Prompted by Nigel showing how he managed to calculate an approximate 10:1 ratio gearing (which I considered far too complex) and the exchange of opinion, I shared an old project of mine with him. This led to his suggestion that a small article about my project would be useful here.

I’m sure that there are many excellent treatises on gearing ratio calculation but suspect that most are simply theoretical/mathematical explanations rather than solutions to real-world problems. This article is intended to show a solution to a real problem which initially seemed insurmountable.

JG

Being disappointed that an Orrery I saw at age 11 was not ‘to scale’, I have long held the ambition to address that, so I’ve had an interest in modelling the solar system for over 60 years. I have also designed a number of clocks. When a small electric motor running at 2 revs per minute came into my possession I had the idea to build a ‘Tellurium’ – a model of the Earth & Moon. A full Orrery to scale is simply not practical but a Tellurium is.

The first thing I did was to select a ‘comfortable’ scale distance between Earth and Moon which I considered was 500mm (giving a 1m Ø circle). Now, ‘life is a compromise’ so this figure was never sacrosanct! Wanting to work as accurately as possible, I had also decided to take account of the fact that any two body system actually rotates about their ‘Barycentre’ which means that the 500mm would need to be increased (a little). Since I am using a fixed centre of rotation – rather than attempting to create a true elliptical path – I must work with the ‘mean’ distances.

The Barycentre is dependent upon the relative masses of the two bodies and the distance between them. In the case of the Earth and Moon this amounts to a shift of 4677.98km, that means that rather than the Moon rotating at a mean distance of 385,000km I have to move the centre of rotation to 380,322.02km. Now I have this distance I can work out what ‘scale’ would put this at 500mm (or thereabouts). I can of course choose whatever scale I wish and simply fix the distance at 500mm regardless but I already have a spread-sheet with a great deal of solar system data using a ‘round’ figure for the size of the Sun at 1m & 4m Ø. Using the 4m data, the mean orbit of the Moon works out at 552.889241mm and the scaled Barycentre distance to 13.435876mm making the nominal 500mm objective 539.453365mm. None of these figures are what you might consider comfortable as far as manufacture is concerned but they can easily be calculated using a spread-sheet! At this scale the Earth model needs to be 36.638mm Ø and the Moon 9.9629mm.

I had already found some 300mm lengths of small square brass tubes which were designed to fit inside each other so two of those could give me 530/550mm variable, allowing for some overlap, so that made sense.

Making the models of Earth & Moon at (say) 37 and 10mm Ø is no big deal, nor putting them on the ends of 2.4mm & 1.6mm tubes, but the point of this discussion is more to do with getting the system to rotate (about the Barycentre) once – in a period of 27.32166 days! (A ‘Tropical’ or ‘Siderial’ Month)

Obviously, the whole project would essentially be a clock, with the added ‘complication’ of a model Earth rotating about its own axis – tilted at 23.4 degrees – once per 24 hours as well as the model Moon taking its 27+ days. Here was my seemingly impossible (certainly difficult) task! Starting with a motor providing a steady 30 second revolution, normal clock gearing is nothing extraordinary – but wanting the Tellurium to be also aesthetically pleasing does add some constraints. I would be disinclined to add an extra spindle to the clock centre spindle carrying the minute and hour hands and having a 37mm Ø sphere offset from that by 13.4mm, as it would be (I thought) clumsy. This led me to place the Earth/Moon above the clock-face centre but naturally ‘in line’. Now came the need to determine the clock-face diameter and how far it should be above the centre spindle. Somewhat arbitrarily, I settled upon 150mm and 120mm, so I now had all the major design constraints in place which influenced the decision to add a ‘Seconds’ dial at 120mm below the centre spindle.

This shows the basic layout of the clock-face.

Because I want the model Earth to rotate once per day I thought it prudent to create a 24 hour rather than 12 hour dial. This would mean that I could use a direct take-off from the hour hand to drive the Earth. All other locations (bearing positions etc.) are dependent upon the gears needed to achieve, first, the minute and hour progression, then ultimately the Moon movement.

