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Performance of Hybrid Band-Worm Drives A rebuttal to DFM's article Abstract Melsheimer's article, "Comparing Telescope Drive Technologies", features an analysis of the theoretical performance of a band drive system. Melsheimer examined a band drive example which is not representative of the band drive systems used by ObservatoryScope. Thus the performance of Melsheimer's band drive example is significantly inferior to the performance of ObservatoryScope's band drive systems. NOTE: The terms "band" and "belt" refer to the stainless steel band component of a band drive. The terms "disk" and "pulley" refer to the disks about which the band is wrapped. Melsheimer's Assumed Geometry for ObservatoryScope's Band Drive
Melsheimer used smaller diameter pulleys rather
than the 28" and 4" diameter pulleys used on our 20" and 24" aperture
telescopes or the larger 36" and 6" diameter pulleys used on our 32" and 36"
aperture telescopes. The smaller pulleys limited the maximum recommended belt
thickness and the overall performance of his example band drive. Melsheimer
quotes material data for
Melsheimer states that 1,000,000 cycles is about 50 years of use at a good site. We assume a band life of 750,000 cycles and that the telescope is slewed back and forth throughout the observable sky up to 40 times per night every single night of the year. This exceeds the normal "real world" use of our band drive on an astronomical telescope, with the exceptions of LIDAR and NEO tracking platforms. The math to calculate the longevity of the band is pretty simple:
Thus the stainless steel bands in our band drives are good for about 50 years of use on a telescope which is used every single night of every year. ObservatoryScope's Hybrid Band-Worm Drive System Geometry ObservatoryScope uses two different versions of its band-worm drive system depending on the aperture class of the telescope. Specifications for each version are shown below:
ObservatoryScope specifically uses 301 High Yield Stainless Steel (301HY) bands since 301HY offers the highest yield and tensile strengths of all 300 series stainless steels. Its an extra cold rolled version of 301FH, 301HY being less ductile and more magnetic than 301FH. High Yield material is used in applications with smaller than ideal pulley diameters, due to the resulting high bending stress. Applications with high loading are also ideal candidates for 301HY material due to 301HY's inherently high tensile and yield strengths. Melsheimer correctly notes that the bending stress of the band is greatest around the smaller pulley. ObservatoryScope uses either 4" or 6" diameter smaller pulleys compared to the 2.857" diameter smaller pulley of his example band drive. The very small pulley in his example band drive severely limits the maximum permissible band thickness to a mere 5 thousandths of an inch. This in turn imposes serious performance limitations for his example band drive. Thus Melsheimer's example band drive is not representative of ObservatoryScope's band drive performance. Recommended Belt Thickness Melsheimer's formulas for calculating the maximum permissible belt thickness are correct. However, we need to plug in the correct diameters for ObservatoryScope's pulleys and the correct values for Belt Technologies 301HY full hard stainless steel belts. First, we will calculate the bending stress within the belt by simply bending our belt around the smaller pulley of our band drive. The bending stress (S) is:
Now we will find the ratio of the band's bending stress relative to the band's yield stress. Although Belt Technologies belts are designed for use with smaller than ideal pulley diameters, longevity of the belts is paramount. Since we haven't yet taken into account the additional stress caused by tensioning the belt about the pulleys, we want to verify that the bending stress alone is less than 50% of the yield stress for the belt. The ratio of bending stress (S) versus yield stress (Sy) is easily calculated:
We see that the bending stress on the bands about the smaller pulleys in our band drive systems is less than 50% of the yield stress for the bands. This is well within acceptable limits, as confirmed by Belt Technologies, for the following reasons:
Now we must take into account at the additional stress caused by tensioning the bands about the drive pulleys. This is the band's preload tension. In our band drive systems, the preload tension is either 250 or 500 pounds depending on the drive system. The preload tension in psi is easily calculated as follows, where Area is the cross sectional area of the band:
