aircraft Motion and the Stability Axes 125 of Motion in Wind Axis Coordinates, Equations VT,ocandp Reduced-Order Modelling: The Short Period Approximations Trimmed Equations of Motion Non-Linear Equations of Perturbed Motion Linear Equations of Motion 140 Chapter Highlights 141 Exercises 142 References Small Perturbations and the Linearised, Decoupled Equations of Motion Introduction Small Perturbations and Linearisations 145

5 x Contents 5.3 Linearising the Aerodynamic Forces and Moments: Stability Derivative Concept Direct Formulation in the Stability Axis Decoupled Equations of Motion Case I: Motion in the Longitudinal Plane of Symmetry Case II: Motion in the Lateral Direction, Perpendicular to the Plane of Symmetry Decoupled Equations of Motion in terms of the Stability Axis Aerodynamic Derivatives Addition of Aerodynamic Controls and Throttle Non-Dimensional Longitudinal and Lateral Dynamics Simplified State-Space Equations of Longitudinal and Lateral Dynamics Simplified Concise Equations of Longitudinal and Lateral Dynamics 181 Chapter Highlights 182 Exercises 182 Reference Longitudinal and Lateral Linear Stability and Control Introduction Dynamic and Static Stability Longitudinal Stability Analysis Lateral Dynamics and Stability Modal Description of Aircraft Dynamics and the Stability of the Modes Slow-Fast Partitioning of the Longitudinal Dynamics Slow-Fast Partitioning of the Lateral Dynamics Summary of Longitudinal and Lateral Modal Equations Phugoid or Long Period Short Period Third Oscillatory Mode Roll Subsidence Dutch Roll Spiral Aircraft Lift and Drag Estimation Fuselage Wing-Tail Estimating the Wing's Lift and Moment Coefficients 219 Interference Effects 220 Maximum Lift Coefficient Drag Estimation Estimating the Longitudinal Aerodynamic Derivatives Estimating the Lateral Aerodynamic Derivatives Perturbation Analysis of Trimmed Flight 238

6 Contents xi Perturbation Analysis of Longitudinal Trimmed Flight Perturbation Analysis of Lateral Trimmed Flight Control Settings for Steady Sideslip Control Settings for Turn Coordination and Banking Perturbations of Coupled Trimmed Flight Simplified Analysis of Complex Manoeuvres: The Sidestep Manoeuvre 250 Chapter Highlights 252 Exercises 255 Answers to Selected Exercises 263 References Aircraft Dynamic Response: Numerical Simulation and Non-Linear Phenomenon Introduction Longitudinal and Lateral Modal Equations Methods of Computing Aircraft Dynamic Response Laplace Transform Method Aircraft Response Transfer Functions Direct Numerical Integration System Block Diagram Representation Numerical Simulation of Flight Using MATLAB /Simulink Atmospheric Disturbance: Deterministic Disturbances Principles of Random Atmospheric Disturbance Modelling White Noise: Power Spectrum and Autocorrelation Linear Time-Invariant System with Stochastic Process Input Application to Atmospheric Turbulence Modelling Aircraft Non-Linear Dynamic Response Phenomenon Aircraft Dynamic Non-Linearities and Their Analysis High-Angle-of-Attack Dynamics and Its Consequences Post-Stall Behaviour Tumbling and Autorotation Lateral Dynamic Phenomenon Flat Spin and Deep Spin Wing Drop, Wing Rock and Nose Slice Fully Coupled Motions: The Falling Leaf Regenerative Phenomenon 311 Chapter Highlights 312 Exercises 312 References 330

7 xii Contents 8 Aircraft Flight Control Automatic Flight Control Systems: An Introduction Functions of a Flight Control System Integrated Flight Control System Guidance System: Interfacing to the Automatic Flight Control System Flight Management System Flight Control System Design Block Diagram Algebra Return Difference Equation Laplace Transform Stability of Uncontrolled and Controlled Systems Routh's Tabular Method Frequency Response Bode Plots Nyquist Plots Stability in the Frequency Domain Stability Margins: The Gain and Phase Margins Mapping Complex Functions and Nyquist Diagrams Time Domain: The State Variable Representation Solution of the State Equations and the Controllability Condition State-Space and Transfer Function Equivalence Transformations of State Variables Design of a Full-State Variable Feedback Control Law Root Locus Method Root Locus Principle Root Locus Sketching Procedure Producing a Root Locus Using MATLAB Application of the Root Locus Method: Unity Feedback with a PID Control Law Optimal Control of Flight Dynamics Compensating Full-State Feedback: Observers and Compensators Observers for Controller Implementation Observer Equations Special Cases: The Full- and First-Order Observers Solving the Observer Equations Luenberger Observer Optimisation Performance Criteria Good Handling Domains of Modal Response Parameters Cooper-Harper Rating Scale 400

