Consider a torque-free cylindrical satellite spinning about its major axis, as shown in Figure 11.35. If the angular momentum, angular velocity, and symmetry (spin) axis are all aligned, it will spin uniformly about the b2 axis, ensuring that all instruments stay pointed. However, should it be disturbed (by a micrometeorite impact, perhaps) so that the angular momentum is moved, the satellite body frame will precess, as seen in Tutorial 11.1. Many satellites are equipped with what is called a nutation damper to remove the small nonspin-axis angular-velocity components. This device consists of a small wheel free to spin about its symmetry axis at rotation rate Ω but fixed to the satellite about the other two axes, as shown in Figure 11.35. The wheel is equipped with a viscous damper, so that it is subjected to a moment about its spin axis of magnitude -DΩ
Suppose the satellite (including the wheel) has principal moments of inertia (IT , IS, IT ), where IT S, and the wheel has a moment of inertia about its
spin axis of IW . Find the equations of motion for the three angular-velocity components of the satellite, ω1, ω2, and ω3, and the spin rate Ω of the wheel. Using initial conditions show, numerically, that the system does indeed damp the initial precession.