This preload tension is nearly 3 times the recommended tension for high speed process belts. High speed belts are subject to significant amounts of vibration and should be tensioned moderately to prevent vibrational induced stresses from approaching the yield strength of the belts as the belts travel around the pulleys. We can use this higher preload tension since the band doesn't experience any significant vibration when running at only 1/2 to 1 revolution per minute. Why does ObservatoryScope tension the bands to 3 times the recommended tension? Because this tension prevents sag, due to gravity, in the unsupported segments of the bands between the pulleys. Now that we have calculated the additional stress due to tensioning the band, we can proceed to calculate the ideal maximum thickness for the band. The calculation for the ideal maximum band thickness must take into account the band's bending stress about the smaller pulley, the band's additional stress due to the preload tension, and the fact that we do not want the overall stress within the band to exceed 50% of the band's yield stress for a band lifetime of at least 750,000 revolutions. Thus the maximum allowable bending stress, due to wrapping the band around the smaller pulley, must be reduced by subtracting the stress caused by the band's preload tension. We simply solve for "t" in the following formula:
ObservatoryScope uses 0.015 and 0.025 inch thick stainless steel bands which keep the overall band stress around the smaller pulleys to less than 50% of the yield stress for 301HY full hard stainless steel. Thus these bands are theoretically good for at least 50 years, even if we assume a reduced life-span of 750,000 cycles, since our band drives experience virtually zero vibration during operation. Nevertheless, ObservatoryScope ships its instruments with a spare set of bands should either band ever need to be replaced. The bands are easy to replace should one ever fail decades from now. Stiffness of ObservatoryScope's Band Drive Now that we have addressed band longevity and band thickness, lets move on to calculating the effects of wind torque upon the band drive. Then we can proceed to calculate the "stiffness" of ObservatoryScope's band drive component. The stiffness of our band drive is directly related to any band stretch which occurs along the segment of band between the two pulleys. On our band drive systems, the length of band between the contact tangent points of the pulleys is shown in the table below. Melsheimer assumed a 1 foot pound wind loading in his calculations. We shall use a 1 foot pound loading in the following calculations as well.
Under a 1 foot pound loading, the belt will experience a change in tension based on the ratio of the foot pound telescope loading versus the radius of the primary pulley. This ratio is 12 inch-pounds divided by the radius in inches of the primary pulley:
Now that we know the change in belt tension due to a 1 foot pound wind loading, we will use the well known formula for Young's modulus to calculate the change in strain (or stretch) of the band due to the change in stress (or tension) on the band.
Now we ask, what does this change in belt stretch translate to in terms of angular deflection (rotation) of the telescope? To obtain the angular rotation or deflection, Theta, we simply divide the change in belt length by the radius of the larger pulley and multiply by 206,265 (the number of arc seconds in a radian).
Why is the band drive for the 32" and 36" scopes so much stiffer than the band drive used for the 20" and 24" scopes? Because the larger 32" and 36" scopes have considerably more mass and because the wind torque upon their larger and longer OTAs is greater. ObservatoryScope takes into account wind loading when designing its optical tube assemblies (OTAs). Its one of the reasons why we only use "open" truss OTA designs regardless of whether we use our band drive system or whether we were to use a friction drive system. The torque caused by wind upon the top of the OTA is approximately opposite to the torque caused by wind upon the bottom of the OTA. We adjust the size of the secondary's spider vanes such that the wind induced torques at each end of the OTA are opposite of each other to within about 20%. This is why the secondary spider vanes in our designs generally are larger than average. An additional benefit of the larger spider vanes is significantly increased stability in supporting the secondary mirror assembly. Lets assume the "worst case" scenario where just the upper half of the OTA is exposed perpendicular to the wind. The only really large surfaces found at the top of the OTA are a pair of spider vanes, the secondary baffle tube and the OTA front ring. We want to find out how much wind force acting upon these surfaces is required to produce the torque (calculated above) about the DEC axis to produce a 1 arc second deflection. First, we need to know the area of the surfaces involved. We have separately integrated these areas, taking into account that some of the surfaces are round and experience less force due to wind, to arrive at the cross sectional area of an equivalent flat rectangular plate which would experience the same force due to wind. We also integrated to find the distance from the DEC axis for the equivalent flat rectangular plate to produce the same wind torque as the actual surfaces involved. It turns out that the spider vane distance is almost exactly identical to the calculated rectangular plate distance. The values for 24" and 36" RC telescopes are:
