Nutational stability of an axisymmetric body containing a rotor

T7K)R many satellite and space-probe missions, full three-F axis attitude control is not required, and simple (passive) spin stabilization is sufficient to insure satisfactory system performance. In many applications, in fact, the need for stabilization is not dictated by primary mission requirements at all but arises indirectly, perhaps as a means of minimizing or removing thermal and antenna design problems. Spin control is ideal for this type of mission. In other caess, considerable improvement in system performance is afforded, e.g., in a high-altitude communication satellite, increased antenna gain can be obtained using an antenna with a toroidal radiation pattern. Fully passive techniques for three-axis attitude control have only limited application, and it is generally necessary to employ far more sophisticated, active stabilization methods such as mass expulsion or flywheel control systems. However, there is one active, but fairly simple, three-axis control technique that is a natural extension of spin stabilization. In this technique, control of the third axis (control about the spin axis) is achieved by "despinning" a portion of the spinning spacecraft. That is, instead of the vehicle being a single body, it is constructed of two bodies (I and II in Fig. 1) constrained to rotate about a common axis. Suppose that II is the despun portion. A motor (not shown) is included which can change the relative rate about OZ. If the motor is servo-controlled, say by a sensor mounted on one of the bodies, the rate of body II can be maintained at zero or can be varied to permit a given axis in II, normal to OZ, to track an external reference. Thus, in effect, the despun component is three-axis stabilized by means of a single-axis active controller. Control of spin-axis attitude, if required at all, would normally be by ground command to activate, for example, pulse jets, or, in the case of earth orbiters, a magnetic coil mounted on the spinning component. A wide range of configurations can be devised which use this basic two-body construction. One application is id the use of a small directional antenna as the despun portion of a spin-stabilized communications satellite. In this application, the spin axis would be maintained normal to the orbit plane, and the antenna controlled to point along local vertical, allowing maximum gain to be obtained. Another type of single-axis control can be provided if the major part of a spacecraft is despun, and a high-speed internal rotor is employed to provide the gyroscopic stiffness. In fact, this case can be considered a derivative of a three-flywheel system from which two wheels are removed and the third operated about a high bias rate instead of null. Because of the high speed of the internal rotor, spin-axis pointing can again be controlled by ground command. The two-body configuration does appear to offer definite advantages as a means of achieving simplified three-axis attitude control of spacecraft; for this reason, it has received, and is receiving, serious attention in the industry. Indeed, one spacecraft of this construction, OSO I (built by Ball Brothers for NASA), was launched over two years ago and has operated successfully in orbit. In this paper the nutational stability of the two-body system is considered, and the constraints that the requirement for stability places on the moments of inertia of the configuration are established. SYMBOLS: O X , , O Y , , O Z ARE A X E S FIXED IN BODY I