Rotational stability equations
The torque T applied to any rigid vehicle with moving c.g. is in general determined by
T = ∂HVE/∂t|E + uvE ×MugE
where E is the inertial axis HVE = I ∗ ωVE is the absolute angular momentum, uvE is the translational velocity of the body relative to Earth, ugE is the translational velocity of the c.g. relative to Earth and M is the total body mass. With the c.g. fixed relative to the body 6 then ugE × uvE = 0 and so T is determined solely by the rate of change of absolute angular momentum that is by
T = ∂HVE/∂t|E = dI/dt * ωV E + I ∗dωV Edt
Since the representation of the inertia tensor I is constant only relative to axes fixed on the body , it is convenient to express this equation in these moving axes. Such axes are known as body axes. There are several different possible ways of fixing the axes in the body. The most common method is the so called stability body axis where we align the X axis with the trimmed steady state air speed, the Y axis pointing starboard (to the right) and the Z axis pointing down. the equation of motion above is then