Question

have sq mics , McGran 2. A spac cecraft must sometimes be rotated about the intermediate inertia axis, although this orque-free rigid body. For initial conditions is unstable. Write Euler's equations for a t an ar an a)20, a constant, and o (not small) and ai, du3 (very small), show that Euler's equations imply (C-Baho] ( ω): 0 (5.94) These are a pair of linear, constant-coefficient equations. Substitute an assumed solution (01 76. 0 C ω3 (B-A)a)20 of the form col lt (5.95) W1 N.s and show that, as expected, rotation about the intermediate inertia axis is unstable. 162 Spaceflight Dynamics ues of the spacecraft rates on, as. Uset according to the feedback laws Now, assume that the spacecraft has rate gyros, capable of supplying measured val os. Use these measured rate s to produce control torques ah, (5.96) where k is a gain constant. Show that the augmented system is stable when rotating about the intermediate axis if k/aro is between IC - Bl and lA 3. A dual-spin satellite shown in Figure 5.22, sometimes referred to as a gyrostat, has nortin E R and C about the axes bi, b, and ba. It contains

Answer 1

4G 16:07 Starting from Newton's second law, in an inertial frame of reference (subscripted "in"), the time derivative of the angular momentum L equals the applied torque dt dt where lin is the moment of inertia tensor calculated in the inertial frame. Although this law is universally true, it is not always helpful in solving for the motion of a general rotating rigid body, since both ln and w can change during the motion Therefore, we change to a coordinate frame fixed in the rotating body, and chosen so that its axes are aligned with the principal axes of the moment of inertia tensor. In this frame, at least the moment of inertia tensor is constant (and diagonal), which simplifies calculations. As described in the moment of inertia, the angular momentum L can be written def where MK, k and wk are as above -丹仚

4G 16:07 In a rotating reference frame, the time derivative must be replaced with (see time derivative in rotating reference frame) dL dt rot where the subscript "rot" indicates that it is taken in the rotating reference frame. The expressions for the torque in the rotating and inertial frames are related by where Q is the rotation tensor (not rotation matrix), an orthogonal tensor related to the angular velocity vector by for any vector v In general, L-Ίω is substituted and the time derivatives are taken realizing that the inertia tensor, and so also the principal moments, do not depend on time. This leads to the general vector form of Euler's equations -丹仚

16:07 expressions for the torque in the rotating and inertial frames are related by 4G where Q is the rotation tensor (not rotation matrix), an orthogonal tensor related to the angular velocity vector by for any vector v In general, L - lw is substituted and the time derivatives are taken realizing that the inertia tensor, and so also the principal moments, do not depend on time. This leads to the general vector form of Euler's equations If principal axis rotation def is substituted, and then taking the cross product and using the fact that the principal moments do not change with time, we arrive at the Euler equations in components at the beginning of the article -丹仚