\Galileo Galilei" (GG)

Test of the Equivalence Principle with a Small Spinning Satellite:The Stabilization of its Weakly Coupled Masses

A. M. Nobili, D. Bramanti, G. Catastini

Gruppo di Meccanica Spaziale, Dipartimento di Matematica, Universit�a di Pisa, Italia

A. Anselmi, S. Portigliotti

ALENIA SPAZIO SpA, Torino, Italia

A. Lenti, G. Volpi

LABEN, Milano, Italia

S. Marcuccio

Centrospazio, Pisa, Italia

Abstract

Galileo Galilei (GG) is a project for a small,non-cryogenic, satellite mission aimed at testingthe Equivalence Principle to 1 part in 1017. Thekey novel features of the experiment are that thesignal of a possible violation of equivalence ismodulated at relatively high frequency by spin-ning the read out capacitance sensors, togetherwith the test bodies and the entire satellite, soas to substantially reduce the impact of \1/f"noise, and that the GG spinning bodies are cou-pled by extremely weak, high quality, mechanicalsuspensions so as to be sensitive to tiny di�eren-tial forces and to make use of the self-centringe�ect in weakly coupled rotors. Constructionand testing of a ground prototype is underway atLaben laboratory in Florence. A Pre-Phase-AStudy of GG has been carried out by the italianspace industries Alenia Spazio and Laben incollaboration with the proposing scientists underASI (Agenzia Spaziale Italiana) funding [1]. Inthis paper we focus on the issue of how the GGweakly coupled rotors can be stabilized withouta�ecting the expected sensitivity.

1 Introduction

The scienti�c objective of the Galileo Galilei

(GG) small satellite mission is to test a fun-damental \Principle" of modern Physics |the Equivalence Principle (EP) formulated byEinstein, generalizing Galileo's and Newton'swork| to 1 part in 1017, the best ground experi-ments [2] with laboratory bodies having found noviolation to 1 part in 1012. Testing the Univer-sality of Free Fall, that Galileo pioneered at thebeginning of the 17th century, is the most directexperimental test of the Equivalence Principle, in

fact not a Principle but a basic property of grav-itation, the �rst discovered and yet most intrigu-ing physical interaction. Unlike all other tests ofGeneral Relativity, which check its consequencesand predictions, an experiment on the Equiva-lence Principle tests the foundations of GeneralRelativity and of all metric theories of gravityalike, it probes the very essence of gravity andits uniqueness among the fundamental forces ofNature. A high accuracy, unquestionable, ex-perimental result on the Equivalence Principle|whether it is violated or con�rmed| will be acrucial asset for the Physics of the next century.

The crucial advantage for EP testing in anEarth orbit is that the driving signal due to thegravitational �eld of the Earth (long range) isgiven by the entire value of its gravitational ac-celeration. For a spacecraft orbiting at 520 kmaltitude this amounts to GM�=a

2 ' 840 cm=s2

(G is the universal constant of gravity, M� themass of the Earth and a the orbital radius ofthe satellite) as opposed to a maximum value of' 1:69 cm=s2 due to the �eld of the Earth at 45�

latitude on the ground, and ' 0:6 cm=s2 due tothe �eld of the Sun. Another main advantageof space is not so much the absence of seismicnoise, but weightlessness: the gravitational at-traction of the Earth is largely compensated bythe centrifugal force due to the orbital motion ofthe spacecraft so the main 1 g ' 103 cm=s2 localacceleration of gravity is absent. On board theGG spacecraft the largest acceleration is abouta factor 108 smaller than 1 g, which means thatit is possible to suspend 100 kg there with thesame suspensions that one would use on Earthfor 1milligram.

The GG satellite is shown in Fig. 1 (and itssection through the spin axis in Fig. 2). It hascylindrical symmetry, it is small (1m by 1:3m)

1

Figure 1: The GG small spacecraft with bar antennas (one foldable); solar cells are shown on the left(1m diameter, 1:3m height). Total expected mass is 250 kg.

and stabilized by single axis rotation around theaxis of maximum moment of inertia. The spinrate is about 5Hz and the spin axis is close to thenormal to the orbit plane (precise values are notrequired). Since it is highly desirable to avoid at-titude manoeuvres and to make the satellite sta-bilization totally passive, the inclination of theorbit plane over the equator of the Earth shouldbe close to zero. An almost circular, almost equa-torial low altitude (' 520 km) orbit is the currentbaseline. There are no strict requirements on theorbit injection parameters. The angular phase ofthe spacecraft and its rotation rate are measuredby EES (Earth Elevation Sensor). Essentially noactive attitude control is required.

In Sec. 2 we recall the main features of theGG test of the Equivalence Principle and howthe signal is modulated at the spin frequency of5Hz (' 3 � 104 times higher than the orbital fre-quency). A general presentation of the mission isgiven in [3]. Details on the GG experiment, thecalibration and balancing procedure, the analysisof all known perturbing e�ects and the expectedsensitivity can be found in [1]. In the remainingof this paper we concentrate on dissipation in themechanical suspensions of the GG bodies, on theslow instabilities that this dissipation gives riseto, and on how these can be actively controlledand the system stabilized so as to allow high ac-curacy EP testing.

2 High Frequency Signal

Modulation

In GG the Equivalence Principle is tested by test-ing the Universality of Free Fall for two concen-tric hollow cylinders of di�erent composition inthe gravitational �eld of a given source mass (the

Earth). Even in space a high sensitive test re-quires to be sensitive to extremely small relativedisplacements. For test bodies orbiting aroundthe Earth the signal is always directed towardsthe centre of the Earth, hence it is necessarilyat the orbit frequency: �orb = 1:75 � 10�4Hz atthe altitude of GG. Low frequency signals are af-fected by \1/f" noise which seriously limits theintegration time; furthermore numerous perturb-ing e�ects, both gravitational and non gravita-tional, do act at orbit frequency too. The an-swer to this problem is to modulate the signal ata higher frequency, the higher the better. Onecould rotate the sensors only (capacitance plates)located in between the test masses to detect therelative displacements, or, which is simpler, ro-tate the whole system, masses plus capacitanceplates, with symmetry/spin axes perpendicularto the orbit plane (see Fig. 3). In addition, ifthe rotors are weakly coupled, weakly enough torespond by an appreciable amount to a small dif-ferential force such as the one of an EP violation,their centres of mass will oscillate one with re-spect to the other at their natural frequency; theweaker the coupling, the larger the displacement.This would happen also in absence of spin. Whatthe spin of the sensors provides is a modulationof the signal at the spin frequency. The sensorsdetect a time changing relative distance of theform:

�x = �xEP cos(!st+ �EP ) � F (1)

where �xEP is the expected displacement dueto EP violation (constant in value), !s is thespacecraft spin angular velocity (with respectto the centre of the Earth), �EP is the knownphase of the EP violation signal, and the fac-tor F = cos � + sin � cos (!orbt+ �) depends onthe angle � between the spin axis of the satellite

Figure 2: Section through the spin axis of the GG spacecraft.