We now use the following generic formula to calculate the wind pressure needed to produce the force which we calculated above:
So, we ask, what wind velocity produces "P" pounds per square foot of force? There is a simple formula relating wind pressure to wind velocity:
We mentioned that ObservatoryScope designs its OTAs to experience nearly equal but opposite torques (within 20% or better) at each end of the OTA when exposed to wind. If the telescope is installed inside a dome, then it is fairly well protected from the wind such that wind loading upon the telescope is reduced by at least 50%. What we are still interested in examining is, again, the worst case scenario where the telescope is installed in a roll-off roof observatory with the telescope fully exposed to wind, and with the telescope OTA oriented perpendicular to the horizontal wind (telescope pointed at the zenith). We also need to take into account that instrumentation at the focal plane can vary and that there might be guide telescopes and other equipment attached to the OTA. Focal plane instrumentation is located fairly close to the DEC axis and is partially shielded by the walls of the roll-off roof observatory when the telescope is pointed at the zenith. Guide telescopes will be attached to the OTA center box since this is the strongest structure within the OTA and is least subject to flexure. Thus guide telescopes will produce a nearly zero wind loading torque upon the OTA. Nevertheless, we will add another 10% "fudge factor" to account for "real world" use of the telescope and auxiliary instrumentation and thus assume that torques about the DEC axis are only being cancelled to within 30%. Therefore we again will use the formula, presented in the above table, but will divide the wind pressure (P) by 30% to arrive at the "real world" wind velocity required to produce an angular deflection of 1 arc second for a telescope which is completely exposed to wind:
This is exactly why ObservatoryScope has been able to test and verify 1 arc second or better tracking performance of our telescopes, when housed in roll-off roof observatories, and with 20 mph winds and wind velocity changes of up to 50%. Yes, through proper implementation of our unique Hybrid Band-Worm Drive combined with proper engineering of the OTA along with the rest of the telescope, exceptionally smooth band drive tracking performance is readily obtained even in fairly breezy conditions. Finally, let's calculate the stiffness of our Hybrid Band-Worm Drive System. Actually, we have already calculated this above, but we will calculate it again, below. The stiffness (K) is expressed by the torque divided by the rotational deflection (Theta) in arc seconds. Thus the stiffness, K, is equal to:
We have already shown that our standard band drive systems are perfectly satisfactory for telescopes housed in roll-off roof observatories and fully exposed to continuous winds up to 20 mph, and are more more than satisfactory for telescopes housed within domes. Thus the only real merit Melsheimer's stiffness value has is in calculating the resulting pointing deflection when the telescope is operated in an out of balance condition. When the telescope is out of balance, the stiffness value may be used to calculate the change in pointing position of the instrument. Note that other torques and flexures, caused by out of balance conditions, within the OTA and the fork were addressed by Melsheimer and are not addressed within the context of this rebuttal. We have also shown that our band drive system has a theoretical life-span of 50 years, that the band drive system's stiffness is perfectly adequate, that the band drive system is immune to damage from contaminants, and that ObservatoryScope has already and tested and verified the pointing and tracking performance of its band drive system in 20 mph winds. Note that ObservatoryScope's 20" aperture prototype instrument, after nearly three years of continuous operation, has yet to show even the slightest wear upon the thin black anodization on the worm gears. This is because the inherent reduction of the band drive systems greatly reduces the forces within the worm drive components to a mere few ounces. One can readily conclude that ObservatoryScope's proprietary Hybrid Band-Worm Drive Systems are perfectly adequate for observatory class telescopes and that they are designed to last for decades. Failure Modes of ObservatoryScope's Hybrid Band-Worm Drive Melsheimer states:
In actuality, the belt is being operated at less than 50% of the yield stress for the belt. This is better than the safety factor of 1.67 to 1.92 as recommended by the American Institute of Steel Construction. The odds of belt failure during 50 years of telescope operation are very low. Engineers at Belt Technologies confirmed this statement. We note that the belt would indeed fail nearly instantaneously but not catastrophically as Melsheimer implies. He makes it sound like there would be some sort of "explosive" failure. The belt can't fail "explosively" since it is tensioned to less than 6% of the yield stress of the belt. The belt would simply tear and then possibly snap completely and then fall away. This would be similar to cutting a tensioned steel packing strap as commonly used on wooden shipping crates. Assuming that the telescope is a couple of foot pounds out of balance, then the affected axis would very slowly rotate until either an obstruction is encountered or when the heavy point is down. Note that the built-in safety clutches in the worm drive component of our drive system require that the telescope be balanced to within 5 foot pounds to operate correctly. Thus the telescope would merely rotate slowly, due to the inherent friction within the axis bearings, on the affected axis. Melsheimer also states:
Melsheimer's above statement simply is not true since, incredibly, he didn't take into account the fact that the belt tension is distributed along the entire surface area of contact between the belt and the surface of each pulley, and that the belt is tensioned to less than 6% of the yield strength of the belt. Note that the average pressure between the belt and the smaller pulley is obviously greater than the the average pressure between the belt and the larger pulley. Also note that ObservatoryScope's Hybrid Band-Worm Drive System features built-in spring loaded mechanical temperature compensation mechanisms which also serve to alleviate the strains caused by foreign contaminants passing between either pulley and the belt. Let's calculate the actual average pressure between the belt and the surface of the smaller pulley:
It seems to us that 57.1 and 58.1 psi are extraordinarily low amounts of pressure compared to the very high engagement pressures (literally thousands of psi) needed in a friction drive system! Regarding DFM's friction drives, keep in mind that the surface area of contact between the two friction disks is theoretically ZERO. The extremely high engagement pressure within a friction drive is necessary in order to prevent slippage. This pressure actually slightly deforms the disks at the point of contact to produce a non-zero yet an extremely small surface area of actual contact. We do note that the very small surface area of contact is slightly increased due to the microscopic surface roughness of the disks. Nevertheless the engagement pressures within a friction drive can easily approach the surface deformation limits of even hard materials such as stainless steel, chrome and nickel. In ObservatoryScope's opinion, the now defunct company SciTech is the only company who "got it right" when they designed their friction drive system. SciTech deliberately used huge 51.25" diameter primary disks on their friction drives. Why? Because SciTech could then use a much lower engagement pressure between the roller and primary disk and still prevent slippage within their friction drive. Thus SciTech was able to keep the engagement pressure within their friction drive system significantly below the surface deformation limits of the materials used for their drive disks. Any grit or contaminants within our band drive have a very low probability (less than 1/10 of 1% when compared to a friction drive) of exceeding the surface deformation limits of the stainless steel belts and the underlying aluminum pulleys. Diamond dust, for example, is one material which would exceed the surface deformation limits even with the the very light pressures between the band and the smaller pulley. Would diamond dust cause tracking errors? No, but there would be slight cosmetic damage to the surfaces of the aluminum drive pulleys. Even large amounts of sand thrown into our band drive component won't damage the belt, but could produce some microscopic pitting in the underlying aluminum pulleys. This microscopic pitting won't affect performance or tracking accuracy and would rapidly be smoothed out over time. Thus any pitting is purely cosmetic in nature as it does not affect tracking performance in any way. To produce this microscopic pitting, one would literally have to throw a handful of sand into our band drive system while the telescope is rapidly slewing. We have already tested our band drive using steel wire and steel shims up to 40 thousandths of an inch thick. No damage whatsoever occurred.Of course, in light of the very low average pressure between the band and the pulleys and the built-in "give" within the band tensioning mechanisms, we were not expecting any damage to occur. We do note that very large pieces of grit within ObservatoryScope's band drive can cause slight tracking errors. Testing with 20 thousandths thick wire or steel shims, for example, we did see a slow tracking error of about 2 arc seconds. This was both previously calculated and expected for such a thick piece of contaminant. The reason that the tracking error was "slow" in showing up is that it