A question of beat

[Sam StonesParticipant @samstones42903]

As a one-clock ‘expert’, I have often wondered why the late Mr John Stevens, in designing his skeleton clock with English lever escapement, chose wheels and pinions that produced a beat of 0.893 seconds instead of a very handy 1.0 beat per second.

In a simple spreadsheet for determining the ratios of the existing wheels and pinions while leaving the number of pinion teeth alone, I tested the feasibility.

With little more than intuition I changed three of the wheels from 84, 72, and 56 teeth down to 80, 70, and 54. I was amazed that it took so little effort.

Apart from shifting the arbor centres, slightly increasing the mass of the balance wheel, and/or using a slightly ‘softer’ balance spring, I can see no other obstacle.

Unless I’m off the beam for some reason I could (just for fun) make new wheels and even grab parts from my CAD file.

Any thoughts?

Sam

Edited By Sam Stones on 28/01/2020 02:45:45

[Sam StonesParticipant @samstones42903]

[not done it yetParticipant @notdoneityet]

Perhaps he found the 0.893 seconds per beat was more pleasing, to his ear, than only once per second? Certainly can’t so easily count ticks for a minute period!

As a matter of interest, do all other clock designs have a beat of 1.0 seconds?

Mathematically, the 0.893s is to one more significant figure, so should be more accurate than one only to two significant figures – or did you mean 1.00s?🙂

As this makes absolutely no difference to the time-keeping, does it really matter that much? A design simply needs to accurately mirror the true time of day.

Do these tooth counts require fewer set-ups or cutters than other tooth counts?

I would particularly frown on the use of smaller pinions being sized as a factor of the larger wheel it contacts. Any comment on this?

As only a clock repair ‘bodger’, I’m not particularly qualified to make too many meaningful comments (just to make it clear to those that are ‘funny’ about replies). Last comment is that it does show that a one second beat is not the ‘be all and end all’ of clock making.🙂

[John HaineParticipant @johnhaine32865]

Unlike a pendulum I guess making a balance for a specific period is rather hard unless you have a lot of data on the spring material etc. So maybe he started with the balance and designed the train to suit?

[Michael GilliganParticipant @michaelgilligan61133]

Posted by not done it yet on 28/01/2020 06:40:09:

[…]

I would particularly frown on the use of smaller pinions being sized as a factor of the larger wheel it contacts. Any comment on this?

[…]

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Yes, one comment: … Sam wrote:

“ In a simple spreadsheet for determining the ratios of the existing wheels and pinions while leaving the number of pinion teeth alone, I tested the feasibility. “

[my emboldening, for emphasis]

MichaelG.

[John HaineParticipant @johnhaine32865]

Well, they are always going to be a "factor" surely, since both have integer numbers of teeth? Was the point to try to make sure that they are relatively prime (if possible) to distribute wear evenly, or something like that? In any case you have to have specific ratios between hour and minute, and minute and seconds hand, if it has one.

[Michael GilliganParticipant @michaelgilligan61133]

Posted by not done it yet on 28/01/2020 06:40:09:

[…]

Mathematically, the 0.893s is to one more significant figure, so should be more accurate than one only to two significant figures – or did you mean 1.00s?🙂

[…]

For anything reasonably approximating accurate time-of-day-keeping we need many more orders of magnitude better than either of those representations. [so, of course, they are both very rough approximations]

MichaelG.

[SillyOldDufferModerator @sillyoldduffer]

Just a suggestion as I've only made two clocks, both out of Meccano and one didn't work!

However the principle of all mechanical clocks is the same: an accurate oscillator, such as a pendulum, counts out steps allowing a gear train to translate a power source (spring or dropping weight) into a spindle turning at a known rate useful to humans, such as one rotation per hour.

