Basic Clock Design

Martin, yes, you do reduce the peak velocity during a swing, but that means that the bob doesn't rise so high as it reaches the extermity of its swing as the reduced kinetic energy is traded into reduced gravitational potential energy. Overall, a non-impulsed pendulum at small amplitudes is very nearly isochronous as it runs down.

With an escapement to supply the energy lost and keeping the amplitude constant, the rate (compared to the natural rate of the pendulum) depends on both the amplitude (through circular error) and the escapement characteristics, in particular the phase offset of the mean impulse from the centre of the swing. In fact the escapement "error" is very nearly (phase error)/2Q. If the phase error arises because the escapement action is somewhat asymmetric relative to the swing, it will vary with amplitude, and may act with or against the circular error. Woodward in MORT describes a Brocot spring driven clock where the circular error reduced as the spring ran down, reducing the amplitude; but this was compensated for by increasing escapement error. In "Clock B" the circular error seems to be nearly compensated by the suspension cheeks, and the escapement error and circular error are traded against barometric error and variation of amplitude with air pressure to give overall a very stable clock.

I agree with you but the period is not independent of drag. It's a damped system and the period will increase with increasing air resistance.

As you know Isochronous just means it is independent of amplitude not that it is not independent of other factors.

Martin

Martin, again sorry to return to this. I'm spending quite a lot of time at the moment simulating Harrison clocks and it was easy for me to do some simulations. Here is the period and amplitude of a seconds pendulum with a Q of 5000, zero circular deviation, coasting down from an initial deflection of 3 degrees.

The period remains within a gnat's of 2 seconds until the end of the 10,000 second run. It would be 2s except that the value of pi used in the simulation is only correct to 9 decimal places. Adding circular deviation gives this.

Here you can see the period some microseconds short at the beginning but as the amplitude decays it rises asymptotically to 2s again. Here's a different Q of 3000 again with no circular deviation.

Again the period is almost exactly 2s within nanoseconds and unchanging with amplitude. To the extent that the period does change actually it decreases with lower Q (i.e. increasing air resistance)

You are correct that there is a very very small change of resonant frequency with Q but for most practical values of interest in a clock it is unmeasurable. It is much much smaller than circular deviation.

No need for the apology John it's a discussion not a fight.

Very interesting plots. From plots 1 and 3 (Q = 5000, 3000) you show a variation of about 2.6mS/day which would over 100 days be a little over 1/4 of a second and that is with a small swing so a lot less velocity and therefore friction. For Harrisons error budget of a second in 100 days even this is a quarter of his allowance.

I totally agree that for all practical clock-making it's irrelevant but not to the boys who are chasing the ultimate mechanical clock (I don't include myself by the way). However for the synchronome the rate reportedly changes noticeably with the case door open from shut as the air is more constrained and less free to move out of the path of the pendulum when it's shut. (don't lets get started on bob shapes yet ;0) )

I stand by my original statement that Harrison in the RAS was using controlled circular error (suspension cheeks) to compensate for changes in the atmosphere, primarily air resistance and buoyancy. You are right to say that from your plots that circular error seems dominant but is 2.7 hours a little long for a practical pendulum to keep swinging in air? Maybe you could have a look at your numbers and simulate period against changing Q. Could you perhaps measure the slope of the period at 3 degrees amplitude in you original plots to give a number to the circular error factor at the working amplitude.

The initial reason I brought it air resistance up was as an example of how reducing friction could speed a clock up. It has however prompted this interesting discussion.

Could you maybe do your simulations again with a much larger swing as Harrison used where air resistance would be correspondingly greater. It's going to rise as some power of velocity.

Perhaps this link would help. You seem to be better at maths than me.

**LINK**

best regards Martin

John & Martin

I am very happy to observe and learn from your interesting discussion ...

May I just offer a starting point for the modelling, taken from Harrison's text:

[apologies for the typography ... there may be a 'copy & paste' problem]

[quote]

Edited By Michael Gilligan on 05/05/2016 10:35:48

Not really. I checked the pendulum of my regulator some time ago and swinging in free air it decayed to 37% of initial amplitude in about 1 hour. So that gives a Q of just over 10,000 and would still have 5% of initial amplitude after 3 hours.

Lots of interesting points being raised here. I need to contemplate a bit more but still haven't seen why a clock with a little oil added to a sticky escapement should speed up.

Russell.

Michael, I think that extract is just saying that the low friction of the knife edge supports for the crutch/pallet assembly will allow it to swing freely for ten minutes on it's own.

Russell.

Exactly Russell.

Maybe a clearer 'translation' is:-

As a relevant example, with the movement on a table, the escapement (with crutch attached, but without the pendulum) will swing freely on it’s knife-edges (at an amplitude low enough to avoid contact with the escape wheel) for 10 minutes, without coming to rest. The air (at that rate of oscillation and being so light a matter) may be supposed to cause stoppage in that length of time.

