Satellite spin stabilization stability criteria
I model the satellite as a rigid cuboid and examine its rotational dynamics in free space, where no external torques are present. I focus on understanding the stability of steady-state rotation about the principal axes.
I begin with Euler’s equations for rigid body rotation, expressed with respect to the body’s center of mass:
\begin{aligned} \sum M_{C_x} &= I_{xx}^C \dot{\omega}_x - \left(I_{yy}^C - I_{zz}^C\right) \omega_y \omega_z \\ \sum M_{C_y} &= I_{yy}^C \dot{\omega}_y - \left(I_{zz}^C - I_{xx}^C\right) \omega_z \omega_x\\ \sum M_{C_z} &= I_{zz}^C \dot{\omega}_z - \left(I_{xx}^C - I_{yy}^C\right) \omega_x \omega_y \end{aligned}
Here, I_{xx}^C, I_{yy}^C, and I_{zz}^C represent the principal moments of inertia about the center of mass, and \omega_x, \omega_y, and \omega_z are the angular velocities about the corresponding body axes. For free motion, the external moments are zero.
I investigate the stability of spin about each principal axis by considering small perturbations to the steady-state rotation. For example, if the satellite is spinning primarily about the z-axis with a constant angular velocity \Omega_z, I introduce small perturbations \Delta\omega_x, \Delta\omega_y, and \Delta\omega_z to the angular velocities. I then linearize the equations of motion, neglecting higher-order terms in the perturbations.
My analysis reveals a relationship between the stability of the spin and the moments of inertia. I find that spin about the axis with the intermediate moment of inertia is unstable. This means that any small disturbance will cause the rotation to deviate significantly from the initial state. In contrast, spin about the axes with the largest or smallest moments of inertia is stable. In these cases, small perturbations lead to oscillatory motions, but the rotation remains bounded and does not diverge significantly.
These findings have practical implications for satellite design and control. To maintain a stable spin, I must ensure that the satellite’s primary axis of rotation corresponds to either the largest or smallest moment of inertia. This understanding allows me to predict and control the rotational behavior of the satellite, which is essential for maintaining its orientation and performing its mission effectively.
For more insights into this topic, you can find the details here.