Figure 3: Section across the spin axes of the test bodies with their centres of mass displaced by adistance �xEP because of an Equivalence Principle violation in the �eld of the Earth. The centres ofmass are �xed in their diplaced con�guration while the bodies rotate independently around O1 and O2

respectively. The relative displacement vector always points towards the centre of the Earth and it istherefore detected by the capacitors at their spinning frequency with respect to the Earth.

and the orbit normal, and is maximum (F = 1)for a spin axis perpendicular to the orbit plane.Fast rotation has other important advantages:many disturbing e�ects give only DC output sig-nals, and temperature perturbations, dissipationand thermal noise, are reduced. Mechanical sus-pensions allow the test masses to be electricallygrounded, hence eliminate the major source ofelectrostatic perturbations which can seriouslya�ect small force gravitational experiments.

The elastic properties assumed in the base-line dynamical model determine the natural fre-quency for di�erential oscillations of the testmasses to be ' 1:2 � 10�2 rad=s. This value iscon�rmed by numerical simulations. From this,the displacement to be expected in response toan EP violation (di�erential) signal at the level of10�17 is obtained. We get �xEP ' 5:8�10�11 cm,which can be detected by the capacitance sensorslocated halfway between the test bodies [1].

2.1 Compensation and/or Rejec-tion of Air Drag Perturbations

The residual atmosphere at the satellite orbitingaltitude, and the radiation from the Sun, act onthe outer surface of the GG spacecraft but not onthe test bodies suspended inside (Fig. 2). Sincegyroscopic e�ects are negligible (due to the veryhigh spin energy of all the GG rotors), these non{gravitational e�ects appear as inertial accelera-tions on the centres of mass of the bodies, equaland opposite to the acceleration acquired by thespacecraft. Most of the e�ect (along{track) is at

the satellite orbiting frequency around the Earth;smaller, low frequency variations are also to beexpected.

It is easy to check that the air drag perturbingacceleration along{track, as well as the accelera-tion due to solar radiation pressure, are largerthan the expected signal by several orders ofmangnitude. However, unlike the signal (whichis di�erential), these perturbations are in princi-ple common mode, i.e. they would give no dif-ferential force provided that the two masses wereequal and identically suspended. Since identicalsuspensions are not realistic, a good strategy isto mechanically couple the test masses and bal-ance them like in an ordinary balance, so as toreject common mode forces leaving only a muchsmaller di�erential e�ect to compete with the sig-nal. This procedure is known as Common ModeRejection (CMR). However, the smaller the non{gravitational force, the easier will be the balanc-ing and the better the CMR achieved. It is there-fore desirable, although not a conditio sine quanon, that a space mission to test the EquivalencePrinciple be, to some extent, drag{free, i.e. ca-pable to partially compensate for the e�ect ofnon gravitational forces such as air drag and so-lar radiation pressure. Indeed, originally GG wasdesigned as a non drag-free satellite; it also acco-modated three pairs of test masses rather thanone [4].

In GG drag compensation is performed withFEEP (Field Emission Electric Propulsion)thrusters. Their advantages for high precisionmissions in Fundamental Physics devoted to the

detection of very small forces, are numerous:high speci�c impulse, negligible amount of pro-pellant (few grams of Caesium for a few monthsmission duration), no moving parts, �ne electrictuning and consequent high level of proportion-ality. By comparison, He thrusters (whose pro-portionality is ensured mechanically rather thanelectrically) would need, for the same purpose, avery large amount of propellant whose perturb-ing e�ect on the experiment becomes a problemin itself.

2.2 Abatement of Platform Noise

The signal of an EP violation for test bodies or-biting around the Earth would be at their orbitalfrequency. With the capacitance sensors (in factthe whole GG satellite) spinning at 5Hz withrespect to the Earth the EP signal is detectedat their spinning frequency. So is the perturb-ing signal from the main along-track componentof the residual atmospheric drag, because it alsoacts at the orbital frequency of the spacecraftaround the Earth. As for the e�ects due to airdensity uctuations at low frequencies, they willbe close to the signal. These perturbations, aswell as the signal, are not attenuated by mechan-ical suspensions because their frequencies are toolow even for a very weak suspension in absenceof weight. Low frequency non-gravitational per-turbations can be reduced either by active dragcompensation or by Common Mode Rejection, orboth; as recalled in Sec. 2.1, in GG we rely ona combination of both active drag compensation,by means of FEEP thrusters, and Common ModeRejection by the balanced suspension of the testmasses [1].Instead, vibrational noise at the spin frequency

of the sensors (or close to it), i.e. noise whichacts at 5Hz w.r.t. the Earth, hence at 10Hzor DC in the rotating frame, will be e�ectivelyattenuated by mechanical suspensions. We dothat by locating the test bodies inside a lab-oratory (that we call Pico-Gravity-Box, PGB)mechanically suspended inside the spacecraft bymeans of weak helicoidal springs. As seen in the�xed frame, the system is transparent to DC andlow frequency e�ects (like the signal, the residualatmospheric drag and its low frequency uctua-tions...) but is very e�cient in attenuating vibra-tional noise above its threshold frequency, par-ticularly around the spin frequency. The trans-fer function of the system, when viewed in therotating frame, shows a sharp peak of value 1at 5Hz (Fig. 4), meaning that the system isperfectly transparent at the signal frequency. In

this way the signal is not a�ected, in amplitude,by the spin; also the low frequency drag e�ects(of the �xed frame) are up-converted to high fre-quency with no ampli�cation (and no reductioneither, of course). The only di�erence, and in-deed big advantage, with respect to the non ro-tating case being that the detecting instrumentswork much better at higher frequency. So thesharp peak at 5Hz is a key feature of GG. Buthow can the peak at 5 Hz be so sharp? Sim-ply thanks to the fact that the PGB providesgood attenuation at its sides, at lower and higherfrequencies, i.e. around 10Hz and 0Hz (w.r.tthe rotating frame). This means good attenua-tion of perturbations which are at 5Hz w.r.t the�xed frame. The need to attenuate these per-turbations should not be neglected. Althoughin space we obviously don't need a motor, wecannot forget that the FEEP thrusters will actat about 5Hz (to reduce the main along tracke�ect of drag at the orbit frequency, and alsoits low frequency components). Since the FEEPthrusters compensate low frequency drag e�ectswhile spinning at 5Hz, any mismatches and im-perfections in their �ring will give rise to space-craft perturbations at 5Hz w.r.t. the �xed frame(hence at 10Hz and 0Hz in the rotating frame).Sonic noise of the GG spacecraft structure ispeaked at much higher frequency, but a tail atthe spin/signal modulation frequency should notbe excluded, and will be attenuated.