Getting the required human rate is achieved by altering the period of the oscillator and the ratios of the gear train to get the required result. Designing the gear train is a little tough: clocks call for big ratios, ie small pinions engaging big wheels, and this is a challenge because there's a mechanical limit. Pinions with less than, say, 12 teeth are in the danger zone, and 6 teeth is probably the practical minimum. Friction rises and pinion teeth have to be undercut. Then, because gears only do integer division, there are limits to the number of gear combinations that achieve the desired ratios.

Sam's picture of Mr Steven's gear train shows a sensible combination of gears, in particular he avoids small pinions. But I think the reason for the 0.893s period lies in the Balance Wheel, which isn't shown.

Not mechanically difficult to make a pendulum oscillate at any reasonable period. The disadvantage is they take up a lot of space, and dislike vibration. Balance wheels can be made much smaller, and they don't mind if the clock is moved. But they're much more complicated to design. They also have mechanical limits: small and light is best for timing accuracy and low friction but they have to store enough energy to work the escapement. A big, heavy balance wheel will happily drive the escapement, but it will hammer the bearings, and need more effort to maintain oscillation which will spoil the time-keeping.

Sam said 'I have often wondered why the late Mr John Stevens, in designing his skeleton clock with English lever escapement, chose wheels and pinions that produced a beat of 0.893 seconds instead of a very handy 1.0 beat per second.' More the other way round I think. 0.893s is in the the zone that allows a sensible balance wheel and there's a sensible integer gear train to bring it to 1 revolution per hour. I think Mr Steven's clock brings together a practically sized balance wheel with an equally practical gear train. Not necessary for him to start with a neat round number oscillator, or even desirable to do so. Once the maths are understood, it's possible to design clocks so the mechanical details are optimised as a whole to improve timing, reduce wear, and simplify the movement.

Dave

Edited By SillyOldDuffer on 28/01/2020 10:50:25

[Michael GilliganParticipant @michaelgilligan61133]

Posted by John Haine on 28/01/2020 09:36:31:

Well, they are always going to be a "factor" surely […]

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I’m not sure if that ^^^ was intended as a comment on my response to ndiy, John

If so, I must emphasise that Sam computed his new train for the existing pinions.

If not … please ignore this interjection

MichaelG.

[not done it yetParticipant @notdoneityet]

Factors are whole numbers which will divide into the other number by a whole number. All primes, other than two, have only two factors.

Using factors is bad practice with gear sets, as the same teeth will be engaging every revolution. Far better for even wear is to have the gears wearing with any of the teeth wearing against all the teeth on the other gear.

You never had an exact 4.00 differential in your car – it would wear much better and evenly if the ratio were the pinion teeth multiplied by an integer, then plus (or minus) one for the crown wheel. A much higher rotational speed and power, but the same physical principles still apply – but clearly ignored in the clock movement exampled.

[John HaineParticipant @johnhaine32865]

Yes I do know what a factor is, thanks. It was just the expression that puzzled me. Quite common I think to see a 60:1 ratio as an 8:60 x 8×64 teeth = 7.5 x 8 = 60.

[Michael GilliganParticipant @michaelgilligan61133]

Posted by not done it yet on 28/01/2020 11:49:55:

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Factors are whole numbers which will divide into the other number by a whole number. […]

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Apologies … I evidently misinterpreted your usage of the word ‘factor’ in your first response.

I wrongly thought that you were saying that the number of of teeth on the pinions would be a relevant factor [rather than a mathematical one].

MichaelG.

[not done it yetParticipant @notdoneityet]

Let me put it another way.

It is bad practice to make gears pairs with whole number ratios.

Is that more clearly understood? I don’t design gear boxes or anything like that and maybe clock makers ignore the principle (that gears pairs should not run on the same teeth for the whole life of the machine). Comments? I’d like to know if they do (and if they do, why?).

[Michael GilliganParticipant @michaelgilligan61133]

Posted by not done it yet on 28/01/2020 15:47:58:

Let me put it another way.

It is bad practice to make gears pairs with whole number ratios.

[…]

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Agreed …

MichaelG.

