The goals of our engineering analysis were to determine how light we could make the mount for an acceptable amount of flexure, to design a light support structure for the secondary mirror stiff enough to perform in the telescope yet small enough not to interfere with the telescope enclosure, and to determine the modes of vibration of the mount and to make their frequencies as high as possible.

Since the major parts of the telescope would be weldments of fairly thin steel plate, we represented the structure primarily with plate elements in the FEA, although the mirrors were a series of mass elements attached to the mirror cell, and link elements represented the secondary-mirror supports and the azimuth bearing in some calcultions of the whole telescope.

Many of the results of these calculations were entirely conventional. For instance, we found the well-known deflection pattern of the fork (see Gunnels 1990, SPIE vol. 1236, p. 854), an outward rotation of a tine where it attaches to the base and a corresponding inward rotation of the vertical part of the tine. We also found how deep the box sections of the fork and mirror cell weldments must be for acceptable deflections under gravity. We also determined that the major plates of the fork and tube could be 3/16-inch steel, perhaps even thinner, especially if we were using a light-weight mirror. In the telescope we have built, which will be primarily of aluminum to use an existing mirror cell, we simply substituted 1/2-inch aluminum for these parts.

In the following sections we give the results for calculations of individual components of the telescope structure, more or less in the order in which we did them.

(1) SECONDARY-MIRROR SUPPORT STRUCTURE.

This piece of the telescope is critical to our design, since it determines both the balance of the tube and how high the torques will be from wind loading. One goal of our design was to reduce the size, weight, and cross section of this structure as much as possible. The original conception of the secondary support was a four-legged (quadrupod) structure similar to the secondary-mirror supports on the 30-inch automatic photometric telescopes at Fairborn Observatory. This approach has the advantage of giving a rather light structure (of the order of 220 pounds) with a low cross section. It has the disadvantage of blocking a moderate amount of light with 1-inch struts (the same size as in HST) and having an obvious torsional mode (17 cps) about the optical axis (which, incidently, had an 8-14 cps resonance in the FEA models of WIYN). This quadrupod design also gives a secondary support structure that could be lifted off the telescope as a unit for maintainence.

a) Static Stability. In its basic form, the four-leg structure we've discussed gives moderately large deflections (of order 0.04 inches) from gravitational sag when pointed toward the horizon. Wind loading would also contribute a moderate amount of deflection (of order 0.01-0.02 inches). Using guy wires (or equivalent solid straps), as L.J. Boyd recommended to counter vibration from wind loading, will reduce these deflections significantly. The main effect is from a link between the base of the top leg to a point about 2/3's the way up the bottom leg. This link pulls up the bulge in the lower leg by about 125 lbs. and straightens out that leg. In the FEA calculations of this strengthened quadrupod, the mirror support points are decentered by 0.004 inches and tilted by 0.4 arcmin. These values are well within the restrictions on the optical system.

b) Dynamic Stability. We originally farmed out the calculations of the dynamical properties of the top end to the Manufacturing Center at Tennessee Tech. We had them calculating resonant frequencies for the structure, and they set the problem up in two or three ways, with different students using different approaches. In their initial modal analysis (Sept. 1996) of a simplified model of the quadrupod, they got frequencies around 1 cps, which would be easily driven by wind gusts and be too low for a useful telescope mount. We did our own dynamical calculations at TSU at about the same time, some in which we apply a force to the top of the quadrupod and calculate the oscillations and some modal calculations with ANSYS. We got modes with frequencies starting at 17 cps, the lowest frequency indeed corresponding to the torsional mode about the optical axis. The very different frequencies with respect to Tenn Tech probably resulted from TTU's using inconsistent units for the mass density or from simplifications in the way they attached the legs to the mirror cell. Tests on the prototype top end showed lowest frequencies even higher than those we calculated at TSU, while TTU has since found consistent results, long after the design of the mirror support had been decided. Indeed, we constructed a simplified prototype of this top-end structure to use to test the design experimentally. The material for it cost about $220, and the physical plant at TSU welded it up for us. We did FEA calculations specifically for the prototype design to compare with test loadings. One role of this prototype is to measure the amount of wind shake and torsional oscillation to be expected in the quadrupod design. The prototype has roughly the same vibrational modes and frequencies in FEA calculations as the more advanced design for the actual secondary support structure. We conducted field tests with an accelerometer in October 1996, finding lowest frequencies near 40 cps.

