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What theoretical advantage, if any, does a duo-in-uno hairspring possess over a helical hairspring?

Clint Geller

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I am curious. Greater ease of positional adjustment? More isochronous? Something else? What justified the added effort?
 

gmorse

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Hi Clint,

One advantage claimed for it was the use of a smaller collet which reduced positional errors. There was some dispute between McLennon, Hammersley and Walsh over who originated the concept, but I wonder if it was partly a matter of 'I can do this spectacularly difficult thing, which very few others can'.

For those here who may not have seen one of these, this is a Hammersley.

DSCF5465.JPG

Regards,

Graham
 

MartyR

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The Duo in Uno spring was first shown to the public on the Charles Frodsham stand at the Great Exhibition of 1862 in Kensington, London.

Vaudrey Mercer's book The Frodshams says the following of the spring:

This is praised on Page 22 [of the Exhibition catalogue], where he [Frodsham] says that it ensures "almost certainly correct performance in all the various positions with a perfectly poised balance".

This type of balance spring is said to have been invented by Mr Mairet, of Baker Street, and was shown by Charles and Mr McLennan in the 1862 Exhibition, but had been used by Arthur Paul Walsh since about 1860.


I have a copy of the appropriate section of the Exhibition catalogue, but it is out on oan right now. As soon as I get it back, I will post again and further reference to the duo-in-uno which it may contain.
 

Clint Geller

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The Duo in Uno spring was first shown to the public on the Charles Frodsham stand at the Great Exhibition of 1862 in Kensington, London.

Vaudrey Mercer's book The Frodshams says the following of the spring:

This is praised on Page 22 [of the Exhibition catalogue], where he [Frodsham] says that it ensures "almost certainly correct performance in all the various positions with a perfectly poised balance".

This type of balance spring is said to have been invented by Mr Mairet, of Baker Street, and was shown by Charles and Mr McLennan in the 1862 Exhibition, but had been used by Arthur Paul Walsh since about 1860.


I have a copy of the appropriate section of the Exhibition catalogue, but it is out on oan right now. As soon as I get it back, I will post again and further reference to the duo-in-uno which it may contain.
Frodsham’s spiel sounds like typical sales hype, but I suppose hyperbole is sometimes built on a kernel of truth.
 

gmorse

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Hi Clint,

It was certainly a very difficult thing to make. The spring had to be wound onto a former before being hardened and tempered, which then posed the question of how to extract the former afterwards without destroying the spring. I believe they used a former composed of a stack of thin disks which could be dismantled and extracted between the coils when the spring was completed. If a spring was heat treated with only the helical and spiral portions on a former and the top terminal curve was formed afterwards, the stresses created meant that the spring could take a long time to settle down into a steady rate, sometimes a year or more, so it was important that it was all formed at the same time.

Regards,

Graham
 

Ethan Lipsig

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I have a free-sprung C. Frodsham (made by Daniel Buckney) with what was described as a double overcoil hairspring. I assume that's a different breed of hairspring from the duo-in-uno, but how so, and what were its perceived advantages?
 

Clint Geller

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I have a free-sprung C. Frodsham (made by Daniel Buckney) with what was described as a double overcoil hairspring. I assume that's a different breed of hairspring from the duo-in-uno, but how so, and what were its perceived advantages?
Do you have a picture, Ethan?
 

Clint Geller

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Hi Ethan,



A normal helical balance spring has 'overcoil' terminals at top and bottom.

Regards,

Graham
Ah, so just a nonstandard term for a helical hairspring, perhaps? Though wouldn't one of those terminals actually be an undercoil? :)
 

SKennedy

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A duo in uno has several coils shaped like a helical spring and of constant diameter the same as the outer diameter of the flat spiral part of the spring. Some (most?) did then also have an Arnold style terminal curve towards the pinning point on the balance cock.
A double overcoil would be a flat spring, pinned at the centre as usual but with, instead of a single turn (or 3/4ish turn) of overcoil, a gradually rising overcoil that makes getting on for two whole turns as it rises and then curves in towards the pinning point.

