Effects of Asymmetries in the Rotor and Flexible Joint of a Dual-Spin Spacecraft

Most past investigations of dual-spin spacecraft have assumed that the platform and rotor are symmetric and rigidly connected. Recently, Cretcher and Mingori, Wenglarz, Willems, and Bainum have considered the effects of flexibility in different parts of dual-spin spacecraft. Scher and FarrenkopP have considered the effects of an asymmetric rotor rigidly connected to the platform. However, the effects of unsymmetric transverse inertias in a dual-spin spacecraft with a flexible joint have not been published. They have been rigorously investigated in rotors and gyroscopes by Brosens and Crandall and by Foote, Poritsky, and Slade. This paper presents the results of an investigation of these effects on the attitude motion of dual-spin spacecraft. It includes only those cases for which the equations of motion can be transformed into linear equations without periodic coefficients. The model, shown in Fig. 1, consists of a spinning rotor which is connected to a nonspinning platform by a flexible joint. The axes XQYQZQ are fixed m space with the origin at the CM of the system. The axes X2Y2Z2 are the principal inertia axes of the platform and are described with respect to axes X0Y0Z0 by rotations 9X about the X0 axis and 0y about the Y1 axis. The orientations of X4Y4Z4, where Z4 is the drive axis of the rotor, are described by rotations x about the X2 axis and y about the Y3 axis. Axes X^ Y4Z4, which will be the principal inertia axes for a balanced rotor, are fixed in the rotor and rotating at the spin rate about the Z4 axis. For an unbalanced rotor, the principal axes are X6Y6Z6, which are defined by rotations Qx about the X4 axis and Qy about the Y5 axis. The center of mass offset of the rotor is g. Both the rotor and the platform are assumed to have unequal transverse inertias and the rotor is assumed to be dynamically and statically unbalanced. The equations of motion of this system are obtained by using Lagrangian formulation. Two cases are considered for unsymmetrical stiffness.