ABSTRACT
Test masses coupled by weak mechanical suspensions are
sensitive to differential forces such as the force due to a possible violation of the
Equivalence Principle (EP). If in addition they are put in rapid rotation, the
differential signal is modulated at high frequency, which is beneficial for noise
reduction. GALILEO GALILEI (GG) is a proposed space experiment for testing the Equivalence
Principle to 1 part in 1017 based on
these concepts. Paper [1] claims that GG can only
reach 10-14. We show that the analysis of [1] is flawed (by several orders of magnitude)
because of two misconceptions: one on the physical nature of mechanical damping and the
other on active control methods for the stabilization of spinning bodies.
1. INTRODUCTION
Paper [1] has
been devoted to the GG space experiment [2-4] addressing
the issue of the stabilization of whirl motions that weakly coupled rotors are known to
develop because of non zero dissipation between rotating parts of the system. The
conclusion of [1] is that the required stabilizing forces overcome by far
the weak passive forces of the mechanical suspensions (springs) on which the GG experiment
relies, thus making it inadequate for a very high accuracy EP test. We show that [1] is affected by two serious misconceptions which invalidate in full its
conclusions: a misconception on the physical nature of mechanical damping (Sec. 2) and a misconception on the active control of spinning bodies
(Sec. 3).
GG is a small satellite project
aiming at testing the Equivalence Principle to 10-17 with concentric hollow
test cylinders in rapid rotation around their symmetry axes. The test bodies are suspended
and coupled by very weak mechanical suspensions; the corresponding frequencies of natural
oscillations are much smaller than the spin frequency. The experiment is run at room
temperature; the spacecraft is spin-axis stabilized and no active attitude control is
needed. In the current version, non gravitational forces acting on the spacecraft surface
are largely compensated by FEEP ion thrusters needing only a few grams of propellant for
the entire mission duration. If the test bodies fall differently in the field of the Earth
because of an EP violation their centres of mass will show a relative displacement,
pointing to the centre of the Earth, whose amplitude depends on the stiffness of the
differential mechanical coupling. Such mechanical displacement is transformed into an
electric potential signal via a capacitance read-out system whose plates are located
halfway in the gap between the coaxial test cylinders. Since the plates spin with the
system (at @ 5Hz), the signal is modulated at this frequency. In the
original torsion balance experiments by Eötvös the signal was DC. Subsequent experiments
with better results (finding no violation to level of @ 10-12) were based on modulation frequencies at least 5
to 4 orders of magnitude smaller than that proposed in GG: 24 h in [5,6] and 2 h in [7]. As of this writing,
GG is 1 of 6 projects selected and funded for Phase A Study by the Italian Space Agency
(ASI) [8]. Information on GG is available on the Web [9].
2.
MISCONCEPTION ON THE PHYSICAL NATURE OF MECHANICAL DAMPING
GG is constructed of rigid bodies
coupled by weak suspensions of high mechanical quality (particularly those of the test
bodies) which moreover undergo only minute deformations (a few mm at most). The
suspensions are carefully clamped so as to avoid parts sliding one against the other,
which is the main cause of mechanical losses in the clamps and in the whole system. There
are no bearings, since, after spin up is completed, there is no need of a motor. There are
no viscous materials: no fluids, no oils, no greases. Therefore the main loss factors
(inverse of quality factor Q) are those due to the very small internal dissipation of the
mechanical suspensions as they undergo minute deformations at the spin frequency. The only
other cause of dissipation are the electrostatic sensors/actuators used to damp the whirl
motions, since all other parts are rigid and have no losses. Calculation of thermal noise
in the active dampers shows that the corresponding losses are by far negligible compared
to those achievable with mechanical suspensions [4,10]
(assuming all parameters as for the GG experiment and a very conservative value of 10 for
the electric quality factor). Crandall [11] has
calculated (using [12]) the back-reaction force on the
plates of the capacitors from the high-frequency measurement voltage, finding that the
electrical contributions to the mechanical stiffness and damping are negligible. Losses in
the dampers will be measured with the GG prototype on the ground after active rotating
control, similar to the GG active control, has been implemented [13,
Chap. 3].