Thanks for a confirmation of realistic decay time too.

Martin

Russell,

I think it true to say that the performance of the oils available in Harrison's time was far below what we have available now ... and that much of 'CSM' is about the [claimed] superiority of his escapement detailing over Graham's. [i.e. it's more about the escapement efficiency than the pendulum]

Mudge's remarks, and their context, [cited here] are quite telling.

MichaelG.

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Edit: The two preceding posts were made whilst I was composing this [delayed by a Coffee break] so I had not seen them.  ... Yes, I am quite aware that the quote concerns the escapement without pendulum.

Edited By Michael Gilligan on 05/05/2016 11:37:54

Going back the the oily pallets issue.

Harrison said:

In the case of Mr Graham’s clocks, with a small pendulum amplitude and other poor characteristics described earlier (mainly pallet action at a large fraction of the pendulum length . . . text inserted by me) most especially when the oil is foul, a touch of fresh oil will cause the amplitude to increase and the clock to go faster *. Dr Bradley . . . reported a variation of approximately 2 seconds a day when the escapement was freshly oiled.. When the pendulum amplitude is larger, a touch of oil will still as just described cause an increase in amplitude but the effect would be to cause the clock to run more slowly. This is a more satisfactory outcome being more in accordance with the nature of a pendulum.

The way I read the text is Harrison was complaining about pallet action taking place so far along the pendulum rod making variation in friction a much greater effect than it need be. Withe the Graham Dead beat the pendulum is constrained by the pallets and escape wheel for most of it's swing resulting in the frictional forces of the pallets across the escape wheel teeth damping the oscillation and slowing the period. Reducing the friction, reduces the damping and shortens the period. The comment at * is an observation of what the clock does and not that the amplitude is the cause of the increase in rate. The last two sentences state that the clock behaves as expected when friction is low and then gets slightly lower. Friction has become the less dominant effect at that point.

Just as a thought all this would indicate that Harrison thought a great deal about the effect of drag on the action of his pendulums and although I have no evidence to prove this, maybe thinking about the shortfalls of Graham's clocks lead him to understand the effect sufficiently to use it (as atmospheric drag) balanced out with controlled circular error to create a system which was largely immune to atmospheric changes.

Martin

Agreed, Martin

'CSM' argues the case for Harrison's mechanism being superior to Graham's

1. Geometrically, and
2. For its lack of reliance upon the 'state' of applied oil.

The combination results in something where the pendulum [as he says] "has dominion" whereas [implicitly], the Graham design responds unpredictably to small changes.

... anyone for Chaos Theory ?

MichaelG.

Thanks for pointing out that link Martin, one I hadn't seen. Yes, it gives a formula for the resonant frequency corrected (effectively) for Q. Applying that with Q values of 3000 and 5000 gives corrections of 3.3 nS and 2.5 nS respectively! The simulation results can't really be relied on to nS periods because of the way it calculates periods, but in mechanical clock terms the difference is irrelevant. The difference is 0.8 ns per 2s, which is 3.5 ms in 100 days.

Somewhere I have seen that the impulse from a deadbeat escapement (such as the Graham) is slightly imbalanced, that is it's slightly off-centre from the pendulum swing. This means that the impulse is slightly out of time phase with the pendulum velocity, resulting in a small time error. If the impulse phase is in advance of the pendulum it speeds it up; if behind it slows it down. The imbalance is geometric, but the amount of resulting phase shift depends on the pendulum amplitude. Suppose the imbalance is slowing the pendulum down. You add a spot of oil, reduce the friction, increase the amplitude, reduce the phase shift, and the pendulum speeds up.

You asked about the circular error at working amplitude. The fractional reduction in rate is just (amplitude in radians)squared/16 assuming no correction. At 1 degree, or about 1/60 radians, this is 1/(3600x16) = about 35 microseconds for a seconds pendulum. This is exactly what the simulation gives as well. You can scale up or down from that for other amplitudes.

Air resistance law has to be specified in advance, linear or square law. For normal amplitudes of a degree or so it seems to be about linear, but at Harrisonian amplitudes of more like 6 degrees it's probably square law.

Your last paragraph -

"Just as a thought all this would indicate that Harrison thought a great deal about the effect of drag on the action of his pendulums and although I have no evidence to prove this, maybe thinking about the shortfalls of Graham's clocks lead him to understand the effect sufficiently to use it (as atmospheric drag) balanced out with controlled circular error to create a system which was largely immune to atmospheric changes."

- there seems to be no doubt that this is exactly what he did, though not just circular error but escapement error too.

Thanks for that link Martin. I'm now beginning to understand what's happening. If his equation for the frequency change with damping factor v is expanded using the binomial theorem we get the frequency being reduced by the damping by a factor of 1- v^2 /8wo plus other higher order terms which are insignificant. As the slowing down depends on the damping factor squared the effect will be very small for low friction cases. So I think Michaels remark about the quality of the oils in Harrison's time makes sense and that the effect Harrison observed may not occur with modern oils.

