Sloan Digital Sky Survey Telescope Technical Note 19941130_04
The tracking uniformity goals for the SDSS 2.5-m telescope are quite demanding so that 50 milliarcsecond root-mean-square (mas RMS) astrometric accuracy can be obtained. The azimuth drive and idler rollers, along with the azimuth drive disk, form the upper bearing of the telescope azimuth axis and are crucial to meeting the tracking performance goals.
Consequently, on November 9-10, 1994, the runout of the azimuth disk was measured. At the time, the telescope azimuth structure was assembled to the level of the azimuth disk and included the spherical roller bearing at the apex of the azimuth cone (Figure 1). The azimuth disk was positioned by four equally-spaced roller assemblies (Figure 2). Two diagonally opposite assemblies are motor driven and were connected to a variable DC power supply for our tests. This allowed the azimuth structure to be rotated.
The data acquisition system is described in SDSS Technical Report 19940306. For most measurements, two electronic indicators were used. One electronic indicator measured the runout of the outer cylindrical surface of the drive disk. A second measured the runout of a 25.4 mm diameter steel ball located on the azimuth rotation axis. The ball was supported by a beam clamped across a diameter of the disk. The grade 5 ball was specified to have 125 nm sphericity and 20 nm surface finish, i.e., almost negligible error.
For other measurements, the two indicators were placed to measure runout of the ball in the southeast (the idler assembly diagonal) and southwest directions (the drive assembly diagonal). These measurements were used to characterize the two-dimensional runout of the upper end of the azimuth assembly.
In addition to the two indicators, the output of a Hall effect sensor was digitized. A magnet was placed on the azimuth disk so that the Hall effect sensor would provide a signal at approximately the instant that the mark "UP" on the disk was aligned with the outer indicator. Typically, a revolution took about 40 to 50 seconds. Most data were digitized at 10 ms intervals. Successive detection of the magnet by the Hall effect sensor were used to scale the data in angle. The azimuth rotation rate was assumed constant, a somewhat dubious assumption since the DC motor speeds were controlled by adjusting the voltage to the motors. However, the rotation rate appears to have been adequately uniform for our purposes. To check for electronic and/or mechanical noise, a few measurements were made with the rotation rate slowed to about 80 seconds/revolution. Only a small feature at 18.5 Hz (2.2 cycles/deg at 8.4 deg/sec) appeared to be due to noise.
Related measurements include the runout of 2.5-m azimuth drive disk during the grinding of the disk (SDSS Technical Report 19940306) and measurements of the runout of the azimuth rollers ( SDSS Technical Report 19941104).
Fig. 1: Runout measurement of azimuth disk. On the left, an electronic indicator measures the runout of the outer cylindrical surface of the disk. At the center, a second indicator measures the runout of a Grade 5 steel ball. A squat cylindrical drive motor is visible in the left foreground.
Fig. 2: Upper azimuth disk assembly. The drive roller assemblies (3 and 6) are located in the northeast and southwest corners of the square frame. The idle roller assemblies (2 and 4) are on the other diagonal. Each roller assembly contains a ø102 mm and a ø51 mm roller. The roller numbers are the same as those in SDSS Technical Report 19941104.
The runout of the azimuth axis at the level of the drive disk was measured by indicators mounted southeast and southwest of the 25.4 mm steel ball. The runout was similar in both directions, about 4 µm peak-valley or 920 nm RMS. This would cause a two-dimensional pointing error of 99 mas RMS (Figure 3). This will contribute negligible pointing error since the total error is likely to be 1 arc second RMS .
Fig. 3: Runout of the upper azimuth axis. Channel 1 is the southeast runout. Channel 2 is the southwest runout. The one-dimensional runout is about 920 nm RMS. This would correspond to a two-dimensional pointing error of 99 mas RMS.
Our tracking error budget (SDSS Technical Report 920608-01) allocates 21.0 mas RMS to the upper azimuth bearing at frequencies between 3 and 300 mHz. This corresponds to 276 nm RMS total runout or 195 nm RMS in x and y separately. To convert the frequency range to angular frequencies for the azimuth axis, it is necessary to know the tracking rate. In azimuth, the tracking rate along survey strips varies from zero to (potentially) many times the sidereal rate. The rate depends on elevation and, to a lesser extent, on azimuth. The rate is high near the zenith and tracking near the zenith should be minimized. It appears that limiting the azimuth tracking rate to less than three times the sidereal rate (15°/hour) would have little impact on the operation of the survey. At three times sidereal the angular frequency range would be 0.24 to 24 cycles/degree. We should note that the low frequency limit is not well established. Thus, in the following, we consider the effect of different low frequency limits.
