Sloan Digital Sky Survey Telescope Technical Note 19941104_01
The tracking uniformity goals for the SDSS 2.5-m telescope are quite demanding so that 50 milliarcsecond (mas) 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 tracking performance goals. Less critical but still important is the performance of the azimuth and altitude encoder rollers. These rollers drive the incremental encoders that measure the angular position of each axis.
Consequently, on October 10-11, 1994, the runout of the drive, idler and encoder rollers for the SDSS 2.5-m telescope were measured. 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 upper end of the partially assembled telescope azimuth structure ( SDSS Technical Report 19941130).
The drive assemblies as well as the measurement techniques are described in the report on the new 3.5-m drive assemblies, SDSS Technical Report 19940919. The drive and idler rollers were ground assembled with their bearings prior to measurement. The encoder rollers are much narrower than the drive rollers, about 13 mm wide (Figure 1). Only a single track was ground on the encoder roller and the track is slightly crowned. Prior to measurement, the encoder rollers were ground and subsequently assembled on their bearings. They had not been ground on their bearings. Therefore, the measurements reported herein provide a baseline for using future measurements to determine the effectiveness of grinding the rollers on their bearings.
Fig. 1: Runout measurement of an encoder roller. An electronic indicator is shown measuring the runout at the center of an encoder roller. On the left, a shaded pole electric gearhead motor drives the roller with a period of revolution of 47 seconds. The setup was similar for the drive and idler roller measurements.
Each roller assembly includes a 102 mm (4") diameter roller and a 51 mm (2") idler roller. For convenience in the following discussion, the 102 mm rollers are referred to as "drive rollers" even though not all are motor driven. The 51 mm rollers are referred to as "idler rollers". None of the 51 mm rollers are driven.
The runout of each roller was measured concurrently on the lower (ch 1) and upper tracks (ch 2). Lower and upper refer to the orientation of the installed azimuth drives. A linear plus first and second harmonic equation was fit to the data. The results for drive roller #1, as assembled in its housing, are shown in Figure 2. The first harmonic, with an amplitude of 0.852 µm, was the largest one measured.
Fig. 2: Runout of the lower (ch 1) and upper (ch 2) tracks of drive roller #1 assembled with its housing. The spikes at 0° and 360° are due to a narrow strip of transparent tape used as an angle fiducial. Both the original data and the residuals of the removal of the first two harmonics and the linear trend are shown.
To examine the consistency of amplitude between tracks and between the bare rollers and their condition as assembled, we fit a second order harmonic to the data (Table 1 and Table 2). The correlation coefficient for amplitude was 0.68 and 0.78 for the first and second harmonics between the two tracks for all rollers. It was between 0.64 and 0.77 for bare roller vs. assembled for both harmonics and both the drive and idler rollers. (A correlation coefficient of 1 implies perfect correlation; 0 implies no correlation; and a correlation coefficient above 0.5 is usually considered a significant correlation.) These results suggest that one measurement of each track is adequate to characterize the runout of a roller. Also, they imply that a measurement of a bare roller is a good predictor of its quality once assembled.
The data indicate that the two tracks have the same phase to better than +/-80° for the first and second harmonics when the amplitude of the first harmonic is 185 nm or larger. This, together with the correlation in amplitude, indicate that most of the larger first and second harmonic amplitudes that we measure will affect telescope tracking, i.e., will not be reduced much by averaging over the line contact between the roller and the drive disk.
Table 1: Coefficients of harmonic fits to drive roller surfaces. The coefficients a1 and a2 are defined by the equation, y = a1*cos(x+ø1) + a2*cos(2*x+ø2). The measured coefficients of the bare and assembled rollers are given along with the file numbers.
drive # bare a1 bare a2 assm a1 assm a2 bare fn assm fn (µm) (µm) (µm) (µm) 1 0.407 0.124 0.852 0.050 45 85 2 0.102 0.126 0.209 0.051 43 120 3 0.197 0.134 0.163 0.056 47 110 4 0.130 0.120 0.204 0.082 41 106 5 0.274 0.178 0.096 0.072 52 88,92 6 0.391 0.105 0.506 0.050 49 98 average 0.250 0.131 0.338 0.060 stdev 0.130 0.025 0.289 0.014 max 0.407 0.178 0.852 0.082 min 0.102 0.105 0.096 0.050
Table 2: Coefficients of harmonic fits to idler roller surfaces. The coefficients a1 and a2 are defined by the equation, y = a1*cos(x+ø1) + a2*cos(2*x+ø2). The measured coefficients for the bare and assembled rollers are given along with the file numbers.
idler # bare a1 bare a2 assm a1 assm a2 bare fn assm fn (µm) (µm) (µm) (µm) 1 0.182 0.125 0.160 0.072 64 96 2 0.174 0.309 0.340 0.501 55 118 3 0.073 0.052 0.046 0.052 63 102 4 0.396 0.196 0.280 0.555 60 108 5 0.424 0.171 0.362 0.125 67 90,94 6 0.340 0.250 0.509 0.285 57 100 average 0.265 0.184 0.283 0.265 stdev 0.141 0.091 0.163 0.220 max 0.424 0.309 0.509 0.555 min 0.073 0.052 0.046 0.052
The first and second harmonics are responsible for most of the non-random runout of the rollers. This is seen clearly in Table 3 where the amplitudes of the first 6 harmonics are tabulated. The root-mean-square (RMS) runout as calculated from the harmonic amplitudes is tabulated. We suspect that these values are a good estimate of the RMS error that will be experienced in use. The higher frequencies that are present in the raw data are not likely to be well correlated along the line contact between the roller and the drive disk and will not contribute significantly to the tracking error.