Converting the 30 second revolution provided by the motor to a dial/hand indicating a 60 second or one minute revolution is a very simple 1:2 ratio but naturally there are many possible combinations which will provide this so other factors will have a bearing upon the selection.

One factor is the direction of rotation of the motor. Because it is natural to expect clock hands to progress in a clock-wise direction, if the motor turns clock-wise then we must use an ‘idler’ gear to make the second gear rotate in the same direction as the motor.

The next factor is, what size the gears need to be, and this is dependent upon the metrology – the MOD or DP – being used. In my case I do have a choice but my preference is 0.5MOD primarily because I have a cycloidal form Hob of that size and that form is preferable (for clocks) to the more usual Involute form used where the gear train is intended to transmit ‘power’. A clock will always be moving in one direction so there is no concern about ‘backlash’ either.

Arguably the most important dimension on any gear is the Pitch Circle Diameter [PCD] since this (along with any ‘clearance’) will determine the centre distance between mating gears. Using MOD 0.5 the PCD is easy to determine as it will be half the number of teeth in mm – ie. a 100T gear will have a 50mm PCD.

Notwithstanding the ease with which the PCD can be determined, it is still very worthwhile setting up a spread-sheet to calculate the centre distance of any two gears given the tooth count because you should also add in a ‘clearance’ so that there is no risk of ‘binding’. With 0.5 MOD I usually set a clearance of 0.1mm.

The spread-sheet fragment shows the only formula needed to provide the centre distance for any pair of gears. Once the MOD & clearance figures are known, it only needs the two tooth numbers to be entered and the centre distance will be shown in B5.

There are many other calculations that could easily be added to this fragment; probably the most obvious would be the ratio between the gears.

Coming back to the question in hand, – – I need a pair of gears – – in fact I actually need three gears since the motor runs clock-wise, so I will need to use an idler between the two gears to reverse the direction of rotation – to some extent this fact frees me from some of the usual constraints. The size of the idler gear is quite flexible which means I can place the motor wherever most convenient. One gear will have to be at the location of the ‘Seconds’ dial and the other will be on the motor spindle. The motor size has a bearing upon the size of this first pair of gears and as that is 50mm Ø, I arbitrarily used the same dimension for the distance between the motor spindle and the seconds dial. Now I had a target measurement for the maximum centre distance between the two 1:2 ratio gears and working down from 120:60 in 10s, the first pairing that gives a large gear smaller than 50mm dia. is 90:45. This has a centre distance, with clearance, of 33.85mm. Making the idler gear also 45T has the advantage that I can make two gears using the same setup so I had to check that the centre distance between 45T-45T – plus that between 45T-90T, was greater than 50mm, otherwise the idler could not link the two.

45-45 = 22.6 and 45-90 = 33.85 so, yes, 22.6+33.85 at 56.45 is over 50 so I now had my first three gears solved!

For the avoidance of any doubt about what I mean, I’ve drawn this first gear-train as [Fig. 4].

The dotted blue lines depict the PCD of each gear and I’ve omitted the OD to limit any possible confusion. The fact that the motor spindle is in line with the seconds dial is arbitrary but does make manufacture easier.

It would be possible to create a 24hour clock gear-train driven from the 90T gear which rotates once per minute but there is no compelling reason to do so and in this case I went back to the motor spindle which I think does have an advantage since this will ultimately be a long train.

Now I have fixed the location of the motor relative to the clock centre spindle (minute & hour hands) I have a framework upon which I can build the ‘Clock’ gear-train. The challenge is to convert the one revolution in 30 seconds supplied by the motor to one revolution in 1 hour (minute hand) and one rev in one day (hour hand) – and of course those two hands must run coaxialy. It was obviously going to be more than a ‘pair’ of gears so it seemed logical to arrange a gear-train on a carrier pivoting about the motor spindle with the final drive gears being fixed about the main dial centre spindle.

