Stability derivatives
Stability derivatives, and also control derivatives, are measures of how particular forces and moments on an aircraft change as other parameters related to stability change (parameters such as airspeed, altitude, angle of attack, etc.).
For a defined "trim" flight condition, changes and oscillations occur in these parameters.
The collection of stability and control derivatives as they change over a range of flight conditions is called an aero model.
That is, the forces and moments on the vehicle are seldom simple (linear) functions of its states.
Air vehicles use a coordinate system of axes to help name important parameters used in the analysis of stability.
:[1] Aircraft (usually not missiles) operate at a nominally constant "trim" angle of attack.
The angle of the nose (the X Axis) does not align with the direction of the oncoming air.
So, for many purposes, parameters are defined in terms of a slightly modified axis system called "stability axes".
If it is assumed that the vehicle is roll-controlled, the pitch and yaw motions may be treated in isolation.
Furthermore, it is assumed that thrust equals drag, and the longitudinal equation of motion may be ignored.
The aerodynamic forces are generated with respect to body axes, which is not an inertial frame.
In order to calculate the motion, the forces must be referred to inertial axes.
Resolving into fixed (inertial) axes: The acceleration with respect to inertial axes is found by differentiating these components of velocity with respect to time: From Newton's Second Law, this is equal to the force acting divided by the mass.
, which will be written more compactly as the yaw rate r. There is one force and one moment, which for a given flight condition will each be functions of
For typical missile configurations the forces and moments depend, in the short term, on
is the force corresponding to the equilibrium condition (usually called the trim) whose stability is being investigated.
In aircraft, the directional stability determines such features as dihedral of the main planes, size of fin and area of tailplane, but the large number of important stability derivatives involved precludes a detailed discussion within this article.
The missile is characterised by only three stability derivatives, and hence provides a useful introduction to the more complex aeroplane dynamics.
generates a lift force in the opposite direction to the motion of the body.
At low angles of attack, the lift is generated primarily by the wings, fins and the nose region of the body.
explains why arrows and darts have flights and unguided rockets have fins.
: The qualitative behavior of this equation is considered in the article on directional stability.
, it also contains a term which effectively determines the angle of attack due to the body rotation.
The distance of the center of lift, including this term, ahead of the centre of gravity is called the maneuver margin.
This damped oscillation in angle of attack and yaw rate, following a disturbance, is called the 'weathercock' mode, after the tendency of a weathercock to point into wind.
These simplifications of the problem based on a priori knowledge, represent an engineer's approach.
Mathematicians prefer to keep the problem as general as possible and only simplify it at the end of the analysis, if at all.
Aircraft dynamics is more complex than missile dynamics, mainly because the simplifications, such as separation of fast and slow modes, and the similarity between pitch and yaw motions, are not obvious from the equations of motion, and are consequently deferred until a late stage of the analysis.
Subsonic transport aircraft have high aspect ratio configurations, so that yaw and roll cannot be treated as decoupled.
However, this is merely a matter of degree; the basic ideas needed to understand aircraft dynamics are covered in this simpler analysis of missile motion.
