The dynamic stability condition

Abbreviations

c_D	Drag coefficient
c_La	Lift coefficient derivative
c_Mpa	Magnus moment coefficient derivative
c_mq+c_ma	Pitch damping moment derivative
s_g	Gyroscopic (static) stability factor
s_d	Dynamic stability factor

More Abbreviations

Explanation

A projectile is said to be dynamically stable, if its yawing motion of nutation and precession is damped out with time, which means that an angle of yaw induced at the muzzle (the initial yaw) decreases.

A dynamic stability factor s_d can be defined from the linearized theory of gyroscopes (assuming only a small angle of yaw) and the above dynamic stability condition can be formulated. An alternate formulation of this condition leads to the illustrative stability triangle.

s_d however depends on five aerodynamic coefficients. Because these coefficients are hard to determine, it can become very complicated to calculate the dynamic stability factor, which varies as a function of the momentary bullet velocity.