There are always ‘constraints’ and one of the most significant is how large the gears can be. A 200T gear would be 101mm Ø, which, whilst not massive, and certainly within the capacity of my Gear Hobber, didn’t offer any particular benefit in the initial stages. I’d determined that the length of a ‘carrier’ could be about 150mm which would put the last gear (driving the minute hand), close to the Dial centre line. Therefore I could probably get a three stage reduction. There is also a lower limit to the gear size that it is comfortable to make and, though I have made 10T 0.5MOD gears, there are other factors that make this less than optimal.

A quick ‘back of a fag packet’ sketch/calculation brought me to work on a 1:4, 1:5, 1:2 & 1:3 consideration. To keep the train simple and as narrow as possible, each progression needed to use a larger gear than the previous pair, so that the centre distances were always larger than the gear radius. This doesn’t apply to the last gear in the train because that will mesh with a gear not on the carrier.

With a 30T gear on the Motor spindle (in parallel with the 45T driving the seconds hand) a 120T gear would take two minutes per revolution, follow this with a 30T driving a 150T to get 10 minutes per revolution, then a 30T to 60T = 20 minutes and finally the 60T driving a 180T would drive the minute hand at one revolution per hour . The ‘Carrier’ needs to be a little longer than the train so that it can be anchored to one of the clock plates.

I now have a slightly more complex problem to solve. The Hour hand must lie coaxially with the minute hand and also travel in the same (clockwise) direction. This means that the gear-train must be ‘folded’ and have a 1:24 ratio. These constraints needed some thought to reach a sensible solution.

As the factors of 24 are 4 & 6, I could create the 1:24 in two steps 1:4 & 1:6. I then needed to decide how much space I had available for a ‘folded’ train which would determine the tooth count. Again some trial sketches suggested the 1:4 should be 20T & 80T – (and, just as it was efficient to make two 45T and three 30T gears) using 20T for the second pair was also most efficient. That led to the pairing of 20T & 120T for the 1:6 set. Having set the 1:24 ratio I could only solve the ‘fold’ by drawing it. That also showed whether or not I needed to add an ‘idler’ to correct the direction of travel.

Since the minute & hour hands needed to run coaxially, and the 180T gear was already in place running at 1 revolution per hour, that had to be the drive for the ‘day’ train. The start of the 1:24 train – a 20T gear – should therefore be fixed to the 180T gear so would also rotate once per hour. This then drove the 80T gear which also had a 20T gear fixed to it, driving the 120T gear which then ran at one revolution in 24 hours.

Although I then had a gear rotating once per day, it was not running coaxially with the ‘minute’ gear. The fact that the motion had gone through three gears meant that it was rotating anti-clockwise so it needed reversing. No more speed adjustment was needed so, as long as the next gear in the train was the same tooth count as the gear that finally drove the hour hand, the actual tooth count is irrelevant so can be determined purely on size.

This shows my first attempt at ‘folding’ the train.

The cyan circles are the PCDs of the 70T gears, the smallest diameter that will mesh to bring the final gear into alignment. Once I’d looked more closely at this arrangement, I realised that it would be sensible to make all three transfer gears the same tooth count which would facilitate the feed to the next gear-train (the Moon drive) with a known rotation of one day. This arrangement was impractical since the centre of the idler was not clear of the 80T gear so would need to be supported via an inconvenient sub-plate.

With just a small revision, Fig. 7 shows a much better ‘fold’ and leaves the idler gear (blue dotted line) in a much better location to connect the ‘Moon’ train.

At long last I’m at the point where I can discuss how I approached the basic problem of calculating the gearing to get the ‘Moon’ rotating once in 27.32166 days!

This train starts with the 70T gear running anti-clockwise over the period of one day – so what was needed was a set of gears giving a ratio of 1:27.32166 and reversing the direction of rotation!

From past experience, I knew that I was not going to do that in a ‘one step’ action so it needed breaking down. Certainly into two sets of 4 gears but maybe three sets.

In addition there were other constraints, such as the position of the basic infrastructure of the clock plates and supporting pillars, which would influence the choice of gear diameter/tooth count. This meant that the approach of ‘guessing’ what combination of gears would give the ratio in a straight line and later adjusting each tooth-count to ‘fit the space’ was less than efficient.

