On the motion of a symmetric rigid body with a "yo-yo" despin device attached
A novel method of reducing the spin of a rotating symmetric body, similar to many earth orbit satellites, is by allowing small, despin weights to unwind from about the satellite so that they absorb some, or all, of the satellite angular momentum. This technique which has been used successfully on several U.S. satellites is commonly referred to as yo-yo despin. Several studies of the motion of a system such as this have been published where it was assumed that the motion was two-dimensional (i.e., without coning).
This dissertation presents a comprehensive study of the yo-yo despin problem which includes a derivation of two-dimensional results as well as a three-dimensional or exact solution. The results presented are sufficient for rudimentary design computations and provide examples of the corrections necessary to apply to two-dimensional computations for their applications as estimates for the general motion. An approximate solution of the three-dimensional equations of motion is also presented along with an example of the accuracy obtained by the approximation.
The equations of motion are derived in a straightforward manner using the vectorial methods of Newtonian mechanics. The Euler equations for a rigid body are used to describe the motion of the rigid body itself. The moment acting on the body through the tension in the yo-yo cables is unknown and it is necessary to apply the second law of Newton to a despin weight so that sufficient independent differential equations are available for the solution of the problem variables. These relations give three first-order differential equations and two second-order ones. An expression for the cable tension is also obtained. This system of equations is integrated numerically by a standard Runge-Kutta process. Two singularities require special attention: first, at the initial instant the fundamental inversion matrix for the Newton equations is singular; and second, special care must be taken at a point in the integration where a discontinuity is found to occur. Outside of these special points, the integration process is quite routine although some cases require precautions near the end of the despinning process in order that the integration is stopped before violent tumbling occurs. In order to discuss the motion relative to a fixed reference axis the Euler angles, and Euler angle rate equations are also integrated. A point of interest concerning the derivation of the equations of motion is that the Lagrange technique cannot be used without modification due to internal constraints which do work.
After a numerical study of several typical examples, one concludes that for initial coning angles of less than 10° a two-dimensional analysis is sufficient for determining many important design variables such as maximum cable tension and despin time, although the cable length is somewhat overestimated and problems may occur in the release of the weights if only a two-dimensional analysis is considered. If one desires information on the angular trajectory of the body in inertial coordinates, a study of the problem must be made using the exact three-dimensional relations or the approximate three-dimensional relations. The approximate expressions save the investigator a great deal of effort and apparently provides excellent results.