The first 20 modes of vibration of the quadrupod are in this table. Plots for some of these modes are: mode1, mode2, mode4, mode5, mode6, mode9, mode12, mode13, mode15,mode16, mode17, and mode19. This other view of mode19 shows the three-dimensional behavior of the quadrupod a little better. The Structural and Dynamics Testing Group of Marshall Space Flight Center tested the vibrational response of the telescope on 27 January 2000 and measured the resonances of the four-legged structure to support its secondary-mirror. Three examples of the data they collected (at the top side of the quadrupod) are (1) parallel to tilt axis (sensitive to torsional oscillations), (2) perpendicular to the tilt axis (sensitive to lateral swaying, or nodding modes), and (3) parallel to the optical axis (sensitive to squatting modes). We sincerely appreciate the efforts of Timothy Driskill and Chuck Seal in conducting these tests for us. They also ran tests to detect vibrations excited by the drives. The drive rates they tested were 1500 & 500 steps/sec in azimuth and 150 steps/sec in tilt, as well as approximate slewing rates of 50,000 steps/sec in both axes. These tests detected few vibrations driven at the resonances of the quadrupod, and those detected were at frequencies above 60 cps with amplitudes of the order of 0.01 g, excited at the faster drive rates for azimuth. The largest velocities expected in these vibrations would be of the order of 0.03 cm/s with displacements of the order of 0.1 microns. As might be expected, the strongest vibrations detected were in the top of the quadrupod in the direction of the motion. Interestingly, the drives did not seem to excite the torsional modes of the quadrupod. Vibrations during slewing were greater, but the largest effect there was probably vibration driven by instabilities in the PID algorithm, especially during deceleration. Settling times for decelerating from a slew were of the order of 1 second for both axes.

(2)THE TELESCOPE BASE AND PIER.

The base we've designed is a two-foot tall cylinder about 60 inches in diameter. It consists of two 2-in thick torroidal plates separated by a cylinder on the inside and a series of gussets. This design looks pretty stable. deflections are less than about 0.001 in. with reasonable assumptions about loading. Details of its design will include apertures for sucking air through the telescope, places for mounting the azimuth bearings, and details of mounts for various other parts of the mount (e.g, brakes, drive motors, limit switches, and shock absorbers for the hard limits).

The pier is a concrete slab anchored to the rock so near the the barren surface of our telescope site. The accompaning picture shows the levelled site and hole for the pier. We modeled the pier as a square slab of concrete loaded over the ring of the original telescope base. Deflections were of the order of 10^-5 inches, hardly a controlling source of source of guiding errors. Analyses of differential thermal expansion of pier and base showed possible problems for some kinds of azimuth bearings. Specifically, limestone-based concrete, which has a deciedly different coefficient of thermal expansion than steel, would stretch a base cylinder attached rigidly to it, canting the upper bearing-support surface. Thie effect would not be good for slewing-ring bearings but could probably be controlled.

(3) THE FORK.

The fork might have been adequate as originally designed, but we were able to reduce its weight significantly from that design of May, 1996. In intermediate designs, the total deflection of the tines under load was of the order of 0.01 inches. In the final designs with deeper box sections in the tines, the deflections are of order 0.003 inches. In all the models we considered the base behaved essentially as a solid unit, with most of the deflection under load resulting from the bending of the tines. Deflection of the base of the fork around its periphery is of the order of 0.001 inches, or less, roughly the smallest variation we could expect from the machine shop. It might have been desirable to reduce the deflections in the fork further, but we decided that was not practical. The structure of the latest designs for a steel fork weighed about 4020 pounds, with a drive ring included, (vs. 5000# + 600# for drives originally) and would be made primarily of 3/16-inch steel plate on a base plate of 1-inch steel plate. These calculations have been crucial in deciding how big to make ventilation holes in the base plate.

FEA calculations of both the fork+base as a unit (with masses attached to represent the tube) and fork+base+tube give frequencies near 20 cps for the principal modes of vibration (vs appx. 8-10 cps in WIYN). The main modes are a lateral swaying of the fork tines at 20 cps, a front-to-back swaying of the tines in phase at 20 cps, and a front-to-back swaying of the tines out of phase at 30 cps. Higher frequencies include overtones of these tine-bending modes, as well as modes in which various plates flex. The plate-flexing modes should be less prominent in the aluminum fork we have actually built because of the larger moment of inertia of the thicker plates. All the lower frequencies depend primarily on the scale of the structure and are changed very little by using thicker material in the tines or by making the cross sections bigger. This is the primary cause of the different resonant frequencies with respect to WIYN.

