A satellite is spinning about the z axis 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

Question

A satellite is spinning about the z axis of its principal body frame at 2π radians per second. The principal moments of inertia about its center of mass are

[latex]\begin{matrix} A = 300 kg · m^{2}& B = 400 kg · m^{2}& C = 500 kg · m^{2} \end{matrix}[/latex] (a)

For the nutation damper, the following properties are given

[latex]\begin{matrix}R = 1 m& µ = 0.01& m = 10 kg& k = 10 000 N/m &c = 150 N·s/m\end{matrix}[/latex] (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.

Accepted Answer

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

[latex]\begin{matrix}a_{4}=(1-μ)mAB \ a_{3}=cA[B+(1-μ)mR^2 ] \a_{2}=k[B+(1-μ)mR^2]A+(1-μ)m[(A-C)(B-C)-(1-μ)AmR^2]ω^2_0 \quad \quad (10.92) \a_{1}=c{[A-C-(1-μ)mR^2](B-C)}ω^2_0 \a_{0}=k\left\{[A-C-(1-μ)mR^2](B-C) ight\} ω^2_0 +[(B-C)(1-μ)^2]m^{2}R^{2}ω^2_0 \end{matrix}[/latex]

[latex]\begin{matrix}r_{1}=a_{4} & r_{2}=a_{3} &r_{3}=a_{2}-\frac{a_{4}a_{1}}{a_{3}}&r_{4}=a_{1}-\frac{a_{3}a_{0}}{a_{3}a_{2}-a_{4}a_{1}}&r_{5}=a_{0} \end{matrix}[/latex] (10.93)

[latex]r_{1} = +1.188 × 10^{6} kg^{3}m^{4}[/latex]
[latex]r_{2} = +18.44 × 10^{6} kg^{3}m^{4}/s[/latex]
[latex]r_{3} = +1.228 ×10^{6} kg^{3}m^{4}/s^{2}[/latex] (c)
[latex]r_{4} = +92 820 kg^{3}m^{4}/s^{3}[/latex]
[latex]r_{5} = +8.271 ×10^{9} kg^{3}m^{4}/s^{4}[/latex]
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,

[latex]\begin{matrix}A = 500 kg · m^{2}& B = 400 kg · m^{2}& C = 300 kg · m^{2}\end{matrix}[/latex] (d)

For these moment of inertia values, we obtain
[latex]r_{1} = +1.980 × 10^{6} kg^{3}m^{4}[/latex]
[latex]r_{2} = +30.74 ×10^{6} kg^{3}m^{4}/s[/latex]
[latex]r_{3} = +2.048 ×10^{6} kg^{3}m^{4}/s^{2}[/latex] (e)
[latex]r_{4} = −304 490 kg^{3}m^{4}/s^{3}[/latex]
[latex]r_{5} = +7.520 ×10^{9} kg^{3}m^{4}/s^{4}[/latex]
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,

[latex]\begin{matrix}A = 300 kg·m^{2}\quad B = 500 kg·m^{2} \quad C = 400 kg·m^{2}\end{matrix}[/latex] (f)

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

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

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