Fig. 4 shows the transfer function (in the ro-tating frame) for a quality factor Q (see Eq. (2)below) of the PGB suspensions of 100. It isworth to point out that a Q value higher thanthis could be obtained, were it necessary in orderto reduce noise around the spin/signal modula-tion frequency (so as to increase the sharpness ofthe peak shown in Fig. 4). The lower Q value ofthe PGB suspension springs (as compared to thesprings which suspend the test bodies; see Sec.3.1), is due to the fact that they have to carrywires and their insulation; experimental mea-surements of the dissipation (in the ground testfor horizontal oscillations at ' 5Hz) in the PGBprototype springs, carrying the required numberof wires and insulation, have yielded Q ' 90.An alternative solution is possible with helicoidalsprings made of separate wires insulated at theclamping; these would have a lower dissipationand therefore a higher Q.

The weak mechanical suspensions of the PGB,which are possible only thanks to weightlessness,provide an e�ective, passive means of isolationfrom the (relatively) high frequency vibrations

Non rotating system, Q=10

5 Hz rotation, Q=20, supercritical rotating damping

5 Hz rotation, Q=100, supercritical rotating damping

10−4

10−3

10−2

10−1

100

101

102

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

Frequency (Hz)

Noi

se r

educ

tion

fact

or

Transfer function of the GG s/c − PGB system in the rotating frame

Figure 4: Transfer function of the GG spacecraft-PGB system in the rotating frame. If the spin rate isput to zero (dashed curve) the transfer function goes back to the one typical of a non rotating system(with Q = 10 in this case). In supercritical rotation two curves are plotted for two di�erent values ofQ (20 and 100, top and bottom curve respectively). It is a peculiarity of supercrtical rotation that thehigher the Q, the better the noise reduction. The peak with value 1 at the spinning frequency showsthat the passive noise attenuator does not reduce vibrations at very low frequency w.r.t. the �xed frame,particularly the DC ones; the observer corotating with the system sees these DC perturbations as 5Hz,and �nds that the attenuator does not reduce them, or better that it is transparent to 5Hz e�ects.Perturbations which are seen at 5Hz by the non rotating observer (and attenuated), have frequencies0Hz and 10Hz for the body �xed observer, and in fact he too �nds that they are attenuated.

around the spin/signal modulation frequency.

3 The GG Bodies as Weakly

Coupled Rotors

If the concentric spinning test cylinders of theGG experiment were free, totally unconstrained,rotors, each one would spin smoothly with thecentre of mass perfectly aligned on the spin axis,and the centre of mass would respond freely toany applied force, albeit small. A constrained ro-tor is indeed very close to a free one provided itsspin angular velocity !s is larger (possibly muchlarger) than its natural frequency of oscillation!n (\supercritical" rotation). Which amounts tosaying that the mechanical suspension is weak,possibly very weak; as it is indeed the case inspace thanks to weightlessness. In this case thecentre of mass of the rotor reaches equilibriumbyaligning itself very accurately on the spin axiswith an extremely small rotation radius; moreimportantly, if external forces are applied to it,it responds simply by moving to another equilib-

rium position while still spinning around its ownaxis.

\Supercritical" rotation is commonly used onthe ground for rotating machines to exploit thee�ect of self alignment. If the centre of massof the suspended body is located, because ofinevitable errors in construction and mounting,with an o�set ~� from the rotation axis, equilib-rium is established on the opposite side of theunbalance vector ~� (�xed in the rotating frame)with respect to the rotation axis, where the cen-trifugal force due to rotation and the restoringelastic force of the spring equal each other. Itcan be shown that this happens at a distancefrom the spin axis smaller than the original un-balance by a factor (!n=!s)2. In space, due tothe absence of weight (i.e. small !n) this ratiocan be very small. In GG it is about 10�6. Thus,it is a physical characteristic of the system thatan equilibrium position exists, �xed in the ro-tating frame, very precisely aligned on the spinaxis. In the presence of a force the equilibriumposition changes but the body continues to spinaround its own axis in the displaced position. If

Figure 5: Mathematical model of a rotor made of two bodies, each of mass m, coupled by weak springs.The coupling constant is k and the natural frequency of oscillation !n. Both bodies are spinning at thesame angular velocity !s around their respective centres of mass O1 and O2. In turn, O1 and O2 arewhirling around the centre of mass O of the whole system, at a distance rw from it, and at the naturalangular velocity !n. In the GG case !n ' 10�3!s.

the rotor is made, as in the GG case, by fourconcentric cylindres all weakly suspended, eachof them will have its own rotation axis.

3.1 Dissipation, Whirl Motionsand Destabilizing Forces

No matter how good is the mechanical qualityof the suspensions and how accurately they areclamped, as the system rotates they will undergodeformations and, in this process, will dissipateenergy. The only energy that can be dissipatedis the spin energy of the rotor, which means thatthe spin angular velocity must decrease. As aconsequence, the spin angular momentum alsodecreases, and since the total one must be con-served, the motion of the two bodies one aroundthe other (in the same direction as the rotationalmotion) will grow in amplitude. Which is whatis observed, and is referred to in the literatureon Rotordynamics as whirling motion. Frictionbetween rotating parts of the system (i.e. fric-tion inside the suspension springs) is the physi-cal cause of the whirl motion, and it is referredto as rotating damping. Except in the case of ro-tating machines with very viscous bearings, therotor whirls at essentially its natural frequency(with respect to a �xed frame of reference). InGG there is no motor, there are no bearings, no uids, no oils, no greases; only carefully clampedsuspensions of high mechanical quality (partic-ularly for the test bodies) which undergo onlyminute deformations. Hence, the GG bodieswhirl at their natural frequencies of oscillation.

