Stability and Control: Introduction

The methods in these notes allow us to compute the overall aircraft drag. With well-designed airfoils and wings, and a careful job of engine and fuselage integration, L/D's near 20 may be achieved. Yet some aircraft with predicted L/D's of 20 have actual L/D's of 0 as exemplified by any paper airplane contest. Many aircraft have been dismal failures even though their predicted performance is great. In fact, most spectacular failures have to do with stability and control rather than performance.

This section deals with some of the basic stability and control issues that must be addressed in order that the airplane is capable of flying at all. The section includes a general discussion on stability and control and some terminology. Basic requirements for static longitudinal stability, dynamic stability, and control effectiveness are described. Finally, methods for tail sizing and design are introduced.

The starting point for our analysis of aircraft stability and control is a fundamental result of dynamics: for rigid bodies motion consists of translations and rotations about the center of gravity (c.g.). The motion includes six degrees of freedom: forward and aft motion, vertical plunging, lateral translations, together with pitch, roll, and yaw.

Definitions

The following nomenclature is common for discussions of stability and control.

Forces and Moments

Quantity	Variable	Dimensionless Coefficient	Positive Direction
Lift	L	CL = L/qS	'Up' normal to freestream
Drag	D	CD = D/qS	Downstream
Sideforce	Y	CY = Y/qS	Right, looking forward
Roll	l	Cl = l / qSb	Right wing down
Pitch	M	Cm = M/qSc	Nose up
Yaw	N	Cn = N/qSb	Nose right

Aircraft velocities, forces, and moments are expressed in a body-fixed coordinate system. This has the advantage that moments of inertia and body-fixed coordinates do not change with angle of attack, but a conversion must be made from lift and drag to X force and Z force. The body axis system is the conventional one for aircraft dynamics work (x is forward, y is to the right when facing forward, and z is downward), but note that this differs from the conventions used in aerodynamics and wind tunnel testing in which x is aft and z is upward. Thus, drag acts in the negative x direction when the angle of attack is zero. The actual definition of the coordinate directions is up to the user, but generally, the fuselage reference line is used as the direction of the x axis. The rotation rates p, q, and r are measured about the x, y, and z axes respectively using the conventional right hand rule and velocity components u, v, and w are similarly oriented in these body axes.

Basic Concepts

Stability is the tendency of a system to return to its equilibrium condition after being disturbed from that point. Two types of stability or instability are important.

A static instability	A dynamic instability

An airplane must be a stable system with acceptable time constants. In general we want the dynamics to be acceptable, actually more than just stable -- we need appropriate damping and frequency. To assure this, a careful analysis of the dynamic response and controllability is required. The dynamic equations of motion are shown below, expressed in body axes. The top six equations are just forms of F=ma and M=I dW / dt for each of the coordinate directions. The bottom three equations are kinematic expressions relating angular rates to the orientation angles Q, F, Y, angles describing the airplane pitch, roll, and heading angles.

In general, we must solve these nonlinear, coupled, second order differential equations to describe the dynamics of the airplane. Many simplifying assumptions are often justified and make the analysis more simple.

If we linearize the equations we find that there exist 5 interesting modes of dynamic motion. These are discussed further in the section on dynamic stability. But one of the useful results is that we usually obtain sets of nearly independent modes: those associated with symmetric, longitudinal motion , and those related to lateral motion. The modes are, of course, coupled for asymmetric aircraft such as oblique wings and the motion can be coupled by nonlinear effects such as pitching moment produced by large sideslip angles or alpha-dependent yawing moments that appear on fighters at high angles of attack, but for many cases the approximate decoupling is useful.