Dome Rail-Roller Optimization

Dome Rail-Roller Optimization

Synopsis A spreadsheet written in AppleWorks v5 & Excel v6 has been developed that enables the deflection and rolling resistance of any rail-roller configuration to be evaluated. The deflection of the rail may be determined for either a tee, angle or circular cross section rail midway between supports. The torque applied to one of the rollers needed to overcome the rolling resistance may be evaluated, providing information useful in choosing the number of rollers, their size and cross section, and the size of motor necessary to power dome rotation.

Introduction Many of you may be aware that for several years I have been involved in a marathon project to refurbish, modernise and install my 10-inch Calver c1894 in my resited 4m domed observatory. This project is as I write approaching completion. Arising from the work I have been doing concerning the design of the observatory, is some original mathematical investigations into dome rail-roller design. None of this information is to be found in the specialist literature, and I intend including it together with a programme in Java script on my website.

When I began designing and building my dome in 1983/84, I did not have access to anything like the computational facilities I really needed in order to investigate curved beam deflections and rolling resistance. I largely went on gut instinct, based on my experience as a mechanical systems engineer. It is now relatively straightforward (if somewhat tedious) to write a spreadsheet programme to unravel all the algorithms involved in working out the deflection of a curved beam loaded normal to its plane of curvature ( a statically indeterminate case), when simply supported at fixed regularly spaced intervals, and to calculate rolling resistance between elastic bodies (i.e. materials which deform under pressure).

At the time I designed my 4m diameter duralamin monocoque dome I became aware of the monkey see, monkey do tradition followed by fellow travelers.  I did not care for the purblind approach so, being a fellow driven by curiousity (which is after all why I got into astronomy in the first place), I decided to find out how the details of rail-roller systems affected ease of dome rotation. It seemed from what I had seen in the previous 15 years that amateurs designed the rail & rollers around what they judged to make for ease and economy of manufacture, installation and assembly, but with scant regard for the resulting weight and rolling resistance. There also seemed to be literally nothing in the available literature describing how to determine rolling resistance, roller design and loadings. Most rail-roller systems furthermore betrayed an innate tendency by their designers to see the problem in two dimensions only. There was evidently no comprehension of the kinematics involved in rotating a dome.

So I began looking at the problem from first principles. I will begin by describing not what I found, but by describing what "everybody knew". Rolling resistance is treated as a classic problem of static friction. The friction at starting from rest, or statical friction is greater than the friction of motion, and depends on the hardness of the bodies and the length of time during which they have been in contact.

The so-called laws of friction are:-
(i) The force of friction is directly proportional to the pressure between the surfaces in contact.
(ii) The force of friction is independent of the extent of the surfaces in contact.
(iii) The force of friction is independent of the velocity of sliding.

Classic Newtonian mechanics treats the bodies as infinitely hard (i.e. there is no distortion due to compression), and that they operate in the elastic range (i.e. well within their ultimate compressive and shear stress limits).

Rolling resistance is caused by rolling friction. When a body rolls on a surface the force resisting the motion is termed rolling resistance, and is directly proportional to the load and inversely proportional to the roller radius. The coefficient of rolling resistance is typically given for prescribed circumstances and material combinations from a table. The coefficients are determined by experiment and are therefore empirical. It is impossible to determine rolling resistance by following this inelastic approach. If you do, then the following scenario ensues:

the rolling resistance is directly proportional to the load per roller, and the number of rollers (assuming each roller shares an equal proportion of the dome's weight). Therefore the rolling resistance is independent of the number of rollers, the roller cross section and the rail cross section.

This, by and large, from my conversations with those amateur astronomers who have designed their own domes, appears to be their thinking. Unfortunately in my case it flies directly in the face of experience. How?

I will now describe what I have found out about "other domes" during my perambulations around British observatories over the past 30 years. These are only a representative selection of all the different domed observatories I have seen and used.

The first was a timber C19th Cooke conical dome, on the Assheton Observatory in the grounds of Rossall school, Lancashire. I curated the 6-inch f/13.5 Cooke refractor c1870 housed therein for seven years, and during that time helped maintain the dome and restore the telescope. The rail was a heavy rolled steel flat section, approximately 150mm deep by 25mm wide, mounted under the dome, and supported by a dozen 150mm diameter flanged cast iron rollers let into in a 200mm square section ash beam surmounting the timber wall. To the ash beam at equal intervals were let in eye screws, into which the hook on the end of a heavy 4 to 1 naval style wooden block and tackle wound with 20mm hemp rope, could be hung. The other end of this block and tackle was shackled to the steel rail. By heaving on the free end of the hemp rope, the heavy timber dome could be painfully inched round. Sometimes it took the combined efforts of either four boys, or two men (lets say about 1000N [250lbsf]) to start the dome turning.

