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3. Self-centering in supercritical rotation

An experiment which aims at testing the equivalence principle must be capable to detect tiny relative displacements of the test masses with respect to one another. However, spurious relative motions would appear, because of gravity gradients, were the centres of mass of the test bodies not accurately centred on one another. This is why the test masses must be concentric, as they are in STEP. In GG their cylindrical shape is not only a construction advantage; it is clearly dictated by the symmetry of the one axis rotation. The crucial question is: how, and to what accuracy, is mass centering obtained? The answer comes from a careful exploitation of weightlessness, which makes the mechanical system (spacecraft, PGB and test masses) a rotor in supercritical rotation with the spin frequency  much larger than the natural oscillation frequency  of the suspended test masses (and the PGB laboratory as well). Since the end of last century such rotors are known to have an equilibrium position very close to the rotation axis, which is pivotal in reducing the otherwise destructive effects of centrifugal forces in high-speed machines such as turbines, centrifuges and ultra-vacuum pumps (see Whitley (1984) for a review). In simple terms, the rotor tends to spin around its centre of mass, i.e. it behaves more like a free rotor rather than a constrained one. If the centre of mass of the suspended body is located, by construction, at a distance  from the rotation axis, equilibrium is established on the opposite side of  with respect to the rotation axis, where the centrifugal force due to rotation and the restoring elastic force of the suspension equal each other. It can be shown that this happens at a distance from the spin axis smaller than the original unbalance  by a factor  (see, e.g., Ch. 6 of Den Hartog, 1985). Thus, at equilibrium, the distance  of each suspended test mass inside the GG satellite from the spin axis - and therefore from one another - is:

Since the pioneer work of Gustaf De Laval about a century ago this relationship has been widely demonstrated in both theoretical and experimental work on high speed rotors. It shows that space offers an important advantage, because in absence of weight the natural frequencies of suspended bodies can be very low, about  smaller that the spin frequency in this case. For an original unbalance  this means that the equilibrium position is only  away from the spin axis. It is important to stress that this equilibrium position slightly displaced from the rotation axis is fixed in the rotating frame of the spacecraft while the signal is modulated at the frequency of spin. Possible imperfections on the surfaces of the bodies would also give a DC effect. The actual direction of the miscentering in the rotating system depends only upon the location of the unbalance and is of no importance for the experiment. Perturbations such as air drag and solar radiation pressure acting on the external surface of the spacecraft produce a nongravitational acceleration of its centre of mass. In the reference frame of the spacecraft the bodies will therefore be subject to inertial forces in the opposite direction, which will move the masses to new displaced positions of equilibrium (along the direction of the perturbation) where the perturbation is balanced by the restoring force of the spring. It is worth noticing that, because of the supercritical state of rotation, the displaced body will always spin around its own axis, which means that no centrifugal force due to the spin will result because of this displacement. The only centrifugal forces due to the spin come from the  miscentering given by Eq. 3 and are balanced by the restoring force of the suspension springs.

It is however well known that any rotating system operating in the supercritical regime is unstable - owing to its internal damping - unless an adequate "non-rotating damping" is applied to it, that is damping caused by friction of the non rotating - or slowly rotating - parts of the bearings (on which the spinning shaft is mounted) against their supports. See Bramanti et al. (1996), Nobili et al. (1996) for an analysis of the various types of friction. Since the spacecraft as a whole is rotating, there is no way of obtaining the required non-rotating damping except by increasing the complexity of the system with the introduction of a fixed or slowly rotating portion of it. A simpler solution is to place in between the various masses active elements able to simulate the behaviour of a non-rotating damper. In the language of automotive active suspension technology the device can be defined as a "skyhook damper" since it acts as a damper which in a way follows an inertial reference frame. In Section 3.1 we show the main properties of supercritical rotation and the rôle of internal and non-rotating damping in a simplified mathematical model of two masses connected with springs; then report the results of a finite element numerical simulation of the system in the 3-chamber setup of Fig. 1 , giving all the unstable whirling modes of the system and showing how they are damped (Section 3.2 ). The electrostatic damper is discussed in Section 3.3 .