In GG it is important that the test cylinders(of equal mass and di�erent composition) be veryweakly coupled in the plane perpendicular to thespin axis, so as to be sensitive to tiny di�erentialforces. The mathematical model typically usedin Rotordynamics literature to describe the sys-tem is shown in Fig. 5, where rw is the radiusof whirl of each body around the common centreof mass which, for the purpose of the presentdiscussion on whirl motion, is shown to coin-cide with the equilibrium position (\perfect" selfalignment).

Can the whirl motion be damped and the ro-tating system be stabilized?

If there is nothing else in the system but ro-tating damping there is nothing to prevent theamplitude of the whirl motion from growing, andtherefore the system is unstable. In rotating ma-chines on the ground whirl motions are usuallydamped by non-rotating damping, namely by suf-�cient friction occurring between two parts ofthe system, both non-rotating as shown in Fig.6 (i.e. between two parts of the non-rotatingsupports, for example friction between the non-rotating part of the bearings and their �xedsupports). This friction generates non-rotatingdamping forces which are e�ective in dampingtransversal translational oscillations of the ro-tor's axis of rotation (such as the whirl motions),and they do so without slowing down its rota-tion. In the ground rotors in which non{rotatingdamping is provided by tipping the non{rotatingpart of the bearing in oil, this is essentially vis-cous damping. Non-rotating friction should not

Figure 6: Sketch of a ground rotating machine showing where rotating damping, non-rotating dampingand friction in the bearings are localized. The di�erent roles they play in the dynamics of the systemare discussed in Sec. 3.1

be confused with friction in the bearings, alsoshown in Fig. 6. This is the friction (mostly vis-cous) between the rotating body and the non ro-tating parts, which is obviously e�ective in slow-ing down the rotor but almost completely inef-fective at damping whirling motions (and alsoat producing them). An important advantage ofthe GG space experiment is the absence of bear-ings, hence of bearing friction at all. From aphysical viewpoint the most important charac-teristic distinguishing the e�ects of rotating andnon{rotating friction on one side from the e�ectsof friction in the bearings on the other is thatthe former produce forces on the rotor, while thelatter produces torques. They are therefore es-sentially independent from one another and in-teract only to a second order, namely because ofconstruction errors, asymmetries, misalignmentsetc. From this it follows that if one were in factusing the forces generated by friction in the bear-ings in order to stabilize the whirling motions(rather than the forces due to non-rotating fric-tion), he would inevitably need extremely largeforces.

In the GG space experiment where there are nonon{rotating parts an equivalent non{rotatingdamping must be provided by an active controlsystem of sensors and actuators �xed with therotating bodies. Before any such device can bedesigned, it is obviously necessary to establishthe magnitude of the forces which destabilize thesystem and which will therefore need to be coun-teracted actively. The actual implementation ofthe active control forces in the real space mis-

sion, with realistic errors in all components of thecontrol system, can only be faced once the mag-nitude of the destabilizing forces has been �rmlyestablished. In turn, this requires to �rmly es-tablish the amount of energy losses in the GGrotating system. The only components wherethere can be losses are the mechanical suspensionsprings and the active dampers themselves (inGG the dampers are electrostatic). Since lossesin the dampers depend in turn on the magnitudeof the control forces they are required to provide,we must �rst evaluate the mechanical losses inthe springs. All the other parts are rigid and nolosses are expected in them.

It is very important to establish at which fre-quency the mechanical suspensions do dissipateenergy. If we consider the two body model ofFig. 5, the only frequencies involved are the spinfrequency !s (of each body around its own cen-tre of mass) and the frequency of whirl (of thetwo bodies around their common centre of masswith a radius of whirl rw), which is equal to thenatural frequency of oscillation !n. In GG theratio !s=!n is very high (' 103); thus, if we ne-glect !n with respect to !s, it is apparent fromFig. 5 that the suspension springs are draggedat the spin frequency through a \corridor" of un-even width. As a consequence, in their own ref-erence system, the springs contract and expandwith amplitude 2rw at the spin frequency andtherefore dissipate energy at this frequency [5].It can be easily shown that, in the general casein which !n is not negligible compared to !s,energy is in fact dissipated at frequency !s�!n.

Mechanical losses in the GG suspensionsprings can therefore be measured experimen-tally by setting them in oscillation at the fre-quency at which the GG spacecraft will spin,with a clamping similar to the one to be usedin the space mission, in realistic conditions oftemperature and vacuum. The (adimensional)quantity to be measured is the quality factor Qof the suspensions at the relevant frequency [5];this is routinely done by monitoring the decreasein time of the amplitude of oscillation, just recall-ing that the timescale for its exponential decay isQ=(��), � being the relevant frequency at whichlosses occur (see Eq. (2) below).Helicoidal springs very similar to the ones that

can be used to suspend the GG test bodies dur-ing the space mission have been manufactured inCuBe and their Q at oscillation frequencies closeto 5Hz has been measured, at room tempera-ture and in vacuum, in the laboratory of Labenin Florence. Care has been taken in consideringtransversal oscillations in a horizontal plane, sothat they are not a�ected by the local gravity,like in space. The Q values obtained so far arebetween 16; 000 and 19; 000.Since energy losses in the mechanical suspen-

sions are responsible for the onset of the whirlmotion (in order to conserve the total angularmomentum of the system) (see Fig. 5), the whirlmotion (at the natural frequency !n w.r.t. the�xed frame) is nothing but an oscillatory motionof growing amplitude, i.e. with a negative qualityfactor equal and opposite to the measured qual-ity factor Q of the suspensions. Hence, the ra-dius of whirl will grow, from a given value rw(0),according to the equation:

rw(t) = rw(0)e!nt=(2Q) (2)

That is, the larger the Q, the slower the growthof the whirl instability. For large Q, as in thiscase, we can write:

��rw

�Tn

r'

�

Q(3)

where��rw

�Tn

is the increase in the amplitudeof whirl in one natural period of oscillation Tn.For the GG test bodies, taking the measuredvalue Q ' 16; 000 and having Tn ' 2m, the am-plitude of the whirl motion will need about oneweek to double: such a very slow growth of thewhirl instability is obviously very important formaking the control forces required to damp itvery small.The increase in amplitude

��rw

�Tn

(during

one natural period) is due to an increase of the

along-track velocity of the bodies, which in turnis caused by an average destabilizing accelerationad, also along-track, such that:

