# Azimuth encoder rack gear run error.

#### 21jan00

The azimuth encoder rack gear is mounted on the inside of the azimuth ring girder. The azimuth encoder turns on this gear as the azimuth moves 0 to 720 degrees. The angular distance that the azimuth moves is determined by the arc length along this gear. The gear itself is not completely circular. The run out is a measure of how far the rack gear deviates from a circle of the correct diameter (127 feet). The software assumes that this arc length along the encoder rack gear is correct (at a constant radius of 127/2 feet). The error in the radius of the rack gear becomes an azimuth error.

To measure the error in the azimuth encoder rack gear, jon hagen built a proximity sensor. It consisted of two plates of a capacitor. One plate was mounted on the azimuth encoder mounting bracket (this is fixed relative to the azimuth arm). The second plate was mounted on the spring loaded mechanism that pushes the encoder into the encoder rack gear (this part moves). Measuring the voltage as the distance between these two plates changed gives the radial distance that the encoder rack gear is moving. The device was calibrated by mounting it in a milling machine in the lab and measuring the inches per ohm (.000425 inches/ohm).

We measured the run out error by moving at .02 deg/sec azimuth velocity (using the encoder arc length) and sampling the voltage at a 5 hz rate.

To convert from radial error to angular we need to define some terms:

• ds :  The differential arclength the encoder covers as it moves.
• R_avg is the average radius of 127/2=63.5 feet.
• R_T: The true radius when the encoder moves through the length ds (actually it is the average true radius along ds).
• dR   = R_T - R_avg.  Error in radius at each point (actual - ravg).
• dTh_M: The  differential angle theta measured by the encoder as it moves though the arc ds. The computation used is:
• dTh_M= ds/R_avg .
• dTh_T: The true angle we moved through when the encoder moved along the path ds.
• dTh_T=ds/R_T = dTh_M * R_avg/R_T = dTh_M * R_avg/(R_avg + dR)

•     To compute the az encoder runout error:
1. Measure the resistance with jon hagens device as it move through 360 degrees.
2. Subtract off the average resistance. The resistance is then measures dr.
4. Compute dTh_T from dth_M and dr.
5. integrate dTh_T from 0 to 360 (in steps of constant dTh_M
6. azErr = Th_T - Th_M .. If you add azErr to the measured the, you'll get the correct azimuth.

The measurement used a clockwise azimuth spin from 70 to 710 degrees followed by a counter clockwise spin from 710 down to 340 degrees.

The plots (.ps)  (.pdf) show the results of the runout measurement:

• Top:  shows the resistance versus azimuth. The dark lines are the clockwise spin while the red lines are the counter clockwise spin. The measurement technique is repeatable.
• Middle:  converts the resistance measurements to dr. We've subtracted off the average resistance from each spin and then mulitplied by the inchesPerOhm factor. It is negative since an increasing resistance gave a shorter radius.Between az = 140 and az=160 degrees the rack gear moves in and out radially by over 1 inch. The other oscillation corresponds to the length of the rack gear segments (i think they are every 10 or 15 degrees). There is a bracket holding the two pieces together forcing them to have the same radius at that point. This bows the sections between the connection points.
• Bottom: The azErr vs azimuth (azTrue - azMeasured). The units are arc seconds at the encoder. The green line is an eleventh order harominc fit to the data (saved in coefIRunout.sav). The idl routine fitsinneval() can evaluate this fit for you.

• The 2nd page shows the frequency spectrum of the runout (.ps) (.pdf):
The function Ai*sin(i*az - ph(i)) (i=0..50) was fit to the measured resistance vs azimuth.
• Top plot: The ohms data vs azimuth. Black is the cw spin, red is the ccw spin. The green lines are the fits.
• Center: The fit residuals for the cw and ccw spins.
• Bottom: The amplitude of each frequency (1..50) .
• The largest term is 3 which has a period of 360/3=120 degrees. This is the period of the triangle. We know that the vertical motion on the track has a strong 3 azimuth term caused by the hard and soft spots between the corners and middle of the triangle beam. It is a bit surprising that this is also the largest term in the perpendicular (radial) direction.
• The next largest terms are: 7,12,21, and 24. The rack gear is welded to the ring girder in sections. Between the welded hard spots the section may have a non optimum curvature. This period will probably be one of these frequencies.
A separate measurement of the error in the rack gear was made using a theodolite mounted at the main bearing of the azimuth arm ( i need to add that data to this page).

The measured azimuth errors  were applied to the pointing error residuals from model11 but they did not reduce the residual errors (this needs to be looked into).

Another check would be to look at the difference in the azimuth encoders (from both sides of the azimuth arm). Their difference should equal the difference in the error at the two positions.

`processing  : x101/000121/doit.pro. the analyz code is in x101/phil/analzy/anz.21jan00t`
` home_~phil`