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https://www.mobiusinstitute.com/site2/item.asp?LinkID=8062&iVibe=1&sTitle=Gearbox

Edited By Michael Gilligan on 28/01/2020 16:08:03

[BazyleParticipant @bazyle]

Posted by not done it yet on 28/01/2020 15:47:58:

gears pairs should not run on the same teeth for the whole life of the machine). Comments? I’d like to know if they do (and if they do, why?).

Yes they do in the motion work. Getting that 12:1 ratio is not easy in two stages when limited to pinions probably of 8, 9, 10, 11, 12 so tend to use 3:1 x 4:1.
There are combinations with factors of, 2.2 or 3.3 that work with 2.4, 2.5, and 3.2 as runners up.

[Michael GilliganParticipant @michaelgilligan61133]

Posted by Bazyle on 28/01/2020 18:02:18:

Posted by not done it yet on 28/01/2020 15:47:58:

gears pairs should not run on the same teeth for the whole life of the machine). Comments? I’d like to know if they do (and if they do, why?).

Yes they do in the motion work. Getting that 12:1 ratio is not easy […]

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Which is, of course, one of the reasons why clocks in the ‘Regulator’ class typically omit motion work.

MichaelG.

[John HaineParticipant @johnhaine32865]

Well a pair of gears is bound to have a whole number ratio surely!

In the motion work as opposed to the going train the gears carry very little load.

[Neil WyattModerator @neilwyatt]

Maybe he made a balance wheel, tested it, then mcame up with a train that worked for its practical range of adjustment.

Neil

[Michael GilliganParticipant @michaelgilligan61133]

Posted by John Haine on 28/01/2020 22:24:13:

Well a pair of gears is bound to have a whole number ratio surely!

 

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So … What are the various whole number ratios between the wheels and pinions in Sam’s first picture ?

Ratio being, by convention, expressed as N to 1

[ tried to use a colon, but that generated a stupid smiley thing ]

MichaelG.

Edited By Michael Gilligan on 28/01/2020 23:40:43

[Sam StonesParticipant @samstones42903]

What a wealth of responses and information. Where do I start?

I think it might be easier (for me at least), to address some of these in separate postings.

I'll be back!

Sam

[Sam StonesParticipant @samstones42903]

NDIY – I enjoyed your replies, so here are my thoughts.

Q. Pleasing to his ear? Perhaps it was. Maybe he didn’t have a preference for the pace of a military beat 120bpm (shades of square-bashing). Personally, I enjoy a good brass or military band.

The ratios the designer selected were 8.4:1, 8:1, and 4:1. The calculated beat (per ‘scape wheel tooth) is 0.892666(recurring) seconds. Whatever reason crossed my mind, I resisted the temptation to quote #1 followed by a string of zeros.

Unintentionally, the three ratios emerging from my choice of wheel teeth were …

8:1, 7.777778:1, and 3.857143:1. Not bad for little effort

Q. Do ALL other clock designs beat at 1.0s?

My guesses …, in not knowing much about clocks would be …

Antique clocks? … not many … apart from longcase (pendulums close to 1 metre).

Electric clocks? … Hummm? … locked into mains frequency at 50 or 60cps.

Quartz clocks? … most of them that I’ve heard ticking … [sourced from a 2^{^15} crystal.]

I agree that none of it (i.e. the ratios) really matters other than that the clock keeps accurate time which, as I found out while building and setting up the clock, is another story altogether.

I chose to refrain from ‘messing’ with the pinion count. Other than a 6-leaf pinion onto a 72 tooth (motion work) wheel, the drive train for the Stevens clock uses lantern pinions.

It seems I've overrun the number of characters … I'll be back.

[Sam StonesParticipant @samstones42903]

Continued …

Re your second response NDIY

As long as the prime number falls within the scope of the available equipment, the idea of using prime numbers is valid. It seems there are few limits when stepper motors are involved.

I’m reminded in your third paragraph (and also your third posting) of what was (once) referred to as a hunting tooth, i.e. an extra tooth that lessened wear. Personally, I have no hard evidence as to whether this method is used.

