and the nXn matrix M22 is diagonal and can always be reduced to the identity matrix by proper formulation of the equations of motion. The 6Xn matrix Mn provides the interaction between the flexible modes and the rigid spacecraft and M2l = Mj2. The matrices M, C22, and Kn are obtained from a dynamic analysis and derivation of the equations of motion for a flexible spacecraft (see, for example, Heinrichs and Fee [1972]).

In terms of the quantities above, the equations of motion for a flexible spacecraft are r = v x=y2

where the 4x4 matrix R(«) is defined by Eq. (16-26). The 6-vector h, refers to the moving parts of the spacecraft with respect to its rigid frame of reference. If the only moving parts are the wheels, hj has the form h7=(0,0,0,hT), where the 3-vector h is the angular momentum of the wheels with respect to the rigid spacecraft. From Eq. (16-144) we see that P, depends on the spacecraft total angular momentum, Llo„ which is given by

'flex, modes

where

^fiex. modes

Here, (A/l2), is the ith row of the matrix Mn. Note that Lrot consists of three terms: the first term gives the angular momentum of the rigid spacecraft, where the moment of inertia, /, includes the mass of the wheels, the solar panels, and the antenna; the second term is the angular momentum of the wheels with respect to the spacecraft; and the third term is the angular momentum due to the flexible modes. Note that Eq. (16-146b) is identical with Eq. (16-3) and Eq. (16-147a) is an extension of Eq. (16-52) for the case of a flexible spacecraft.

Equations (16-146) and (16-147) form a complete set of equations of motion for the flexible spacecraft. In this representation, the state vector, of the flexible spacecraft is

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