Azimuth Axis Bearing Configuration
Figure 1
The pointing and tracking specifications must be broken down into their numerous different contributors. Detailed error budgets for both pointing and tracking are attached as Appendices 1 and 2. This report will address one aspect of these error budgets, the contribution of azimuth axis wobble due to upper and lower azimuth bearing runout. The pointing tolerance due to azimuth bearing runout is 0.4 arc seconds rms and the tracking tolerance is 0.034 arc seconds rms.
Initially, the factor of 10 difference in these two specifications may be somewhat confusing, after all, the cause of both of these errors is the same, azimuth axis bearing wobble. How can they be more than a factor of 10 different? The reason is that the pointing specification must be met at all points in the sky, the tracking specification only needs to be met for short intervals during tracking. A more detailed breakdown of these specifications is discussed in the next section.
The wobble is measured by a single sensor mounted on the azimuth axis. Measurement at only one position will only detect one axis of wobble. We will make the assumption that the two axes are statistically independent and equal in magnitude. (This is a worst case assumption. Some of the errors, such as the 11 c/rev error of the bogeys is very well correlated and therefore the measurement of only one axis is required to characterize the rms error). If the two axes are independent and equal, the magnitude of the error measured on one axis is 1/root(2) of the total rms error. Therefore the axis wobble measured on one axis must be less than 0.28 arc second rms.
It could be argued that 40x earth rate is a very high tracking rate which will not occur very often in actual operation and that 10x or 20x earth rate might be more reasonable. On the other hand, it may often not be possible for the guide camera to update at 10 second intervals because of the lack of sufficienty bright guide stars. In those situations the guide update rate my drop by a factor of 2 or 4. Taking all of these variables into consideration, a rotation frequency of 100 cycles/revolution appears to be a good compromise.
As with the pointing error, bearing runout will only be measured on one axis. Assuming that the errors are statistically indepedent, the error measured on one exis must be less than 0.034/root(2) or 0.024 arc seconds rms.
Azimuth
Axis Bearing Configuration
Figure
1
Referring to Figure 1, there are two main contributors to azimuth axis wobble, the upper azimuth bearing consisting of 5 cam followers riding on the precision ground azimuth ring and the lower azimuth bearing consisting of a spherical roller bearing. We will postulate that runout of the upper azimuth bearing causes a wobble of the azimuth axis about the center of the lower spherical bearing (point A) and runout of the lower azimuth bearing will cause a wobble of the azmuth axis about the plane of the upper azimuth axis bearing (point B). The validity of this assumption will be established later in this report.
Since there are two unknown disturbances effecting the motion of the azimuth axis, two measurements should be sufficient to resolve the magnitude of the two seperate disturbances. For the first measurement a one inch tooling ball was mounted on a 5 ft steel tube attached to a steel base plate. This tube was inserted through the center hole of the azimuth axis so that it protruded above the top of the horizontal fork member. The tooling ball was mounted on the post with a 2 axis cross slide so that it could be accurately centered with respect to the rotation axis. A lever type Mitutoyo electronic indicator head was mounted to the rotating fork using a magnetic base and adjsuted to measure the runout of the fork relative to the tooling ball. (Note that the indicator rotated and the tooling ball was stationary.)
For the second measurement, the steel tube was removed and the tooling ball was mounted directly on the steel base plate below the azimuth axis. The indicator was mounted on a plate attached to the bottom of the azimuth axis. Once again, the indicator rotated with the fork relative to the stationary tooling ball.
For both measurements the fork was rotated at a constant rate of 1 revolution in a period of from 2 - 8 minutes by a velocity servo. The data from the indicator and the Farrand position output was digitized by a computer . In all cases 4096 points were digitized at rates from 8.7 Hz to 90 Hz. The resulting data was plotted and Fourier transformed using the Mathcad worksheet in Appendix 3. This worksheet could also compute the rms motion and pointing error between any two frequencies.
Azimuth Axis Measurement Geometry
Figure
2
Using the statistical runout data for the upper and lower tooling ball positions, the runout of the upper and lower bearings can be computed. (refer to Figure 2) It is assumed that the motions generated by the upper and lower bearings are statistically indpendent and therefore their effects must be added in quadrature. This should be a safe assumption in the higher frequency regions above 100 cycles per aperture. The measured motions at the upper and lower tooling ball can be written as:
Where Mxx are motions at the various points in the model and are indicated in Figure 2.
Substituting in the values for the constants from Figure 1:
2.
Since Mut and Mlt are results we can measure, the rms disturbances caused by the upper and lower bearings Mub and Mlb can be determined. Knowing Mub and Mlb the rms azimuth axis wobble can be computed using equation 3.
Azimuth
Axis Runout, Upper Tooling Ball.
Figure
3a.
Azimuth Axis Runout Frequency Spectrum, Upper Tooling Ball
Figure 3b.
Azimuth
Axis Runout, Lower Tooling Ball.
Figure
4a.
Azimuth
Axis Runout Frequency Spectrum, Lower Tooling Ball
Figure 4b.
There is an anomoly in Figure 3a at about 159 degrees. This is most likely due to a small scratch in the tooling ball. There is also an anomoly in Figure 4a at about 65 degrees which is due to the an interference between indicator cable and the tooling ball mount. Neither of these anomolies will have a measureable effect on the results.
There are two common characteristics which are present in both Figures 3a and 4a. First, each has a periodicity with a period of about 33 degrees. This is related to the rotation frequency of the bogey wheels on the upper azimuth bearing. An examination of figures 3b and 4b shows a peak in the frequency data at a corresponding 11 cycles/revolution. Secondly, both graphs have high frequency noise. This high frequency noise is related mechanical resonances between the mount and the tooling ball. The resonances have temporal frequencies of approximately 15 and 17 Hz. The frequency peaks related to these vibrations occur slightly below and slightly above 2,000 cycles/revolution.
