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.

A stability derivative. This is an example of a common shorthand notation for stability derivatives. The “M” indicates it is a measure of pitching moment changes. The {\displaystyle \alpha }\alpha indicates the changes are in response to changes in angle of attack. This stability derivative is pronounced “see-em-alpha”. It is one measure of how strongly an aircraft wants to fly “nose first”, which is clearly very important.

1 Stability derivative vs. control derivative

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 derivative
measures how much change occurs in a force or moment acting on the vehicle when there is a small change in a flight condition parameter such as angle of attack, airspeed, altitude, etc. (Such parameters are called “states”.)
Control derivative
measures how much change occurs in a force or moment acting on the vehicle when there is a small change in the deflection of a control surface such as the ailerons, elevator, and rudder.

2 Uses

2.1 Linearization (simplification) of stability analysis

2.2 Use in flight simulators

3 Names for the axes of vehicles

3.1 Body-fixed axes

3.2 Stability axes

4 Names for forces, moments, and velocities

4.1 Forces and velocities along each of the axes

4.2 Moments and angular rates around each of the axes

5 Equations of motion

6 Stability derivative contributions

7 Response

8 Comments

9 Control derivatives

10 Examples

11 See also