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A question of beat

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Sam Stones29/01/2020 03:53:51
764 forum posts
304 photos

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.


not done it yet29/01/2020 07:31:24
4889 forum posts
20 photos

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.

SillyOldDuffer29/01/2020 10:08:17
6191 forum posts
1345 photos

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.


John Haine29/01/2020 13:34:41
3272 forum posts
175 photos

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.

Bill Davies 229/01/2020 14:06:13
197 forum posts
11 photos

Dave, at the risk of being labelled a pedant, 32768 is a power of two.


Sam Stones29/01/2020 23:23:36
764 forum posts
304 photos

Inexplicably overstepping the thread character limit, I was obliged to split my last offering.

In answer to your suggestion John H, that … the designer might have started with the balance and designed the train to suit?... this did pass through my mind on occasions. In Part 1 of the five-part Model Engineer article he abandons the idea of using platform escapements as looking like an afterthought rather than an integral part of the design.

Mr Stevens appears to have based his escapement design along lines similar to already established principles of escapement. On Page 268 in the third edition of Model Engineer (17 March 1972. Vol 138, issue 3437), he refers to methods in the late W.J.Gazeley’s book …“Watch and Clock Making and Repairing”.

Whether he developed his own set of balance wheel and balance spring details is unclear. I imagine he would have to. He certainly chose to use flat-section spring wire for the balance spring, whereas I switched to round-section wire, driving myself into a gloomy backwater and many failed attempts at producing a suitable (just passable) balance spring.

In contrast, the (bimetallic) balance wheel design provides timing screws around its periphery.

The Cage

Despite the bimetallic facility however, I have never tried to adjust the beat for variations in temperature. There were other imperfections that I wasn’t willing to fix; for example, the fusee/main spring calibration, versus the sensitivity to power variations of the English Lever Escapement.

More to come,


Edited By Sam Stones on 29/01/2020 23:27:58

Sam Stones30/01/2020 23:29:10
764 forum posts
304 photos

This thread has invited a useful selection of entertaining comment. I trust others have been similarly amused. Thanks Michael for your watchful eye on detail, and your attention to the motion ratio aspect Bazyle.

On that score, it’s possible that he thought the design less cluttered when Mr Stevens was considering the motion’s 12:1 ratio. Instead, and with minimal stress to contend with (thanks for your input there John), he used a wheel with 72 teeth meshing with a 6-tooth pinion, the only solid pinion in the clock. However, this choice shoved the smallest of the pinions over to the side, thus requiring a 1:1 pair of wheels to bring the movement back to centre.

Dave’s chosen description of a clock and its variants while delving into the depths of atomic clocks and GPS was quite unexpected. Thanks for that Dave.

We’ll probably never know whether Mr Stevens started with a known (oscillator) source of timing, e.g. a variant of a platform escapement, or the alternative … a fixed stack of crossed out wheels and lantern pinions, each identified according to traditional terminology.

If the former, then I would imagine (as Neil suggests), he may have experimented with varying sizes of balance wheel or, as is quite feasible, calculated variants of the wheel’s 2nd moment of area. In the original article, brass was the material specified for the balance wheel. Whereas, having read somewhere (Gazeley’s book, perhaps) I digressed making the rim bimetallic (brass on steel). As I recall, the suggestion was 2/3 brass fused onto 1/3 steel. By the time I was into about the fifth attempt, having produced one without clearly visible blowholes in the brass, it was nearer 45/55.

Alongside this, he may instead have chosen to calculate the section properties of the wire, in particular the MI (moment of inertia). Classified in Wikipedia as a chronometer helix … **LINK**, the spring was supposed to be of flat section wire, giving the spring substantially greater stiffness in the vertical direction. Ideally, it would appear to ‘breath’, whereas my switch to round wire stood little chance. Wobble would be a better description.

In hindsight, it appears to matter little if the beat is 0.89266667 secs or 1.00000000 sec so long as we have the time (to wonder).


PS Have you tried locating a cicada (or a cricket) guided only by the sound they make?

Edited By Sam Stones on 30/01/2020 23:32:10

Grindstone Cowboy31/01/2020 00:02:43
325 forum posts
27 photos

Slightly off-topic, but including time, temperature and crickets...

To get a rough estimate of the temperature in degrees fahrenheit, count the number of cricket chirps in 15 seconds and then add 37.

John Haine31/01/2020 06:54:24
3272 forum posts
175 photos

Ah, but how do you estimate the 15 seconds?

Michael Gilligan31/01/2020 08:52:10
16202 forum posts
706 photos

Tinkerers [whether practical or armchair] might find this helpful: **LINK**

Although it doesn’t answer Sam’s original question, it does allow one to experiment.



Edit: Yes, it does work as a check on the illustrated train : we just need to allow for the different actions of pendulum and lever escapements.

Edited By Michael Gilligan on 31/01/2020 08:58:19

SillyOldDuffer31/01/2020 09:56:34
6191 forum posts
1345 photos
Posted by Bill Davies 2 on 29/01/2020 14:06:13:

Dave, at the risk of being labelled a pedant, 32768 is a power of two.


