page 1 | page 2 | -->page 3<-- | page 4 | page 5 | acknowledgements | figures | references |

**Measurements and error corrections**

Diameter Measurements

Because of a 0.001 inch (25 mm) gap in the suspension of the sphere in the gyroscope, the diameter tolerance did not need to be better than ± 20 micro inch (0.5 mm). For production purposes a relative diameter measurement with respect to a standard fused silica sphere was performed by using a specially constructed instrument (made by students in the Precision Engineering class, (ref.11 ). The instrument uses an LVDT (Linear Variable Differential Transducer) displacement sensor with a minimum scale division of 5 micro inch (127 nm), adequate for the specification. For the silicon spheres, corrections were made for the difference in thermal expansion coefficient between silicon and fused silica. The temperature was measured to a few tenths of degree C. Absolute measurements of the standard fused silica sphere were also performed as a project of the Precision Engineering student class, (ref.12 ). For this purpose students built an apparatus which is used in conjunction with an interferometer donated by the Hewlett Packard company. The size of the standard sphere was found to be 1.4959" ± 45 micro inch (37.996 ± 0.001 mm). The error is the geometric sum of statistical and estimated systematic errors.

Sphericity Measurements

Measurements of sphericity were performed using a Talyrond roundness measuring instrument manufactured by the Rank Taylor Hobson Company (Fig.5 ). The main component of the instrument is a precision spindle, driven by a synchronous motor. Measurements were performed at the slow speed of 6 revolutions per minute. The spindle carries a rail, on which a displacement measuring instrument (LVDT) is mounted. During the measurement, the tip of the stylus is brought in contact with the sphere at a low force of 2-3 g. The tip of the stylus is made out of diamond for low friction and has a radius of about 9 mil (0.22 mm). The instrument has a good provision for precision centering. The resulting centering error (as determined by a least squares fit for each measurement) was of the order of 0.1 micro-inch (2.5 nm). The instrument is equipped with an analog amplifier and an active band pass filter. Usually the filter cutoff was set at 150 undulations per revolution and the amplification at 20,000 times.

The precision spindle upper bearing is conical and the lower one is semi-spherical, both floating on a thin oil film. Deviation of the spindle motion from pure rotation causes a measurement error which is a function of the rotation angle and reaches about 1 micro inch. As the motion of the spindle is quite repeatable over long periods of time, this error (Fig.6 ) can be subtracted from the measurements.

The sphere to be measured is supported by three small spheres. A thin kapton foil is inserted between the supports and the sphere, so as to avoid scratching the sphere. The incomplete rigidity of sphere support contributes slightly to the measurement errors. We estimate this contribution to be less then 0.1 micro inch (<2.5 nm)

Mechanical and electrical drifts also contribute to the measurement errors. The mechanical drifts are mainly due to temperature changes and appear as relaxations. Electrical drifts tend to be of more continuous nature. Usually 2 or 3 measurements are taken in a sequence and averaged. If the drift exceeds 0.1 micro inch (2.5 nm) the measurement is rejected.

Determination of Talyrond spindle error

The largest correction to the measurements was the subtraction of the Talyrond spindle error. This error was determined using the following methods:

Averaging method: One can measure the same great circle, in a series of rotated positions of the sphere. When taking the average of such measurements for a very good sphere, deviations from a perfect circle tend to average out, and spindle error (always the same in all measurements) can be determined. This method was used to check error correction obtained by other methods, as the corrected consecutive measurements should average very close to a circle. As an example, for a Talyrond error of 1 micro inch ( 25 nm) peak to valley (p/v), and a sphere error of 0.5 micro inch (12.5 nm) p/v, the error of the average of 12 or 15 measurements in a series of rotated positions, could be 0.1 micro inch (2.5 nm). The method is straightforward and does not require manipulation of the instrument.

Inversion method: The inversion method, (see Lipa, J. and Siddall, G.J., ref.13 ), takes advantage of the Talyrond instrument design, which allows the insertion of the displacement measuring assembly from either side of the rail connected to the spindle. After rotating the sphere 180 degrees around the z-axis and inverting the arm, one can measure the same point of the sphere again but with the opposite spindle error. By subtracting the two measurements one obtains twice the spindle error.

Fourier method: In this method the data is taken in the same way as
in the method of averaging, say 15 consecutive rotated positions. The data
is analyzed using a Fourier series expansion in order to detect and separate
the component which rotates with the sphere (sphere error) from the component
which is stationary (spindle error) (see DiDonna, B., ref.14
).

For our apparatus, the spindle error was found to be about 1.1 micro inch (28 nm) maximum, (see Fig 6 .)

**Data acquisition and processing**

The analog displacement signal is amplified, filtered and sampled synchronously with the spindle rotation. It is then converted into a digital signal in a Lawson 12 bit A/D converter. Sampling is done for 600 points per circle corresponding to a ~128 samples per linear inch (50 samples per cm). The start and stop signals for sampling are obtained from a micro switch which is actuated by a small cam on the spindle, corresponding with the stylus at the north pole position. Sixteen meridian circles are measured. After each measurement the sphere is rotated by 180/16 degrees. The equator is then measured as well.

Software for Reconstruction of the Sphere shape and study of sphericity.

The original software was developed under the guidance of Prof. J.Turneaure and written by Stanford University students. It was further modified to improve the display and allow easier measurement of a sequence of 17 circles (see ref.15).

The reconstruction can be broken into the following steps. For each meridian great circle, 2-3 measurements of 600 data points are taken, corrected for spindle error and stored in separate measurement files. Averages of consecutive measurements of the same circle are centered using least squares. Each circle is re-aligned and the standard deviations at the poles and the equator are checked. Two meridians are formed from the single great circle measurement by inverting the second half of the sequence. The data is stored, (32 data points for each parallel for 126 parallels and two poles). Next, the sphere is represented by an expansion in spherical harmonics, Y(l,m,J,j). The integration is performed for each l and m: j angle first, using trigonometric functions and J angle next, using associated Legendre functions up to Lmax=16. The result is a set of coefficients representing the sphere. The expansion is later used for calculating global peak to valley deviations from the ideal sphere and for plots. The coefficients are also used for the study of the gyroscope motion.