1

2adT

2n ' 2�

��rw

�Tn

; ad '1

Q!2nrw (4)

which means, an average destabilizing force(along-track) of magnitude

jFdj '1

Qm!2

nrw (5)

Since Fc = m!2nrw is the centrifugal force, equal

and opposite to the elastic force of the springFspring = �krw, the destabilizing force whichgenerates the unstable whirl motion depicted inFig. 5 turns out to be

jFdj '1

QjFspringj (6)

i.e., only a fraction of the elastic spring force [5]:the higher is the Q of the suspensions (at the spinfrequency) the smaller is the destabilizing forcewhich needs to be damped in order to stabilizethe system.Note that the quality factor Q in Eq. (6) ex-

presses the total losses in the system, regardlessof their physical nature, e.g. structural (alsoknown as hysteretic) or viscous, and also regard-less of where in the system they occur (e.g., inGG, inside the suspension springs and at theirclampings). Once the Q has been measured ex-perimentally, at the relevant frequency and inrealistic conditions, this gives the total dissipa-tion in the system and is therefore the value tobe entered in Eq. (6) in order to get the inten-sity of the destabilizing force. Were the mea-sured Q of the system to be interpreted as if en-tirely due to viscous dissipation, the correspond-ing value of the viscous quality factor would haveto be, according to current models which give aratio of !s=!n for the structural-to-viscous Q,correspondingly larger. In GG this would be 103

times larger than the measured values, meaninga very small viscous dissipation. Indeed, thisresult had to be expected because GG has nobearings, no uids, no oils, no greases; it hasonly carefully clamped suspensions of high me-chanical quality (particularly for the test bodies)which, moreover, undergo only minute deforma-tions. Therefore, viscous losses must be veryvery small. Nonetheless, they are included inthe value of the quality factor measured exper-imentally. Losses in the electrostatic dampersthemselves have not been measured yet, but the-oretical estimates show that they are negligi-ble [1, 4, 6].

It must be pointed out that in the analysis ofGG performed at ESTEC [7] the destabilizingforce is taken to be a factor !s=!n ' 103 largerthan in Eq. (6); this amounts to assuming theGG system as dominated by viscous damping,which is ruled out by the values of Q measuredin the laboratory. Q values of 16; 000 to 19; 000,as obtained for the total losses in the helicoidalsuspensions of the test masses, clearly show thatviscous losses {if any{ must be very very small,and therefore the assumption made in [7] is in-correct.The following question becomes relevant at

this point: how much energy (per unit time) isgained by the (destabilizing) whirling motion asfraction of the energy lost by the spinning rotor?Let us consider the rotor in the two body modelof Fig. 5. The spin energy of the rotor is:

Erotor =1

2I!2

s (7)

with I the total moment of inertia of the two bod-ies with respect to the spin axis (perpendicularto the plane of the Figure). The energy (kineticplus elastic) of the whirl motion, at the constantnatural frequency !n with respect to the �xedframe and with radius of whirl rw is:

Ew = 2m!2nr

2w (8)

The time derivatives of Ew and Erotor are:

_Ew = 4mrw!2n _rw ; _Erotor = I!s _!s (9)

From the conservation of angular monetum it ispossible to relate _!s to _rw. The total spin angu-lar momentum of the rotor is:

Lrotor = I!s (10)

The angular momentum of the whirl motion is:

Lw = 2mr2w!n (11)

Since the total angular momentumhas to be con-served it must be:

_Lrotor + _Lw = 0 (12)

from which, since !n is constant, it follows:

_!s = �4mrw!n

I� _rw (13)

Using Eq. (13) for _!s, we get from (9):

_Ew

_Erotor

= �!n!s

(14)

which is a very important result. In the GGcase, where the frequency of the whirl motionis very small compared to the spin frequency(!n ' 10�3!s), Eq. (14) tells us that the energygained by the whirling motion is one thousandtimes smaller than the energy lost by the rotor.All the rest, that is 1�(!n=!s) ' 99:9%!!, is dis-sipated as heat inside the springs; which meansthat it is not transferred to the (destabilizing)whirl motion. In simple terms one can say thatin \supercritical" rotation the energy balance isessentially between the rotor and the springs; therotor looses spin energy and the springs dissipatealmost all of it as heat: the faster the spin, thelarger is the energy dissipated inside the springs,as it can be seen from Eq. (9). On the otherhand, the springs do not enter at all in the bal-ance of the angular momentum; the onset of thewhirl motion is inevitable for the total angularmomentum to be conserved, but the energy itgains from the rotor is only the small fractiongiven by Eq. (14). So, the idea one might havethat the faster the spin (as compared to the nat-ural frequency) the higher the energy gained alsoby the whirl motion (by which argument the GGsystem would be highly unstable), is proved tobe incorrect. As a consequence, also the objec-tion raised in [7] that the GG system should bemuch more unstable than ground based rotatingmachines due to the smaller ratio !n=!s (by oneorder of magnitude or more) does not hold, be-cause the fraction of destabiling energy is alsocorrespondingly smaller, as shown by Eq. (14)

3.2 Active Stabilization and Con-trol with Rotating Dampers

All the GG bodies spin at the same rate. Thereare no non-rotating parts, and therefore therecan be no non-rotating friction to damp the whirlmotions. They must be damped actively, withdampers necessarily �xed to the rotating bodies.We have shown (Eq. (6)) that the destabiliz-ing forces are only a small fraction of the pas-sive spring forces, which in turn are very smallbecause the suspensions are designed to be ex-tremely weak (k ' 10 dyn=cm), as it is in factpossible in space in spite of the fact that the testbodies have masses of 10 kg each. Small capaci-tance plates (see Fig. 7), with surfaces of 1 cm2,turn out to be su�cient to provide the requiredactive forces.