Not at all Bill, it's a fair cop guv!


At least I got the number right...


SillyOldDuffer31/01/2020 10:31:48
6191 forum posts
1345 photos
Posted by Sam Stones on 29/01/2020 23:23:36:

Inexplicably overstepping the thread character limit...

I think this is because Sam's posts are pasted into the forum after being typed in a Word Processor. (A sensible precaution when preparing long posts.)

Unfortunately Sam's Word Processor font (Calibri) is exported as part of the post so it can be used in place of the forum's built-in default when the rest of the world reads it. The actual payload is higher than the visible character count because the font definition is part of the message too.

This might help. After pasting words into the forum editor, select all, and press the blue rubber button in the forum's edit tool-bar. 'Remove Format' should revert the font back to bog-standard forum issue, hopefully releasing space for links to photos etc.


Bazyle31/01/2020 10:59:05
5393 forum posts
206 photos

Going right back to the beginning and looking at the wheel counts in the first post. It could be the designer just used some numbers he was familiar with out of habit. Alternatively if he was using a direct dividing wheel engine to make his wheels it might have been governed by what his plates had. I only have Chronos plate DDE62 which has 72 but not 70 though has both 84 and 80, and nothing below 62. Another plate does perhaps give both 54 and 56 but anyway it could come into the thinking.

Sam Stones31/01/2020 21:53:09
764 forum posts
304 photos

Thanks for the explanation Dave.

Despite having used or had access to word processors since the 70’s, I never knew that.

Several times during the course of my ten years of M.E. forum ‘threading’, without realising why, I’ve lost text or found myself logged out.

I know I’m not the only one who has experienced this frustration after spending considerable time composing.

Unless I have only a remark or two, I compose in MS Word. Then, for no reason that I can recall, I switch to 12pt Times New Roman. All this while avoiding being logged into the forum editor.

This will be a good test.


It appears to have worked!!!

Many more thanks Dave

not done it yet31/01/2020 22:37:31
4889 forum posts
20 photos

However, this choice shoved the smallest of the pinions over to the side, thus requiring a 1:1 pair of wheels to bring the movement back to centre.

Does this mean he could have used a non-integer ratio and then run them in the opposite way to get back nearer to the centre? Just being a nuisance here, as I have given it no more consideration myself.

duncan webster31/01/2020 23:11:16
2734 forum posts
40 photos
Posted by Sam Stones on 29/01/2020 03:52:41:

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.


It's 100 bpm when you're playing for a Remembrance Sunday Procession. Why? Because old guys walk more slowly. This is also the recommended rate for doing chest compressions in CPR, so just do it in time to 'Boys of the Old Brigade'. Never mind the funny looks.

You learn a lot of useless facts on this forum!

Sam Stones31/01/2020 23:38:03
764 forum posts
304 photos

Bazyle, I think you may have cracked it!

Here’s what happened in a similar vein when I was making the wheels of Mr Stevens’ clock.

My second hand ML7, bought in about ’65, came with a dividing head and three indexing plates.

The design for both the great and the maintaining had 96 teeth. I set up the 40:1 homemade dividing attachment selecting the appropriately drilled index plate. As with the other gears, I used a single-toothed fly cutter made from hardened silver steel. All seemed to be going well until I decided to check that the result was correct by dropping into the first groove I had cut.

To my dismay, that didn’t happen.

Instead, the first tooth began to disappear as the cutter traversed the edge of the wheel. It was too late to do anything but start all over, but not until I had discovered the source of my error. Either I had miss counted the number of holes during indexing, or I had made a mistake in the arithmetic. Then I discovered that I didn’t have the correct number of holes in the index plate either, so I ended up machining a new wheel with 94 teeth.

The Great & Maintaining Wheels

It was fortunate too, that `give or take a couple’, the number of teeth on the great wheel is not critical in the actual operation of the clock providing that the depth of engagement is taken into consideration. They simply transmit power to the gear train. The only real damage was to my ego.

It is now clear (if you know what you are doing), that the rest of the ‘going train’ wheels don’t have to conform exactly to the number of teeth specified.

Thanks Bazyle!

Thanks for the CPR tip Duncan.

When I was blowing clarinet (badly) in the RAF Auxiliary Band I recall the Cameron Highlanders had to march at about 180bpm. Oh to be so fit now!


Edited By Sam Stones on 31/01/2020 23:42:38

Sam Stones01/02/2020 00:09:57
764 forum posts
304 photos

Does this mean he could have used a non-integer ratio and then run them in the opposite way to get back nearer to the centre?

Guilty as charged NDIY.

I gave it even less thought, other than it might look better embarrassed


Sam Stones01/02/2020 00:14:47
764 forum posts
304 photos

That's a nice piece of kit, Michael.

Tinkerers [whether practical or armchair] might find this helpful: **LINK**

Thank you.


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