How odd that I was reading about cicadas, and how two species have evolved. Spending many years of their life underground, they emerge from the ground after a specific period. One species, after thirteen years, and another after seventeen … both, as you can see are prime numbers. This, it would seem, reduces the propensity for their predators to match their emergence.

Sam

[not done it yetParticipant @notdoneityet]

Re the cicadas, My first thought was of the Fibonacci series, but that doesn't fit. Most nature series, but not exclusively, fit that series.

Clocks are still much of a mystery to me. So many interacting sections – all needing to work together and separately.

[SillyOldDufferModerator @sillyoldduffer]

Posted by Sam Stones on 29/01/2020 03:52:41:.

…

Q. Do ALL other clock designs beat at 1.0s?

…

No, far from it. Beating in seconds simplifies the maths and the resulting gear train(s) are mechanically satisfactory. (Not too complicated, and no need for extreme ratios.) Apart from that, it's not a constraint.

This table is from Britten's Watch & Clock Makers Handbook. Just to keep us on our toes it tabulates 'Vibrations of Pendulum per Minute' rather than seconds, but it gives the clockmaker a choice of 32 different periods.

The two times marked with an * are for Turret Clocks with a rotation time of 3 hours.

Another table explains lever clocks run at 16,200 or 18,000 vibrations per hour (4½ and 5 vibrations per second), and gives 7 gear trains for each. Three trains are given for "Non-Seconds" clocks running at 19,800 vibrations per hour and two more for clocks running at 21,000 vibrations per hour.

I like to think of clocks as consisting of an oscillator followed by a divider chain. In a mechanical clock the oscillator is a slow pendulum, balance wheel or verge. The divider is a gear train, essentially counting ticks and translating them into h:m:s. In an electric clock the oscillator is faster – mains frequency – and the divider is a synchronous motor turning a few gears. A quartz clock has an electronic crystal oscillator and an electronic divide chain. As it is extremely easy to reliably divide frequencies by two, the oscillator usually runs at a power of two, 32678Hz being common. A quartz clock ticking at least 650 times faster than mains at 50Hz makes it difficult to build a mechanical divider, so electronics are needed. The oscillator frequency is divided down repeatedly by 2 to drive either a digital display or analogue hands with a simple synchronous motor plus a few gears. One second out is quite common.

Although quartz crystals are excellent time-keepers, they're not the best that can be done. At the moment, this is the average of several Caesium oscillators running at 919263177Hz with human-useful frequencies derived by electronic dividers. Human useful can be almost anything: there are radio standards locked to atomic time at 10MHz, 5MHz, 198kHz, 60kHz and several other radio frequencies. (Almost any broadcast radio station can be made a reference.) Today, accurate time is mostly got from GPS satellites and the Internet.

Sadly, although electronics have dramatically improved the quality of time-keeping for rock-bottom prices, they aren't half as aesthetically pleasing as mechanical clocks. Just like diesels vs steam locomotives, one is a brutally efficient tin box on wheels, while the other, despite being mechanically out-matched, is a treat for sore eyes.

Dave

[John HaineParticipant @johnhaine32865]

Once people needed time resolution to 1 second but displaying hours minutes and seconds the die was cast. For example if you were taking a sight of the sun on board a ship a 1 second error at the equator would amount to a quarter of a nautical mile, the difference between shipwreck and safety. So you would want your chronometer to register seconds, even if the balance would beat faster. Then you needs a ratio of 60:1 (or a quotient of 0.016666….) between the seconds and minute hand. Not a lot of choice of ratios to get this given 60 has prime factors 2x2x3x5. Similarly for the minute to hour hand, and possibly worse with fewer prime factors. The fact that so many clocks have run for so many hundreds of years without having many relatively prime ratios perhaps means that wear from this cause isn't so much of a problem.

Once you get into electric and quartz clocks where the whole train is "motion work" it carries hardly any power and the forces are minimal.

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