Question 10.6: A satellite is spinning about the z axis of its principal bo...

The data in (a) are for a major-axis spinner. Substituting into Equations 10.92 and 10.93, we find

r_{1} = +1.188 × 10^{6}  kg^{3}m^{4}
r_{2} = +18.44 × 10^{6}  kg^{3}m^{4}/s
r_{3} = +1.228 ×10^{6}  kg^{3}m^{4}/s^{2}                                           (c)
r_{4} = +92 820  kg^{3}m^{4}/s^{3}
r_{5} = +8.271 ×10^{9}  kg^{3}m^{4}/s^{4}
Since the rs are all positive, spin about the major axis is asymptotically stable. As we know from Section 10.3, without the damper the motion is neutrally stable. For spin about the minor axis,

For these moment of inertia values, we obtain
r_{1} = +1.980 × 10^{6}  kg^{3}m^{4}
r_{2} = +30.74 ×10^{6}  kg^{3}m^{4}/s
r_{3} = +2.048 ×10^{6}  kg^{3}m^{4}/s^{2}                                         (e)
r_{4} = −304 490  kg^{3}m^{4}/s^{3}
r_{5} = +7.520 ×10^{9}  kg^{3}m^{4}/s^{4}
Since the rs are not all of the same sign, spin about the minor axis is not asymptotically stable. Recall that for the rigid satellite, such a motion was neutrally stable.
Finally, for spin about the intermediate axis,

We know this motion is unstable, even without the nutation damper, but doing the Routh–Hurwitz stability check anyway, we get
r_{1} = +1.485 × 10^{6}  kg^{3}m^{4}
r_{2} = +22.94 ×10^{6}  kg^{3}m^{4}/s
r_{3} = +1.529 ×10^{9}  kg^{3}m^{4}/s^{2}
r_{4} = −192 800  kg^{3}m^{4}/s^{3}
r_{5} = −4.323 ×10^{9}  kg^{3}m^{4}/s^{4}
The motion, as we expected, is not stable.