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. Equations of motion are used to analyze these changes and oscillations. Stability and control derivatives are used to linearize (simplify) these equations of motion so the stability of the vehicle can be more readily analyzed.

Stability and control derivatives change as flight conditions change. The collection of stability and control derivatives as they change over a range of flight conditions is called an aero model. Aero models are used in engineering flight simulators to analyze stability, and in real-time flight simulators for training and entertainment.

Stability derivatives and control derivatives are related because they both are measures of forces and moments on a vehicle as other parameters change. Often the words are used together and abbreviated in the term "S&C derivatives". They differ in that stability derivatives measure the effects of changes in flight conditions while control derivatives measure effects of changes in the control surface positions:

Stability and control derivatives change as flight conditions change. That is, the forces and moments on the vehicle are seldom simple (linear) functions of its states. Because of this, the dynamics of atmospheric flight vehicles can be difficult to analyze. The following are two methods used to tackle this complexity.

In addition to engineering simulators, aero models are often used in real time flight simulators for home use and professional flight training.

Air vehicles use a coordinate system of axes to help name important parameters used in the analysis of stability. All the axes run through the center of gravity (called the "CG"):

Two slightly different alignments of these axes are used depending on the situation: "body-fixed axes", and "stability axes".

Body-fixed axes, or "body axes", are defined and fixed relative to the body of the vehicle.:

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. The difference in these directions is the angle of attack. So, for many purposes, parameters are defined in terms of a slightly modified axis system called "stability axes". The stability axis system is used to get the X axis aligned with the oncoming flow direction. Essentially, the body axis system is rotated about the Y body axis by the trim angle of attack and then "re-fixed" to the body of the aircraft:

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