Stability Derivatives - Equations of Motion

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 (psi) with respect to inertial axes. The body is oriented at an angle (beta) with respect to the velocity vector, so that the components of velocity in body axes are:

where U is the speed.

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 into 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. 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, is a small quantity, so the small perturbation equations of motion become:

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; and, 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, r and their time derivatives. For typical missile configurations the forces and moments depend, in the short term, on and r. The forces may be expressed in the form:

where is the force corresponding to the equilibrium condition (usually called the trim) whose stability is being investigated. It is common practice to employ a shorthand:

The partial derivative and all similar terms characterising the increments in forces and moments due to increments in the state variables are called stability derivatives. Typically, is insignificant for missile configurations, so the equations of motion reduce to: