Sloan Digital Sky Survey Telescope Technical Note 19950209-01
The plug-plates of SDSS project are responsible for locating the optical-fiber plugs spatially and for defining the plug tilt with respect to the surface of best focus. The plates are 787 mm (31") in diameter and 3.2 mm (0.125") thick. Approximately 700 holes will be drilled in each plate. For drilling, the plate is held by a drilling fixture that deforms it elastically so that its upper surface is convex. The center of the drilling region is about 10 mm higher than the edge. The hole axes are drilled vertical. In the telescope, the plate is deformed to match the surface of best focus. When this is done, the hole axes are aligned with the principal rays from the optics.
Two plates were drilled at D-Velco Manufacturing on December 21-22, 1994. A horizontal milling machine, a Mitsui Seiki HR-7A #134, was used. The machine has a travel of 1750 mm (69") in x and 1500 mm (59") in y. This was more than adequate to reach the entire drilling region.
Fig. 1. Histogram of the x location error for dv100. The distribution is non-normal. The errors are dominated by systematic effects, especially a difference in scale in x and y.
D-Velco plate 100 (dv100) took 125 minutes and D-Velco plate 102 (dv102), 120 minutes. (There was no dv101.) This did not include the time required to set up the plate for drilling. It did include the time to measure the coolant temperatures (5 measurements).
The hole drilling order for dv102 was not well optimized whereas the order for dv100 was optimized using the simulated annealing travelling salesman algorithm (from Numerical Recipes, William H Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery, Cambridge, New York, 1994). This may account for the 4% difference in drilling time. However, possible variations in the time spent making manual measurements make it difficult to conclude anything other than that the drilling order does not have a large effect on drilling time. This is not surprising since the machine moves quickly between holes.
The drilling time is more than twice that estimated in SDSS Technical Note 19941206, i.e., 44 minutes. Dv100 took 70 minutes to set up and 30 minutes to unload. Dv102 took >15 minutes to set up and 5 minutes to unload. The load/unload operations were considerably more difficult on the horizontal milling machine at D-Velco Manufacturing than on the vertical milling machine at the University of Washington. D-Velco Manufacturing generated the CNC program from the table of hole locations and depths.
Fig. 2. Histogram of the hole diameter for dv100.
Fig. 3. Histogram of the hole diameter for dv102. The standard deviation of the distribution is similar to that for dv100.
The plates were drilled at 3500 rpm. A different 9.5 mm long spade drill bit was used to drill each plate. The diameter of each bit was specified to be 0.0867+0/-0.000,50" (2.202 +0/-0.001 mm). The bits were made of carbide steel by Johnson Carbide Produces, Inc., Saginaw, Mich. A custom drill bit holder (described in SDSS Technical Note 19940412-01) was made by D-Velco Manufacturing to minimize bit runout.
The temperature of the coolant was measured at 30 minute intervals during the drilling. For dv100, the temperature was 21.1 +/- 0.2°C. For dv102, the temperature was 21.15 +/- 0.25°C. A new bit was used for each plate. Drill runout was measured prior to drilling and was 1.3 µm (0.000050") for each plate.
Before shipping the plates to Fermi National Accelerator Laboratory (FNAL), the plates were cleaned and power flushed. Robert Riley (FNAL) reports, "The holes in the D-Velco plates looked very clean, with very little contamination."
The plug-plates were measured on February 2, 1995 at FNAL. A Giddings & Lewis-Sheffield Measurement, Inc., Apollo RS-50 coordinate measuring machine (CMM) with an accuracy specified at +/-2.5 µm (0.0001") was used for the measurements. The CMM was checked by Giddings & Lewis technicians on January 25 and 26 and found to be within calibration.
The plates were measured flat on the CMM. The two 4.76 mm (0.1875") locating pin holes that are at a radius of 349.3 mm (13.750") and define the x-axis were used to center and orient the plate. Twenty-one points were measured on the top of the plate and the average of these became z = 0. Dv100 flatness was 0.36 mm (0.0142") peak-valley. Dv102 flatness was 0.50 mm (0.0198") peak-valley. Since the maximum hole tilt is about 30 mrad at a radius of 230 mm, the maximum hole location error due to the lack of flatness of the plate is 15 µm peak-valley.
The CMM extracts hole location, diameter and non-circularity from measurements at eight points equally spaced in angle at the same value of z. Non-circularity is defined as the difference in radius between the points closest to and farthest from the center of the hole. Consequently, non-circularity is quite sensitive to contamination of the hole. These parameters were recorded at three different heights; -2.5375, -1.5875 and -0.3810 mm (-0.1000", -0.0622" and -0.0148").
The hole locations at the three heights were averaged to obtain a mean hole location, x and y. The desired hole locations (the drilling machine coordinates) were subtracted from these values to get hole location errors. The functions f(x) = dx + b1*y + (a1 + a3*r^2 + a5*r^4)*x and g(x) = dy - b1*x + (a1 + a3*r^2 + a5*r^4)*y were fit to the x and y errors respectively. The coefficient a1 includes the effect of thermal expansion between drilling and measurement and the lowest order effect of bending the plate for drilling. The coefficients a3 and a5 account for higher order effects due to the drilling fixture. The coefficients dx and dy are the offset of the plate center between drilling and measurement. The coefficient b1 is the rotation of the plate between drilling and measurement.
During operation, guide stars on 5 arc-second diameter coherent fiber-optic bundles will be used to determine the actual value of a1, dx and b1 and the telescope scale, pointing and rotator angle will be adjusted accordingly. The telescope scale is adjusted by moving the primary axially and refocusing. Consequently, errors in these coefficients may affect the initial acquisition of the guide stars, but will not affect the ability of the telescope to center the targets in the spectrograph fibers.
The holes for the coherent fiber-optic guide bundles are the same diameter as the holes for the spectrograph fibers and will be drilled intermingled with the spectrograph fiber holes. Consequently, we expect that the coherent fiber-optic guide bundles will share the same mean location and orientation statistics as the spectrograph fibers.
The fit coefficients are given in Table 1. The a3 and a5 coefficients cannot be determined separately for each plate during operations without measuring each plate. Since this is not envisioned, these coefficient were set using finite element model results. As for the UW plates, the plate center and rotation offsets would correspond to 1 to 2 arc seconds (the scale is 60 µm/arc second).
Table 1: Hole location fit coefficients for each plate
Plate a1 a3 a5 b1 dx dy (mm/mm) (mm/mm^3) (mm/mm^5) (mrad) (mm) (mm) 100 -0.273 9.26E-06 -4.65E-11 -0.039 58.5 64.6 102 -0.280 9.26E-06 -4.65E-11 -0.001 51.9 7.2
Table 2 summarizes the results of the hole measurements. The histogram of Figure 1 is not normal since distribution is dominated by systematic errors, i.e., primarily a difference in scale in the x and y axes. The histograms for the y-axis and dv102 tend to be asymmetric and/or non-normal also.
Histograms of the hole diameters are shown in Figure 2 and Figure 3. The standard deviation of hole diameter for dv102 is similar to that for dv100. Figure 4 shows that small hole diameters are correlated with large noncircularity. A similar but larger effect was reported in SDSS Technical Note 19941206 and was attributed to contamination in the holes.
Fig. 4. Non-circularity is plotted vs. diameter for the mid-level data for dv100. These data are typical of those for other levels and for dv102. Small non-circular holes may be due to contamination.
The radial components of the hole location at the top and bottom of the each hole in combination with the separation of the two measurements were used to calculate the tilt of each hole. The hole tilt as a function of radius is compared to the ideal tilt from the optical design (kent005) and to the finite element model of the plug-plate in its drilling fixture (drl42) is shown in Figure 5 and Figure 6.
Table 2: Results for each Plate
Plate Pos Error Diameter Error Non-Circ Tilt RMS mean std dev RMS RMS std dev RMS (mm) (mm) (mm) (mm) (mm) (mrad) (mrad) 100 16.6 6.5 3.2 7.3 8.3 1.5 2.3 102 13.4 10.0 3.6 10.6 13.3 1.4 2.4
Fig. 5. Hole tilt is plotted as a function of radius for dv100 (open circles). The filled squares are the optimal tilts from the optical design. The filled diamonds are the tilts calculated from the finite element model. The RMS error is calculated with respect to the kent005 curve. The standard deviation is that of residuals to a 7th-order odd-polynomial fit (as shown).
Fig. 6. Hole tilt is plotted as a function of radius for dv102 (open circles). The other symbols have the same meaning as in Figure 5.
The temperature of each plate was monitored during measurement by taping a thermocouple probe to the plate. Thermally conductive grease was used. Each plate took 3 hours to measure and the temperature was recorded every half hour. During measurement of dv100, the temperature was in the range of 18.7 to 19.4°C. For dv102, the temperature was in the range of 18.9 to 19.2°C.
Fig. 7. Mean hole diameter is plotted vs. time-ordered quartiles for each plate. The error-bars are plus and minus one standard deviation in the mean diameter. These data do not show a significant trend, e.g., due to drill bit wear.
The large residual errors in the measured locations of the holes drilled by D-Velco Manufacturing are due to large-scale effects, not random errors. The most important is a difference in the scale in the x and y axes. To explore this in more detail, we fit an eight parameter model to the data. For comparison purposes, we fit the same model to the data for University of Washington (UW) plate 100 (uw100) and UW plate 102 (uw102). The functions f(x) = dx + rx*y + (sx + a3*r^2 + a5*r^4)*x and g(x) = dy - ry*x + (sy + a3*r^2 + a5*r^4)*y were fit to the x and y errors respectively. The parameters dx and dy are the mean offsets of the holes in x and y. The mean of rx and ry is the mean rotation of the holes about the plate center (positive counterclockwise). The difference of rx and ry is the non-perpendicularity of the x and y axes. The parameters sx and sy are the scale factors errors in x and y. Finally, a3 and a5 are the high order distortion coefficients due to the bending of the plate for drilling.
The residual 2-d errors indicate that the random error of the measurements is very comparable for the UW and D-Velco Manufacturing plates (Table 3). The Table indicates that the main systematic error in the D-Velco Manufacturing measurements is that sx and sy are different by 160 and 210 µm/m for dv100 and 102 respectively. The non-perpendicularity indicated by the rx and ry coefficients is comparable for the UW and D-Velco Manufacturing measurements. This non-perpendicularity in the UW data was noted but not removed in SDSS Technical Note 19941206. The uncertainties in a3 and a5 are such that the differences in these coefficients between plates are not very significant.
Table 3: Results of an eight parameter fit to each plate.
Plate dx dy rx ry sx sy a3 a5 error µm µm µrad µrad µm/m µm/m µm/m^3 µm/m^5 µm RMS uw100 19.9 16.6 54 0 -253 -257 8557 -46900 6.1 uw102 20.1 36.8 170 115 -225 -232 8859 -50800 5.7 dv100 58.9 63.5 -30 52 -134 -53 5460 -30364 7.7 dv102 51.7 7.6 -4 2 -182 -102 6096 -32354 5.8
A minor objective of this study was to learn a bit about how many holes could be drilled with each drill bit. (Also, see SDSS Technical Note 19941206.) The hole diameter data was divided into quartiles. The first quartile is the first quarter of the holes drilled, and so on. The diameter data for all three levels is averaged and plotted for each quartile (Figure 7). Since the diameter of a given hole is likely to be correlated along its length, the error bars are calculated by dividing the standard deviation for each quartile by the square root of the number of holes in each quartile, not by the square root of the number of measurements. These data indicate no strong trend in hole diameter, nor in the standard deviation of the hole diameter.
The error budget that we proposed in SDSS Technical Note 19930430 allows 9 mm root-mean-square (RMS) for hole location and 8 mm RMS for plug/hole concentricity. The position error measured for the D-Velco plates is not consistent with our error budget. It is not clear whether the scale error difference reported above occurred during measurement or drilling.
One possibility is that the scale difference is due to an anisotropic strain that was imposed on the plate during drilling or measurement. The scale difference is 81 µm/m and 80 µm/m for dv100 and dv102 respectively. The elastic modulus of the aluminum alloy is 73.1 GPa and its thickness is 3.2 mm, so a stress of 5.9 MPa and a force of 19 kN/m (108 lb/in) would be required. The clamping force needed to deform the plate in the drilling fixture is only 3 kN/m (17 lb/in). It seems unlikely that friction between the plate and the bending fixture during the clamping operation could cause a stress of the required magnitude. Turning the argument around and assuming a friction coefficient of 0.5 between the plate and the bending fixture, the maximum expected strain would be 6 µm/m, much too small to be a concern.
In the absence of deformation of the plate during drilling or measurement, it would appear that either measurement or drilling machine error is the source of the scale difference. However, this is surprising given the care with which both machines are calibrated and maintained. We plan further tests to determine the source of the scale difference.
The RMS diameter error continues to be a concern, but would be alleviated if bit to bit variations were better controlled. For example, the mean diameter for dv100 was +10 µm (as compared with the nominal drill diameter). With such a large departure from the nominal diameter, it will be very difficult to achieve a good fit of plugs in the holes. The standard deviations of the hole diameters for both plates were smaller than those reported for uw100 and uw102 (5.7 µm and 3.7 µm, see SDSS Technical Note 19941206).
The tilt results are more than adequate, and very comparable to those measured for the UW plates.
We are grateful to our colleagues at FNAL, Paul Mantsch, Robert Riley and Charles Mathews for their help with the measurement of the plates and their interest in and assistance with various aspects of plug-plate drilling. We thank Steve Quick and John Hance of the D-Velco Manufacturing for their interest and expertise.
