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Stabilization of Weakly Coupled Rotors:
a General Derivation of the Required Forces

Donato Bramanti, Anna M Nobili and Giuseppe Catastini
Gruppo di Meccanica Spaziale, Dipartimento di Matematica, Universitdi Pisa, Italia
Pisa November 1996


A system of two weakly coupled rotors with natural frequency of relative oscillation tex2html_wrap_inline585 much smaller than the spin frequency tex2html_wrap_inline587 and non zero dissipation (due to friction inside its rotating parts, referred to as rotating damping) is known to develop a forward whirling motion of increasing amplitude. The force necessary to stabilize the system (by preventing the whirling motion from growing) is derived on the basis of general physical principles with no assumption on the nature and the amount of dissipation in the system. It is found that, even in the presence of very high viscous friction the stabilizing force is smaller than the elastic spring force which couples the system. In all other cases (structural damping only, or structural damping plus small-to-medium viscous damping) the frequency of the destabilizing whirling motion is essentially the natural frequency of the system and the stabilizing force is only tex2html_wrap_inline589 of the spring force, tex2html_wrap_inline591 being the quality factor of the springs which accounts for all dissipation (structural plus viscous), measured at the spin frequency. In the GG case, where the spin frequency is tex2html_wrap_inline593 at which very high tex2html_wrap_inline591 can be achieved (as we have already found in laboratory tests), the required stabilizing force is therefore smaller than the spring force by far. Application of the right amount of force by means of active electrostatic dampers which spin together with the rotors does in no way change the amount of stabilizing force to be provided; this has also been checked in a thorough numerical simulation of the GG system carried out by Alenia Spazio, including drag disturbance and implementation errors, and with a very conservative assumption on the amount of dissipation in the system. We conclude that: i) the stabilizing forces required in the GG experiment (provided by electrostatic actuators spinning with the system) are a factor about tex2html_wrap_inline597 smaller than the value claimed by Y. Jafry and M. Weinberger in their Appendix to the ESTEC Technical Assessment of GG; ii) even in the presence of a very large amount of viscous damping the stabilizing forces would be smaller than the spring forces and never dominate the system. We notice that Y. Jafry and M. Weinberger have assumed the damping coefficient of the system (whose physical dimensions are mass/time) to depend on the reference frame, which is clearly incorrect in Galilean mechanics. Moreover, they appear to have misunderstood non rotating friction with friction in the bearings, claiming that the system is stabilized by friction in the bearings (or by an active simulation of it), while the most efficient way of stabilizing the system is well known to be the non rotating friction (or an active simulation of it). From the fact that the active damping forces are much smaller than the forces of the springs it follows that the essentially passive nature of the GG space experiment at a rotation rate of tex2html_wrap_inline593 is confirmed. Extremely weak mechanical coupling and good balancing of the test masses (a common mode rejection of tex2html_wrap_inline601 has already been achieved with a ground prototype) are very advantageous for testing the Equivalence Principle, as it is the signal modulation at tex2html_wrap_inline593 .


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