Question 10.6: A satellite is spinning about the z of its principal body fr...
A satellite is spinning about the z of its principal body frame at 2π radians per second. The principal moments of inertia about its center of mass are
A = 300 kg · m² B = 400 kg · m² C = 500 kg · m² (a)
For the nutation damper, the following properties are given
R = 1 m μ = 0.01 m = 10 kg k = 10,000 N/m c = 150 N-s/m (b)
Use the Routh-Hurwitz stability criteria to assess the stability of the satellite as a major-axis spinner, a minor-axis spinner, and an intermediate-axis spinner.
The data in (a) are for a major axis spinner. Substituting into Equations 10.92 and 10.93, we find
\begin{aligned}&a_{4}=(1-\mu) m A B \\&a_{3}=c A\left[B+(1-\mu) m R^{2}\right] \\&a_{2}=k\left[B+(1-\mu) m R^{2}\right] A+(1-\mu) m\left[(A-C)(B-C)-(1-\mu) A m R^{2}\right] \omega_{o}^{2} \\&a_{1}=c\left\{\left[A-C-(1-\mu) m R^{2}\right](B-C)\right\} \omega_{o}^{2} \\&a_{0}=k\left\{\left[A-C-(1-\mu) m R^{2}\right](B-C)\right\} \omega_{o}^{2}+\left[(B-C)(1-\mu)^{2}\right] m^{2} R^{2} \omega_{o}^{4}\end{aligned} (10.92)
r_{1}=a_{4} \quad r_{2}=a_{3} \quad r_{3}=a_{2}-\frac{a_{4} a_{1}}{a_{3}} \quad r_{4}=a_{1}-\frac{a_{3} a_{0}}{a_{3} a_{2}-a_{4} a_{1}} \quad r_{5}=a_{0} (10.93)
\begin{aligned}&r_{1}=+1.188 \times 10^{6} kg ^{3} m ^{4} \\&r_{2}=+18.44 \times 10^{6} kg ^{3} m ^{4} / s \\&r_{3}=+1.228 \times 10^{9} kg ^{3} m ^{4} / s ^{2} \\&r_{4}=+92,820 kg ^{3} m ^{4} / s ^{3} \\&r_{5}=+8.271 \times 10^{9} kg ^{3} m ^{4} / s ^{4}\end{aligned} (c)
Since every r is 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,
A = 500 kg · m² B = 400 kg · m² C = 300 kg · m² (d)
For these moment of inertia values, we obtain
\begin{aligned}&r_{1}=+1.980 \times 10^{6} kg ^{3} m ^{4} \\&r_{2}=+30.74 \times 10^{6} kg ^{3} m ^{4} / s \\&r_{3}=+2.048 \times 10^{9} kg ^{3} m ^{4} / s ^{2} \\&r_{4}=-304,490 kg ^{3} m ^{4} / s ^{3} \\&r_{5}=+7.520 \times 10^{9} kg ^{3} m ^{4} / s ^{4}\end{aligned} (e)
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,
A = 300 kg · m² B = 500 kg · m² C = 400 kg · m² (f)
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 \times 10^{6} kg ^{3} m ^{4}
r_{2}=+22.94 \times 10^{6} kg ^{3} m ^{4} / s
r_{3}=+1.529 \times 10^{9} kg ^{3} m ^{4} / s ^{2}
r_{4}=-192,800 kg ^{3} m ^{4} / s ^{3}
r_{5}=-4.323 \times 10^{9} kg ^{3} m ^{4} / s ^{4}
The motion, as we expected, is not stable.