The uniform, monolithic 10 000 kg slab, having the dimensions shown in Figure 10.30, is in a circular LEO. Determine the orientation of the satellite in its orbit for gravity gradient stabilization, and compute the periods of the pitch and yaw/roll oscillations in terms of the orbital period T.

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According to Figure 9.9(c), the principal moments of inertia around the xyz axes through the center of mass are
[latex]A =\frac{10000}{12}(1^{2} + 9^{2}) = 68 333 kg·m^{3}[/latex]
[latex]B=\frac{10000}{12}(3^{2} + 9^{2})=75 000 kg·m^{3}[/latex]
[latex]C=\frac{10000}{12}(3^{2} + 1^{2})=8333.3 kg·m^{3}[/latex]
Let us first determine whether we can stabilize this object as a minor axis spinner. In that case,

[latex]\begin{matrix}I_{pitch} = C = 8333.3 kg · m^{2}& I_{yaw}= A = 68 333 kg·m^{2} & I_{roll}=B = 75 000 kg·m^{2}\end{matrix}[/latex]

Since [latex]I_{roll} > I_{yaw}[/latex], the satellite would be stable in pitch. To check yaw/roll stability, we first compute

[latex]\begin{matrix} K_{Y}=\frac{I_{pitch}-I_{roll}}{I_{yaw}}=−0.97561&K_{R}=\frac{I_{pitch}-I_{yaw}}{I_{roll}}=−0.8000 \end{matrix} [/latex]

We see that [latex] k_{Y} k_{R} > 0[/latex], which is one of the two requirements. The other one is found in Equation 10.181, but in this case

[latex]3k_{R} + k_{Y} k_{R}+1>4\sqrt{k_{Y} k_{R}}[/latex] (10.181)
[latex]1 + 3k_{R} + k_{Y} k_{R}−4\sqrt{k_{Y} k_{R}}=−4.1533 < 0[/latex]
so that condition is not met. Hence, the object cannot be gravity-gradient stabilized as a minor axis spinner.
As a major axis spinner, we must have

[latex]\begin{matrix}I_{pitch} = B = 75 000 kg·m^{2}& I_{yaw} = C = 8333.3 kg·m^{2} &I_{roll} = A = 68 333 kg·m^{2}\end{matrix}[/latex]

Then [latex]I_{roll} > I_{yaw}[/latex], so the pitch stability condition is satisfied. Furthermore, since

[latex]\begin{matrix}K_{Y}=\frac{I_{pitch}-I_{roll}}{I_{yaw}}=0.8000 & K_{R}=\frac{I_{pitch}-I_{yaw}}{I_{roll}}= 0.97561 \end{matrix}[/latex]

we have
[latex]k_{Y} k_{R} = 0.7805 > 0[/latex]
[latex]1 + 3kR + kY kR − 4\sqrt{k_{Y} k_{R}}= 1.1735 > 0[/latex]
which means the two criteria for stability in the yaw and roll modes are met. The satellite should therefore be orbited as shown in Figure 10.31, with its minor axis aligned with the radial from the earth’s center, the plane abcd lying in the orbital plane, and the body x axis aligned with the local horizon.

According to Equation 10.171, the frequency of the pitch oscillation is
[latex]\omega_{f_{pitch}}=n\sqrt{3\frac{I_{roll}-I_{yaw}}{I_{pitch}} }=n\sqrt{3\frac{68 333 − 8333.3}{75 000}}= 1.5492 n[/latex]
where n is the mean motion. Hence, the period of this oscillation, in terms of that of the orbit, is
[latex]T_{pitch}=\frac{2\pi }{\omega_{f_{pitch}}}=0.6455\frac{2\pi }{n}=\underline{0.6455 T}[/latex]
For the yaw/roll frequencies, we use Equation 10.184,
[latex]\omega_{f_{yaw/roll}})_{1,2}=n\sqrt{\frac{1}{2}(b\pm \sqrt{b^{2}-4c)}}[/latex] (10.184)

[latex]\omega_{f_{yaw/roll}})_{1}=n\sqrt{\frac{1}{2}(b+\sqrt{b^{2}-4c)}}[/latex]
where
[latex]b = 1 + 3k_{R} + k_{Y} k_{R} = 4.7073 and c = 4k_{Y} k_{R} = 3.122[/latex]
Thus,
[latex]\omega_{f_{yaw/roll}})_{1}=2.03015 n[/latex]
Likewise,
[latex]\omega_{f_{yaw/roll}})_{2}=n\sqrt{\frac{1}{2}(b-\sqrt{b^{2}-4c)}}= 1.977 n[/latex]
From these we obtain
[latex]\begin{matrix} T_{yaw/roll_{1}} = \underline{0.5058 T} & T_{yaw/roll_{2}} =\underline{ 0.4345 T} \end{matrix}[/latex]
Finally, observe that
[latex] \frac {I_{roll} - I_{yaw}}{I_{pitch}}= 0.8 [/latex]
so that we are far from the pitch resonance condition that exists if the orbit has a small eccentricity.

Question 10.14: The uniform, monolithic 10 000 kg slab, having the dimension...

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