On the Stabilization of the GG System
Stephen H. Crandall and Anna M. Nobili
MIT, January 23 1997

  1. The relevant loss factors (inverse of quality factor Q) are those of the mechanical suspensions at the spin frequency. For a firm estimate these must be measured experimentally, setting the springs in oscillation under realistic operating conditions (oscillation frequency, vacuum, temperature, clamping)
  2. The effects of such dissipation are unstable forward whirl motions whose frequencies are close to the natural frequencies of the system.
  3. The destabilizing forces which generate the whirl motions are equal to the passive spring forces divided by the Q (defined and measured according to point 1 above). The magnitude of the forces is the same in the stationary and in the rotating frame; only their frequencies change. The forces required to achieve neutral equilibrium are equal and opposite to the destabilizing forces. They never exceed the passive spring forces as long as Q is larger than 1, as in GG. If the measured Qs at the spin frequency are large, then the destabilizing forces, as well as the active ones required for stabilization, are much smaller than the passive spring forces. This also means that the instabilities to be damped grow very slowly.
  4. The system can be stabilized by rotating active dampers which apply (internal) control forces of the same magnitude as required in the stationary system. To do that, the rotation velocity of the dampers must be subtracted away. This requires the spacecraft rotation to be measured independently. Since the frequencies of whirl in the rotating frame are very close to the spin frequency (within a factor between 10-4  and 10-3 ) the spacecraft rotation has to be measured much more accurately than that, to avoid an inadequate knowledge of rotation to result in the excitation of the (otherwise stable) backward whirl motions. If such measurement is available, and there is sufficient computing power on board, the task can be met. Crucial to success is that the instabilities grow slowly, i.e. that the Qs are large and the destabilizing forces are small.
  5. When applying damping forces of order 1/Q times the passive spring forces, with Q larger than 1, thermal losses in the electrostatic dampers themselves are much smaller than losses in the suspensions. They would become appreciable only if the dampers were required to apply much larger forces.
  6. As long as backward modes are not excited, all implementation errors will give perturbation forces smaller than the control forces themselves, which in turn are smaller than the passive spring forces. A numerical simulation based on the subtraction of the dampers' rotation must be performed to confirm that this approach leads to perturbations on the test masses which are compatible with the experiment goal.
  7. The two movable rods that the test bodies are suspended from, although like pencils in shape, can be made to behave like disks spinning around their axes of maximum moment of inertia by having high density rings in the planes of their centers of mass. Their conical modes are stable. The rods may not need to be stabilized on their own for whirl motions because each rod is connected to both test masses (via helicoidal springs) and the PGB (via flat gimbals), and these are all stabilized individually. The rods should be included in the numerical simulations to verify that the system as a whole is stable.

Research Papers Available Online

       (Anna Nobili- nobili@dm.unipi.it)