We will see below that the fundamental frequency of the ø102 mm drive/idler rollers dominates the power spectral density at moderate frequencies. This effect occurs at 25 cycles per revolution (0.069 cycles/degree) of the azimuth axis and does not contribute to tracking error in the range cited above.
Table 1: Amplitudes of the harmonics of the upper azimuth bearing runout. The amplitudes are for two independent measurements, 247 and 248, and in two directions, x1 (southeast) and x2 (southwest). The amplitudes are quite repeatable between measurements. The changes in the amplitudes of the 25th harmonic are likely real and are due to the phase drift in diagonally opposite rollers.
Amplitudes (nm) Harmonic 247x1 248x1 247x2 248x2 2 668 669 348 323 3 789 791 1169 1252 4 402 391 182 239 5 359 327 305 311 6 57 56 48 57 7 107 95 181 151 8 33 29 72 51 25 327 425 59 120
If the drives and azimuth assembly were perfectly symmetric, the even harmonic components would not contribute to axis runout. The two drive assemblies are located across one diameter and the two idler assemblies are located across a diameter perpendicular to the first. An even harmonic component should cause symmetric deflection of the drive disk and roller assemblies but no net runout of the axis. In Table 1, the even harmonics are systematically smaller than the odd harmonics, but not negligible.
The dominance of the 25th harmonic can be seen clearly in the residual data after the lowest 8 harmonics have been removed. In Figure 4, channel 1 is the southeast indicator and channel 2 is the southwest indicator. The residual runout for the two directions is 289 nm RMS and 174 nm RMS respectively.
Fig. 4: Residual runout of the upper azimuth bearing with the lowest eight harmonics removed. Much of the residual runout in channel 1 is due to the 25th harmonic, i.e., the fundamental of the ø102 mm drive and guide rollers. The two curves have been separated by 1.5 µm for clarity.
After the lowest eight harmonics were removed from the data, the power spectral density (PSD) of the residual runout was computed (Figure 5 and Figure 6). The power is normalized so that twice the area under a spectral feature is the mean square amplitude of the data at that frequency. Two different methods of PSD estimation were used. One was based on the fast Fourier transform (FFT) while the other used the maximum entropy method (MEM). (See chapter 13 of Numerical Recipes, William H Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery, Cambridge, New York, 1994). Agreement between the two methods is excellent.
Most of the power is below 0.2 cycles/degree (2.5 mHz at 3 times the sidereal rate of 15 degrees/hour). Note the logarithmic y-axis. Above 0.2 cycles/degree, the runout is 123 and 118 nm RMS for the idler and drive directions respectively. These values are well within the amount budgeted above.
Fig. 5: Power spectral density (PSD) of the runout in the idler (southeast) direction. Two different methods of PSD estimation were used, the fast Fourier transform (FFT) and the maximum entropy method (MEM). At frequencies above 0.2 cycles/degree, the total runout is 123 nm RMS.
Fig. 6: Power spectral density (PSD) of the runout in the drive (southwest) direction. At frequencies above 0.2 cycles/degree, the total runout is 118 nm RMS.
The 25th harmonic of the axis is the fundamental of the ø102 mm drive and idler rollers. These rollers are friction coupled to the drive disk. Since their diameter is not precisely 1/25 of the drive disk diameter, the phase of the 25th harmonic drifts slowly with continuous rotation of the axis (Figure 7). The drift rate is consistent with the ø102 mm rollers being 0.11 mm (0.004") larger than 1/25 of the disk diameter.
Of more importance, the phase of a roller will slowly drift with respect to its counterpart across the disk diameter. As the rollers drift in and out of phase, the amplitude of the 25th harmonic will change. Figure 8 shows this effect although the data are too sparse to be completely convincing. The modulation is a factor of two in amplitude. This suggests that it may be useful to mark the rollers and to periodically adjust them so that the amplitude of the 25th harmonic is minimized.