The existence of the terminal curve on the duo in uno springs I've seen does away with the supposed intended advantage that Graham mentions of the quicker settling down of rate due to the stresses introduce in making the usual overcoil/curve. There was a recent BHI article about them though, by David Boettcher, which may explain further.
 

Clint Geller

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Hi Seth,



Yes, this was in the July and August issues in 2018, and I had some correspondence with David regarding this.

Regards,

Graham
Could you possibly summarize the main relevant point(s) of that article, for those of us who do not subscribe?
 

gmorse

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Hi Ethan,

Here are the best photos I could take of my double overcoil C. Frodsham's hairspring.
Thanks for posting these, they illustrate the structure nicely. It appears that the 'double' refers to the extra turn at the outer end before the stud, and the slight step in the cock table accommodates the resulting small increase in height.

Regards,

Graham
 

gmorse

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Hi Clint,

I'll dig out the article and see what I can do.
David begins by stating that John Hammersley announced his invention of the tria-in-uno spring in the February 1860 edition of the HJ, with the stated objective of eliminating the problem of 'acceleration' in new chronometers, and that this has not been disputed. However the origins of the duo-in-uno spring were disputed with Hammersley by both McLennon and Walsh. Acceleration is the effect seen in chronometers with balance springs formed by hardening and tempering the helical portion and only afterwards forming the terminal curves at each end by bending. Such springs can take a long time in use to settle into a stable rate. Hammersley's invention allowed the spring to be completely formed into its final shape before hardening and tempering, without resorting to any manipulations afterwards. He includes diagrams of tria and duo from Rupert Gould's famous work on chonometers. David goes on to mention Hammersley's design for a 'double flat isometrical balance spring' which corresponds exactly with the example shown by Ethan in post #17, and which was contested by J.F. Cole who stated that he had made such springs 30 years earlier, which Hammersley had subsequently to accept.

In 1894 Hammersley wrote in the HJ that he accepted that McLennon had been the first to apply a duo-in-uno spring to a watch, although he had been making the claim earlier on his own watches for some time. Hammersley did however claim that the duo-in-uno was merely a tria-in-uno with one spiral cut short.

In a letter to the HJ of September 2018, David expanded on his article with an image of a US patent, (2,457,631 from Dec. 28 1948), concerning a complex demountable former for cylindrical balance springs, by W.O. Bennett Jr. for Hamilton's Model 21. He supposes that Hammersley must have used a similar former to avoid damage to the finished spring when the former was removed. He also clarified a point I had brought to his attention regarding the exact configuration of duo-in-uno springs.

Regards,

Graham
 

Clint Geller

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Hi Clint,



David begins by stating that John Hammersley announced his invention of the tria-in-uno spring in the February 1860 edition of the HJ, with the stated objective of eliminating the problem of 'acceleration' in new chronometers, and that this has not been disputed. However the origins of the duo-in-uno spring were disputed with Hammersley by both McLennon and Walsh. Acceleration is the effect seen in chronometers with balance springs formed by hardening and tempering the helical portion and only afterwards forming the terminal curves at each end by bending. Such springs can take a long time in use to settle into a stable rate. Hammersley's invention allowed the spring to be completely formed into its final shape before hardening and tempering, without resorting to any manipulations afterwards. He includes diagrams of tria and duo from Rupert Gould's famous work on chonometers. David goes on to mention Hammersley's design for a 'double flat isometrical balance spring' which corresponds exactly with the example shown by Ethan in post #17, and which was contested by J.F. Cole who stated that he had made such springs 30 years earlier, which Hammersley had subsequently to accept.

In 1894 Hammersley wrote in the HJ that he accepted that McLennon had been the first to apply a duo-in-uno spring to a watch, although he had been making the claim earlier on his own watches for some time. Hammersley did however claim that the duo-in-uno was merely a tria-in-uno with one spiral cut short.

In a letter to the HJ of September 2018, David expanded on his article with an image of a US patent, (2,457,631 from Dec. 28 1948), concerning a complex demountable former for cylindrical balance springs, by W.O. Bennett Jr. for Hamilton's Model 21. He supposes that Hammersley must have used a similar former to avoid damage to the finished spring when the former was removed. He also clarified a point I had brought to his attention regarding the exact configuration of duo-in-uno springs.