Let the whirling motion be damped using elec-trostatic sensors/actuators �xed to the rotatingbodies. The capacitors are required to provide a

Figure 7: Four capacitance plates, at 90� from one another, rotate with the system at angular frequency!s. They provide an electrostatic force F in order to prevent the growth of the whirl motion of thebodies (at a frequency !w equal to the natural one !n). The reaction of the active force on the plate hasa small tangential component fa which spins up the rotor by transferring to it the angular momentumof whirl, which would otherwise grow. The �gure is not to scale; the distance 2rw between the centresof mass O1 and O2 of the two bodies is in reality many orders of magnitude smaller than R.

force at the whirling frequency !n while spin-ning at angular frequency !s; therefore, theymust actuate at frequency !s � !n. By pro-viding forces internal to the system they can-not possibly change its total angular momentum:they can only transfer the angular momentum ofwhirl to the rotation angular momentum of therotor by spinning it up. This is what happensif they are made to provide a stabilizing force ofthe same intensity as the destabilizing one givenby Eq. (6) (in fact a slightly larger one). Thisforce must always act along the vector of relativevelocity of the centres of mass of the bodies intheir whirling motion, as seen in the �xed frameof reference. Since the centres of mass of thebodies are displaced from the centre of mass ofthe system by an amount rw (see Fig. 7), theelectrostatic plates will necessarily apply also asmall force fa tangent to the surface of the rotoramounting to a fraction rw=R (R ' 10 cm is thelinear dimension of the rotor) of the the activeforce F , of intensity F ' (2=Q)jFspringj (for bothbodies), which will damp the relative velocity ofwhirl (see Fig. 7). Of the corresponding reac-tion components on the electrostatic actuatorsonly the reaction to the small tangential com-ponent fa ' (1=Q)jFspringj(2rw=R) will produce

a non zero angular momentum by spinning upthe rotor at the expense of exactly the angularmomentum of whirl:

faR '1

QjFspringj�2rw '

1

Q2m!2

nr2w = _Lw (15)

which will therefore increase the spin angularmomentum of the rotor Lrotor = I!s in sucha way that the total angular momentum of thesystem is conserved. That is:

I _!s '1

Q2m!2

nr2w (16)

thus producing a spin up of the rotor at the rate_!s given by Eq. (16). By integrating _!s for theentire duration of the mission Tmission = tf �ti, from intial to �nal epoch, the ratio !f=!i of�nal{to{initial spin angular velocity of the rotoris obtained:

!f!i

= 1 +!iQ

2mr2wI

!2n

!2i

Tmission (17)

In the case of the GG test bodies, taking the(smallest) measured value of 16; 000 for the qual-ity factor Q at the spin frequency of 5Hz, rw '10�6 cm and !2

n=!2spin ' 2:5 � 10�6 we get, for a

6-month duration of the mission:

!f!i

� 1 ' 10�15 (18)

The corresponding angular advance is:

�� '1

2Q

2mr2wI

!2nT

2mission ' 4 � 10�2 arcsec

(19)which is clearly negligible. The amount of spinenergy gained by the rotor at the end of the mis-sion is vanishingly small:

�Erotor ' 2 � 10�15Erotor ' 2 � 10�6 erg (20)

The corresponding (negative) quality factorQrotor for the spin up of the rotor (in absenceof any external disturbances to its spin rate) isobviously huge (in modulus):

Qrotor = �2�Tmission

Ts

Erotor

�Erotor' �2:5 � 1023

(21)(Ts is the spin period). The amount of spin an-gular momentumgained by the rotor in 6 monthsis, similarly to the spin energy, vanishingly small:

�Lrotor ' 10�15Lrotor ' 6 � 10�8 g cm2 s�1

(22)The contribution to the energy of the system bythe electrostatic dampers is obtained by comput-ing the work done by both components (Fa andfa, shown in Fig. 7) of the active force theyprovide. While only fa will transfer angular mo-mentum from the whirl motion to the rotor, bothcomponents of the active force will provide en-ergy. It can be easily demonstrated that fa sup-plies the rotor with the energy necessary to in-crease its spin rate, while the energy suppliedby Fa balances exactly the energy dissipated bythe springs as heat, which would otherwise spindown the rotor.The capacitors must also act as sensors in or-

der to recover, from relative position measure-ments, the linear relative velocity of the whirlingmotionwhich needs to be damped to stabilize thesystem; or, for example (see Fig. 7), two of thecapacitors can be used as sensors and the othertwo as actuators. In GG the capacitors spin (to-gether with the bodies) at a frequency about 103

times larger than the frequency of whirl. Is itpossible, by means of rotating sensors, to recoverthe slow relative velocity of whirl as with non-rotating sensors? This question can be answeredin a totally general way. The answer wouldbe \No", if the rotating observer had no othermeans of \looking outside" to measure indepen-dently (i.e. by means of some other instrument)

its own phase and rotation rate; the answer is\Yes", if the rotating observer can do that, whichis the case in GG using the Earth Elevation Sen-sors. An independent knowledge of the phaseand the rotation rate of the sensors makes it pos-sible to subtract from the measurements of therotating sensors their own velocity of rotation,hence to recover the (much slower) relative ve-locity which would be obtained with �xed sen-sors. The problem amounts to that of comput-ing numerically a time derivative, because thecapacitors measure relative displacements whilethe electrostatic active dampers require to knowthe relative velocity of the bodies in the �xedframe. If the velocity were computed by takingsuccessive measuremets of the sensors and com-puting their di�erence, it would be dominated bythe rotation velocity, i.e. it would be larger thanthe velocity to be damped by a factor about 103

(i.e. the ratio spin{to{natural velocity). If thensuch a velocity output were used to drive theelectrostatic actuators which must damp it, therequired force would be (correspondingly) about103 times larger than necessary. Clearly this pro-cedure would not be appropriate for a systemlike GG where the spin{to{natural velocity ratiois very large. The analysis of GG performed atESTEC [7] has been carried out following thisprocedure; as a result, their �nal value of theactive control forces has been oversized by an-other factor 103. In addition to the 103 factorby which the same authors have overestimatedthe magnitude of the destabilizing forces to bedamped, this further ampli�cation of the activecontrol forces brings their total oversizing to afactor of 106, thus making their analysis inappli-cable to the GG system.

Instead, the problem of reducing the magni-tude of the active control forces close to thatof the destabilizing ones (given by Eq. (6)), issolved as follows. A reference signal is built froma 5Hz oscillator (clock) synchronized by the EESoutput averaged over many spin periods of thespacecraft. Since the spin period is 0:2 s, it cancertainly be considered constant over time inter-vals of the order of many times the spin perioditself. Note that the reference signal is conti-noulsy synchronized to the output of the EarthElevation Sensors so that any error in their mea-surement of the spin rate will not build up. Thereference signal is needed in order to performthe transformation from the rotating to the �xedframe (and back) as accurately as possible. A rel-ative displacement of two bodies due to the grow-ing unstable mode is measured by the rotating

capacitance sensors. Instead of taking succes-sive measurements of the sensors and comput-ing their di�erence, the di�erence is computedbetween measurements which are 1 spin periodapart, using the reference signal. The sensor sig-nal is sampled a certain number of times per spinperiod (e.g. 10 to 20), at regular intervals, andfor each sampled point the di�erence is computedwith respect to the sampled point 1 spin periodlater. In this way the rotation velocity of thesensors is subtracted from their measurements;a best �t to the sampled data points (each oneis a relative position di�erence after 1 spin pe-riod) gives the relative velocity vector ~v betweenthe two bodies in the �xed frame. To this veloc-ity corresponds a whirling motion of amplituderw ' v=!n, and then, taking into account Eq.(5), we can obtain the required stabilizing force,antiparallel to the velocity ~v,

~Fstab = �1

Qm!n~v (23)

This force will prevent the whirl motion fromgrowing, i.e. it will maintain the system in asteady condition. To actually damp a whirlingmotion one has to apply a slightly larger force:for instance, a force twice as large will dampthe whirling motion at the same rate at whichit would grow in the absence of ~Fstab.

