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θ5.4
HCC small angle evaluation
5.4.1
Background and experimental setup
By necessity, the numerical evaluation of the LEC is divided into two categories; the small angle and the full circle evaluation. The small angle refers to displacements not greater than ±300 arc-seconds. For angle displacements greater than approximately one degree, the other
plateau motions (i.e. tilts about the x and y axes and the translation movements along the x,
y and z axes) begin to influence measurements made by the instruments evaluating the
LEC. To avoid these influences altogether we examine very small angle displacements. For an idea of the magnitude of HCC spindle motions, the standard deviation and peak to peak values
of the motions shown in Figure 5.19 are less than 0.08 μm/degree and 0.2 μm/degree
respectively in all cases30.
A second reason to make small angle evaluations is that capacitive probes can be used. Given the experimental set up shown in Figure 5.20, and the uncertainty in capacitive probe
readings (0.16 μm for a period up to 24 hours – refer to section 6.4.2 and Table 6.6
specifically), we can expect to potentially resolve down to 0.16 μrad or 0.031 arc seconds of
angle motion. This is considerably better than any other instrument presently available at the ESRF.
To this end an experiment was made on the LEC, using a Möller-Wedel ELCOMAT 3000 autocollimator measuring to a 12 sided polygon mirror, and four capacitive probes measuring to a 1 m long bar installed on the HCC plateau. A number of temperature sensors were also employed. The experimental setup is shown in Figure 5.20
30 Referring ahead to the discussion of the HCC collimation error HCC
CE at the end of this chapter, these spindle motions translate to peak to peak motions of less than ±0.1 arc seconds per degree intervals in all cases. Thus the maximum error expected in the ‘small angle’ evaluation over the range of HCC displacements of ±300 arc seconds due to HCC collimation error influences is considerably less than 0.1 arc seconds. Assuming quasi- linearity over one degree, this error will be in the order of
( )
Figure 5.20 The small angle experimental setup. On the left hand side is a schematic while on the right hand side are two photos. The cut A is along the bar at the position of capacitive probes number 1 and 2. The bar was moved in small displacements of up to ±150 arc seconds or ±365µm at the capacitive probe positions (i.e. ~0.5 m from the centre). Angle
displacements were also measured by the ELCOMAT 3000 to a 12 side polygon mirror.
Positions of the temperature sensors are denoted by T1"T4 in the drawing to the left. A fifth
temperature sensor which is not shown was installed in the LEC encasement.
5.4.2
Experimental results
The experiment consisted of randomly moving the plateau in steps so that it was never more than ±150 arc seconds from its original position. The position of the plateau is shown in graph c) of Figure 5.21. The actual movements that were made are simply the differences between adjacent values in this graph. The top left hand graph shows the raw temperature over the study period. The readings from the four thermocouples shown in Figure 5.21 are clustered around 21 ºC while readings from a fifth thermocouple installed on the shaft linking the two RON 905 encoders (d in Figure 3.5) are clustered around 26.5 ºC. Filtered temperature
differences (i.e. Matlab filter command31) with respect to the first reading in the test series are
shown in graph b).
It is clear from this graph that there is considerable temperature variation at the LEC. One could speculate that this variation is due to its continuous rotation. However, there may be another, more banal reason for these temperature variations. The manner in which the cable connected to the RON 905 encoder is installed in the RV350 plateau causes it to ‘rub’
31 Temperature values in Figure 5.21 are the derived using a running average filter whose window size
is 15 minutes. The running average filter uses the Matlab filter command filter(ones 1, ws / ws, data)
(
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unpredictably against the continually rotating shaft of the LEC (see c in Figure 3.5). At the time these tests were made, this was not known. As we shall see, this can be accounted and indeed corrected for. However this is not the most elegant solution and the manner in which this cable is managed is the subject of a foreseen amelioration to the system.
Figure 5.21 Experimental conditions for the small angle tests. Graph a) shows the raw temperature. Graph b) shows the filtered temperature evolution. The blue line (i.e. largest displacements) is the temperature on the continuously rotating shaft of the LEC. Graph c) shows the plateau position over the experiment.
At first glance one is temped to correlate the LEC temperature variation graph b) with plateau position in graph c). Indeed this was done, as well as a correlation between the first differences (i.e. Ti+1−Tiand Pi+1−Pi where T and Prepresent the filtered temperature
evolution and the position respectively andPi+1−Pi is the plateau movement), and a cross
correlation between the temperature and position and their first differences. In all cases, there is no correlation whatsoever. Thus, although there is significant temperature variation, it is not correlated with either the magnitude or the direction of the plateau movement.
Figure 5.22 shows the results of measurements made using 10964 small angle
displacements over a period of just over 72 hours. The top graph a) shows the differences between angles measured by the ELCOMAT 3000 and the LEC (blue line) and the capacitive probes and the LEC (green line). These differences are moderately correlated (graphs b of this
figure;R=0.4 to 0.5) with the changes in temperature of the continually rotating LEC shaft
temperature (blue line in graph b of Figure 5.21). Applying these temperature models give the results of graph c) of this figure.
Figure 5.22 Experimental results of the small angle tests. The top graph a) shows the differences between measured angles for the ELCOMAT 3000 and the LEC (blue) and the capacitive probes and the LEC (green). The middle graphs show the temperature models for the capacitive probes and the ELCOMAT 3000 versus the LEC shaft temperature (graph b) of Figure 5.21. The bottom graph shows the temperature modelled differences between angles for the ELCOMAT 3000 and the LEC (blue) and the capacitive probes and the LEC (green).
The overall standard deviations for the full period of the test for graphs a) and c) of Figure 5.22 are summarized in Table 5.3. We remark a number of things. First, the capacitive probes
are in better agreement with the LEC than the ELCOMAT 3000. This is not surprising. After all, the manufacturer’s stated uncertainty for the ELCOMAT 3000 is 0.2 arc seconds.
Secondly, there is a modest improvement in the standard deviations of approximately 0.01 arc seconds using the temperature correction for both the capacitive probes and the ELCOMAT 3000.
Table 5.3 Overall standard deviations of the uncorrected and temperature corrected
differences between; the capacitive probes measuring to the 1 m long bar, and the ELCOMAT 3000; and the angles determined by the LEC.