Regards,

Graham
Thank you, Graham. That was very informative.
 

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I now have a scan of the 1862 catalogue page containing i nformation on the Charles Frodsham stand. It says of the duo-in-uno harispring (at the bottom right of the page attached):

New "Duo in Uno" balance springs for perfecting the adjustments of high-class watches and chronometers in their various positions.

Not very specific or intelligible, I suspect, but Charles was "the greatest showman" I think.
 

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I now have a scan of the 1862 catalogue page containing i nformation on the Charles Frodsham stand. It says of the duo-in-uno harispring (at the bottom right of the page attached):

New "Duo in Uno" balance springs for perfecting the adjustments of high-class watches and chronometers in their various positions.

Not very specific or intelligible, I suspect, but Charles was "the greatest showman" I think.
Despite the vagueness, Frodsham nevertheless is claiming that duo-in-uno hairsprings confer some kind of an advantage regarding positional adjusting. An interesting claim. Pocket chronometers were serious working watches, not just rich men's toys, so if duo-in-uno hairsprings were not actually perceived to provide some kind of a genuine advantage to justify their additional expense, you would think they would have been no more than a quickly passing fad.
 
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Dr. Jon

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Theory was evolving then as now, but the primary theory was that of Huriet who showed that a balance spring is isochronous if and only if the force is exactly in the radial direction. This placed the focus on terminal curves. The thinking was that spiral was a kind of lower end terminal curve allowing a smaller collet and possibly a thinner watch than fully helical spring with Arnold type terminal curves.

Another possibility is due it being a longer spring than would be practical in a pocket timepiece so thicker wore could be used. A thicker spring is less affected by inclusions in the steel making it more likely to behave according to theory.

From the standpoint of a watchmaker adjusting beat it combines the worst of both worlds. You have to take the balance out to adjust it's beat position. Perhaps this difficulty was seen as a benefit too.
 

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Theory was evolving then as now, but the primary theory was that of Huriet who showed that a balance spring is isochronous if and only if the force is exactly in the radial direction. This placed the focus on terminal curves. The thinking was that spiral was a kind of lower end terminal curve allowing a smaller collet and possibly a thinner watch than fully helical spring with Arnold type terminal curves.

Another possibility is due it being a longer spring than would be practical in a pocket timepiece so thicker wore could be used. A thicker spring is less affected by inclusions in the steel making it more likely to behave according to theory.

From the standpoint of a watchmaker adjusting beat it combines the worst of both worlds. You have to take the balance out to adjust it's beat position. Perhaps this difficulty was seen as a benefit too.
Interesting. So Huriet's theory would imply that a simple volute hairspring would be the most theoretically isochronous, would it not? But conversely, it may also be the toughest to adjust to positions. So perhaps the duo-in-uno hairspring was envisioned to enable a superior trade-off between isochronism and positional adjustment?
 
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Dr. Jon

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No, the hairspring has to open and close such that the spring force at the stud and collet are normal (perpendicular) the the radius. This was and is the defining requirement for terminal curves. This condition is met when the spring opens and closes concentrically. Perhaps a single properly curved volute would work were the material strong enough that a single turn would could have the right frequency. Huriet advocated and used spherical springs.
 

Clint Geller

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No, the hairspring has to open and close such that the spring force at the stud and collet are normal (perpendicular) the the radius. This was and is the defining requirement for terminal curves. This condition is met when the spring opens and closes concentrically. Perhaps a single properly curved volute would work were the material strong enough that a single turn would could have the right frequency. Huriet advocated and used spherical springs.
Ah, normal to the radius. That would be called "axial" in most situations. But you mean that the restoring force points along the line of the spring at the terminal points, right? But wouldn't a spring with a single turn need to be weaker, not stronger than a multi-turn spring, because the strain (dL/L) for a given angle of balance wheel rotation would be greater?
 