Were the capacitance sensors perfect, thiswould be enough to stabilize the GG bodies, as itcan be also checked with numerical simulations.However, the sensitivity of the sensors in realistic ight conditions has to be taken into account be-cause, being the growth rate of the whirl instabil-ities very slow, the relative displacement duringone spin period is so small to be dominated bythe noise of the sensors. As a result, the recov-ered velocity vector is also very noisy, and muchlarger than its actual value in the �xed frame.If the actuators were programmed to damp thevelocity vector recovered in this way, the result-ing active force would be driven by the noise,i.e. it would be much larger than the force whichdestabilizes the system. Such a controlled sys-tem would be dominated by the active controlforces rather than by the very low sti�ness pas-sive mechanical suspensions. This is in contrastwith the basic physical design of the GG experi-mental test of the universality of free fall wherebythe test bodies must be very weakly coupled soas to be sensitive to tiny di�erential forces.

A better active control can be devised in whichactive damping is not applied until the (slow) rel-ative velocity of the whirl motion of the bodies

has been recovered from the noise after appro-priate averaging. To do this, the relative velocityvector is averaged over a few spin periods (typ-ically a fraction 1=100 of one period of whirl),and then �tted to the whirl period (the naturalperiod of oscillation) in order to reconstruct itsrotation in the �xed frame. Since the growthrate of the instability is slow, this averaging pro-cedure can be carried on over a large numberof whirl periods (particularly for the test bod-ies), thus making the determination of the ve-locity to be damped more and more accurate.Active control starts only once the small rela-tive velocity of the bodies w.r.t. the �xed framehas been reconstructed and separated from thenoise. Indeed we have found that a velocity vec-tor obtained from averaging over 10 spin periodsonly (which amounts to a small fraction of onewhirl period), if used to drive the dampers withno further averaging/�tting, already reduces theintensity of the control force by one order of mag-nitude. The control forces provided by the elec-trostatic dampers can be further tuned to ad-just for remaining phase errors. Again, such ad-justments are successful because the instabilitiesgrow very slowly: the dampers spin very fast (1turn every 0:2 s), but the instabilities take sev-eral days to grow (for the test bodies), so plentyof time is available for the active control forces toachieve stabilization by counteracting the phys-ical destabilizing forces very accurately, withoutovershooting. Which is very important, becauseit means that active forces only slightly largerthan the destabilizing ones can stabilize the sys-tem, no e�ort being wasted in attempting todamp ill determined relative motions.

Although the GG bodies can be stabilized bysmall active control forces, it is very important toensure that the application of these forces doesnot a�ect the expected sensitivity of the EP test-ing experiment. If the stabilizing forces wereapplied perfectly antiparallel to the actual rel-ative velocity vector of the bodies, they wouldnot change their relative position of equilibrium,hence they would not a�ect the EP experimentat all. However, there will be a phase error, andtherefore also a component of the active controlforce which will perturb the equilibrium positionthus a�ecting the EP experiment. In our esti-mates of these perturbations we take 1:5 deg forthe phase error and 10�6 cm for the minimal dis-placement (r.m.s) that can be sensed by the ca-pacitance sensors; note that this �gure refers tothe small capacitors used for active control, notto those of the read out system, which are lo-

cated in between the test bodies, are much moresensitive. The largest stabilizing forces are thoseapplied to the PGB laboratory inside which thetest bodies for the EP experiment are located.This is due to the lower Q value of the PGBsuspension springs (as compared to the springswhich suspend the test bodies), as discussed inSec. 2.2. Although an alternative solution is pos-sible to get a higher value of Q, with helicoidalsprings made of separate wires insulated at theclamping, this is not crucial as far as the per-turbations from the PGB control forces on thetest bodies are concerned. Since the test bodiesare suspended inside the PGB laboratory, anyforce applied to the PGB is the same for boththe test bodies, that is, it is a common modeforce. And since the test bodies are arrangedin a balanced suspension in order to reject com-mon mode forces (to 1 part in 50; 000 in the cur-rent baseline [1]), this means that only a fraction1=50; 000 of the PGB control forces will a�ect thetest bodies di�erentially and therefore competewith the expected signal. It can be checked eas-ily that such a perturbation is well below the ex-pected signal. As for the perturbation due to theactive forces which stabilize the test bodies them-selves, since each one is stabilized with respectto the PGB laboratory (as envisaged above), theperturbation is di�erential (rather than commonmode) and therefore competes directly with theexpected signal. With Q = 16; 000 (the lowervalue measured for the suspension springs of thetest masses), and the errors given above for thecapacitance sensors and the phase, this perturba-tion too is below the expected signal. It is there-fore concluded that not only all the GG bodiescan be stabilized, but also that such stabilizationis compatible with the scienti�c goal of the mis-sion to test the Equivalence Principle to 1 partin 1017.