A firm estimate of the losses in the
GG mechanical suspensions requires them to be measured experimentally, by setting the
springs in oscillation under realistic operating conditions (oscillation frequency,
vacuum, temperature, clamping); note that there is no need to perform this measurement
with the system rotating [14]
In order to measure, for a given
mechanical system, the quality factor Q (defined as the ratio of the total energy stored
in the system to the amount of energy dissipated in one cycle) the system is made to
oscillate and then the oscillation amplitude A(t) is recorded as it decays with time. Q
can also be defined as follows:
(1)
where w is the frequency of the oscillation
and A(0) its amplitude at initial time. Hence:
(2)
which yields the value of Q
from measurements of A1, A2 at times t1, t2. Consider a
helical spring with its (unavoidable) clamping and the attached mass necessary to obtain
the oscillation frequency of interest. Oscillations performed in the horizontal plane
avoid pendulum-like motion due to local gravity which would yield a higher Q because
gravity contributes to the total energy but not to the dissipation. In vacuum (@ 10-5 torr) at room
temperature and for oscillation frequencies from 2 to 10 Hz, the measured Q values of the
prototype springs manufactured for the suspension of the GG test masses were between
16,000 and 19,000. Oscillations were excited with a small electromagnet and their
amplitudes were measured optically [15, 9].
Although further improvement is possible, these values are quite good because of how the
suspensions are made: they are helical springs carved out of a single piece of material (Cu-Be) by electroerosion in 3-D,
followed by an appropriate thermal treatment. The Q measurement procedure (by recording
the decaying oscillation amplitude) is a standard one, which obviously does not require
the system to be taken into space, even though in this case it is designed for use
in space, no scaling is necessary either.
Energy is dissipated because of
different types of losses (structural or viscous, in the spring material as it undergoes
deformations, because of imperfect clamping or because of resistance of residual air) and
the oscillation amplitude decay is due to all of them. Consequently, the measured Q
is the Q of the whole system and gives a quantitative measurement of all losses in
it: whatever their physical nature. Once dissipation has been measured
experimentally, model-dependent estimates of it are no longer needed and, in any case,
should be consistent with experimental results. To the contrary, speculations in [1, Appendix] that dissipation in the GG system should be amplified by a
factor ws/wn (the ratio spin-to-natural
frequency; @ 103 in GG) over the measured
value are proven to be wrong by experimental measurements.
The dissipation discussed above --in
the springs and their clamping as they are deformed at the frequency of spin, referred to
as "rotating damping"-- is known to give rise to unstable whirl motion at the
natural frequency wn with respect to the non-rotating frame. If Q quantifies all losses
at the spin frequency, the fractional variation of the radius of whirl rw in one natural period of
oscillation Tn =2p/wn is:
(3)
In GG the ratio ws/wn is 630 for the test masses and 1600 for
the PGB (Pico Gravity Box) suspended laboratory inside which the test masses are in their
turn suspended. The time-scales for doubling the whirl radius are 2.5 weeks for the test
masses (with Q=16,000) and 2.5 h for the PGB (with Q=90), that is the whirl motions grow
very slowly, which makes it easier to keep them under control and to damp them.
The forces required to damp
the slow whirl motions are (in modulus) slightly larger than the destabilizing forces
which give rise to the whirl, whose value is known to be smaller than the passive spring
forces by a factor 1/Q (see e.g. [16], Eq. (35)), where Q
quantifies all losses at the frequency of spin and must be measured as discussed
above. Hence, the required stabilizing force, anti-parallel to the slow velocity of whirl 
, is
(slightly larger than):
(4)
where m is the reduced
mass of the system. Because of the misconception on the nature of damping, [1]
erroneously gets the time-scales of whirl
motion to be a factor ws/wn shorter (5.6 s for the PGB and 0.7 h for the test masses) and the forces (4) a
factor ws/wn larger; hence, also the effects of imperfections and errors in these forces are
amplified by the same factor.
3.