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Dr. Jon

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Touche! you are right. Such a short spring would have to be very thin and possible not strong enough to hold its shape, which is what I was thinking. The mainpoint is that duo in uno came when theory was in its infancy and not mathematically rigorous. Philips developed the theory of the overcoil a bit later.

Also, FWIW, Parkinson and Frodsham sent two duo and uno spring watches to the open Geneva trials of 1876 and both did respectably and were arguably among the ten most accurate watches of their time. The trial results were a bit if a surprise in the the Parkinson and Frodsham/Walsh lever got a first and detent got a lower award. The best entry was a Nardin that was so good they had to give it a special prize so that the other ten or so entries could still get a first prize. Whatever the theory and reasons were, the duo and uno performed well in the most demanding trial of the time.
 
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Philip Poniz

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The initial reason was to decrease the thickness. In the description of his tria-in-uno, Hammersley, originally (Feb 1, 1860), claimed that the new shape helped him eliminate the “cause of the acceleration of rate in new chronometers”. A month later, however, he had a new reason pointing out that his springs “will secure all the advantages of the cylindrical spring without increasing the thickness of the watch, which cannot be avoided in the use of cylindrical springs”.

Hammersley’s tria-in-uno (a sketch below), clearly shows that he was trying to save on the thickness. His argument of Feb 1 refers to the fact that, in time, springs change their elasticity due to the changes in the arrangement of atoms in hardened steel subjected to bending.

Hammersley tria-in-uno 1867.jpg

In his watches and chronometers Hammersley used duo-in-uno springs (as shown by Graham earlier in the thread) too but never wrote about them. In fact, we do not know much about them at all.

Duo-in-uno’s explanation, given by Tony Mercer in his The Frodshams, was taken almost word by word from a eulogy of A. P. Walsh given by Robert Gardner in 1893. Gardner wrote that Walsh told him that the duo-in-uno was invented by Mairet.*

There is compelling evidence that this might not be true. In 1860, there was an argument between James Fergusson Cole and Hammersley about who invented the trio-in-uno (which, at the time, was called a “double flat balance spring”).
Cole, claiming priority, knew Mairet well. He used Mairet’s movements in quite a few of his watches. Had Mairet invented the duo-in-uno, Cole would have known about it and, logically, he would either have mentioned the fact in his arguments or he would have not argued at all. But instead he did argue.

My impression has always been that the tria-in-uno was invented first (first by Cole circa 1840 without the helical section, then by Hammersley about 1858** with the helical section) and then the duo-in-uno was a logical simplification.

Had the duo-in-uno been invented before 1860, the heated argument between Cole and Hammersley would not have made sense. Someone, even Mairet, would have objected that both of them stole the idea from him.

Having said that I cannot resist but mention that there are known watches from circa 1820 with duo-in-uno springs. Like John Roger Arnold No. 2175 (British Museum) or Pennington No. 130/596 (hallmarked 1810).

Another advantage of the duo-in-uno and the tria-in-uno over helical is that the sagging effect is smaller.

* Interestingly, in the Chronometer Makers, the same Mercer says that it was John Osborn McLennan that was “reputed to have invented the Duo-in-Uno balance spring”. This is probably due to the fact that the McLennans exhibited them in 1862 Exposition. But in June 1860 McLennon, writing on the subject of double flat springs, did not mention anything about himself as a precursor of the idea.

** In May 1860 Hammerslay claimed he introduced them "more than two years since in London".

Philip Poniz
 
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Clint Geller

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Thank you for that, Philip. I must beg to differ with you, however, only on one point, from my perspective as a materials physicist. You stated that, "... in time, springs change their elasticity due to the changes in the arrangement of atoms in hardened steel subjected to bending."

Hardness and elasticity are distinctly different material properties often conflated by laymen. Hardness is resistance to permanent deformation. Elasticity is the ability of a material to deform without losing its original shape when unloaded. Glass will scratch steel because glass is harder than steel. But steel will return to its original shape after it has been stretched under a load, as long as the elastic limit of the steel is not exceeded. The elastic limit of glass is virtually nil. In elastic deformation, the atoms in a material will move slightly apart or together, depending on whether the load is tensile or compressive, but they do not "rearrange." Permanent, non-recoverable shape changes upon loading are referred to as "plasticity," not "elasticity." Plastic deformation does rearrange the atoms in a material, and "hardness," not "elasticity," is resistance to plastic deformation. Plastic deformation often becomes progressively more difficult as it continues, a phenomenon often referred to as "work hardening."

