High precision measurement of gyro rotor sphericity

High precision measurement of gyro rotor sphericity J.A. Lipa* and G.J. Siddall t Quartz g y r o rotors with a s p h e r i c i t y o f b e t t e r than one p a r t p e r m i l l i o n are needed f o r the R e l a t i v i t y Gyroscope E x p e r i m e n t . This p a p e r describes the measurement techniques t h a t are e m p l o y e d to ensure t h a t this r e q u i r e m e n t is met. The basic measurement is a surface i r r e g u l a r i t y map, generated w i t h the help o f a c o m p u t e r - a i d e d roundness i n s t r u m e n t with a measurement u n c e r t a i n t y approaching 2. 5 nm. Analysis and p l o t t i n g o f the data, a n d the i m p a c t o f the results on g y r o performance, are discussed

In an earlier paper I we reviewed the NASA/Stanford Relativity Gyroscope Experiment and placed it in context with respect to other tests and current research in gravitation. The aim of this experiment is to test two major predictions of gravitation theory: the motion of a gyroscope in orbit about the Earth should cause a precession of about 7 arc-sec/ year, and the rotation of the Earth should cause another precession of about 0.05 arc-sec/year.lt is hoped to test these predictions to an accuracy of 10 -3 arc-sec/year. Since the best conventional gyroscopes used for navigation can drift by this amount in as little as a few seconds it is clear that powerful new techniques are needed to achieve the required performance, By operating the gyroscope in a spacecraft environment with a residual acceleration of 10-1°g, where g is the gravitational acceleration on the surface of the Earth, the desired performance can be obtained if certain other requirements are met. Two of these requirements are directly related to the fabrication of the gyro rotor: for a 38 mm diameter rotor, it is necessary to achieve a sphericity of about 30 nm a, and a density homogeneity of the quartz from which it is fabricated of 3 x 10-7 , assuming a spin speed of 10 000 rev/min. Using the lapping machine described in the preceding paper in this

issue, it has recently been possible to achieve the required sphericity, and it is expected that further improvements can be made. Also, it now appears possible to achieve the required degree of homogeneity by careful selection: homogeneity levels of about 1 or 2 x 10-7 over adequately large volumes have recently been reported 2 . Once the quartz rotor is fabricated it must be coated with a thin film of superconducting material to allow suspension and readout. While the requirements on this coating are severe, they are much less demanding than those on the quartz since the coating thickness is less than 1 part in 10 s of the rotor diameter. In most of what follows the techniques are applicable to either coated or uncoated rotors. In this paper we describe a measuring system which we have developed, to measure the surface shape of the rotors to an accuracy approaching 2.5 nm, and discuss some of the results obtained with the system and their impact on gyro performance prediction.

Measuring system

*W.W. Hansen Laboratories of Physics, Stanford University, Stanford, California 94305, USA

To obtain roundness information from a rotor we use a Talyrond x rotating stylus roundness instrument to measure radial deviations in a great circle plane. This data is then preprocessed using a Talynova x on-line computer to remove set-up and instrument errors before being transferred to a data cartridge using a data acquisition unit. The ball is then indexed through a series of equi-angular steps about an axis perpendicular to the instrument

tNow witL HP Labs, Hewlett-Packard Company, 1501 Page Mih Road, Palo Alto, California 94304, USA

XManufactured by Rank Taylor Hobson, Leicester, UK

aln our previous paper ~ we specified 10 nm; i t now appears that we can use spacecraft roll averaging to relax this constrain t somewhat

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spindle to allow great circle measurements to be obtained over the complete ball surface. The multi-plane data is then transferred from the data cartridge to an off-line computer for further analysis. A schematic block diagram of the system is shown in Fig 1. In the remainder of this section we describe details of the measuring techniques and equipment.

