PLANAR LIBRATIONS OF AN EXTENSIBLE DUMBBELL SATELLITE

If a gravitationally stabilized satellite has a "tip mass" connected to it through a long elastic arm, the librational motion about the local vertical will excite longitudinal or "pumping" vibrations of the elastic arm. If the "pumping" motions are damped out by some type of internal (passive) friction, the librational motion will be damped also. In this paper, the equations of planar motion for such a system are derived and then linearized for the case of viscous damping in a circular orbit. The complete solution is given in terms of two nondimensional parameters, which are a measure of spring stiffness and viscous damping, respectively. It is shown that, if the internal friction arises from "material damping" within the spring, there will be relatively little damping of a viscous nature ; however, there is a nonlinear timeindependent type of hysteretic damping which could be significant. It is shown how the latter type of damping may be analyzed by a technique of "equivalent viscous damping." A configuration of practical interest is examined in a numerical example.