In essence, the driving principle of activedamping in GG is not to activate the dampersuntil the (small) relative velocity of the bodieshas been reconstructed from the noisy measure-ments of the rotating sensors through appropri-ate averaging. Otherwise, active forces drivenby a poor analysis of the sensors measurementsdo themselves generate instabilities, instabilitieswhich do not exist in the real system and whichin turn would require large forces to be damped.The active control of the GG bodies as describedin [7] does just that, thus transforming the sys-tem into a di�erent one entirely dominated bythe active control forces. In contrast, our nu-merical simulations show that it is possible to im-

plement control laws in which the magnitude ofthe control forces is of the order of the value de-rived theoretically (Eq. (6)) for the forces whichdestabilize the system. This analysis is carriedout with the DCAP [8] software package devel-oped at Alenia Spazio and now running also atthe University of Pisa.In addition, numerical simulations con�rm the

expected linear behaviour of the GG system.This is in agreement with what is reported inthe literature on Rotordynamics. From [9] wecan quote: "Most of the components that com-prise a rotordynamic system can be quite accu-rately modeled as linear. The strongly non linearcomponents of the system, if present, are usu-ally localized in the sense that they are directlycoupled to only a small number of the systemcoordinates. Such components include uid-�lmbearings, squeeze-�lm dampers, piecewise linearsupports (e.g. deadbands, rubs) working uid in-teraction..". We can also quote from [10]: "..The system as a whole may however be stronglynon linear because of the behavior of the bearings,in the case of uid-handling machines, becauseof the non linear uid mechanics in the bladepassages and in the seals. Both uid-�lm bear-ings and rolling elements bearings severely limitthe total relative radial displacement of the rotat-ing journal with respect to the stationary bear-ing. Nevertheless, the dynamic characteristicsof the bearing reactions change dramatically overthis small range of permitted displacement so thateven very small radial displacements of the rotorintroduce nonlinearities". GG has neither bear-ings nor any of the recognized sources of non lin-earities mentiones above. This means, in partic-ular, that no jumps phenomena or bifurcation ofthe solutions occur as the system parameters arevaried, and that the method of superposition isvalid, whereby all excitation sources need not tobe considered simultaneously.

4 Conclusions

Attempts at modulating the EP signal at highfrequency are almost as old as EP experimentsthemselves. The GG modulation frequency is byfar the highest attempted so far (by several or-ders of magnitude). The underlying feature isthat of weakly (mechanically) coupled test bod-ies rotating much faster compared to their nat-ural periods of oscillations. The azimuthal sym-metry and fast spin reduce thermal e�ects so thatthe experiment can be performed at room tem-perature.

The speci�c issue of active stabilization of suchweakly coupled rotors has been brought up in theESTEC analysis [7] of GG as a very critical issue:having derived the need for control forces over-sized by a factor of about 1 million this analysiscouldn't but conclude that the GG proposed ex-periment was totally inadequate for testing theEquivalence Principle. This conclusion has beenendorsed by the Fundamental Physics AdvisoryGroup (FPAG) of ESA (chaired by M. Jacob),which made it the key element of its resolutionabout GG [11] (points 2,3 and 5 of the resolu-tion). The scienti�c analysis of this controversialissue carried out by members of the GG ScienceTeam, whereby the GG system can be stabilizedwith control forces 1million times smaller thanin [7], was made available to the FPAG [12, 13];the opinion expressed by a leading scientist inthe �eld of Rotordynamics was also made avail-able [5].Theoretical analysis, numerical simulations

and experimental measurements have con�rmedso far the validity of the GG concept for highaccuracy EP testing (1 part in 1017 being thetarget); all e�orts are now being focused on theGGG ground test at the laboratory of Laben inFlorence. Once accounted for the smaller drivingsignal on Earth, and for the di�erent coupling ofthe test bodies, the target signals to be detected{on the ground and in space{ are comparable.As a result, key features of the space experimentcan be validated on the ground. Among them,the stabilization of the whirl motions with ro-tating electrostatic dampers and the rejection ofcommonmode forces by mechanical balancing ofthe test bodies. We have no doubt that a valida-tion of the GG concept on the ground is going tomake the case for a small satellite mission muchstronger.

5 Acknowledgments

We are grateful to Bernard Sacleux (ONERA,France), Vladislav Sidorenko (Keldysh Institue,Russia), Giancarlo Genta (Politecnico di Torino,Italy), Erseo Polacco and Piero Villaggio (Uni-versity of Pisa, Italy) for interesting discussionson dissipation and stabilization in the GG sys-tem. Special thanks are due to Stephen H Cran-dall (MIT, USA), a worlwide recongized expertin the �eld of Rotordynamics, for devoting a con-siderable amount of his time to a careful analysisof this controversial issue. We acknowledge �-nancial support from ASI (Agenzia Spaziale Ital-iana), contract ARS/96/171.

References

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[2] Y. Su, B.R. Heckel, E.G.Adelberger., J.H.Gundlach, M. Harris, G.L. Smith and H.E.Swanson, Phys. Rev. D 50, 3614 (1994).

[3] A.M. Nobili, D. Bramanti, G. Catastini, E.Polacco, A. Milani, L. Anselmo, M. An-drenucci, S. Marcuccio, A. Genovese, F.Scortecci, G. Genta, E. Brusa, C. Del Prete,D. Bassani, G. Vannaroni, M. Dobrowolny,E. Melchioni, C. Arduini, U. Ponzi, F.Curti, G. Laneve, D. Mortari, M. Parisse,F. Cabiati, E. Rossi, A. Sosso, G. Zago, S.Monaco, G. Gori Giorgi, S. Battilotti, L.D'Antonio and G. Amicucci, J. Astronau-tical Sciences 43, 219 (1995).

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[5] S.H. Crandall and A.M. Nobili,On the Stabilization of the GGSystem (1997), available on line:http://adams.dm.unipi.it/~catasti/crandall/crandall.html

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[7] J.W. Cornelisse, Y. Jafry, M. WeinbergerTechnical Assessment of the GALILEOGALILEI (GG) Experiment, ESTEC (1996)(We refer only to the part on dissipation andactive control written by Y. Jafry and M.Weinberger)

[8] DCAP (Dynamics and Control AnalysisPackage), ALENIA SPAZIO DATA PACK-AGE, ESA Contract No. 7971/88/NL/JG,Release 7.0, (1994)

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Chapter 2 in Handbook of Rotordynamics,F.E. Ehrich Ed., McGraw Hill, (1992)

[10] S.H. Crandall Rotordynamics, pp. 1{44in Nonlinear Dynamics and StochasticMechanics, W. Kliemann and N.S. Na-machchivaya Eds., CRC Press, Boca Raton,Florida (1995)

[11] FPAG(96)4, ESA, October 9 (1996)

[12] D. Bramanti, A.M. Nobili and G. Catas-tini Stabilization of Weakly CoupledRotors: A General Derivation of theRequired Forces (1996), available online: http://adams.dm.unipi.it/~catasti/pap1/pap1.html

[13] G. Catastini, A.M. Nobili and D. Bra-manti Passive Vibration Isolation in aSpinning Spacecraft (1996), available online: http://adams.dm.unipi.it/~catasti/pap2/pap2.html