In fact, I seldom ‘guess’ at any particular pairing except in the very simple 1:2, 1:4 situations. I usually resort to a ‘Gear Calculator’ created by Duncan Munro – http://metal.duncanamps.com/software.php – a very versatile piece of software (bottom of the page) which I came across when I first bought my small gear hobber.

Having to start somewhere, it seemed sensible to select two factors of 27 to split the problem into two ratios which would combine two sets of 4 gears each. Nine & three are the only factors so keying in ‘9′ as the ratio required in Duncan’s Gear Calculator gave me about 350 perfect combinations. The first in the list which looked very promising was 44-22 / 90-20 but this is naturally only a starting point. The actual tooth count will be affected by the space available but it does provide a basis to work from and shows essentially a 2:1 – 9:2 pairing for the first part of the train.

The initial drawing of the train immediately brought to light the fact that I would have to use an ‘idler’ to move the centre of the 2:1 pair beyond the outer rim of the 70T gear. You will see in [Fig.8] that the centre-line is inside even the PCD.

Idler gears are no real problem, they just facilitate a re-positioning – they also cause a change of rotation direction but that is of little import – what size the idler needs to be, in this case, is controlled by the OD of the 70T gear. It has to be large enough for the spindle about

which it rotates to be outside the purview of that gear by at least the radius of the spindle plus some clearance.

The OD of the 70T gear is 36mm and a sensible spindle needs to be at least 4mm Ø so the distance from the centre of the idler to the centre of the 70T gear must be, at least, 21mm

((36+4)/2 + 1 clearance)

Now, the PCD of the 20T gear is 10mm so the idler PCD must be at least

(21 – 5) x 2 = 32mm

This leads to the selection of a 64T gear as the idler.

That is not the only factor that had a bearing upon where I needed to position this train of gears. As with all clocks, there has to be a means to fix the spindles in position on the ‘plates’. This means that there are also ‘pillars’ to hold the plates apart (or together if you prefer!) so [Fig.9] adds in these factors.

The location of the pillar meant that the idler and the 2:1 pair had to be moved away from being in line with the 70T gear. As I prefer to work with round numbers, I moved the Idler 15° clockwise. That was insufficient to also move the 2:1 pair clear of the pillar so they were moved a further 10°.

The final location of the 90T gear is as yet undetermined, I’ve simply placed it in line with the 2:1 pair. There are more constraints which will affect it’s final position and they have yet to be calculated.

What is known now is that the 90T gear takes 9 days to make one revolution and I now have to convert that to the 27.32166 days so the new ratio that I need to key into Duncan’s program is 3.03574.

The first ‘solution’ that appears shows an error of 0.000012% using 121 – 65 : 106 – 65 gears and when I key those figures into the spread-sheet it calculates that the number of days that the final gear – and consequently the ‘Moon’ – takes to make one revolution is 27.3216568 days. I suspect that if I were to take another iteration, adding another four gears, I might get closer but, for now, I consider being within 0.0000031953 of a day or 0.276 seconds over a tropical or sidereal month (less than 4 seconds per year) to be adequate, particularly since I was not taking into account the fact that the motor is notionally running at 2 rpm, and there will be mains voltage fluctuations beyond my control!

I did still have to determine the relative locations of these new gears that complete the train though.

With the final gear having to be directly above the centre of the main dial and 120mm distant, there is a limited arrangement of the other gears which can satisfy the link to the start of the ‘Moon’ gear-train. It wasn’t possible to move the 90-65 gearing pair to the right far enough to put the first 121T gear in the necessary position so a second equal tooth gear needed to be incorporated. The rotation period is still the same, the direction is reversed and, with hindsight, I now realise that in fact that is the correct direction, being prograde to the direction of the model Earth. This simply means that the model is being viewed from the South pole rather than the normally accepted view – from the North pole!

[Fig.10] shows the final arrangement from the 70T idler along with the minute and hour hand gears which set the position of the clock centre-line.

After some consideration I decided to change the 44-22 pairing to 42-21 to give more clearance for the pillar.