The proportionality constant that relates the recoverable linear elongation of an isotropic material to the magnitude of an applied stress is called the "elastic modulus" of that material. In anisotropic materials, which include all crystalline materials, Hooke's Law becomes an equation between tensors, and the scalar elastic modulus becomes a fourth rank "stiffness tensor" relating the components of the stress tensor to the components of the strain tensor. (Stress and strain are both second rank tensors, meaning that one must specify the directions of two vectors to specify each component of stress and strain.) Several factors can affect the stiffness tensor of a material: crystalline phase changes, wherein the basic symmetry of the crystal lattice is altered; mass diffusion that changes the composition of the crystal lattice; and temperature changes, which cause the crystal lattice to expand or contract. An inadequately tempered spring might conceivably undergo a diffusionless phase change in service, but I think that would be unusual. At the service temperatures of interest here, oxidation might be the only mass transport process that would occur rapidly enough to matter, but that process is not accurately described as an atomic rearrangement. Similarly, the small lattice expansions and contractions associated with heating and cooling do not "rearrange" atoms. The point here is that hardening processes do not change the elastic modulus or the stiffness matrix of a material. The details of Hooke's Law remain unchanged for small deformations, regardless of hardening.

In ductile materials, which include all metals, plastic deformation, often referred to as yielding, occurs by the production and movement of line defects in the crystal lattice called "dislocations," which can be thought of as extra or missing half planes of atoms. The movement, i.e. "glide," of dislocations in a ductile material is impeded by any other imperfection in the geometric order of the crystal lattice a dislocation encounters, including other dislocations. Large applied stresses can activate dislocation sources, causing the number density of dislocations to increase, giving rise to hardening behavior. However, any spring well designed for its purpose will not be asked to stretch beyond its elastic limit, so these processes should not occur. And in any case, the elastic properties of the spring would not be changed.
 
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With respect to A.P. Walsh, his major contribution in the mid 19th century was to develop a process for hardening hairsprings. For a time that was his major business venture.

I believe the reason he was so admired by his peers was that he could go from concept to implementation in a matter of a few days. After first seeing the duo in uno in 1862, he displayed a collection of them on his stand two days later according to Gardner.
 
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Clint, your analysis of the distinct differences between hardness and elasticity took my mind back more than half a century to a pre WW2 wooden hut at the Royal Air Force 'No1 School of Technical Training'. Having not done particularly well at school I joined the RAF as an apprentice and it was here, especially during the 'Materials and Structures' lectures, that I discovered the intrigue of education. Masters in gowns and boards teaching in much the same way as their Victorian forbears held no appeal for me whatsoever, but put practicality and context into the process and I was hooked. Physics and maths had been my favourite subjects at school and now it all had meaning. Thank you for that very succinct description and for the memories. BTW, I spent my working life in the RAF as an aircraft mechanical and systems engineer and finished off as a lecturer in the discipline. Apologies for deviating off topic.
 

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[QUOTE=".. I must beg to differ with you, however, only on one point, from my perspective as a materials physicist. ...Elasticity is the ability of a material to deform without losing its original shape when unloaded. [/QUOTE]

Hi Clint,

This discussion has been going on ever since Arnold started hardening (and tempering) his hairsprings.

The process was simple starting with a soft steel helical spring with alraedy formed terminal curves which was hardened, tempered, and installed in a chronometer.
And this is when the problem started - regardless of how well the chronometer was adjusted it started accelerating about one second per month - after the first month it was 1 sec fast, after the second month 2 seconds, etc.

In other words, new chronometers went faster day after day. To make sure that there is no room for ambiguity in the explanation; after the first month, the chronometer could be re-adjusted perfectly, and a month later it would gain a second again, then re-adjusted again, and again would run one second fast the next month and so on for at least twelve months.