Unlike GG, the STEP experiment has been based on the fact that its attitude be actively maintained fixed with respect to inertial space as well as drag-free (in angle) to a level compatible with the requirement on orbital drag-free, which turns out to be a very demanding requirement. In this way, with a considerable effort in accurate active control, an EP violation signal from the Earth would appear at the satellite orbital frequency (). However, since the signal frequency is the frequency of orbital motion around the Earth it is bound to be also the frequency of a number of dangerous perturbing effects (e.g from the South Atlantic Anomaly and from the on board Helium used to make the experiment cryogenic). More recently it has been proposed that the STEP spacecraft be spinning too, but only slightly faster than its orbital revolution around the Earth. The benefits of fast satellite spin could not be incorporated either in the first proposed rotating experiment (Chapman & Hanson, 1970). In that experiment centrifugal forces remain a major limitation because the test bodies are constrained to move along one diameter of the rotating platform, and it is well known that any such rotating system is always strongly unstable above the critical speed (Ch. 6, p. 228 of Den Hartog, 1985).

3.1. Simplified mathematical model

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Fig. 5. A mathematical model of two axially symmetric bodies of masses  and  coupled by springs of stiffness k and  are the inertial and the rotating plane respectively. z and  are the complex variables in the two planes.

Let us first study the system in the simplified model of two axisymmetrical rigid bodies of masses  and , length , moments of inertia  and  with respect to their symmetry (polar) axis and  with respect to any transversal axis x, y. The bodies are coupled by two identical springs, each of radial stiffness k as shown in Fig. 5 . By introducing the complex coordinates  and , where  and  are the rotation angles around the y and x axes (the minus sign allows to simplify the metrical form of the equations), the equation of motion for the lateral dynamics of the system is (Section 4.6 of Genta, 1993):


is the vector of the generalized coordinates of the system (subscripts 1 and 2 distinguish the two bodies) and  are known as the mass, gyroscopic and stiffness matrices:

Because of possible construction errors each body will have the centre of gravity located a distance  away from its rotation axis, and the symmetry axis tilted by an angle  with respect to its rotation axis. These unbalances will result in the forcing terms at the right hand side of the equation of motion (Eq. 4 ) which contain the vector 

where  defines the direction of the vector  in the rotating frame and  is the phase angle of the couple due to the unbalance. In the general case of the Jeffcott rotor (Jeffcott, 1919) the system is subject to nonconservative forces (damping forces) which can be of two kinds: either of fixed direction in the inertial frame (non-rotating damping) or of fixed direction in the rotating frame (rotating damping). The latter occurs in the parts of the system which spin at speed  (e.g. the springs), while the former is linked with the nonrotating parts of the machine. They are usually expressed as matrices  and  respectively, with  the total damping matrix of the system. In the case of GG the entire system is spinning and therefore there is no non-rotating damping, i.e.  and . The matrix  is given by:

where c is the damping coefficient. For internal hysteretic damping the value of c, for the translational modes, can be approximated as , with Q the quality factor and .

Because of symmetry Eq. 4 can be split into two different sets of uncoupled equations, one for the translational modes

and one for the rotational modes

Let us consider the equation for the translational dynamics of the system (Eq. 7 ). Assuming a solution of the type  for the free whirling, the characteristic equation of the homogeneous system is

whose solutions (i.e. the complex frequencies of the system) are 0 and the solutions of the equation:

which is the characteristic equation of a Jeffcott rotor with mass  in the absence of non-rotating damping. It is well known that this device is unstable at all speeds exceeding the critical speed (see, e.g., 4.8.3 of Genta, 1993)

Thus, without non-rotating damping operation in supercritical regime (i.e. above the critical speed) is not possible because the rotor is necessarily unstable. From a more physical point of view, the springs - because of their internal dissipation - will necessarily transfer the spin angular momentum of each body to their rotational motion around one another, giving rise, in the inertial reference frame, to a circular forward motion of increasing amplitude of each axial end of the rotation axis of each body around the equilibrium position. There is a cylindrical whirl if the two ends move in phase and a conical (also called precessional) one if they move  out of phase. The natural position of equilibrium with the two axes very accurately aligned still exists but whirling motions around it necessarily grow in time, inevitably bringing the system to instability. It is worth stressing that if the suspension springs are very tiny, with very low stiffness k and relatively high Q values, the timescales of these instabilities are large, as numerical simulations confirm (Section 3.2 ). This is very important in devising an efficient active damper.