MISCONCEPTION ON ACTIVE CONTROL METHODS FOR THE STABILIZATION OF SPINNING BODIES
The GG bodies are stabilized
actively, by means of small capacitance sensors/actuators rotating with the system at a
velocity @ 103 times higher than the velocity of whirl they are required
to damp. In order to recover and damp this slow (and slowly growing) velocity with much
more rapidly rotating sensors/actuators we have developed a control strategy [15,
17,13] in which:
the relative velocity of the
bodies is computed from differences of measurements taken by the rotating displacement
sensors 1 spin period apart
the relative velocity is averaged
over several spin periods ( @10)
the relative velocity data are
best-fitted to a vector rotating at the known angular frequency of whirl.
A reference signal at the spin frequency (@ 5 Hz) is
constructed continuously (so as to avoid accumulation of errors; averaging over a few
minutes) from the output of commercial Earth Elevation Sensors which measure the angular
phase (hence also the spin rate) of the spacecraft. Note that for a time interval as short
as the whirl period the rotation of the system (whose spin energy is very large) can be
regarded as constant. Instead, in [1] the relative velocity is computed
by taking differences of successive measurements from the sensors and without making use
of the reference signal. In this way they fail to recover the correct value of this slow
relative velocity since it is overwhelmed by the much larger velocity of spin of the
sensors themselves (by a factor ws/wn @103). As a consequence, also their control forces are a factor ws/wn larger than in (4).
This is strikingly apparent already in the simple case of the two-body system made of the
GG outer spacecraft and the PGB when the two control strategies are compared (Figure 1).
Figure 1. Trajectory of the relative motion of the centers of mass of the GG
outer spacecraft and the PGB in the plane perpendicular to the spin axis in a 2-body model
(coupling constant 0.02 N/m, Q=90). The Y axis is pointed to the center of the Earth,
hence the largest effect of the residual atmospheric drag, assumed of 5× 10-9
N, is a constant displacement along the X axis (of @ 0,08 mm); its second harmonic (assumed 40% of it) appears in
this system as a variation at the orbital period (5,700 s). This is the dashed circle,
showing -in both plots- the stationary state that the system would reach if the whirl
motion were perfectly damped. The plot on the left is obtained with the control laws of
the GG Team assuming the following errors:
initial bias of 10 mm linear and 1° angular; fractional error in spin rate measurements Dws/ws=10-4; offset (by construction and mounting) of
10 mm; errors in the
capacitors of 0.1 mm RMS. Whirl oscillations with the natural period of 314 s
(around the points of the dashed circle) and of decreasing amplitude are apparent as the
system is brought to its stationary state in 8,000 s only. Note that at this point the
relative distance of the two centers of mass is below 5 Å . These results have been
obtained independently using DCAP software package (of Alenia Spazio) and Matlab. The plot
on the right shows, for the same system, but under much more ideal assumptions (perfect
knowledge of spin rate; perfect centering of the rotor; an initial linear bias of 1m m and
no angular bias; an error in the sensors/actuators 10 times smaller, i.e. of 10-2
mm) the results obtained by applying the control laws proposed in paper
[1]. It is apparent that even in a much more favorable situation the same
system has been unwittingly transformed into one dominated by very large active forces for
which there is in fact no need, as the plot on the left demonstrates. Note that the
dissipation has been assumed to be the same in both cases (Q=90), hence failure to
stabilize the whirl motion (right hand plot) has to be ascribed only to the control laws
implemented in that case. Regarding the plot on the left note that the assumptions for the
various error sources are conservative. For instance, small capacitors like those designed for GG can be shown in the laboratory to be sensitive to relative
displacements of 10-2 mm.
Since the control laws used in [1] fail so heavily already in the simpler 2-body model, they are certainly
useless for the scope claimed in the paper, that is to evaluate the sensitivity achievable
in EP testing by the full 6-body GG system.
How the full GG system (4 bodies plus 2 small coupling arms) is
stabilized by controlling all whirl motions at the same time is shown in Figure 2. The resulting relative distance between the test bodies is
shown in Figure 3, while Figure 4 gives the
intensity of both the passive elastic force and the control force.