takes several seconds for the wire or shim to "get caught up under the band and disk" as the telescope slowly tracks at sidereal rate. This "slow" tracking error is far more desirable compared to the instantaneous tracking errors seen due to grit within a friction drive system. Let's now look at the effects of grit and contaminants within a friction drive system. Note that grit within a friction drive produces nearly instantaneous and sometimes quite large tracking errors, and causes permanent damage to the friction disks. This is because the grit instantaneously changes the effective diameter of whichever disk within the friction drive which is constructed of the softer material. The grit also becomes embedded within the softer friction disk. Most of the grit is pulverized due to the high engagement pressure, but not necessarily all of it. Usually a small "bump" of the grit remains permanently embedded in the softer disk. This "bump" will be surrounded by a raised ridge of the disk material due to the grit literally being forced into the disk. The end result is a small surface deformation in the disk. Over time these small surface deformations, along with the surrounding raised ridges of disk material, will get worn away. Unfortunately, small amounts of the disk's surface are being lost because they get worn away along with the embedded grit. Also note that the extremely high engagement pressures within a friction drive result in surface hardening (compression) of the surface of the softer disk. This is equivalent to the loss of disk surface material. The result is that the disk no longer has a consistently uniform diameter about its edge. Eventually the friction disk can develop an oval shape or even some other unpredictable shape depending on how the telescope is routinely used and how much grit gets into the friction drive. Safety Concerns of ObservatoryScope's Hybrid Band-Worm Drive Melsheimer states:
The minimum band thickness used by ObservatoryScope is 15 thousandths of an inch thick. Also note that the bands feature edges which have been ground round and smooth by the manufacturer for safety and to prevent metal fatigue at the edges. Our prototype 20" doesn't feature the protective band drive covers used on our production units since the prototype is installed at a private facility and since we have already proven that contaminants can't damage the band drive system or affect its tracking performance. Everything from large insects to dirt and cut grass has gotten into the prototype's band drive systems without causing damage or any visible tracking errors. Even without protective covers, it would be extremely unlikely that an observer would get cut on the smooth and rounded edges of the relatively thick bands used in ObservatoryScope's band drives. Melsheimer also states:
This has already been addressed, above. The scope, assuming it is properly balanced to within a few foot pounds, would simply proceed to slowly rotate to a "heaviest side down" position. The built-in safety clutches within our drive system will not allow the telescope to operate correctly unless the user has properly balanced the telescope to within a few foot pounds. The safety clutches are a necessary design feature to ensure that the telescope is operated in balance and to assure that the worm drive gears will wear evenly and become more accurate over time. Conclusions Melsheimer concluded that:
In reality, the band drive may be used for moving very heavy loads where stiffness is important. All that really matters is that the loads are well balanced about the driven axes. We've also shown that our band drive, unlike DFM's friction drive, is virtually immune to contaminants. Are there better ways to drive a telescope than with our proprietary Hybrid Band-Worm Drive? We think not. Is a friction drive capable of performing every bit as well as our band drive? Absolutely yes, assuming that the manufacturer has fully enclosed the friction drive to prevent contaminants from damaging the drive and addressed surface deformation limit issues. The major differences are cost and performance under extraordinarily windy conditions. We have proven that a properly designed band drive system used in conjunction with a properly designed OTA provides observatory class performance even in fairly breezy conditions. We can manufacture customized versions of our band drive systems which incorporate an additional band drive reduction stage. By doing this, we can use substantially thicker bands for the primary reduction stage and increase drive stiffness four-fold. Such customized instruments, designed to be housed in roll-off roof observatories, are be capable of tracking to 1 arc second accuracy in 42 to 47 mph continuous winds! All drive system performance specifications found throughout our web site are based on our engineering calculations which were then been verified by the proven results obtained with our 20" prototype instrument. — Michael Marcus |
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