The gyroscopic stability condition

Abbreviations

c_Ma	Overturning moment coefficient derivative; c_Ma(B,Ma)
s_g	Gyroscopic stability factor

More abbreviations

Explanation

A spin-stabilized projectile is said to be gyroscopically stable, if, in the presence of a yaw angle d, it responds to an external wind force F₁ with the general motion of nutation and precession. In this case the longitudinal axis of the bullet moves into a direction perpendicular to the direction of the wind force.

It can be shown by a mathematical treatment that this condition is fulfilled, if the gyroscopic stability factor s_g exceeds unity. This demand is called the gyroscopic stability condition. A bullet can be made gyroscopically stable by sufficiently spinning it (by increasing w!).

As the spin rate w decreases more slowly than the velocity v_w, the gyroscopic stability factor s_g, at least close to the muzzle, continuously increases. An practical example is shown in a figure .
Thus, if a bullet is gyroscopically stable at the muzzle, it will be gyroscopically stable for the rest of its flight. The quantity s_g also depends on the air density r and this is the reason, why special attention has to be paid to guarantee gyroscopic stability at extreme cold weather conditions.

Bullet and gun designers usually prefer s_g > 1.2...1.5, but it is also possible to introduce too much stabilization. This is called over-stabilization.

The gyroscopic (also called static) stability factor depends on only one aerodynamic coefficient (the overturning moment coefficient derivative c_Ma) and thus is much easier to determine than the dynamic stability factor. This may be the reason, why some ballistic publications only consider static stability if it comes to stability considerations.

However, the gyroscopic stability condition only is a necessary condition to guarantee a stable flight, but is by no means sufficient. Two other conditions - the conditions of dynamic stability and the tractability condition must be fulfilled.