(4) THE TUBE.

The tube consists of a bucket-like structure to hold the primary mirror. This design assumes we will remove the mirror by lifting it out of the cell vertically with a crane built into the telescope enclosure. Such a strategy greatly simplifies design of the tube over such telescopes as WIYN in which the mirror is dropped out from under the tube with an hydraulic hoist. The lifting fixture in this design will be incorporated into the cell. Our original design for the tube was not stiff enough to resist sagging, and we have subsequently redesigned the tube with somewhat deeper sections. The FEA calculations also show this structure ccould be built of thinner steel, which would save some weight. The weight of the advanced design (Sept. 1996) is roughly 2200 pounds. There will also be about 800 pounds of levers to support the primary mirror, and this weight will be important for balancing the telescope. The tube has very different amounts of deflection when pointed at zenith and horizon (and, of course, in between). This is a necessary consequence of the very different moments of inertia resisting the gravitational loading in the two cases. A possible way of redusing this difference is using structural braces attached to a central drive wheel to increase the moment of inertia of the structure when pointed at the zenith. Conversely, any such braces must be strong enough to survive the sag expected and may have problems with fatigue in a constantly flexing telescope. The sagging of the base plate of the tube means the axial hard points must be very carefully designed and stiffened.

(5) THE PRIMARY MIRROR.

We made several FEA calculations to determine limits on variations in support of the primary mirror. These show that the existing axial and lateral support positions (36 points in three concentric rows) on the surplus mirror we have bought for the telescope will be adequate and that a 32-point axial support system, better suited to the new steel tube design, would also work. These FEA calculations have greatly clarified the relative forces required at the various support points and shown that our original estimates of those forces, based on the original 1970 engineering report, were wrong. We will lift the mirror with a lifting fixture through the central hole, which gives moderate stresses in the glass.

We have decided to use the original mirror cell with its 36 existing axial and 6 existing lateral support points. Consequently, we have conducted further FEA analyses of the mirror supports as a project that Allen Keel did as a project for a science fair. This study has clarified the forces required to support the mirror adequately and the toleration on setting up the forces at each of the support points.

(6) THE SECONDARY MIRROR.

The analysis of the secondary mirror and its support structure are somewhat more critical than that for the primary mirror, and we have completed the first of these, which shows the effect of mounting the 11-inch, 10-pound secondary mirror on a 2-inch stud glued into a hole in its center. The stud in this design will attach to a three-legged lever supported by links attached to the ends of the legs. Analysis of this structure is left to a later detailed phase of the design.

(7) AZIMUTH BEARING.

We considered several different types of large-diameter bearings to support the vertical axis of the telescope. Candidates were (a) a 60-inch four-point-contact ball bearing from such manufacturers as Kaydon or Rotek, (b) a six-segment curved equivalent of a linear-motion bearing with two rows of recirculating balls on each side of a track from THK. This device would have only two-point contact for any ball and should therefore have lower friction than the four-point slewing rings. (c) A large three-row roller slewing ring, possibly from Kaydon. These have lower friction than the four-point-contact ball bearings, especially with precisely machined foundations. (d) An air bearing. Tenn. Tech pursued this option extensively, and we found several ways of applying an air bearing to an astronomical telescope. Presumably there is a telescope in Hawaii that uses them. This option requires the base of the fork to be very precisely machined to achieve the clearances required for an air bearing to work properly. (e) A hydrostatic (oil-pad) bearing. These have been used on many telescopes in the past, including the coude-feed, 4-m, and McMath-Pierce at Kitt Peak, the VATT at Mt Hopkins, and numerous huge telescope like Magellan and Gemini. This kind of bearing has the challenge of using a fluid whose viscosity varies by an order of magnitude over the operating temperature of the telescope and that might cause cause troublesome mechanical and optical problems by being at a different temperature than the telescope.

