Ia

where M = 27— imi i® total mass of the body. If O' is taken to be the center of mass of the body,

2 mfii=0

by definition, so the second and third terms on the right side of Eq. (16-23) vanish identically. We will always choose this reference point for rigid body dynamics, giving

where the first term on the right side represents the angular momentum of the total mass considered as a point located at the center of mass, and the second term,

is the contribution of the motion of the n mass points relative to the center of mass.

Fig. 16-2. The position Of mass point, i, relative to the origin, O, of an inertia! reference frame is the vector sum of its position relative to a reference point, O', fixed in the body and the vector R from O to O'.

A similar separation between center-of-mass motion and motion relative to the center of mass occurs for the kinetic energy

The middle term in Eq. (16-27) vanishes identically if O' is the center of mass of the body, so

where

is the kinetic energy of motion relative to the center of mass.

Spacecraft rigidity has not been assumed to this point. If the body is not rigid, a reference point other than the center of mass is often used, in which case all the terms in Eqs. (16-23) and (16-27) must be retained. If we now assume that the spacecraft is rigid, i.e., that all the vectors p, are constant in a reference frame fixed in the spacecraft, then all the vectors may conveniently be resolved into components along a spacecraft reference system. In this section, the subscripts I, 2, 3 will be used for components along spacecraft-fixed axes. This should be distinguished from the notation in Section 12.1, where the subscripts I, 2, 3 referred to an arbitrary reference system, and u, t\ w to the spacecraft reference system. The attitude dynamics problem is only concerned with motion relative to the center of mass, and thus only with the angular momentum, L, and kinetic energy, Ek, defined by Eqs. (16-26) and (16-29), in the rigid body case.

Although the components of p, in the spacecraft frame are constant, the components of dp,/d/ are not zero if the spacecraft is rotating with instantaneous angular velocity to, because the vector dp,/d/ is the rate of change of p, relative to inertiaI coordinates, resolved along spacecraft-fixed axes. All time derivatives must be evaluated in an inertial reference frame if Newton's laws of motion are to be applied directly. Equation (16-19) with a=p, gives dp,

where p, and to are understood to be resolved into components along spacecraft-fixed axes.

Substituting Eq. (16-30) into Eq. (16-26) yields

We define the symmetric 3x3 moment of inertia tensor, I, by

Then, Eq. (16-31) can be written in matrix form as

Substituting Eq. (16-30) into Eq. (16-29) and using Eq. (16-31) for L yields n n

0 0

Post a comment