Error separation techniques Roundness measurements with an accuracy of better than about 50 nm are only possible if careful allowance is made for the errors associated with the rotating spindle of the instrument 3. Special error separation techniques have been developed to identify and remove the effects of spindle error, and an accuracy of about 2 nm has been achieved4 . This is well below the level required for the gyroscope experiment. The two main error separation methods, the multistep and reversal techniques, are illustrated in Fig 2. The multi-step technique, Fig 2(a), is based on taking a series of roundness measurements, in each of which the component is stepped or rotated relative to the instrument spindle. A series of equi-angular steps totalling 360 ° is usually chosen from which the component errors, which move with each step, can be readily separated from the systematic spindle errors, which remain stationary. The reversal technique, Fig 2(b), requires only two orientations, in the second of which the orientation of both the component and transducer is reversed relative to the spindle. In the two orientations the relative position of transducer and component is unchanged, while the effect of the radial spindle error on the transducer at any position is equal and opposite.

0 1 4 1 - 6 3 5 9 / 8 0 / 0 3 0 1 2 3 - 0 6 $02.00 © 1980 IPC Business Press

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not suffer from the inherent harmonic suppression of the multi-step method 3 . In our measuring system, the reversal technique is used to calibrate the spindle error. The Talyrond 73 instrument, on which the system is based, has been modified to enable the stylus and carriage to be easily reversed and the spindle has been statically balanced for a component radius of 19 mm. This latter measure ensures that changes in spindle error, due to changes in loading after reversal of the stylus carriage, are minimised. The only additional equipment needed to effect the error separation is a simple fixture for rotating the ball through 180 ° . Initially an inexpensive plain bearing rotary table was used but better repeatability was obtained when this was replaced by a more stable, kinematically designed fixture. The upper part of this fixture can be reversed and relocated kinematically to w i t h i n a few arc seconds of 180 °. A onesigma repeatability of 1 nm has been

The component error can be extracted simply as the mean of the two traces while the difference gives the spindle error. A basic difference between the two techniques is that the reversal technique can only be used to remove radial spindle errors since axial error effects do not cancel on reversal. With the multi-step method, however, the 'spindle error' is interpreted as that component of the transducer signal which does not rotate when the component rotates. This means that the effects of spindle error can be removed even when the component surface is inclined to the spindle axis as in the case of a latitude circle measurement on a hemisphere. For the majority of measurements on cylindrical orspherical parts, however, only radial spindle errors are involved and the two methods give substantially the same results, particularly for errors of low harmonic order. For higher harmonics the reversal method is potentially superior since it does

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Equipment description A general view of the measuring system is shown in Fig 3. The heart of the system is a high quality roundness measuring instrument, the Talyrond 73. This is a rotating stylus instrument with a hydrodynamic spindle of proven accuracy and repeatability 4 . A maximum radial spindle error of 50 nm peak-to-valley deviation from the least squares circle is specified by the manufacturer, but our particular instrument has been found to have a peak-to-valley radial error of only 14 nm. Mounted on the table of the Talyrond is the ball indexing fixture. This device provides high precision movement of the ball from one great circle plane to another and extends the operation of the Talyrond by allowing us to generate a complete ball map as a series of longitude circles. The fixture is shown in more detail in Fig 4. During roundness measurement the rotor is held kinematically on three hardened steel balls. After measurement, the Talyrond spindle is stopped and the rotor is lifted vertically from its kinematic support: by a smaller three-ball holder operated by a handwheet-driven cam. Further rotation of the cam first brings two diametrically opposed cups with their axis horizontal into contact with the rotor and secondly lowers the rotor support. The ball is then free to be

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Fig 3 Genera/view of measuring system

