Consider the gyrostat in Problem XXXXXXXXXXAssume initially that the rotor is spinning with Ω and the spacecraft does not have any angular motion. A small perturbation is applied and the spacecraft...

Consider the gyrostat in Problem 11.18. Assume initially that the rotor is spinning with Ω and the spacecraft does not have any angular motion. A small perturbation is applied and the spacecraft acquires angular velocities of 1, 2, and 3 about b1, b2, and b3, respectively. Linearize the equations of motion and conduct a stability analysis on the motion. Can the rotor stabilize the spacecraft if b1 is the intermediate axis of inertia (that is, it is unstable when the rotor does not rotate)?

Problem 11.18

A gyrostat is a spacecraft that has a rotor attached inside it. The rotor is placed in the gyrostat for providing additional stability and for maneuvering (rotating) the spacecraft. Consider the gyrostat in Figure 11.31a that is inside a body with principal moments of inertia I1, I2 = I, and I3 along the b1b2b3 axes. External moments N1, N2, and N3 act on the spacecraft. The disk has mass moments of inertia J1 and J2 = J3. The free-body diagrams are shown in Figure 11.31b. Obtain the equations of motion of the spacecraft, using the reference frame attached to the body as the shape frame. Neglect any translational motion. A servomotor keeps the angular velocity of the gyrostat constant