Non-rotating damping can be simulated by an active device which exerts on the mass  a force

where  is the overall gain of the device and  is the complex displacement measured in the rotating reference frame   as shown in Fig. 5 . The device exerts a force of the same intensity and direction but opposite sign on the mass . In the inertial reference frame  the equation of motion of the system is now given by Eq. 4 by adding the term

to its left hand side. Assuming again a solution of the type  for the free whirling, the characteristic equation of the homogeneous system for translational motions is

whose solutions are 0 and the solutions of the equation:

which is the characteristic equation of a Jeffcott rotor with mass , non-rotating damping  and rotating damping c. It is well known that this device is stable at all speeds below the maximum value 

It follows that rotation at supercritical speed  which were to ensure very good centering (i.e. ) as well as stability necessarily requires . The case of the test masses in the GG satellite is a very favourable one because the tiny suspension springs have very low internal damping c due to the very low stiffness k and relatively high Q. Hence a small amount of active damping  is sufficient to guarantee stability even at a spin frequency  much larger than the critical frequency . Thus, the active dampers are neither required to provide large forces nor to operate with small response times, since the unstable modes of the rotor are characterized by low frequencies (see Section 3.2 ). The nice fact about supercritical rotation is that the equilibrium position, with the axes closely aligned, is a physical property of the system and unstable rotational motions around this equilibrium position take place very slowly. This makes relatively easy centering to the position of equilibrium by means of active damping, clearly much easier than it would be in absence of such a naturally provided position of equilibrium. In Section 3.3 we present an electrostatic damper that appears to be suitable for our purposes. Here we wish to stress that in the rotating reference frame (to which the electrostatic plates of the damper are attached as shown in Fig. 2 ) it must supply a force with components:

Thus, since the effect produced by the damper must be at the frequency of the whirling motions, the unstable ones being at the (slow) natural frequency of oscillation, the damper must actuate at the spin frequency minus the natural one, which is different from the frequency of an EP violation. For the electrostatic plates to be able to recover and damp the slow velocity of whirl while spinning much faster, the control software must be able to subtract away their own velocity of spin, and this requires that either star trackers or Earth elevation sensors provide a reference signal synchronized with the spin. (For a detailed discussion on unstable whirl motions and active rotating damping in space see Bramanti et al., 1996; GALILEO GALILEI, 1997; Nobili et al., 1996).

3.2. Finite element model (FEM) numerical analysis

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Fig. 6. Sketch of the FEM (Finite Element Method) model of GG. The figure shows the final FEM model used to analyse the rotordynamics of the active controlled system with DYNROT. The beam elements have been drawn on the left side of the picture: the white parts correspond to zero mass beams with structural stiffness. The nodes are shown on the right side of the sketch, each node corresponding to two translational and two rotational degrees of freedom which describe the lateral dynamics. In order to provide an understandable overview each node has been located on the corresponding beam element, instead of on the rotation axis of the satellite, as it actually is. Since each active damper has been connected to the central rod there are shorter beam elements near the gimbals and two nodes very close to each other.

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Fig. 7. Schematic representation of the various components of one experimental chamber placed next to one another in order to show the different kind of connections between them (springs, gimbals and electrostatic dampers). The outer and the inner test mass are respectively connected by spring elements (continuous line) to the pair of movable supports. Gimbals join the movable supports to the central rod, while the electrostatic dampers act between the test masses and the central rod (dotted line).