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rotated about the horizontal axis to a new position, after which further rotation of the camhandwheel reverses the sequence and carefully lowers the rotor back onto its kinematic support. A wheel with 32 accurately spaced indents is used to determine the relative orientations of the great circles. Observations with marked balls showed that the device returned the rotor to its starting orientation to within 0.4 ° over a full sequence of 32 steps. In actual use, however, only 16 positions are needed, giving 180 ° rotation, since each great circle is measured twice in the full rotational sequence. To the right of the Talyrond in Fig 3 are the Talynova processor and the teletypewriter used for control purposes. The Talynova, with 16k store, provides on-line operation of the Talyrond and pre-processing of the roundness data, including automatic compensation of eccentricity, computer magnification of the profile, error separation by reversal, data averaging, drift checking and the plotting of best-fit and envelopebased reference circles. A limacon reference is used in preference to a circle in all computation as it gives a more accurate modelling of the distortions caused by radius suppression s . Data are sampled from the Talyrond at a rate of 512 points per spindle revolution and, when working to high precision, are averaged over a number of revolutions. During measurement, checks are made at the same point in each revolution to identify and reject drift-affected traces 4 . Both the number of revolutions and the maximum allowable drift can be set by the operator. Once data have been acquired there is no further need for Talyrond operation. A record of the component profile is stored in the Talynova and this is used for analysis with output via the visual display, polar recorder and teletypewriter. The display, transducer amplifier and polar recorder can be seen to the left of the Talyrond in Fig 3.

Rotor measurement For measurements of quartz rotors, the Taylnova is used in the high precision operation mode. This requires that the spindle be previously calibrated using an error separation technique. The spindle error profile is then stored in the computer memory and subtracted, point for point, in

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Fig 4 Ball indexing fixture subsequent measurements. In our system this calibration is effected using the reversal technique. When measurements of the very highest accuracy are required the reversal method can be applied to give the component error directly. This limits the requirement for spindle error stability to the 30 rain or so required for measurement. In practice, the spindle stability of the Talyrond 73 is such that very little loss in accuracy, but much faster measurements, are possible using the stored spindle error. This approach is used for all routine measurements of ball sphericity. A complete set of 16 longitude circle measurements can usually be measured in 3 - 4 h after overnight thermal stabilising, of the rotor and ball indexer. Each measurement is made at a magnification of x 20 000 with a filter of 1 - 5 0 upr and is averaged over 10 revolutions with a maximum allowable drift error of 5 nm. After least squares centring ahd spindle error compensation the data is computer-magnified to x 200 000 and is then checked for integrity on the visual display. Should

there be evidence of dust contamination at this stage the rotor is removed to the clean bench and re-cleaned before re-commencing the series of measurements. If the data appears satisfactory a signal from the teletypewriter transfers the data onto magnetic tape using the data acquisition unit (shown on the extreme left of Fig 3). This unit redigitizes the great circle data at the polar recorder input point and records at 1° intervals on a data cartridge. At an effective radial magnification of x 200 000, the maximum bit resolution is 0.62 nm which significantly exceeds the expected repeatability of the measuring instrument from trace to trace. After the 16 planes have been measured the basic set of 16 x 360 words of ball profile information is then transferred via the data cartridge and a reader to a remote mini-computer for off-line processing.

Surface mapping The Talynova system provides three types of visual output: a data listing

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on the teletype, and polar plots of a great circle either on a crt display screen or on a chart recorder. For visualisation of the complete rotor shape, these forms of data presentation are inadequate. What is needed instead is some form of map which ,presents all the great circle informatioR together to allow the experimenter to obtain a coherent view of the surface features. It is only via this type of presentation that we can hope to obtain information that can be fed back to the lapping process, to produce better rotors. Probably the simplest map that can be generated from the data is similar to a Mercator projection of a globe, but with uniform latitude spacing: we simply lay out each great semi-circle in a linear fashion and plot them adjacent to each other with the radial ordinate vertical and the angular ordinate horizontal. An example of such a plot is shown in Fig 5. This type of plot is useful for getting a general idea of the surface contours in the equatorial region but distorts the region near the poles. A more useful plot is a view of the data as it would appear on the rotor, but projected onto the plane of

