Roll Derivatives

There are only two important derivatives in roll degree of freedom (DOF). Lp is the usual damping derivative in the roll, a change in the rolling moment due to a change in the roll rate. As usual it is negative in sign. It is often considered to have a constant value for a given Mach number and altitude. Due to positive roll rate, the change in the rolling moment is such that it will oppose this change in the roll rate and hence provide (increased) damping in the roll axis. It is usually defined as

In an aircraft the major contribution to this derivative is from wing; however, in the case of a missile this is not the case.

LSa is the aileron control surface effectiveness derivative. It is defined as

1.4.1.1 Yaw Derivatives

Effect of change in side speed v/AOSS:

(Yv, Np)

Yv is given as

It must be noted here that for a symmetrical missile the normal force coefficient for yaw has the same value as the Cz. This lateral force derivative should be calculated from the total force from the wings, body, and the control surfaces. It is presumed that the control surfaces are in a central position. Np is the force derivative Yv times the distance of center of pressure from the CG. As in the case of an aircraft this distance is known as the static margin. It is a measure of the static stability of the missile. In the usual manner the derivative is defined as

It will change sign for an unstable missile configuration.

Effect of change in yaw rate r:

(Yr, Nr)

Yr is a difficult derivative to calculate and measure and hence it is usually neglected. Nr is given as

n = rUSb2 c

1У r y~-nr

with the usual definitions as in the previous case. It is a damping derivative in yaw. It has a small value.

Effect of change in rudder control surface deflection:

(YSr, NSr)

Ys is the side force derivative due to deflection of the rudder control surface. Nd is the yawing moment derivative due to deflection of the rudder control surface. It has the opposite sign for a missile with canard control surface. It is given as Yd times the distance of the center of pressure of the rudder from the CG. The formats of these derivatives are usually similar to the ones for aircraft.

Table 4.9 gives the most important missile aerodynamic derivatives. It can be observed that a missile has very few derivatives that are essential to describe its dynamics, especially from the control point of view. It will be a good idea and exercise to establish the equivalence between the formulations of Tables 4.5 and 4.8. We further illustrate this as follows since acceleration = force/mass:

day dFy @8 =

d(qSCy) pU2SdCy PU 2SCyb Y m@8 2m@8 2m 8