Russell.

Edited By Russell Eberhardt on 05/05/2016 20:06:13

To me it is evident that the effect he observed has nothing to do with the very small influence of damping on resonant frequency (measured in nanoseconds!) but much more to do with escapement error changing as the amplitude increases when oil is applied.

In fact, Rawlings (Science of Clocks and Watches) on pp 110/111 of the 3rd edition confirms that for the Graham dead-beat in practice the impulse has to be slightly late so the pendulum runs slightly slow (but of course this is regulated out). With more pallet friction reducing the effective impulse when the oil is dry, the pendulum amplitude will decrease, and in effect the impulse will be later and the clock run slower. When oil is applied friction reduces amplitude increases impulse gets earlier and pendulum speeds up.

For info.

Here is an article I've not seen before, about 'clock B'

Slightly different slant on the story, and a rather nice picture.

MichaelG.

Hi John and Russel

Russell you are correct. The old oils were animal based and did degrade very quickly.

The point I was making was any damping always acts to slow a pendulum. The example of air resistance was just to illustrate the point although at the time I was not sure of the magnitude of the effect. I'm still not convinced it is as small as you state but that's slightly off the point. With a Graham Dead beat friction will be present in the escapement for virtually the entire swing so increasing the friction at the pallets by virtue of the foul oil increases the damping and slows the clock. Better or newer oil has the opposite effect With very degraded oil this is going to be significant enough to cause the effect as observed. I agree with John re phasing which just makes things worse. With long pallet arms the friction has an excessive effect which is what Harrison was complaining about.

best regards Martin

Interesting one Michael and a little clearer than some. To me it gives a good clue - the circular suspension cheeks can account for more than what people might generally suppose including barometric air pressure changes. I suspected as much because they must account for the variations from the remontoire he used. Harrison is way past basic clocks and is a pretty deep subject much like Hendrix's guitar and effects pedal there can be many miss interpretations about.

Notices where has basic clocks gone comments - note that there may have been a conclusion that a Brocot escapement would be a good direction to go in as it's easier to make accurately. The main reason it interests me though is that it can be oil free.

The Harrison debate hinges around what pendulums do in practice. Ideally the swing time would be identical irrespective of the angle swung. It isn't so precision clocks may reduce it to as little as 2 degrees. 3 and below is reckoned to have negligible effect, in other words behaves as expected with minimal error from the ideal. Important as there will be minor variations even from the mechanics. Air density raises it's head as well, Usual fix very heavy pendulums.

I mentioned that a Graham must add friction throughout the pendulum swing. It's also generally lightly oiled. It is possible to make the pallets accurately. Rather than cut pallets out of some "turned up tube" it's also possible to turn them directly. This finish up sticking out just like the the version where the "tube" pallets are fastened into a frame. The frame though needs to be in 2 parts. The inner diameter can be measure via a mic with ball attachment usually used for measuring tube wall thickness's.

Here is a shot of an escapement actually made by Brocot. The clock is virtually held together with clock pins but still no sign of a reamer.

Looks like very slight flats on the end of the teeth to me. From the split in the escapement it also looks like the pallet spacing can be adjusted. Probably in part to account for wear over the years..

John

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John,

[a] ... forgive the pedantry, but at this level it's important: The cheeks are cycloidal, not circular ... If you haven't already done so, I recommend reading Huygens' Horologium Oscillatorium. [thankfully, there is an excellent English translation available, by Dr Ian Bruce.]

[b] ... Thanks for the Brocot photo.

[c] ... I really don't think you will find a commercial reamer for Clock Pins; if a broached hole isn't good enough for you, I would suggest making your own reamer from a steel clock-pin.

Welcome back, by the way.

MichaelG.

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P.S. ... The anchor on a Brocot escapement is almost always adjustable in some way ... There's not much chance of re-shaping the jewell pins to make fine adjustments, so it's much simpler to change the distance between their centres.

Edited By Michael Gilligan on 07/05/2016 18:33:10

You mean T=4pi sqrt(a/g) where a is the rad of the curve used to generate the cycloid and is 1/4 the length of a simple pendulum and T is the period and etc Michael?

He didn't use it in his subsequent clock, He used a link arrangement instead. No chops as they are usually called. The pivot point of the crutch didn't coincide with the pendulum pivot. He later went back to chops again - on paper.

In Harrison's case it's probably safer to say curved chops that were adjustable anyway.

John

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Edited By Ajohnw on 07/05/2016 19:17:29

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John,

What I meant was "I recommend reading Huygens' Horologium Oscillatorium." .... For several reasons, but mostly because Huygens was not only a genius, but also a great communicator.

MichaelG.