the paper. Even better is the view shown in Fig 6 which consists of stereo pairs of such projections, to which we have added latitude lines by linear interpolation between adjacent great circles. The great circle data on every other curve has been terminated 20 ~ from the poles to minimise congestion. We developed a computer program to perform arbitrary rotations of the data about three orthogonal axes before plotting, thus facilitating the viewing of various features of the ball surface. The computation and plotting sequence takes about 10 rain for a stereo pair. Fig 6 shows features that are not readily deduced from Fig 5: for example, the upper and lower stereo pairs show the same rotor simply rotated by 180° about the polar axis, and it can be seen that the polar region is not horizontal but tilted towards the back in the upper view. Also, one is able to get a far better idea of the angular relationships between the various surface features.

A p p l i c a t i o n to gyro p e r f o r m a n c e An electrically suspended gyro rotor with an irregular surface shape will

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undergo a precession due to the interactions between the suspension forces and the surface irregularities. The basic source of the precession torque is a mis-alignment between the electric field vector at a rotor surface element and the corresponding radius vector to the centre of support. To estimate the magnitude of these torques we assume that the rotor shape can be averaged about its spin axis and represent the resulting 'symmetric top' shape by a Fourier cosine expansion: z~r(O) : L

~:, cos(nO)

where 0 is the co-latitude angle measured from the spin axis, A r is the radial deviation from the mean spherical radius, averaged around the ball at a given latitude, and an is a co-efficient to be determined. The upper limit, N, of the series depends on the rate of convergence, and for smoothly varying rotor shapes may be as low as five. The averaging assumption is very good:for rotor surface variations in a longitudinal direction the torques cause spin down (or up) rather than precession, and the amount of precession due to the other torques during a single rotation of the bail is exceedingly small, les's than 10 -Js radians, eliminating any significant cross-coupling. The coefficients a n can be evaluated by the standard formula: 2 ~ an = - : J~r(d)cosnd d8 E

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Quartz rotor surface profiles with poles at the sides Fig 5 Planar projections o f r o t o r surface. R o t o r poles are at the sides. Radial deviations from a best fit sphere are p l o t t e d vertically and latitude angle is p l o t t e d horizontally on a linear scale

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In Fig 7(a), we plot the amplitudes {an[ for the rotor shown in Fig 6 for N=20. It can be seen that major contributions exist up to the 5th harmonic. Since there is no pro.. ferred spin axis in the gyro we plan to fly, it is not possible to obtain an exact characterisation of the hap monic amplitudes beforehand, so the best that can be done is to put upper bounds on the various con. tributions to the total gyro torque. It is, therefore, necessary to perform the calculations about a number of different axes and search for a maximum. Fortunately, the ball surface is relatively smooth, and in any case only the lower order undulations are important, so this search need not be too extensive. Further averaging of the shape is also obtained through the polhoding action'of the rotor, during which tl~e spin axis slowly moves on an ellipticai path through the body axis

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reference frame. This motion can result in an effective harmonic amplitude significantly less than the maximum obtained from rotor surface measurements. Given reasonable estimates for the upper bounds to the harmonic amplitudes, plus information on the configuration of the gyro housing support electrodes and the applied voltages, it is possible to calculate the effect of the surface irregularities on the gyro drift. For a perfectly spherical housing with a centred rotor, the general form for the torque F is

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where Vi is the voltage applied to the i th electrode and Si is its surface area, and i,j are unit vectors orthogonal to the spin axis, and q) is the longitude co-ordinate orthogonal to 0. The drift rate, ~ , is related to F by the wellknown formula r~-

