It follows from Eq. (16-146d) that in the above representation, information must be provided on the time derivative of the six-vector h, to solve the equations of motion. However, the available information is often the timfe dependence of h, itself. In this case, it is convenient to work in a different representation where the spacecraft state vector is xI = (rT,iT,LT,xJ,lT) (16-149)

The 6-vector L, and n-vector Lj are

We will show that the equations of motion for xL do not involve the time derivative of h,. Also, in the absence of any generalized forces, the vectors L, and L2 are conserved. Thus, physical meaning can be attributed to the various vector components. In particular, the last three components of L, are the components of the total angular momentum of the flexible spacecraft, Lul (see Eq. (16-147)). To obtain the equations of motion for L, and Lj, Eq. (16-150) must be solved with respect to y, and y2. The result is yI=(MII-MI2^IM21)"l[L1-h1-M11A/2iIL2] (16-151a)

y2=(M22-M21MI7IMI2)-I[-M21M1T'(L1-h1)+L2] (16-I51b) Using Eq. (16-150), Eqs. (16-146d) and (16-146e) reduce to

iCrfi-KvX2 (16-152b)

where y2 is given by Eq. (16-151b), and the angular rate term, <0, in P, (see Eq.

This follows from Eq. (16-147a). We see that the new equations of motion, namely Eqs. (16-146a), (16-146b), (16-146c), (16-152a), (16-152b), together with Eqs. (16-151b), (16-153), and (16-147b) involve only components of the state vector xL, and the time derivative of h, is not present. Note that the equations of motion of the state vectors x^ and xL are equivalent and should lead to identical solutions.

The equation of motion for the total angular momentum can be obtained from Eq. (16-152a) as

This relation and Eq. (16-153) are extensions of Eqs. (16-55) and (16-53) for the case of a flexible spacecraft.

In Eq. (16-151 b>, inversion of the nXn matrix M^— M2lM{]'Af 12 is required. However, the useful matrix identity,

(M22- M21M{] ' = M22'-Mn lM2l(Mtl- MuM{2 'A/2I)~'Mi2M£'

may be used to reduce this to inversion of only 6x6 matrices. Moreover, substituting Eq. (16-155) into Eq. (16-15lb) and noting that M21= 1, we obtain y2= L2 - M2i {[ 1 - (Mt, - M]2M2]y1 M]2M21 ] X Mu '(L, -h.)

Thus, we have replaced the multiplication of an n X n matrix by an n X 1 vector involving n2 multiplications, with multiplication of a 6 Xn matrix by an n X 1 vector and an n x 6 matrix by a 6 X 1 vector. This involves only 12n multiplications. Thus, for complex systems, Eq. (16-157) should take the place of Eq. (16-15lb) in the equations of motion for the state vector xL.

16.4J Characteristics of Various Flexible Spacecraft

Flexibility effects for some past and future spacecraft are summarized in Table 16-2. The satellites are excellent examples of large flexible spacecraft [Blanchard, el a!., 1968] (see Fig. 15-18). They are gravity-gradient stabilized by four 230-m-long antenna booms. The antenna booms have a double-V configuration with a nominal included angle of 60 deg. Librational motions are damped by a libration damper that is skewed a nominal 66.5 deg from the plane of the antennas. The estimated oscillation period of the RAE-1 antenna booms is 91 minutes using linear beam theory. Because the orbital period is 224 minutes, flexibility effects are obviously important. The equilibrium deformation due to gravity-gradient forces is on the order of 50 m. Hence, the linear beam theory is not adequate and detailed simulation is necessaiy. Axial tension due to gravity-gradient forces also increases bending frequencies. During a dynamics experiment performed on RAE-1, two distinct short-period oscillations of 19 and 6 min (corresponding to antisymmetrical

Table 16-2. Typical Spacecraft Flexibility Characteristics. "X" indicates a potentially significant mode.
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