Fig. 3: Power spectral density of the runout of the lower track of drive roller #1 assembled with its housing. Nearly all the power is at frequencies below 1 cycle/degree.
To confirm that the harmonic content of the runout is adequately described by the lowest 6 harmonics, the residual error after removing the fitted harmonics was calculated. Then, the power spectral density (PSD) of the residual error was estimated. The power is normalized so that two times 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, 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.
A typical result is given in Figure 3. The data show no significant spectral features and a rapid decrease of power with frequency. Nearly all the residual power occurs at frequencies less than 1 cycle/degree. We should mention that the PSDs for most of the encoder roller showed a peak between 5.4 and 5.6 cycles/degree. However, the amplitude was less than 6 nm RMS in all cases and of little consequence. The PSDs for one track of assembled drives 2 and 4 showed a feature with similar amplitude at 5.7 to 5.8 cycles/degree. Again, this has no practical effect.
Table 3: Coefficients of harmonic fits to assembled drive roller runout. The coefficients of the first 6 harmonics are given. The Table shows that the 1st and 2nd harmonics dominate the other terms. The RMS is calculated from the coefficients. The two entries for drive #5 are for slightly different locations on the roller surfaces.
Order of harmonic term drive# 1 2 3 4 5 6 RMS file (µm) (µm) (µm) (µm) (µm) (µm) (µm) 1 0.888 0.053 0.026 0.007 0.039 0.033 0.630 86 2 0.217 0.054 0.013 0.018 0.020 0.018 0.160 120 3 0.159 0.052 0.022 0.019 0.014 0.015 0.121 110 4 0.216 0.083 0.014 0.007 0.018 0.032 0.166 106 5 0.076 0.082 0.011 0.027 0.026 0.016 0.084 88 5 0.112 0.059 0.032 0.017 0.025 0.017 0.095 92 6 0.506 0.049 0.022 0.029 0.020 0.022 0.361 98 average 0.310 0.062 0.020 0.018 0.023 0.022 0.231 stdev 0.291 0.014 0.008 0.009 0.008 0.008 0.199 max 0.888 0.083 0.032 0.029 0.039 0.033 0.630 min 0.076 0.049 0.011 0.007 0.014 0.015 0.084
Results for the encoder rollers are not as good (by a factor of two) as those for the other rollers (Table 4). In particular, the 1st, 3rd and 4th harmonics are significantly worse. These rollers were ground and then assembled with their bearings for measurement rather than being ground on their bearings as were the other rollers. Consequently, one would not expect their performance to be as good.
Table 4: Coefficients of harmonic fits to encoder roller runout. The coefficients of the first 6 harmonics are given. Unlike the other rollers, these were not ground and measured on their bearings. The RMS is calculated from the coefficients. The two entries for encoder #6 are for slightly different locations on the roller surface.
Order of harmonic term encoder# 1 2 3 4 5 6 RMS file (µm) (µm) (µm) (µm) (µm) (µm) (µm) 1 0.982 0.162 0.093 0.084 0.034 0.032 0.710 69 2 2.170 0.009 0.083 0.063 0.030 0.027 1.537 71 3 0.937 0.042 0.059 0.045 0.022 0.043 0.666 73 4 0.460 0.078 0.124 0.042 0.018 0.012 0.343 75 5 0.348 0.099 0.046 0.068 0.018 0.005 0.262 77 6 0.796 0.116 0.042 0.093 0.045 0.055 0.576 79 6 0.787 0.066 0.047 0.171 0.061 0.043 0.575 81 average 0.926 0.082 0.070 0.081 0.033 0.031 0.667 stdev 0.597 0.050 0.031 0.044 0.016 0.018 0.417 max 2.170 0.162 0.124 0.171 0.061 0.055 1.537 min 0.348 0.009 0.042 0.042 0.018 0.005 0.262
Our tracking error budget (SDSS Technical Report 920608-01) allocates 50 nm RMS to guide roller radius error at frequencies between 3 and 300 mHz. To convert this to cycles/degree on the drive rollers, it is necessary to know the reduction ratio (1/25) and 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. For declinations less than 30° and greater than 37°, the azimuth tracking rate is never larger than three times the sidereal rate (15°/hour). At three times sidereal, the first harmonic has a frequency of 0.9 mHz and does not contribute to tracking error in the range cited above. The error contribution from the second through fifth harmonics is 44 to 65 nm RMS for the drive rollers. This is more or less consistent with our error budget.
These results are significantly better than the results reported for the new 3.5-m drive assemblies, SDSS Technical Report 19940919. However, the distribution of the runout among the rollers is non-gaussian (Figure 4). This suggests that one or more factors may not be well-controlled in the grinding process. The identification and correction of such factors may result in lower runout.
Fig. 4: Histogram of the amplitude of the first harmonic of all unassembled rollers. The distribution is non-gaussian.
We are greatful for the assistance of Mr. Terry King, Mr. Paul Baird and the personnel of the assembly shop of L&F Industries, Huntington Park, CA.
