Introduction to Aircraft Stability and Control Course Notes for M&AE 5070 David A. Caughey Sibley School of Mechanical & Aerospace Engineering Cornell University Ithaca, New York 14853-7501 2011 2 Contents 1 Introduction to Flight Dynamics 1 1.1 Introduction....................................... 1 1.2 Nomenclature........................................ 3 1.2.1 Implications of Vehicle Symmetry . 4 1.2.2 AerodynamicControls .............................. 5 1.2.3 Force and Moment Coefficients . 5 1.2.4 Atmospheric Properties . 6 2 Aerodynamic Background 11 2.1 Introduction....................................... 11 2.2 Lifting surface geometry and nomenclature . 12 2.2.1 Geometric properties of trapezoidal wings . 13 2.3 Aerodynamic properties of airfoils . ..... 14 2.4 Aerodynamic properties of finite wings . 17 2.5 Fuselage contribution to pitch stiffness . 19 2.6 Wing-tail interference . 20 2.7 ControlSurfaces ..................................... 20 3 Static Longitudinal Stability and Control 25 3.1 ControlFixedStability.............................. ..... 25 v vi CONTENTS 3.2 Static Longitudinal Control . 28 3.2.1 Longitudinal Maneuvers – the Pull-up . 29 3.3 Control Surface Hinge Moments . 33 3.3.1 Control Surface Hinge Moments . 33 3.3.2 Control free Neutral Point . 35 3.3.3 TrimTabs...................................... 36 3.3.4 ControlForceforTrim. 37 3.3.5 Control-force for Maneuver . 39 3.4 Forward and Aft Limits of C.G. Position . ......... 41 4 Dynamical Equations for Flight Vehicles 45 4.1 BasicEquationsofMotion. ..... 45 4.1.1 ForceEquations .................................. 46 4.1.2 MomentEquations................................. 49 4.2 Linearized Equations of Motion . 50 4.3 Representation of Aerodynamic Forces and Moments . 52 4.3.1 Longitudinal Stability Derivatives . 54 4.3.2 Lateral/Directional Stability Derivatives . ...... 59 4.4 ControlDerivatives................................ ..... 69 4.5 Properties of Elliptical Span Loadings . 70 4.5.1 UsefulIntegrals.................................. 71 4.6 Exercises .......................................... 71 5 Dynamic Stability 75 5.1 Mathematical Background . 75 5.1.1 AnIntroductoryExample ............................. 75 5.1.2 Systems of First-order Equations . 79 CONTENTS vii 5.2 Longitudinal Motions . 81 5.2.1 Modes of Typical Aircraft . 82 5.2.2 Approximation to Short Period Mode . 86 5.2.3 Approximation to Phugoid Mode . 88 5.2.4 Summary of Longitudinal Modes . 89 5.3 Lateral/Directional Motions . ..... 89 5.3.1 Modes of Typical Aircraft . 92 5.3.2 Approximation to Rolling Mode . 95 5.3.3 Approximation to Spiral Mode . 96 5.3.4 Approximation to Dutch Roll Mode . 97 5.3.5 Summary of Lateral/Directional Modes . 99 5.4 Stability Characteristics of the Boeing 747 . 101 5.4.1 Longitudinal Stability Characteristics . 101 5.4.2 Lateral/Directional Stability Characteristics . 102 6 Control of Aircraft Motions 105 6.1 ControlResponse................................... 105 6.1.1 Laplace Transforms and State Transition . 105 6.1.2 TheMatrixExponential............................. 106 6.2 SystemTimeResponse.................................. 109 6.2.1 ImpulseResponse ................................. 109 6.2.2 Doublet Response . 109 6.2.3 StepResponse ................................... 110 6.2.4 Example of Response to Control Input . 111 6.3 System Frequency Response . 113 6.4 Controllability and Observability . 113 viii CONTENTS 6.4.1 Controllability ................................ 114 6.4.2 Observability .................................... 118 6.4.3 Controllability, Observability, and Matlab ................... 118 6.5 State Feedback Design . 119 6.5.1 Single Input State Variable Control . 121 6.5.2 Multiple Input-Output Systems . 130 6.6 OptimalControl .................................... 130 6.6.1 Formulation of Linear, Quadratic, Optimal Control . 130 6.6.2 Example of Linear, Quadratic, Optimal Control . 135 6.6.3 Linear, Quadratic, Optimal Control as a Stability Augmentation System . 138 6.7 Review of Laplace Transforms . 143 6.7.1 Laplace Transforms of Selected Functions . 144 Chapter 1 Introduction to Flight Dynamics Flight dynamics deals principally with the response of aerospace vehicles to perturbations in their flight environments and to control inputs. In order to understand this response, it is necessary to characterize the aerodynamic and propulsive forces and moments acting on the vehicle, and the dependence of these forces and moments on the flight variables, including airspeed and vehicle orientation. These notes provide an introduction to the engineering science of flight dynamics, focusing primarily of aspects of stability and control. The notes contain a simplified summary of important results from aerodynamics that can be used to characterize the forcing functions, a description of static stability for the longitudinal problem, and an introduction to the dynamics and control of both, longitudinal and lateral/directional problems, including some aspects of feedback control. 