The second, which I also helped fabricate contemporaneously to employing the Assheton Observatory, was the Fylde Astronomical Society's "Colin Lynch" observatory - now defunct. It had a 2.8m dome made from a 20mm by 10mm tee section framework clad with oil tempered hardboard. The rail, cantilevered off the dome framework, was cold drawn steel strip, 50mm deep by 6mm thick, rolled and fish plated. The rollers were a dozen small steel wheels roughly 50mm O.D. with raised flanges at each end, free to run along 12mm axles supported in "U" channel shoes bolted onto a timber frame ring supported off the timber roof structure. The dome had provision to be wound round via a windlass and an endless wire, but it did not in fact ever rotate because of distortion to the dome, even though subsequently heavily braced.

The third, still in use after two decades, is a 2.5m oil tempered hardboard dome with a routed timber rail supported on the wall, and under the dome, trapped between which are 18, 40mm diameter crowned moulded nylon wheels, slightly narrower than the routed groove. The dome, which is relatively lightweight, is not easy to turn, and requires the effort of one man, shoving with most of his weight.

The fourth, no longer in use although still extant, is a 5.2m aluminium monocoque dome similar in construction to my own 4m dome. It has 12 rollers cantilevered off the dome's lower framework. The rollers are about 200mm OD with a concave profile, designed to run along a rail made from 37.5mm CDS thin walled tube. Although the dome was quite heavy, it was not difficult to turn, although it did not glide.

The fifth, a converted silo top, 6m diameter, has 12, 150mm diameter Nylon 6.6 rollers, with a single flange, running on a rolled scaffold tube rail. Despite its weight, it is easy to turn, and glides.

If what "everybody knows" (yet nobody has ever bothered to write down) is fundamentally correct, how is it that turning these five domes varied so markedly? We all expect a big dome to be harder to push round than a small dome. But why? Simple; we are all Aristotelians at heart. Although we are aware of Newtonian dynamics and his three laws of motion, we pay lip service to them. In reality, all things else being equal, a big dome will be harder to get moving because of its inertia, but once it is moving, the force needed to keep it moving should be nominal. If it isn't, its due to rolling friction.

It has been my experience that regardless of size and weight, some domes are far easier to get started turning and then to keep turning, than others. Those which have a cumbersome structure, a heavy rail on the dome itself, and numerous rollers, especially small rollers, with a wide contact area, are by far the worst. The reason in my opinion why so many amateur domes are awkward to turn, and require a disproportionate effort to get them turning and to sustain turning is because of a rail-roller design concept stuck in Aristotelian physics.

Design Principles  One of the reasons a dome can be difficult to turn is because it distorts when a tangential force is applied to its rim. Many amateur astronomers place the rail on the dome, and the rollers on the wall, in an effort to add rigidity to the dome. They also make provision for slight radial movement of the rollers along their axles, to accommodate circularity errors in the rail. This can work, but the rail either has to be very stiff, or the dome itself must be sufficiently rigid not to distort significantly when the load needed to get it turning is applied. If this is not the case then the rail will deform and either crab, bind or jump off the rollers. It should be born in mind that the dome will not initially rotate if pushed from its rim, tangentially. It will try to move sideways, in the opposite direction to the applied force. Only the rail-roller system obliges the dome to rotate, and the more rollers that are needed to force the dome to turn, the greater the rolling resistance.

Admittedly it is easier to power one of the rollers when they are mounted on the wall than on the dome. But the problem of powering a moving drive roller is trivial compared to ensuring the dome is stiff enough to support the rail adequately without adding unduly to its weight. It is therefore better to mount the rail on the wall and the rollers off the dome. The dome does not need to be as stiff when the rollers are mounted off it because they can follow the rail in any case. The only consideration is that they share the same circumference, otherwise they will ride off the rail at some point and cause the dome to crab. Provision must therefore be made for radial adjustment, but the position of the rollers, once adjusted, can be fixed. This makes driving the dome via one of the rollers, held permanently in contact with the rail, a relatively straightforward piece of engineering.

The rail-roller as an elastic system  Once the dome has been designed and its overall weight calculated, it is possible to design a rail-roller system adequate to take the load and provide easy rotation, without being over-engineered to accommodate Airy's infamous "factor of ignorance".