Figure 2. Evolution of the full 6-body GG system: outer spacecraft,
PGB, two test masses with two gimballed coupling arms. These arms are pencil-like in shape
and have no rings. As individual bodies they would be unstable; in GG they couple two much
more massive test bodies which are individually stabilized by whirl control, hence, also
the arms are found to be automatically stabilized with no need of adding rings around
their midpoints. Only the trajectories of the PGB and of one test mass are plotted (for
9,500 s after the first 2.500 s) showing their distance from the center of mass of the
spacecraft (0,0). The plane of the Figure is perpendicular to the spin axis and the Y axis
is pointed to the center of the Earth. The residual drag acting on the spacecraft has a DC
component equal to 5 × 10-9 N (giving rise to a constant X displacement in
this plot) plus an orbital frequency term which is 40% of the DC component (giving rise to
the dashed circles) and a 10% noise on both components. Whirl motions appear as oscillations
at the natural periods around the points of the corresponding dashed circles: if active
control is effective their amplitude must decrease. This is indeed what happens. Here we
have assumed Q values of 90 for the PGB and 500 for the test masses (a very conservative
assumption for the test masses, since the measured Q for their suspensions is of 16,000¸19,000). The errors
included were: 10-2 mm RMS (tested in the laboratory for capacitance plates of @ 2 cm2 as in
the GG active dampers), 10 mm linear bias, 1° angular bias for the capacitance sensors; Dws/ws = 10-4 RMS for the Earth Elevation Sensors
(doable with EES by "Officine Galileo", Firenze); 1 mm initial offset of the
suspension springs.
Figure 3.
Numerical simulation of the same 6-body GG
system as in Figure 2, with the same residual drag and the same error
sources. Here we plot the relative displacement between the inner and the outer test mass
(differential displacement) as obtained after applying active control of their whirl
motions. Note that in this simulation whirl control is always on, i.e. this is a worst
case simulation because whirl control can in fact be switched off during scientific data
acquisition. This result is impressive in that it shows how active control by means of
electrostatic sensors/actuators can be so accurate as to make the GG macroscopic test
bodies self-center on one another as expected in supercritical rotation in absence of
dissipation (infinite mechanical quality factor, zero-whirl).
Figure 4.
Numerical simulation of the 6-body GG system
with the same error sources as in Figures 2 and 3.
Here we plot the passive elastic force of the suspension springs (in red) for comparison
with the control force acting on the outer test mass (in black). The control force is
clearly much smaller than the elastic force. We recall that, in order to speed up the
simulation, the quality factor of the test bodies suspension springs was taken to be 500
(4 times worse than measured); in addition, the system was controlled with a force 11
times larger than the minimum required theoretically by (4). In point
of fact, we have also run experiments in which a control force only 2.5 times larger than
the minimum could stabilize the system.
Figures 2¸ 4 refer to planar simulations; simulations in 3-D have been carried out
[13, Chap. 6] showing that the dynamical behavior is not affected by the
increased number of degrees of freedom; however, the required computing time increases
significantly.
As for the effect of drag (and of
solar radiation pressure), it is huge compared to the expected signal; however, it is
transferred to the test masses as an inertial acceleration in common mode by its nature,
while an EP violation signal would be differential. This is why the GG test cylinders are
arranged in a coupled suspension similarly to an ordinary beam balance (except for the
fact that the beam is vertical rather than horizontal): by adjusting the length of the
arms with piezoelectric actuators common mode forces can be rejected leaving only a much
smaller differential effect to compete with the signal. In physics experiments this is
known as Common Mode Rejection; the attainable level of rejection depends on the
specific system and mechanism for rejection. With the prototype of GG in the laboratory we
have achieved a rejection level of 1 part in 200,000, which is better than the current
requirement for the GG experiment in space [13] where we assume that
drag is partially compensated (by drag-free control with FEEP mini-thrusters), and
partially rejected. Drag could also be totally rejected (no compensation) [4]
provided the rejection level is improved accordingly. The drag-free control of GG is based
on a notch filter at the orbital frequency [13, Chap. 6]; it has been
tested in combination with whirl control for the full 6-body GG system also in 3-D. No
additional difficulties are encountered in the 6-body case as compared to the 2-body
model, but the computing time required by the simulations is much bigger.