Eventually, between the twelfth and 24th month the acceleration stopped.

It was a serious problem for chronometer makers. Old springs that were already settled, were in high demand. Some chronometer makers installed a light balance and ran the chronometer for six months to make the setting faster.

If a hairspring was changed for an old, already settled one, the acceleration did not occur. This proved the anomaly was restricted to the hairspring only. Clearly, the hardened and tempered spring was getting stronger day by day, the coefficient of elasticity was continuously changing.
Evidently, there must have been changes in the molecular structure of the spring.

Already in 1923 Commander Gould, stated “It is now fairly certain that this acceleration is due to the molecular change which the outer surface of the spring undergoes during the process of hardening”.

To me it was clear that this anomaly proved that Hooke’s law quantifying the elasticity of small deformations, is pretty good, but not perfect. I do not think in other fields but horology, such minuscule anomalies have ever been noticed. In what other field would an experiment be made by bending and re-bending a spring over 126 million times? But this is a question for a materials physicist, not a historian.

Now, let’s go back to the duo- and tria-in-uno springs.
In practice, even if a hairspring is already hardened and tempered with formed terminal curves, the curves need a final adjustment. It was found that manipulation of the tempered curves can make the acceleration even more rapid. This is what Hammersley had on mind in his explanation from February 1, 1860.

Tria-in-uno springs do not need that final curve adjustment, the curvature of their terminals is small. Therefore their acceleration is smaller than helical springs which gives them advantage over the former. The same is true, although obviously to a lesser degree, with duo-in-uno springs.

Duo-in-uno and tria-in-uno springs were installed in pocket watches to save on the thickness (and to a lesser degree on the acceleration) and in marine chronometers to limit the acceleration.

Philip Poniz
 

Clint Geller

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[QUOTE=".. I must beg to differ with you, however, only on one point, from my perspective as a materials physicist. ...Elasticity is the ability of a material to deform without losing its original shape when unloaded.

Hi Clint,

This discussion has been going on ever since Arnold started hardening (and tempering) his hairsprings.

The process was simple starting with a soft steel helical spring with alraedy formed terminal curves which was hardened, tempered, and installed in a chronometer.
And this is when the problem started - regardless of how well the chronometer was adjusted it started accelerating about one second per month - after the first month it was 1 sec fast, after the second month 2 seconds, etc.

In other words, new chronometers went faster day after day. To make sure that there is no room for ambiguity in the explanation; after the first month, the chronometer could be re-adjusted perfectly, and a month later it would gain a second again, then re-adjusted again, and again would run one second fast the next month and so on for at least twelve months.

Eventually, between the twelfth and 24th month the acceleration stopped.

It was a serious problem for chronometer makers. Old springs that were already settled, were in high demand. Some chronometer makers installed a light balance and ran the chronometer for six months to make the setting faster.

If a hairspring was changed for an old, already settled one, the acceleration did not occur. This proved the anomaly was restricted to the hairspring only. Clearly, the hardened and tempered spring was getting stronger day by day, the coefficient of elasticity was continuously changing.
Evidently, there must have been changes in the molecular structure of the spring.

Already in 1923 Commander Gould, stated “It is now fairly certain that this acceleration is due to the molecular change which the outer surface of the spring undergoes during the process of hardening”.

To me it was clear that this anomaly proved that Hooke’s law quantifying the elasticity of small deformations, is pretty good, but not perfect. I do not think in other fields but horology, such minuscule anomalies have ever been noticed. In what other field would an experiment be made by bending and re-bending a spring over 126 million times? But this is a question for a materials physicist, not a historian.

Now, let’s go back to the duo- and tria-in-uno springs.
In practice, even if a hairspring is already hardened and tempered with formed terminal curves, the curves need a final adjustment. It was found that manipulation of the tempered curves can make the acceleration even more rapid. This is what Hammersley had on mind in his explanation from February 1, 1860.

Tria-in-uno springs do not need that final curve adjustment, the curvature of their terminals is small. Therefore their acceleration is smaller than helical springs which gives them advantage over the former. The same is true, although obviously to a lesser degree, with duo-in-uno springs.