1.1 Introduction Flight dynamics characterizes the motion of a flight vehicle in the atmosphere. As such, it can be considered a branch of systems dynamics in which the system studies is a flight vehicle. The response of the vehicle to aerodynamic, propulsive, and gravitational forces, and to control inputs from the pilot determine the attitude of the vehicle and its resulting flight path. The field of flight dynamics can be further subdivided into aspects concerned with Performance: in which the short time scales of response are ignored, and the forces are • assumed to be in quasi-static equilibrium. Here the issues are maximum and minimum flight speeds, rate of climb, maximum range, and time aloft (endurance). Stability and Control: in which the short- and intermediate-time response of the attitude • and velocity of the vehicle is considered. Stability considers the response of the vehicle to perturbations in flight conditions from some dynamic equilibrium, while control considers the response of the vehicle to control inputs. Navigation and Guidance: in which the control inputs required to achieve a particular • trajectory are considered. 1 2 CHAPTER 1. INTRODUCTION TO FLIGHT DYNAMICS Aerodynamics Propulsion M&AE 3050 M&AE 5060 Aerospace Vehicle Design Structures Flight Dynamics (Stability & Control) M&AE 5700 M&AE 5070 Figure 1.1: The four engineering sciences required to design a flight vehicle. In these notes we will focus on the issues of stability and control. These two aspects of the dynamics can be treated somewhat independently, at least in the case when the equations of motion are linearized, so the two types of responses can be added using the principle of superposition, and the two types of responses are related, respectively, to the stability of the vehicle and to the ability of the pilot to control its motion. Flight dynamics forms one of the four basic engineering sciences needed to understand the design of flight vehicles, as illustrated in Fig. 1.1 (with Cornell M&AE course numbers associated with introductory courses in these areas). A typical aerospace engineering curriculum with have courses in all four of these areas. The aspects of stability can be further subdivided into (a) static stability and (b) dynamic stability. Static stability refers to whether the initial tendency of the vehicle response to a perturbation is toward a restoration of equilibrium. For example, if the response to an infinitesimal increase in angle of attack of the vehicle generates a pitching moment that reduces the angle of attack, the configuration is said to be statically stable to such perturbations. Dynamic stability refers to whether the vehicle ultimately returns to the initial equilibrium state after some infinitesimal perturbation. Consideration of dynamic stability makes sense only for vehicles that are statically stable. But a vehicle can be statically stable and dynamically unstable (for example, if the initial tendency to return toward equilibrium leads to an overshoot, it is possible to have an oscillatory divergence of continuously increasing amplitude). Control deals with the issue of whether the aerodynamic and propulsive controls are adequate to trim the vehicle (i.e., produce an equilibrium state) for all required states in the flight envelope. In addition, the issue of “flying qualities” is intimately connected to control issues; i.e., the controls must be such that the maintenance of desired equilibrium states does not overly tire the pilot or require excessive attention to control inputs. Several classical texts that deal with aspects of aerodynamic performance [1, 5] and stability and control [2, 3, 4] are listed at the end of this chapter. 1.2. NOMENCLATURE 3 Figure 1.2: Standard notation for aerodynamic forces and moments, and linear and rotational velocities in body-axis system; origin of coordinates is at center of mass of the vehicle. 1.2 Nomenclature The standard notation for describing the motion of, and the aerodynamic forces and moments acting upon, a flight vehicle are indicated in Fig. 1.2. Virtually all the notation consists of consecutive alphabetic triads: The variables x, y, z represent coordinates, with origin at the center of mass of the vehicle. • The x-axis lies in the symmetry plane of the vehicle1 and points toward the nose of the vehicle. (The precise direction will be discussed later.) The z-axis also is taken to lie in the plane of symmetry, perpendicular to the x-axis, and pointing approximately down. The y axis completes a right-handed orthogonal