The rail is to be supported at equal intervals around its circumference. The point loading at each roller is due to the weight of the dome plus wind loading. This is the other salient factor which despite detailed study during the past couple of decades, has yet to find its way into serious amateur astronomical literature. The attached diagram depicts the pressures on my hemispherical dome in Pascals for lamina, transition and turbulent flow. The loading is a summation of the pressures over their respective areas. The maximum load is taken as the greatest positive pressure over the dome's projected area. Note that in the lamina and transitional flow regimes where the Reynolds number is below 1200, pressures are negative. These negative pressure flow regimes can be used to design rail restraints to prevent the dome being lifted off the rail.

The design load is thus the static weight plus the greatest expected wind load, assumed to be equally distributed over the number of rollers. I wrote a spreadsheet to enable the deflection of the rail between supports and the rail-roller contact stresses and rolling resistance to be evaluated, and to readily see how these varied when the number and diameter of rollers was altered, and the rail section and size altered.

The roller contacts the rail over an area referred to as a contact ellipse (although the term ellipse does not actually imply the contact area is elliptical!). The pressure caused by the applied load over the contact ellipse deforms the roller and the rail. Providing the pressure remains below the materials' yield points, the rolling resistance can be calculated from their Resilience Modulii. The harder the materials the lower the rolling resistance, but at the same time the lower the frictional resistance of the rail to the roller, which if too low causes skidding. It is wiser to use a hard rail material and a more ductile roller material (or fit a tyre to the drive roller).

Determining rail deflections and rolling resistance     The illustrated spreadsheets and chart is for a 47mm CDS schedule 10 railtube and Nylon 6.6 rollers. (NB although only 3 rollers are in permanent contact, the dome has 5 rollers, two intermediate rollers are mounted 3mm off the rail either side of the drive roller to prevent the dome crabbing. A vertical tyred jockey roller is mounted under the shutter transom directly opposite the drive roller, running around the outside diameter of the railtube. Its purpose it to prevent the drive roller riding up off the rail when the drive torque is applied.

The dome load is 1.3kN and rail diameter 3.8m. However because I idiosyncratically prefer to work in the imperial units, all the values in the spreadsheet are in the inch-pounds-second system. To evaluate rail deflection midway between supports I enter the dome load and number of rollers, the rail radius, the rail section sizes, Young's Modulus, the Rigidity Modulus, and the angular separation of the rail supports in radians. To evaluate rolling resistance I enter into the second spreadsheet the dome load, number of rollers, Poisson's Ratios, Young's & Resilience Modulii for rail and roller, rail section and roller section sizes. The spreadsheet gives the rolling resistance per roller and total rolling resistance, from rest and at running speed, and the torque at the drive roller axle at start and at running speed. The algorithms used in the cell equations are taken from "Formulas for Stress & Strain" by Roarke & Young. The spreadsheets were created on a PowerMac 7200-90 using AppleWorksv5. I can e-mail the file on request.

Findings The chart demonstrates how rolling resistance changes with the number and diameter of rollers. Contrary to what "everybody knows", increasing the number of rollers does not make a dome easier to turn. The minimum number of rollers is three, but it would be impractical to make a dome base ring adequately stiff to maintain its shape with only three support points, so in practice either five or six are needed. For lightweight domes, less than 4m diameter, a five to eight roller configuration is ideal. Larger domes will need a dozen or more rollers.

It will assist keeping rolling resistance down if the system avoids the need for lateral guide rollers. To this end, and also because it is an easy section to roll, a rail tube is superior to either angle or tee section. Big rollers with fat axles are also easier to turn, although if the dome is driven via one of them, it will require more torque to rotate. Rollers that have a minimum contact area will also reduce rolling resistance. A flat sided Diablo shaped roller, on a toroidal rail provides all the kinematic stability needed, particularly if the rollers are inclined both downwards and inwards.

Conclusions Designs leading to  increased rolling resistance are, on the basis of my spreadsheet analysis, as follows:

i) a flat or strip section steel rail with flanged rollers & lateral guide rollers
ii) balls or crowned rollers trapped between two grooves
iii) crowned rollers in a groove or on a U section rail
iv) flanged rollers on angle or tee section rail with guide rollers
v) U section rollers on a bull head rail

Designs leading to free running and  decreased rolling resistance are as follows:

i) Diablo or back to back conical rollers on a toroidal rail, preferably inclined inwards and downwards
ii) Conical section rollers running on the apex of an inverted angle section rail
iii) Spherical rollers running within an angle section rail
iv) Fully radiused rollers running within an angle section rail

This page was created by SimpleText2Html 1.0.2 on 30-Jan-102.


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