Duo-in-uno and tria-in-uno springs were installed in pocket watches to save on the thickness (and to a lesser degree on the acceleration) and in marine chronometers to limit the acceleration.

Philip Poniz
Philip, the phenomenon you describe, which I do not doubt was a real phenomenon, might be explained by oxidation of the surface of the spring. If the oxidation process passivates, it will shut off after a while. There may be some other explanation for the phenomenon that has been reported. However, it is certain that hardening processes do not affect the elastic modulus of a material. Some crystallographic phase transformation might, however. Tempering is supposed to eliminate the hard martensite phase that develops upon rapid cooling.

As for Hooke's Law, it is only rigorous for infinitesimal displacements from equilibrium, but that is always true. Hardening has nothing to do with that. In a crystal, as you push adjacent atoms together closer than their equilibrium spacing, there will be a repulsive force between them, whereas if you try to separate them beyond their equilibrium spacing, there is an attractive force between them. In either case, the system tries to restore equilibrium. If you plot the potential energy of the system as a function of atomic spacing, the minimum potential energy occurs at the equilibrium spacing - call that xo. Now if you expand the potential energy function in a power series about xo, as in V(x) = [sum over all n > or = to zero]cn(x - xo)^n, where cn is an arbitrary constant coefficient and "^n" means "to the n'th power," then it is easy to see that c1, the coefficient of the linear term, must be zero, since at the minimum potential position, the slope must, by definition, be zero. That means the leading non-constant term is the parabolic term, (x - xo)^2. For small displacements, this term will dominate. The restoring force associated with that potential is then: F = -(d/dx)V(x) = -2c(x - xo). The linearity of the force law is known as "Hooke's Law." Now for finite displacements, the potential well is no longer parabolic. V(x) rises more steeply as you compress the lattice than when you stretch it. That "anharmonicity" is why most materials expand when they are heated.
 
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MartyR

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So Hooke's Law, like Newton's Law of Gravity (and his Laws of Motion) before it, has now to be consigned to the dustbin of history.

Is nothing sacrosanct? :(
 
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Clint Geller

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So Hooke's Law, like Newton's Law of Gravity (and his Laws of Motion) before it, has now to be consigned to the dustbin of history.

Is nothing sacrosanct? :(
Marty, to paraphrase a famous quote: All theories are "wrong," some theories are useful. My point is that science is progressive. Einstein did not toss Newton's work into the dustbin, he built on it. He showed that Newtonian mechanics was a special case of a more general theory, and an extremely accurate approximation for treating objects moving at speeds less than a substantial fraction of the speed of light. At 10% of the speed of light, which is 30,000 kilometers per second, faster than just about any macroscopic object moves in our part of the Milky Way, nearly all the predictions of Newtonian mechanics are still accurate to within half a percent. At common terrestrial speeds the errors incurred by the Newtonian formulation are typically too small to measure. Someday most physicists expect that General Relativity will be quantized, and then Einstein's theories will likewise be shown to be a special case of an even more general theory - Quantum Relativity.

Similarly, "Hooke's Law" is only a very good approximation for many practical purposes, and I'm guessing even Hooke himself may have realized that. Few material behaviors in nature are truly mathematically linear.
 
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Philip Poniz

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Philip, the phenomenon you describe ... might be explained by oxidation ...
Unlikely Clint, oxidation would have the reverse effect - it would weaken the spring, not make it stronger.

The fact that hardened and tempered balance springs become stronger in time is the most remarkable aspect of the anomaly.
Maybe the molecular friction during the bending hardens the spring? Interestingly, molecular friction in hairsprings was already realized by Breguet before 1820, although he did not elaborate on it.

The restoring force associated with that potential is then: F = -(d/dx)V(x) = -2c(x - xo).
But the restoring force, in this case, would be continuously increasing.

Horological metallurgy is probably the least understood subject of horology. Even the model of basic bimetallic balance, as far as I know, is just made with many assumptions and simplifications (Yvon Villarceau, 1862, the Grossmanns, 1905, Mrugalski, 1972). I have tried to find a math model describing behavior of bimetallic laminae but could find only the basics (associated with bimetallic thermostats).