To determine the rms runout which affects tracking at the upper and lower tooling ball positions the square of the fft should be integrated from 100 cycles/revolution to infinity. Since we have finite data sampling the upper frequency limit in these measurements is restricted to 4,000 cycles/revolution. Unfortunately, if the data is integrated from 100 - 4,000 c/rev the two peaks from the mechanical vibration add a significant error into the measurement. To avoid this problem the integration is only carried out from 100 - 1,500 c/rev. Since the power of the Fourier transform is decreasing continously above 100 c/rev (excluding the two anomolous peaks), the exclusion of the higher frequencies does not make a significant change in the value of the integral.
For each tooling ball position four, half revolution measurements were taken and integrated from 100 - 1,500 c/rev. The resulting integrands were averaged to give an rms error for both positions. The results are shown in Table 1. Table 1 also shows the result if integrating from 9-13 Hz to determine the rms height of the 11 Hz peak.
The 11 cycle/revolution error is known to be introduced at the upper ABA. The ratio of the amplitudes of this signal measured at the two tooling ball positions can be used to determine if the azimuth axis really is wobbling about the spherical cener of the lower bearing. If this is truly the pivot point for the wobble, the ratio of these two measurements should be equal to the ratio of the distances between the top and bottom tooling balls to the center of the spherical bearing. From Figure 2 this is V/(X+W).
4.
The results shown in Equation 4 indicate that disturbances introduced by the upper bearing do cause the azimuth axis to pivot about the sherical center of the lower bearing. It is desirable to perform a similar analysis to prove that disurbances introduced at the lower bearing pivot about the upper bearing but there are no distinct frequency peaks introduced by the lower bearing which can be used for the measurement.
We can now use Equation 3 to solve for the rms runout of the upper and lower bearings. The results are shown in Table 2.
Azimuth Axis Wobble Summary
Table
2.
The specification for this measurement is 0.024 arc seconds indicating that the wobble is 33% to large. At this time we are in theprocess of replacing the upper azimuth axis bogey bearings because of another problem. It is expected that the new bogeys which will incorporate precision bearings (as opposed to the current standard bearings) will reduce the error to below the specification.
Applying this data to the evaluation of pointing requires some some preprocessing. The experimental setup which we used introduces a significant amount of 1 c/rev error because of the inability to perfectly center the tooling ball. It turns out that periodic errors such as these are not relevent. The TCC software has the ablity to experimentally determine the periodic errors in the ponting and tracking hardware up to 99 cycles/revolution. Therefore, prior to analyzing the rms error from 0-infinity c/rev all low order harmonics are removed. A single value decomposition program courtesy of Walter Seigmund was used to remove the first 10 harmonics from the data prior to computing the rms error Rather than compute the rms error in the frequency domain the rms error was computed in the time domain. This eliminates the error due to the windowing function which can significantly reduce the contribution from lower frequency components. It does include the high frequency preiodic error from the mechanical vibration but this makes a small contribution since there is much more energy at low frequencies. The error which is introduced by the high frequency vibrations will bias the measurements to the high side. The resulting rms amplitudes at the upper and lower tooling balls are shown in Table 3.
Tooling Ball Runout (First 10 harmonics removed)
Table
3.
Using Equations 2 and 3 we can compute the upper and lower bearing runout and their contributions to pointing error.
Azimuth Axis Wobble Summary
Table 4.
These results are well within the specification of 0.28 arc seconds rms.
The results of these measurements of the azimuth axis bearing indicate that the pointing specification is easily met, even without removing the ~11c/rev error introduced by the upper azmuth bearing bogey wheels. The new bogey wheels will be moer accurate and the periodic 11 c/rev error will be shifted to exactly 11 c/rev (it is currently slightly greater). This will allow any residual to be removed very accurately using TPOINT.
The tracking performance of the azimuth axis slightly exceeds specification due primarily to the contributions of the upper azimuth axis bearing. The high frequency wobble contributions from this bearing are caused by roughness in the rotation of the bogey wheel bearings and the residual roughness of the azimuth axis drive ring with the major contributor being the bogey bearings. These bearings are being replaced by precision bearings which should improve the upper bearing performance and bring the high frequency wobble within specification.
The data proved that runout introduced by the upper azimuth bearing caused a wobble which was pivoted about the spherical center of the lower azimuth bearing. This result was important because there have been concerns regarding the suitability of this heavy duty bearing for our application which only loads the bearing to less than 1% of its rated load.
These measurements should be repeated once the new bogeys are installed. To determine if the required performance improvement was achieved.
EOST Absolute Pointing Error Budget
Cause rms Variance
pointing
error (arc
seconds)
Secondary mirror pointing uncertainty 0.62 0.3844
Tertiary Rotator pointing uncertainty 0.1 0.01
Tertiary actuator pointing uncertainty 0.1 0.01
Azimuth axis position uncertainty due to Farrand encoder 0.05 0.0025
Elevation axis position uncertainty due to Farrand encoder 0.05 0.0025
Instrument rotator position uncertainty due to Heidenhain 0.1 0.01
encoder
Wind load induced deformation of mount 0.5 0.25
Wind load induced deformation of tower 0.6 0.36
Gravity induced deformations of truss and mount with 0.1 0.01
elevation changes
Gravity induced pointing changes in primary mirror with 0.1 0.01
elevation changes
Gravity induced pointing changes in secondary mirror with 0.1 0.01
elevation changes
Pointing uncertainty due to azimuth bearing runout errors 0.4 0.16
Total RSS Pointing Error 1.10 1.2194
Target 1.00