The amplitude of the 25th harmonic is larger in Channel 1 than Channel 2 by a factor of three or so (Figure 4 and Table 1). This is puzzling since drive roller 6 has the worst runout by a factor of two and should degrade results in Channel 2 so that they are worse than Channel 1. In more detail, we can use the measurements of the 1st harmonics of the ø102 mm rollers from SDSS Technical Report 19941104, Table 3, to predict the amplitude of the 25th harmonic of the azimuth axis, i.e., the mean vector sum of the amplitudes of the 1st harmonics of the diagonally opposite rollers where the angle between the vectors is the phase difference of the rollers. The predicted range for each channel is tabulated in Table 2 as are the measured amplitudes of the 25th harmonic from Table 1. While the overall ranges are consistent with the measured amplitudes, they are inconsistent on a channel by channel basis. This may indicate a problem in our records or a subtle problem in our interpretation of the data.
Fig. 7: The phase of the 25th harmonic is plotted against azimuth revolution count during a period of 1.6 hours. The azimuth axis rotated continuously during the interval. The slope of the linear fit is consistent with the ø102 mm rollers being 110 µm larger than nominal.
Fig. 8: The amplitude of the 25th harmonic is plotted against azimuth revolution count during the same period as Figure 5. A sine wave is fit to the data suggesting it is plausible that the amplitude modulation is due to the phase changes among the drive and guide rollers. About 50 µm of variation in roller diameters would account for the effect observed.
Table 2: Predicted amplitude of the 25th harmonic in the two measurement channels. These are based on the measurements of the first harmonic reported in SDSS Technical Report 19941104, Table 3. For Channel 1, the north roller is 2 and the south roller is 4. For Channel 2, the north roller is 3 and the south roller is 6. The measured amplitude of the 25th harmonic (the last two columns) should be in the range of the out-of-phase amplitude and the in-phase amplitude.
Amplitudes (nm) Channel North South Out-of-phase In-phase file247 file248 1 217 216 1 217 327 425 2 159 506 174 332 59 120
Radial runout of the disk was measured by an indicator mounted on the west side of the disk along an east-west line passing through the azimuth axis. The ball indicator was mounted on a similar east-west line. Neglecting the intrinsic runout and decentering of the ball, the difference in the two indicator readings was the radial runout of the azimuth disk. The effect of ball decentering was corrected by removing the first harmonic from the ball data before further processing.
Measurements were made at five approximately equally spaced intervals in height (Figure 9). The peak-valley runout was as large as 13 µm. This is approximately twice that measured during the grinding of the drive disk (SDSS Technical Report 19940306). Although a couple of light grinding passes occurred after those measurements to improve the smoothness of the disk, it is unlikely that these were responsible. A more likely explanation is that the disk was stressed in a different manner during grinding and after assembly to the azimuth cone. This would explain why little correlation is apparent in the overall shape of the disk measured then and now.
Discontinuities are apparent in some of the data, e.g., at z = -56 mm and 0°. Fortunately, these do not have a visible effect on the axis runout (see Figure 4). As speculated in SDSS Technical Report 19940306, it seems that averaging over the contact rectangle between the roller and the disk is effective in minimizing the impact of localized discontinuities.
Fig. 9: Radial runout of the azimuth disk. The location of each measurement is given with respect to the upper surface of the disk. Axis runout was removed by measuring the runout of a high quality ball mounted on the rotation axis. The curves are spaced at 2 µm intervals for clarity. Discontinuous features are present in individual measurements that are fortunately not present in the measurements of axis runout.
It appears from these data that a two dimensional upper azimuth axis runout of 170 nm RMS has been achieved for angular frequencies above 0.2 cycles/degree. This frequency corresponds to 2.5 mHz at an azimuth tracking rate of 3 times the sidereal rate, i.e., 45 degrees/hour. The resulting contribution to the tracking error would be 13 mas RMS.
At the other extreme, with a low frequency cutoff of 0.022 cycles/degree or 0.28 mHz at 3 times sidereal, the two dimensional runout is 337 nm RMS or 26 mas RMS. This exceeds the 21 mas budgeted. A good portion of the error is due to the fundamental frequency of the ø102 mm rollers. Regrinding or replacing one or two of the worst rollers would allow our tracking error budget to be satisfied even over this larger frequency range.
We are greatful for the assistance of Mr. Paul Baird and the personnel of the assembly shop of L&F Industries, Huntington Park, CA.