Nice discussion Clint!
Philip
 

Clint Geller

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Unlikely Clint, oxidation would have the reverse effect - it would weaken the spring, not make it stronger.

The fact that hardened and tempered balance springs become stronger in time is the most remarkable aspect of the anomaly.
Maybe the molecular friction during the bending hardens the spring? Interestingly, molecular friction in hairsprings was already realized by Breguet before 1820, although he did not elaborate on it.


But the restoring force, in this case, would be continuously increasing.

Horological metallurgy is probably the least understood subject of horology. Even the model of basic bimetallic balance, as far as I know, is just made with many assumptions and simplifications (Yvon Villarceau, 1862, the Grossmanns, 1905, Mrugalski, 1972). I have tried to find a math model describing behavior of bimetallic laminae but could find only the basics (associated with bimetallic thermostats).

Nice discussion Clint!
Philip
Philip, I don't have a definite answer for the empirical fact you state that hairsprings in watches tend to get "stronger" over time. But I can tell you that it is a very well-established fact of 21st century material science that elastic properties are not affected by hardening. Hardening means that the yield stress of the material has increased. Elasticity is what happens before the yield stress is reached. Citing 18th and 19th century sources who speculated on the subject before x-ray crystallography, transmission electron microscopes, or a real understanding of the nature of atomic bonds existed is only of historical interest.

I would also say that if hairsprings are indeed hardening in service, then they are being pulled at least slightly past their elastic limits. That would slowly drive the yield stress up, and since the displacement amplitude is approximately constant in a watch mechanism, the yielding process would tend to be self-limiting and eventually stop. That hypothesis could explain the phenomenon you described, and it would even accord with your suggestion of a certain amount of atomic rearrangement taking place. My only point of contention here was the conflating of elastic deformation with hardening.
 
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Clint Geller

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Unlikely Clint, oxidation would have the reverse effect - it would weaken the spring, not make it stronger.

The fact that hardened and tempered balance springs become stronger in time is the most remarkable aspect of the anomaly.
Maybe the molecular friction during the bending hardens the spring? Interestingly, molecular friction in hairsprings was already realized by Breguet before 1820, although he did not elaborate on it.


But the restoring force, in this case, would be continuously increasing.

Horological metallurgy is probably the least understood subject of horology. Even the model of basic bimetallic balance, as far as I know, is just made with many assumptions and simplifications (Yvon Villarceau, 1862, the Grossmanns, 1905, Mrugalski, 1972). I have tried to find a math model describing behavior of bimetallic laminae but could find only the basics (associated with bimetallic thermostats).

Nice discussion Clint!
Philip
Philip,

As for bimetallic balances, have you tried searching on “bimetallic cantilevered beam”? That’s essentially what the arm of a bimetallic balance wheel is. I think you’ll find quite a lot of relevant material.

Clint
 

Scott Tzorfas

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Is there any difference between a helical hairspring coil in some of the rare American pocket watches that have them and a "duo in uno" hairspring on a European watch?
Scott
 

Tom McIntyre

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A "standard" helical hairspring has an attachment to the balance cock at the top that is roughly in the vertical line of the hairspring with a slight inner curve to the attachment point. The end that is attached to the balance typically has a single curve of one turn that goes to the collect on the balance. Almost all helical springs are made that way including the larger marine chronometers as well as pocket chronometers.

The "duo in uno" hairspring changes the curve from the cylindrical spring to the balance to a flat spiral of several turns.

Other variants have the flat spiral on both the top and bottom of the cylindrical portion or have the entire hairspring formed into a spherical form.

A. P. Walsh had succeeded in "perfecting" the hardening and tempering of hairsprings in the 1850's and was the supplier of the springs to many of the top makers in England, so many feel that his springs and the watches made from them are superior and thus Gardner's praise of his skills.
 
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Scott Tzorfas

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Thank you Tom for your detailed explanation- I appreciate it. It appears that Charles Fasoldt made both types of hairsprings.
Scott
 

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