8 Contents xiii 8.6 Application to the Design of Stability Augmentation Systems and Autopilots Design of a Pitch Attitude Autopilot Using PID Feedback and the Root Locus Method Example of Pitch Attitude Autopilot Design for the Lockheed F104 by the Root Locus Method Example of Pitch Attitude Autopilot Design, Including a Stability Augmentation Inner Loop, by the Root Locus Method Design of an Altitude Acquire-and-Hold Autopilot Design of a Lateral Roll Attitude Autopilot Design of a Lateral Yaw Damper Design of a Lateral Heading Autopilot Turn Coordination with Sideslip Suppression Application of Optimal Control to Lateral Control Augmentation Design Performance Assessment of a Command or Control Augmentation System Linear Perturbation Dynamics Flight Control Law Design by Partial Dynamic Inversion Design Example of a Longitudinal Autopilot Based on Partial Dynamic Inversion Design of Controllers for Multi-Input Systems Design Example of a Lateral Turn Coordination Using the Partial Inverse Dynamics Method Design Example of the Simultaneously Operating Auto-Throttle and Pitch Attitude Autopilot Two-Input Lateral Attitude Control Autopilot Decoupling Control and Its Application: Longitudinal and Lateral Dynamics Decoupling Control Full Aircraft Six-DOF Flight Controller Design by Dynamic Inversion Control Law Synthesis Example of Linear Control Law Synthesis by Partial Dynamic Inversion: The Fully Propulsion-Controlled Example of Quasi-Non-Linear Control Law MD11 Aircraft 462 i Synthesis by Partial Dynamic Inversion: The Fully r Propulsion-Controlled MD11 Aircraft 464 i Full Aircraft Orientation Control Law Design by Dynamic Inversion Aircraft Flight Control Synthesis in Wind Axes ' ) Coordinates, VT, p and a 471

9 xiv Contents Chapter Highlights 474 Exercises 475 Answers to Selected Exercises 484 References Piloted Simulation and Pilot Modelling Introduction Piloted Flight Simulation Full Moving-Base Simulation: The Stewart Platform Kinematics of Motion Systems Principles of Motion Control Motion Cueing Concepts Principles of Human Pilot Physiological Modelling Auricular and Ocular Sensors Human Physiological Control Mechanisms Crossover Model Neal-Smith Criterion Pilot-Induced Oscillations PIO Categories ' PIOs Classified under Small Perturbation Modes Optimal Control Models Generic Human Pilot Modelling Pilot-Vehicle Simulation Spatial Awareness Visual Displays Animation and Visual Cues Visual Illusions 520 Chapter Highlights 522 Exercises 522 References Flight Dynamics of Elastic Aircraft Introduction Flight Dynamics of Flexible Aircraft Newton-Euler Equations of a Rigid Aircraft Lagrangian Formulation Generalised Coordinates and Holonomic Dynamic Systems Generalised Velocities Virtual Displacements and Virtual Work 538 of Virtual Work Principle Euler-Lagrange Equations Potential Energy and the Dissipation Function Euler-Lagrange Equations of Motion in Quasi-Coordinates 545

10 Contents xv Transformation to Centre of Mass Coordinates Application of the Lagrangian Method to a Rigid Aircraft Vibration of Elastic Structures in a Fluid Medium Effects of Structural Flexibility in Aircraft Aeroelasticity Wing Divergence Control Reversal Wing Flutter Aerofoil Flutter Analysis Unsteady Aerodynamics of an Aerofoil Euler-Lagrange Formulation of Flexible Body Dynamics Application to an Aircraft with a Flexible Wing Vibrating in Bending and Torsion Longitudinal Small Perturbation Equations with Flexibility Lateral Small Perturbation Equations with Flexibility Kinetic and Potential Energies of the Whole Elastic Aircraft Kinetic Energy Simplifying the General Expression Mean Axes Kinetic Energy in terms of Modal Amplitudes Tisserand Frame Euler-Lagrange Matrix Equations of a Flexible Body in Quasi-Coordinates Slender Elastic Aircraft Aircraft with a Flexible Flat Body Component Elastic Large Aspect Ratio Flying Wing Model Flexible Aircraft in Roll Estimating the Aerodynamic Derivatives: Modified Strip Analysis 622 Chapter Highlights 627 Exercises 627 Answers to Selected Exercises 648 References 649 Index 651

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