[, Ice

where / is the moment of inertia of the gyro rotor and co its spin speed. Thus the quantity of primary interest from the rotor surface shape analysis is nlanl which is plotted in Fig 7(b). For the electrode configuration we have chosen, it can be shown that values of this parameter up to about 200 nm give ~[Z less than the design goal of 10 -3 arc-sec/year. It can be seen from Fig 7 that we have achieved this goal, at least for the terms considered independently. It is desirable, however, to do better than this, since any improvement in this area can give a relaxation in others, such as the constraint on the residual acceleration of the spacecraft, which enters the torque formula through the term Vg. / The remaining parameters related to rotor fabrication which affect the precession rate are the mass unbalance and the differences between the various moments of inertia. Since these are not related to surface geometry we will only briefly discuss them here. There are two reasons why the centre of support may not coincide with the centre of geometry: the material that constitutes the rotor may be of non-uniform density, and the coating on the rotor may not be concentric with the quartz. The constraints on

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Stereo views of quartz rotor

Fig 6 Stereo pairs of the front and back surfaces of the best quartz rotor so far obtained

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5a

where ~0 is the upper limit on the

precession rate and r is the radius of the rotor, and 2~r is the component of the displacement between centre of support and centre of gravity perpendicular to the applied acceleration a. For our case, we require

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~ r % 5 x 10-7 which implies that the r Ap h o m o g e n e i t y , - - , of the quartz should P be about 3 x 10-7 . While this number implies quartz of very high quality, significantly better homogeneity has been observed in some samples 2 . Since the density of the coating is over three times that of the quartz, the above z~r limit on - - means that it must be r centred on the quartz to within 3 nm. We have been able to achieve a coating uniformity of about 3%, so the upper limit on the thickness is 100 nm which is adequate for readout purposes and suspension in space. It can also be seen that film thickness variations of 3% would have Jittle effect on the rotor surface map obtained from measuring the bare quartz. The limits on the moments of inertia differences are much less severe and are adequately met when the sphericity and homogeneity requirements are satisfied.

Conclusion We have demonstrated that a corn-

puter-aided roundness system can be used to make complete ball maps to an accuracy of about 5 nm using a technique based on discrete great circle measurements. This data can then be used to obtain parameters useful in predicting gyro drift performance. The accuracy we have achieved is sufficient to meet the metrology requirements for a highly sophisticated experiment in gravitational physics. This development in surface mapping could have application in other areas of precision engineering, such as the manufacture of precision bearings and inertial navigation gyroscopes.

Talynova software to our specification. The rotor used to generate the surface maps was produced by R: Taylor and J. Reed at NASA Marshall Space Flight Center as part of the Relativity Gyroscope development program. This work was supported by NASA Contract NAS8-32355.

References 1. Everitt C.W.F, Lipa J.A. and Siddall G.J. Precision Engineering and Einstein: the Relativity G~./roscope Experiment.

Precision Engineering, January t 9 7 9 1(1)~5 1t

2, Siddiqui I.M. and Smith R.W. A Holographic Interferometer for Investigating the Homogeneity of Optical Glass.

Optica Acta, t978, 25(8), 737-749

Acknowledgements We wish to thank J.J. Gilderoy, Jr, for designing and fabricating the ball indexing and reversal fixtures, M. Debiche for assistance with the ball mapping software, R.R. Clappier of Penelco Corporation for design and construction of the data acquisition unit, and D. Kinsey of Rank Taylor Hobson, Leicester, for modifying the

3. Whitehouse D,J. Some Theoretical Aspects of Error Separation Techniques in Surface Metrology. J. Phys. E: Sci Instrum., 1976. 9, 531--6 4. Chetwynd D.C. and Siddall G.J. Irnprov ing the Accuracy of Roundness Measures ment. J. Phys, E: ScL Instrum,, 1976,

9,537-44 5. Chetwynd D.C. Roundness Measurement using Limacons. Precision Engineering,

1979, 1(3), i37 -14!

21st Inkmmtioai likiekkie Teei :heign ikamreh 8 - 12 SeldemlNn1960 Pre-Conference Course 8 -- 9 September 1 9 8 0

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The i n t r o d u c t o r y sessions will be devoted to lectures on the application of the finite elernent method to metal working.

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O r g a n i z i n g Secretary:

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