On March 23, 2000 we installed Heidenhain 2571 encoders on the 3.5-m secondary mirror and used them to measure mirror motions under various conditions. A total of 4 encoders were installed. Three of them were positioned radially outward from the mirror actuators near the edge of the mirror. The forth encoder was positioned to measure the transverse motions of the secondary mirror. Figure 1 shows the location of the encoders and mirror actuators on the back of the 3.5-m secondary mirror. The Heidenhain encoders have 2 µm periods. With 4 pulses per periods and x10 interpolators, these encoders generate a pulse every 0.05 µm. The transverse encoder's plunger is spring loaded against a right angle bracket that is mounted to the top of the transverse support rod on the secondary mirror. The other three encoders have plungers which are attatched to the back of the secondary mirror by means of metal pads which are RTVed to the mirror. The readout of these encoders was done with the Galil controller. The auxillirary port on the Galil was used to obtain position readouts of these encoders at a rate of approximately 20 Hz. This full data set was recorded, but is not analyzed here. The measurements reported on here come from the static sampling that is available directly from the secondary encoder positions reported by the TCC. Note that with the current Galil software, it reports the encoder positions in "microsteps". In this case, this just means that the TCC multiplies the encoder pulses by a factor of 50. Russ calls the microsteps "steps" in the TCC and the raw encoder pulses "ticks". You can get at the pulse counts directly from the Galil controller by issuing a host session "TP" command. That was the method used to acquire the data reported here.
After the encoders were installed on the back of the mirror, but before we replaced the secondary cage on the telescope, we took a series of measurements with the encoders. For these "off telescope" tests we also installed two Mu Checker linear gauges right next to the B and C Heidenhain encoders. The Mu Checkers allowed us to see the high frequency response of the mirror in real time and also allowed us to check the scale of the Heidenhain encoders. Figure 2 shows the results of some of these measurements. The off telescope measurements immediately confirmed two things. First it was seen that the C axis was much more sensitive to shock vibrations of the secondary cage than any of the other axes. Small impules of a few pounds on the secondary vanes would cause the C Mu Checker readings to oscillate by about 2 µm. The amplitudes seen in the other axes were much smaller and tended to dampen more rapidly. Secondly, in the good agreement between the Mu Checker readings and the Heidenhain encoders we can see that we have no scaling problems with the Heidenhains.
After the off-telescope measurements were completed, we reinstalled the secondary on top of the telescope and took a series of measurements. Because we had readout channels for only 3 encoders at a time, we had to accomplish the on-telescope measurements in two stages. The first set of measurements were designed to measure the transverse motion of the secondary as a function of telescope elevation angle. At the same time we measured the hysterisis in the A and B actuator motions. Figure 3 shows a sample of the transverse sag measurements and Figure 4 shows an example of the axial hysterisis measurements. The maximum transverse sag between the secondary mirror and its support cage is about 200 µm. The hysterisis in the transverse sag is approximately 15 µm. The maximum axial sag for both the A and B actuators is about 75 µm. The hysterisis in the A and B actuators is approximately 7 µm. The measurements shown in Figures 3 and 4 were repeated. At any given elevation angle the measurements repeated to within about 2 µm.
I should point out here that there is a significant difference between the A and B actuators which should be kept in mind during these comparisons. Actuator B has the piezo installed while actuator A has an aluminum slug in place of its piezo. The similarities between the sag and hysterisis seen in actuators A and B show that these motions are not coming from the piezos.
The following calculations indicate that the transverse sag of 200 µm that we have measured here is probably not due to flexure in the central post which supports the secondary mirror when the telescope is at the horizon. It appears that some other component of the secondary support system must be in motion in addition to the flexures that we calculate here.
The central post which supports the secondary mirror at the horizon is a stainless steel shaft of 0.5" diameter. This post is held into the secondary cage by the use of two trantorqe bushings. In the middle of the secondary mirror, near its center of mass is a linear bearing which rides on a gimbol. This gimbol allows the mirror to tilt on the linear shaft. The distance between the nearest trantorque bushing and center of the gimbol is 1.73" when the mirror is close to the center of its total range of motion. The total motion of the secondary is limited in software to approximately 0.375".
The mirror on the central post is a catllevered end load. The equation for the maximum deflection of the supporting beam in this situation is given in many engineering texts and is:
y = F l3/ 3EI
where y is the deflection of the beam, F is the load on the end of the beam, l is the distance from the load to the fixed end of the beam, E is the beam's modulus of elasticity, and I is the area moment of inertia of the beam. For a beam with a circular cross section, the moment of inertia is given by
I = źd4 / 64
where d is the diameter of the beam. For a half-inch beam we have I = 0.00307 in4. For 303 stainless steel the modulus of elasticity is 28 Mpsi. If we assume that the mirror is at the middle of its focus range, then we have l = 1.73 - (0.375 / 2) = 1.54". If we assume that the approximate weight of the mirror and the hardware that is directly attatched to it is 100 lbs, then we have y = 0.00142 in = 36 µm. Similarly, if the mirror is at the farthest extent of it's focus travel, we have l = 1.92" and y = 70 µm.
