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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.
Contents
- Stability derivative vs control derivative
- Linearization simplification of stability analysis
- Use in flight simulators
- Names for the axes of vehicles
- Body fixed axes
- Stability axes
- Forces and velocities along each of the axes
- Moments and angular rates around each of the axes
- Equations of motion
- Stability derivative contributions
- Response
- Comments
- Control derivatives
- Examples
- References
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 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:
Linearization (simplification) of stability analysis
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.
Use in flight simulators
In addition to engineering simulators, aero models are often used in real time flight simulators for home use and professional flight training.
Names for the axes of vehicles
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
Body-fixed axes, or "body axes", are defined and fixed relative to the body of the vehicle.:
Stability axes
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:
Forces and velocities along each of the axes
Forces on the vehicle along the body axes are called "Body-axis Forces":
Moments and angular rates around each of the axes
Equations of motion
The use of stability derivatives is most conveniently demonstrated with missile or rocket configurations, because these exhibit greater symmetry than aeroplanes, and the equations of motion are correspondingly simpler. If it is assumed that the vehicle is roll-controlled, the pitch and yaw motions may be treated in isolation. It is common practice to consider the yaw plane, so that only 2D motion need be considered. Furthermore, it is assumed that thrust equals drag, and the longitudinal equation of motion may be ignored.
The body is oriented at angle
where
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. This requires the body components of velocity to be resolved through the heading angle
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. Now forces arise from the pressure distribution over the body, and hence are generated in body axes, and not in inertial axes, so the body forces must be resolved to inertial axes, as Newton's Second Law does not apply in its simplest form to an accelerating frame of reference.
Resolving the body forces:
Newton's Second Law, assuming constant mass:
where m is the mass. Equating the inertial values of acceleration and force, and resolving back into body axes, yields the equations of motion:
The sideslip,
The first resembles the usual expression of Newton's Second Law, whilst the second is essentially the centrifugal acceleration. The equation of motion governing the rotation of the body is derived from the time derivative of angular momentum:
where C is the moment of inertia about the yaw axis. Assuming constant speed, there are only two state variables;
where
The partial derivative
Stability derivative contributions
Each stability derivative is determined by the position, size, shape and orientation of the missile components. 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.
Consider first
At low angles of attack, the lift is generated primarily by the wings, fins and the nose region of the body. The total lift acts at a distance
The need for positive
The effect of angular velocity is mainly to decrease the nose lift and increase the tail lift, both of which act in a sense to oppose the rotation.
Response
Manipulation of the equations of motion yields a second order homogeneous linear differential equation in the angle of attack
The qualitative behavior of this equation is considered in the article on directional stability. Since
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.
Comments
The state variables were chosen to be the angle of attack
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.
Control derivatives
Deflection of control surfaces modifies the pressure distribution over the vehicle, and these are dealt with by including perturbations in forces and moments due to control deflection. The fin deflection is normally denoted
Including the control derivatives enables the response of the vehicle to be studied, and the equations of motion used to design the autopilot.