4. CONCLUSIONS
We have shown that paper [1] overestimates the
required stabilizing forces of the GG system by a factor ws/w n @103 because of a misconception on the physical nature of mechanical damping. In
addition, it overestimates the active control forces to be applied by rotating
sensors/actuators by another factor ws/wn because of a
misconception on the control laws of spinning bodies. Overall this amounts to an error by
a factor @106. This invalidates in full the evaluation of GG as carried out in [1], according to which GG could only reach a sensitivity in EP testing of 1 part in 1014.
Paper [1] is the final version of a
precursor technical report [18] prepared by the same authors in support
of the Fundamental Physics Advisory Group (FPAG) of ESA for its evaluation of GG [19]. Therefore, we can also answer a few questions raised in [19]. In particular: (i) [19] states that "The high
spin rate is not an advantage for the experiment. The advantages conveyed by spin
(suppressing the effects of low-frequency noise) are outweighed by the disadvantages of
having unstable modes around the signal frequency." Instead, unstable modes can be
stabilized and they are so slow that scientific data acquisition can take place while
whirl control is off, hence the advantages of high spin rate can be fully exploited; (ii) [19] states that "The servo forces will dominate the passive spring
forces." Instead, we have implemented control forces which are by far smaller than
the elastic forces and yet can stabilize the whirl motions (Figures 4,2,3); (iii) [19] states that the
gimballed rods (the coupling arms) "..appear to be highly unstable in high-speed
rotation and are a source of significant perturbations." Instead, numerical
simulations of the full GG system show that this is not the case (further details in [13, Chap. 6]), confirming the physical guess made by the GG Team before a
full simulation could be carried out; (iv) [19] states that "The
control forces have to mimic damping forces in the non-rotating frame but must be
synthesized from measurement in the rotating frame. Imperfections in the sensors and
actuators will cause significant disturbances in the differential mode." The first
statement is true, but the second one has been found to be wrong if control forces are
properly computed and applied; which is not the case in [18, 1]. Another
issue raised in [19], that of the usefulness of the PGB laboratory has
not been touched here; it has been answered in [15;13, Fig. 2.6].
REFERENCES
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219-42
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[7] Su Y et al 1994 Phys. Rev. D 50 3614-3636
[8] "Italy selects finalists for satellite
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http://tycho.dm.unipi.it/nobili/ggproject.html
[10] Nobili A M, Catastini G and Bramanti D 1997
"GG: Dissipation by the Electrostatic Dampers" http://tycho.dm.unipi.it/nobili/ggweb/qdampers/qdampers.html
[11] Crandall S H, 1997 private communication
[12] Braguinski V et Manoukine A. 1976 "Mesure de
petites forces dans les expériences physiques" Editions Mir, Moscou
[13]GG Phase A Report, ASI (1998) http://tycho.dm.unipi.it/nobili/ggweb/phaseA/index.html
[14] Crandall S H and
Nobili A M 1997 "On the Stabilization of the GG System" http://tycho.dm.unipi.it/nobili/ggweb/crandall/crandall.html
[15] Nobili A M et al
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[16] Crandall S H 1970
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[17] Addendum No. 1 to
GG Pre-Phase A Report, ASI 1997 http://tycho.dm.unipi.it/nobili/ggweb/addendum/addendum.html
[18] Cornelisse, J. W., Y. Jafry and M. Weinberger
"Technical Assessment of the GALILEO GALILEI (GG) Experiment, ESTEC (here reference
is only to the part on dissipation and active control due to Jafry and Weinberger), 1996
[19] ESA, Fundamental Physics Advisory Group, FPAG(96)4,
1996 http://tycho.dm.unipi.it/nobili/ggweb/referee/fpag.html
