Let the whirling motion be damped by electrostatic sensors/actuators fixed to the
rotor. By providing forces internal to the system they cannot change its total angular
momentum: they can only transfer the angular momentum of whirl to the rotation angular
momentum of the rotor by spinning it up. This is what happens if they are made to provide
a stabilizing force of the same intensity as the destabilizing one (37). This force must
always act along the vector of relative velocity of the centres of mass of the bodies in
their whirling motion, as seen in the inertial frame of reference. Since the centres of
mass of the bodies are displaced by an amount 
, the electrostatic plates will necessarily
apply also a small force tangent to the surface of the rotor amounting to a fraction 
of
the main component 
of the active force, of intensity 
(for both bodies), which will damp the relative velocity of whirl. Of the corresponding
reaction components on the electrostatic actuators only the reaction to the small tangent
component 
will produce a non zero angular momentum by spinning up the rotor at the
expense of exactly the angular momentum of whirl:
which will therefore increase the spin angular momentum of the rotor 
in such a way
that the total angular momentum of the system is conserved. That is:
thus producing a spin up of the rotor at the rate 
.
Except for the sign this is essentially the same as (23) because 
and 
.
By integrating for the entire duration of the mission 
, from intial to final
epoch, the ratio 
of final-to-initial spin angular velocity of the rotor is obtained:
In the GG case, even assuming a very low value for the quality factor at the spin
frequency of 
( 
), 
(for the smallest test
body made of Pt/Ir), 
(the capacitance sensors and actuators
that provide the active damping can in fact maintain the radius of whirl within a smaller
value than this) and 
we get, for a 6 months duration of the
mission:
The corresponding angular advance is:
which is clearly negligible. The corresponding quality factor 
for the spin
up of the rotor (in absence of any external disturbances to its spin rate) would be
obviously huge:
the amounts of energy and angular momentum gained in 6 months being vanishingly small:
Let us now consider the conservation of energy in the presence of rotating active
dampers. The contribution to the energy by the electrostatic dampers comes from the work
done by both components of the active force. The larger one, of intensity and the
smaller one of intensity 
:
While only will transfer angular momentum from the whirling motion to the rotor,
both components of the active force will provide energy. By spinning up the rotor at the
rate 
, will obviously also increase the spin
energy of the rotor by supplying to it the power:
As for the component (always directed along the relative
velocity vector of the centres of mass of the bodies and opposite to it) it will supply
energy at the rate:
because the electrostatic dampers, being fixed to the rotor (hence spinning at 
), are
required to provide a force at frequency 
(in order to damp the whirling motion at ) and
therefore must actuate at frequency 
; in order to give the required stabilizing
force they must necessarily supply energy at the rate (50). This is transferred to the
springs, to be dissipated as heat, while the energy of the whirling motion does not change
any longer. Thus, the springs are provided with a power 
which is exactly the
energy that they dissipate in the presence of a whirling motion of constant radius 
(see
Eq. (27)).
In GG the electrostatic sensors/actuators are fixed to the spinning bodies and actuate
at the frequency , i.e. close to the spin/signal frequency of . This means that they
produce noise close to (with respect to the fixed frame, i.e. close
to 
or DC w.r.t. the rotating frame). We demonstrate in Paper II that the
weakly suspended PGB laboratory inside the GG spacecraft is very effective in attenuating
vibrational perturbations which act at frequencies close to the spin/signal frequency with
respect to the non rotating frame. This is very useful to attenuate noise produced by the
electrostatic dampers, by FEEP thrusters (which also fire close to the spin/signal
frequency), and in general by any other effect which may disturb the experiment at the
spin/signal frequency. For the action of the electrostatic actuators to be successful it
is necessary that the whirling circle be sufficiently small, its centre defining the
position of relative equilibrium of the system given the original unbalance 
of the rotor
because of manufacturing and mounting errors and of its spin-to-natural frequency ratio.
In GG this is achieved after initial unlocking by means of inch-worms equipped with
pressure sensors. Once inch-worms have achieved the initial centring by removing the
unbalance bias , the electrostatic dampers can damp the whirling motions and stabilize
the system (by providing forces as small as we have calculated) also in the presence of
external disturbances such as drag. All this has been confirmed with numerical simulations
(performed by Alenia Spazio) assuming very low quality factors (20 for the suspensions of
the PGB and 500 for those of the test masses) and including drag disturbances as well as
implementation errors. Using forces of the right intensity, and allowing for removal of
the initial unbalance, the work by Alenia Spazio shows that no problems arise due to the
fact that the damping force is provided by spinning actuators. There is no indication
whatsoever that the damping force needs to be amplifyed by a factor 
if applied in
the rotating frame as stated in the ESTEC Report.
Numerical simulations of the GG system do not yet include the small movable rods, pivoted at their centres on flat elastic gimbals, to which the GG test masses are suspended. As of our present understanding it is possible that, due to the differential oscillations of the bodies, they will develop only conical whirling motions, and no cylindrical whirling motion. If so, they will not require any active stabilization, as conical whirls are naturally damped. If a closer analysis will show that this is not the case, they can easily be stabilized similarly to all other bodies of the GG system. Obviously, in introducing them with a non-zero mass in the numerical simulations they will also be given the appropriate values for the principal moments of inertia which will best suit their stabilization. The possibility of adjusting the ratios of the principal moments of inertia in a body of cylindrical symmetry is well known.
Conservation of energy and angular momentum in the presence of spinning actuators definitely demonstrates that whirling motions in weakly coupled rotors (such as GG) can be stabilized by means of extremely small forces, far smaller than the spring forces and never competing with them. It should also be stressed that the fastness of the electronics is not an issue because, although the electrostatic dampers must actuate at a frequency close to the spin frequency, the whirling motion to be damped is much slower, so that corrections and adjustments are possible over several spin periods. Furthermore, this electronics is the same for all the GG bodies, and as a matter of fact it is also similar to the electronics needed for drag-free control with FEEP thrusters. In no way it can be regarded as a critical issue in the space experiment.
Far more important it is to stress the fact that the major source of dissipation in the
GG experiment, namely the springs, dissipate energy at the frequency of spin, not at the
frequency of whirl. This is true only because of the fast rotation as compared to the
small natural frequency of oscillation provided by the weak mechanical coupling. This
means that thermal noise will be dominated by the quality factor of the suspension at , not
at the much smaller natural frequency. This Q is bound to be very high, making
thermal noise small and the integration time short, in spite of the fact that the
experiment is run at room temperature. The advantage of weak coupling and fast spin is
apparent once more.
(Anna Nobili- nobili@dm.unipi.it)
