Introduction to Aircraft Stability and Control Course ...

1 Introduction to Aircraft Stability and ControlCourse Notes for M&AE 5070 David A. CaugheySibley School of Mechanical & Aerospace EngineeringCornell UniversityIthaca, New York 14853-750120112 Contents1 Introduction to Flight Introduction .. Nomenclature .. Implications of Vehicle Symmetry .. Aerodynamic Controls .. Force and Moment Coefficients .. Atmospheric Properties ..62 Aerodynamic Introduction .. Lifting surface geometry and nomenclature .. Geometric properties of trapezoidal wings .. Aerodynamic properties of airfoils .. Aerodynamic properties of finite wings .. Fuselage contribution to pitch stiffness .. Wing-tail interference .. Control Surfaces .. 203 Static Longitudinal Stability and Control Fixed Stability .. Static Longitudinal Control .. Longitudinal Maneuvers the Pull-up .. Control Surface Hinge Moments .. Control Surface Hinge Moments.

2 Control free Neutral Point .. Trim Tabs .. Control Force for Trim .. Control -force for Maneuver .. Forward and Aft Limits of Position .. 414 Dynamical Equations for Flight Basic Equations of Motion .. Force Equations .. Moment Equations .. Linearized Equations of Motion .. Representation of Aerodynamic Forces and Moments .. Longitudinal Stability Derivatives .. Lateral/Directional Stability Derivatives .. Control Derivatives .. Properties of Elliptical Span Loadings .. Useful Integrals .. Exercises ..715 Dynamic Mathematical Background .. An Introductory Example .. Systems of First-order Equations .. Longitudinal Motions .. Modes of Typical Aircraft .. Approximation to Short Period Mode .. Approximation to Phugoid Mode .. Summary of Longitudinal Modes .. Lateral/Directional Motions .. Modes of Typical Aircraft .

3 Approximation to Rolling Mode .. Approximation to Spiral Mode .. Approximation to Dutch Roll Mode .. Summary of Lateral/Directional Modes .. Stability Characteristics of the Boeing 747 .. Longitudinal Stability Characteristics .. Lateral/Directional Stability Characteristics .. 1026 Control of Aircraft Control Response .. Laplace Transforms and State Transition .. The Matrix Exponential .. System Time Response .. Impulse Response .. Doublet Response .. Step Response .. Example of Response to Control Input .. System Frequency Response .. Controllability and Observability .. Controllability .. Observability .. Controllability, Observability, andMatlab.. State Feedback Design .. Single Input State Variable Control .. Multiple Input-Output Systems .. Optimal Control .. Formulation of Linear, Quadratic, Optimal Control .. Example of Linear, Quadratic, Optimal Control .

4 Linear, Quadratic, Optimal Control as a Stability Augmentation System .. Review of Laplace Transforms .. Laplace Transforms of Selected Functions .. 144 Chapter 1 Introduction to Flight DynamicsFlight dynamics deals principally with the response of aerospace vehicles to perturbationsin 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 actingon 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 theengineering science of flight dynamics , focusing primarily of aspects of Stability andcontrol. The notes contain a simplified summary of important results fromaerodynamicsthat can be used to characterize the forcing functions, a description of static stabilityfor the longitudinal problem, and an Introduction to the dynamics and Control of both,longitudinal and lateral/directional problems, including some aspects of feedback IntroductionFlight dynamics characterizes the motion of a flight vehicle in the atmosphere.

5 Assuch, it can beconsidered a branch of systems dynamics in which the system studies is a flight responseof the vehicle to aerodynamic, propulsive, and gravitational forces, and to Control inputs from thepilot determine the attitude of the vehicle and its resulting flight path. The field of flight dynamicscan be further subdivided into aspects concerned with Performance: in which the short time scales of response are ignored, and the forces areassumed to be in quasi-static equilibrium. Here the issues are maximum and minimum flightspeeds, rate of climb, maximum range, and time aloft (endurance). Stability and Control : in which the short- and intermediate-time response of the attitudeand velocity of the vehicle is considered. Stability considers the response of the vehicletoperturbations in flight conditions from some dynamic equilibrium, while Control considers theresponse of the vehicle to Control inputs. Navigation and Guidance: in which the Control inputs required to achieve a particulartrajectory are 1.