The calculations above indicate that the maximum transverse mirror sag which can be explained by the flexure of the central post is only 70 µm. Figure 3 shows a maximum transverse sag of 200 µm! The data in Figure 3 were obtained with the mirror positioned at the home position where the mirror is as close to the support cage as it can get. We would therefore have expected to see only about 36 µm of flexure if this motion were the result of the deflection of the central post. We are therefore seeing flexures of more than just the central post. The trantorque bushings that hold the central post in place and the flex pivots in the central post gimbol are possible sources of this flexure.
After the transverse sag measurements of Figure 4 were collected, encoder C was connected in place of the transverse encoder and the axial sag of actuators A, B, and C were measured simultaneously as a function of telescope elevation angle. Figure 5 shows a sample of these data taken as the telescope was moved from the horizon towards the zenith. The axial sag of the mirror along axes A and B is seen to be very similar to that measured earlier in the evening. The sag seen in axis C is approximately 3 times that seen in axes A and B! There is no ambiguity in these results. Axis C is broken. Figure 6 shows that the axial sag seen in axis C is reproducible to a few microns. In this figure data from two altitude sweeps are plotted on the same figure. This comparison shows that the differential motions between axes A, B, and C are smooth and apparently predictable.
It is clear that the broken C axis is causing a tilt in the secondary mirror which is orthogonal to that which one would normally expect to see owing to sag in the secondary truss. The maximum differential motion between C and the A and B axes is 130 µm. As seen in Figure 1, axis C is 24.52 inches away from the A-B baseline. The maximum differential motion therefore equates to a maximum y-tilt of 43 arcseconds of the secondary mirror.
It is natural to ask whether or not the axial and transverse motions shown here have been detected by the Shack-Hartmann measurements that have been taken at various times since the telescope was put back in operation. We'll deal with the x- and y-tilt motions seperately.
In the nomenclature of the TCC, x-tilts are equivalent to a rotation of the secondary about an axis that runs parallel to the Nasymth telescope axis. X-tilts are therefore what one would expect from transverse motions of the secondary generated by sags due to gravity as it moves in elevation. Y-tilts are by definition orthogonal to these and are not expected to be seen in the secondary.
The transverse motions of the secondary shown in Figure 3 have clearly been seen in the Shack-Hartmann data. On Jan 8, right after the telescope was put back into service, between 74 and 35 degrees elevation angles we acquired a full set of Shack-Hartmann data. These data show coma misalignments which can be interpreted as either x-tilts or a transverse motion of the secondary-primary mirrors. Earlier we reported these measurements in terms of the x-tilts. The conversion between tilts which produce the coma and the equivalent transverse motions are given by the formula
a = l / dn
where a is the x-tilt required to produce a given amount of coma in the optics, l is the equivalent transverse motion, and dn is the location of the "neutral point" from the secondary vertex. For the 3.5-m dn= 1.24765 meters. On Jan 8, between the elevations of 74 and 35 degrees, the SH sensor measured a total coma correction tilt of 47.5 arcseconds. The equation above may be applied to show that this is equivalent to a transverse secondary motion of 287 µm. Figure 3 shows a transverse shift of 120 µm should be expected between the secondary and its cage over a similar range of elevation angles. The motions seen in the x-tilts on Jan 8 were smoothly varying functions of elevation angle. These two sets of results are very consistent. The difference between the 287 µm measured by the SH sensor and the 120 µm of secondary motion seen here can easily be attributed to sag in the secondary truss supports themselves. The suprise in these measurements is that nearly half of the secondary transverse sag is due to motions between the glass and its support cage!
As I have mentioned earlier, the differential motions seen between the C axis and the A and B axes are equivalent to y-tilts in the secondary. The measurements of Figures 5 and 6 indicate that these measurements are smooth and predictable. These data show that between 74 and 35 degrees elevation angles one would expect to see a differential axial motion of 80 µm. Given the geometry in Figure 1, this equates to a differential y-tilt of 26 arcseconds. The Shack-Hartmann data taken on Jan 8 do NOT show a smoothly varying y-tilt in the coma corrections. At that time the y-tilt corrections were quite constant until the telescope reached an elevation angle of 35 degrees. The previous measurement was taken at 45 degrees. Between 45 degrees and 35 degrees a single 17 arcsecond jump in the y-tilt corrections were seen. Subsequent Shack-Hartmann measurements as a function of altitude angle have been taken with the SH sensor at an unknown rotational angle (ever since it was moved to the TR2 port!). As such its impossible to distinguish x- and y-tilts in these data until that orientation is determined.
The discrepencies between the Jan 8 Shack-Hartmann data and the differential axial sags reported here are indications that either the Shack-Hartmann y-tilt data are unreliable for currently unknown reasons or there has been a change in the y-tilt behavior of the secondary between Jan 8 and March of this year. We currently have no other indications that we are getting bad results from the Shack-Hartmann analysis. If the second hypothesis is correct, then it appears that the C axis was not as bad when the telescope was first placed into service and has been deteriorating over time. This may be a very important question to resolve!