A more realistic model was built using the finite element rotordynamics code DYNROT, developed over the years at the Department of Mechanics of "Politecnico di Torino". The model, which includes the satellite body, the PGB laboratory and three pairs of test masses, is shown in Fig. 1 where the numbers 1-8 distinguish the various components of the system for later reference. The test masses are connected to the PGB by very low stiffness springs and movable supports with elastic gimbals at their midpoints as shown in Fig. 2 . We first compute the whirling modes of the system assuming that no active dampers are present. The model consists of 36 beam elements and 20 spring elements (see Fig. 6 , Fig. 7 ). A number of beam elements which is larger than the minimum necessary to model the 8 cylindrical bodies and the central rod with movable supports has been used in order to allow us to define the location of the attachment points of the springs. The stiffness of the beam elements is orders of magnitude larger than that of the springs, so that they behave as rigid bodies in the whole frequency range of interest. Beam elements have been chosen instead of concentrated mass elements in order to use the ability of the code to compute directly the inertial properties from the geometrical parameters. Once the model was built, the number of degrees of freedom was reduced from 98 complex degrees of freedom, related to the displacements and rotations of all 49 nodes, to 16 through Guyan reduction (see, e.g., Section 2.8 of Genta, 1993). The minimum number of degrees of freedom necessary to define rigid-body motion was chosen, thus ensuring that no deflection of the beam elements can occur. The inertial properties of the rigid bodies are listed in Table 1 , where the numbers 1-8 refer to the various parts of the system as shown in Fig. 1 . The ratio  is also listed. The masses of the central rod and of the movable supports have been neglected. The code was run using the same stiffness of , both longitudinal and transversal, for all the springs (2 for each body) and the same torsional constant , for all the elastic gimbals (2 for each pair of test masses). This value can be obtained if the diameter of the wires in the gimbals is about  and their length from  to . The stiffness of the springs is relevant in response to forces in common mode while the gimbals enter into play when a pair of coupled test masses is subject to a differential force, resulting in an equivalent transverse stiffness  (l is the length of the rod from the gimbal to the spring, that is the arm; l is between  and  in the 3-chamber model of Fig. 1 ). The resulting equivalent transverse stiffness are therefore not exactly the same, but they are all smaller than the  stiffness of the springs. In the code the system is simplified in that either the springs or the gimbals respond, depending on whether the motion is in common mode (both masses together) or in differential mode (one mass with respect to the other) respectively. While it is true that the gimbals do not affect common mode motions, the springs play a rôle also in differential motions. However, they respond to differential deformations with a lower elastic constant (the bending constant) than they do in the case of transversal common mode deformations. How much smaller is computed by Den Hartog (Appendix, p. 429 of Den Hartog, 1985), for a beam, and the numerical factor is 4. In laboratory tests with helicoidal springs we have measured a factor 3. Each pair of suspended test bodies will therefore have a lower natural frequency for oscillations in differential mode and a higher one for those in common mode:  and  (where  is the equivalent transverse stiffness and k the stiffness of each spring;  because there are two springs for each mass and m because the reduced mass between the test mass m and the rest of the spacecraft is essentially m). A factor about 2 between these frequencies is reasonable to obtain, and this is the situation simulated with the DYNROT code.


Table 1. Inertial properties of the rigid bodies (as numbered in Fig. 1) which constitute the FEM model

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Fig. 8. Mode shapes of GG satellite. The mode shapes found by the DYNROT FEM code can be divided into three types: the first includes 7 cylindrical modes forward and backward, the second 7 backward mainly conical modes and the third 8 conical forward modes. A sample of each set including the  and  forward modes and the  backward mode is shown by giving the position of the axis of rotation (continuous line) and the location of the nodes (the "" symbols). The x- is a coordinate along the spacecraft axis (in cm); the y- is an adimensional normalized coordinate (the mode automatically scales all the modes to the maximum value).

The frequencies of free whirling are computed for the spacecraft spin rate . Apart from the zero frequency modes, we find (in ): a set of 7 backward mainly conical whirling modes of frequencies  and ; a set of 7 cylindrical modes, forward and backward of frequencies  and ; a set of 8 conical forward whirling modes of frequencies  close to the spin frequency. A graphical representation for some of the computed modes is given in Fig. 8 . With the introduction of some internal damping of the springs which suspend the test masses (e.g.  it is found that only the forward cylindrical modes become unstable, and the e-folding times are a few  (all other modes are naturally damped). As expected, instabilities are there but they build up slowly. As for the PGB laboratory the timescales for instability are shorter because, although the mass is bigger than that of the test bodies, the quality factor  of its springs is smaller. Note that all modes with eigenfrequencies close to the spin/signal frequency are conical (i.e. angular precessions), not cylindrical modes, which means that they do not affect the centre of mass of the bodies whose relative displacement is the observable in this experiment. Moreover, it is well known and the simulations confirm it, that they are naturally damped, i.e they are not unstable. We can conclude that there is no interference between these modes and the signal. As for the frequencies of the 7 cylindrical modes listed above (both forward and backward, the forward ones being those which become unstable in the presence of a nonzero internal damping of the springs) we note that they are the eigenfrequencies of the suspended bodies (two for each pair of test masses plus one for the PGB laboratory) as they have been computed in the numerical simulation of the 3-chamber system. As a matter of fact, since we study the mechanical system as a whole these are the natural frequencies of the system; however, due to the weakness of the springs, they can be still recognized as due to the various suspended masses. In order to avoid mutual mechanical influences between the three pairs of masses it is sufficient that their frequencies be separated by a few times their bandwidths. In this simulation in which all the springs have a stiffness of  and all the gimbals have a torsional constant of  the DYNROT code provides: a frequency of about  to be associated with the oscillations of the PGB and corresponding to an elastic constant ; three frequencies between  and  to be associated with the differential modes; three higher frequencies between  and  to be associated with the common modes. From now on we use  for the common mode and  for the differential mode, noticing that there is enough liberty in the choice of the springs and gimbals to actually obtain these values. The corresponding elastic constants are: , for the response to forces in differential mode and  for the response of each mass to forces in common mode. The fact of having a good separation between the eigenfrequencies in common mode and those in differential mode (the latter being smaller) is very useful because in this way displacements in response to perturbations in common mode (air drag) are reduced while those in differential mode (EP violation) are increased.