We decided to use either an air bearing or an oil-pad system because of the significant torques required to overcome the moderate friction of even the most favorable rolling bearing. We initially decided to use a three-point support with air-bearing pads in contrast to a proposal for a monolithic air bearing developed by TTU. These pads would have been a foot in diameter and cost a moderate $5000 (appx.) each. However, the small air gap in these beraings (appx. 10 microns, or 0.0004 inches) would have required an exceedingly precisely machined fork. So we decided to use three oil pads in the same configuration. This support requires a very stiff fork to resist deflections, and we stiffened it by making a base structure of two concentric cylinders around the bearing surface tied together at the top and bottom with plates. An FEA of this structure shows the value of using concentric cylinders, but the bottom support plate could be stiffer than in our final design. A separate azimuth bearing report will be available eventually.

The fork is constrained laterally by a central pivot that attaches it to the azimuth drive wheel. The bearing in this pivot is a Kaydon KF110XP4A ball bearing large enough to allow the control lines of the telescope to pass through the center of the pivot's hub.

(8) THE DRIVE.

The drive consists of two parts, the motors chosen to turn the telescope and the coupling to the telescope axes. We considered several approaches to the mechanical design of this component: (a) Large-diameter (~40-inch) DC servo motors are an attractive choice. (b) The alternative is smaller motors coupled to the axes through some sort of gearing, such as spur gears, friction-coupled cylinders, or cables would around drums.

Large-diameter servo motors provide a very stiff drive, since they can be coupled directly to the telescope, thus reducing torsional deflections in the drive system. They have been used successfully on military surveillance telescopes as well as on the Vatican Advanced Technology Telescope. Prices of such motors are of the order of $30-40K, which compared favorably with prices of coupling other types of drive motors to telescope axes. They dissipate about the same amount of energy for a given torque at the telescope axis as smaller motors geared down by suitable factors. The only practical source of such large motors seems to be Sierracin Magendyne, which provided motors to VATT. Sierracin would not provide adequate engineering support in designing and building the servo system, however, for a small project like ours. The most compelling reason for not using large-diameter servo motors is the difficulty and expense of mounting such heavy devices in one tine of the fork and stiffening up the altitude axis enough to keep the two halves of the motor aligned. Mounting such a motor on the azimuth axis would be relatively easy, however.

Using smaller DC servo motors has the advantage of motors that are less expensive and presumably easier to replace. The gearing required would give a reduction of 10-200 between the motor shaft and the telescope axis. It could consist of (a) precise spur gears, as used on the MMT, (b) friction-coupled cylinders of different diameter, as used on many recent telescopes, including WIYN, ARC, SDSS, and our existing automatic telescopes., (c) capstan drives in which steel cables couple a small cylinder on the motor shaft to a big drive wheel.

We considered both friction-coupled wheels and steel-cable-coupled capstan drives. The latter reduce somewhat the problems of contact forces in the drive system, and we and our lead consultant from Tenn Tech visited Sagebrush Technology in Albuquerque, a company that makes such drives, on Sept. 19, 1996. We eventually decided to use friction coupled wheels because cable drives stiff enough for our telescope would have been inconveniently large. We also decided to use Compumotor AC servo motors to drive the axes, specifically DM1015B's, which cost about $5200 each. The moderate peak torque delivered by these units (11 lb-ft) make it easier to control contact forces in the drive (700 lbs per drive cylinder), but we will consequently use six of these motors, two for altitude and four for azimuth. The minimum number of motors would be three in any case, one for altitude and two for azimuth separated by 180 degrees to avoid a so-called palm-tree instability (Bruce Truax-private communication).

The drive motors are mounted in tractor units in which each a motor is coupled to a 4-inch diameter roller. The tractors are pushed into the drive wheel by clamps that squeeze on the back side of the drive wheel with roller studs. This arrangement flexes the drive wheel slightly, but the deflections should not be much different from pushing the roller into the wheel with a completely external spring. The base of the motor is attached to the tractor body through a flexure designed to give a maximum angular deflection of 1.9/20=0.1 arc sec at the telescope axis for maximum torque.

(9) DRAWINGS AND OTHER REPORTS.

We will eventually provide all the mechanical drawings for this telescope as a tarred and compressed set of postscript files and the solid models of the mount as tarred and compressed AutoCAD .dwg Release 13 (or .dxf) files on this web site. These will be available when construction of the mount is finished and modification of the design is complete. In addition, we expect to provide the non-proprietary parts of the control system in the same location. These shall include the basic program to calculate the instantaneous pointing and update the servo loop as well as any code written at TSU or elsewhere for the AST project.