6 Introduction TO FLIGHT DYNAMICSA erodynamicsPropulsionFlight dynamics ( Stability & Control )StructuresAerospaceDesignVehicle M&AE 3050M&AE 5060M&AE 5070M&AE 5700 Figure : The four engineering sciences required to design a flight these notes we will focus on the issues of Stability and Control . These two aspects of the dynamicscan be treated somewhat independently, at least in the case when the equations of motion arelinearized, so the two types of responses can be added using the principle of superposition, and thetwo types of responses are related, respectively, to thestabilityof the vehicle and to the ability ofthe pilot tocontrolits dynamics forms one of the four basic engineering sciences needed to understand the designof flight vehicles, as illustrated in Fig. (with Cornell M&AE Course numbers associated withintroductory courses in these areas). A typical aerospace engineering curriculum with have coursesin all four of these aspects of Stability can be further subdivided into (a) static Stability and (b) dynamic Stability refers to whether the initial tendency of the vehicle response to a perturbationis toward a restoration of equilibrium.

7 For example, if the response to aninfinitesimal increasein angle of attack of the vehicle generates a pitching moment that reduces the angle ofattack, theconfiguration is said to be statically stable to such perturbations. Dynamic Stability refers to whetherthe vehicle ultimately returns to the initial equilibrium state after some infinitesimal of dynamic Stability makes sense only for vehicles that are statically stable. But avehicle can be statically stable and dynamically unstable (for example, if the initial tendency toreturn toward equilibrium leads to an overshoot, it is possible to have an oscillatory divergence ofcontinuously increasing amplitude). Control deals with the issue of whether the aerodynamic and propulsive controlsare adequate totrim the vehicle ( , produce an equilibrium state) for all required states in the flight envelope. Inaddition, the issue of flying qualities is intimately connected to Control issues; , the controlsmust be such that the maintenance of desired equilibrium states does not overly tirethe pilot orrequire excessive attention to Control classical texts that deal with aspects of aerodynamic performance [1, 5] and Stability andcontrol [2, 3, 4] are listed at the end of this NOMENCLATURE3 Figure : Standard notation for aerodynamic forces and moments, and linearand rotationalvelocities in body-axis system; origin of coordinates is at center of mass of the NomenclatureThe standard notation for describing the motion of, and the aerodynamic forces and moments actingupon, a flight vehicle are indicated in Fig.

8 All the notation consists of consecutive alphabetic triads: The variablesx,y,zrepresent coordinates, with origin at the center of mass of the lies in the symmetry plane of the vehicle1and points toward the nose of thevehicle. (The precise direction will be discussed later.) Thez-axis also is taken to lie in theplane of symmetry, perpendicular to thex-axis, and pointing approximately down. Theyaxiscompletes a right-handed orthogonal system, pointing approximately out the right wing. The variablesu,v,wrepresent the instantaneous components of linear velocity in the directionsof thex,y, andzaxes, respectively. The variablesX,Y,Zrepresent the components of aerodynamic force in the directions of thex,y, andzaxes, respectively. The variablesp,q,rrepresent the instantaneous components of rotational velocity about thex,y, andzaxes, respectively. The variablesL,M,Nrepresent the components of aerodynamic moments about thex,y,andzaxes, respectively.

9 Although not indicated in the figure, the variables , , represent the angular rotations,relative to the equilibrium state, about thex,y, andzaxes, respectively. Thus,p= ,q= ,andr= , where the dots represent time velocity components of the vehicle often are represented as angles, as indicated in Fig. Thevelocity componentwcan be interpreted as the angle of attack tan 1wu( )1 Virtually all flight vehicles have bi-lateral symmetry, and this fact is used to simplify the analysis of 1. Introduction TO FLIGHT DYNAMICSxyzV uwvFigure : Standard notation for aerodynamic forces and moments, and linearand rotationalvelocities in body-axis system; origin of coordinates is at center of mass of the the velocity componentvcan be interpreted as the sideslip angle sin 1vV( ) Implications of Vehicle SymmetryThe analysis of flight motions is simplified, at least for small perturbations from certain equilibriumstates, by the bi-lateral symmetry of most flight vehicles.

10 This symmetry allows us to decomposemotions into those involvinglongitudinalperturbations and those involvinglateral/directionalper-turbatio ns. Longitudinal motions are described by the velocitiesuandvand rotations about they-axis, described byq(or ). Lateral/directional motions are described by the velocityvand rota-tions about thexand/orzaxes, described bypand/orr(or and/or ). A longitudinal equilibriumstate is one in which the lateral/directional variablesv,p,rare all zero. As a result, the side forceYand the rolling momentpand yawing momentralso are identically zero. A longitudinal equilib-rium state can exist only when the gravity vector lies in thex-zplane, so such states correspond towings-level flight (which may be climbing, descending, or level).The important results of vehicle symmetry are the following. If a vehicle in a longitudinal equilibriumstate is subjected to a perturbation in one of the longitudinal variables, the resulting motion willcontinue to be a longitudinal one , the velocity vector will remain in thex-zplane and theresulting motion can induce changes only inu,w, andq(or ).