At this point we insert into the model a set of ideal active dampers providing the force (Eq. 16 ). In doing so we also refine the model by increasing the number of elements to 82 (48 beam elements, 20 spring elements and 14 active dampers). The model includes 61 nodes and the number of degrees of freedom is reduced from 122 complex degrees to 16 through Guyan reduction. We assume a gain of  for all the 12 elements damping the test masses. It is found that all unstable modes become stable and the timescales for damping are of 1 to 10 hours. For details on the physics of unstable whirl motions and their active control with rotating electrostatic dampers see Bramanti et al. (1996), GALILEO GALILEI (1997), Nobili et al. (1996).

3.3. The electrostatic damper

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Fig. 9. Electrostatic damping of whirling motions. The circular instability motions of the rotation axis of the inner shaft, which have the natural (low) frequency of the suspended masses , can be actively damped by means of the electrostatic force obtained by applying a voltage pulse V of short duration (for example for about one fourth of the spin period, i.e. about ) to each plate rotating at the (rapid) spin frequency  when it is passing through the position  before the point of its nearest approach to the inner shaft. This happens, for each plate, at a frequency which is equal to the spin frequency  minus the natural frequency .

The force acting between the two elements of the electrostatic damper (Fig. 9 ) for an assumed voltage V is (in MKS):

with S the surface of the actuator and x the gap between the equipotential surfaces of the damper. For small displacements and tensions the force can be linearized around a constant voltage  and reference gap  for a superimposed control voltage  yielding

The second term on the right hand side describes the behaviour of a spring element with negative stiffness, and this must be softer than the mechanical springs: those linking PGB to the spacecraft body and those connecting each test masses to the central rod. The actuator is driven by a power amplifier modulated by the controller output signal. It provides the control voltage  to the capacitive load constituted by the pair of electrostatic actuators. The resulting transfer function between the controller output signal  and the control voltage  is (the variable s indicates a Laplace transform with respect to time)

with  the stationary gain and  the power amplifier bandwidth. The transfer function between the displacement from the reference position of the rotor and the sensor output signal , including the conditioning circuitry, is

with  the stationary gain and  the sensor bandwidth. In order to behave as required by Eq. 16 the controller must supply the output signals  and  for the plates acting respectively in  and  directions in accordance with the sensor outputs  and  in the same directions. The input-output relationships of the controller can be written in the form

with  and  the stationary gains, and  the time constant of the causal pole of the derivative term. By comparing Eq. 16 , Eq. 21 it follows that the stationary gains of the controller must satisfy the relationships

It is easy to verify that a gain  can be obtained with  and . The negative stiffness of the electrostatic damper due to  is , lower than the stiffness of the springs.

We have shown the feasibility of an electrostatic active damper with the characteristics required to ensure stability. The approximations introduced, particularly as far as the simulation of the electric circuit is concerned, are quite rough. Also, the dynamics of the sensors, controllers and power amplifiers has been neglected. However, the results are essentially correct because the frequencies at which the system works are quite low, orders of magnitude smaller than the characteristic frequencies of the electronic subsystems.

As for the thermal noise of the electrostatic active damper we have:

where  is the equivalent resistance of the electrostatic damper of capacity  and electric quality factor  operated at frequency . Then,

Since the force exerted by the damper is  ( the potential difference) the perturbing force corresponding to the noise is  . Dividing by the mass  of the test body we get the perturbing acceleration:

A comparison with the mechanical thermal noise (Section 6.2 ) shows that with a reasonable value of the electric quality factor , and  the contribution of each electrostatic damper to the thermal noise is negligible and it remains so even considering that the damping of each test body involves 8 capacitance plates. This result could be guessed from the fact that the force to be provided by each plate is very small. In point of fact we plan to confirm this result with experimental tests of the electrostatic damping system in the framework of a ground experimental test of the equivalence principle based, as GG, on supercritical rotation and mechanical suspension of concentric test cylinders.

Let us now estimate the amount of shot noise to be expected:

where  and  is the potential difference due to the quantized current . With a capacity  for each electrostatic damper we get:

If  (but usually is much smaller) the shot noise is about 5 times bigger than the thermal noise given by Eq. 24 , which we have just seen to be negligible with respect to the mechanical thermal noise of the test masses.

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       (Anna Nobili-