# Blog: Watch mainsprings

Date: 18 May 2016

Copyright © David Boettcher 2006 - 2017 all rights reserved.I make additions and corrections to this web site frequently, but because they are buried somewhere on one of the pages the changes are not very noticeable, so I decided to create this blog section to highlight new material. Here below you will find part of one of the pages that I have either changed or added to significantly.

The section below is from my second page about watch movements.

If you have any questions or comments, please don't hesitate to contact me via my Contact me page.

## Watch mainsprings

The mainspring is the power source for the watch movement. It is a spiral strip of metal contained in a circular enclosure called the barrel. The outer end of the mainspring is hooked to the inner wall of the barrel. The inner end of the spring is hooked onto the arbor, a shaft around which the barrel can turn, or which can turn inside the barrel. By turning the arbor or the barrel the spring can be wound up, storing energy. In watches with "going barrels" the barrel has teeth on its outside that drive the pinion of the centre wheel. To wind the mainspring the arbor is turned so that the spring is drawn off the inner wall of the barrel and wound tightly round the arbor. In watches with fusees the outside of the barrel is smooth and one end of the fusee chain is hooked onto it. As the watch runs the chain is drawn off the fusee onto the barrel. To wind the watch the fusee is turned, drawing the chain back and making the barrel rotate about its arbor, which is fixed, which winds the mainspring.

How to determine the correct dimensions, width, thickness and length, of a mainspring? It might be thought that this is a problem for the designer and manufacturer of a watch movement, and indeed when designing a new calibre a manufacturer will have determined the correct strength and length of mainspring for it, by calculation and from experience, and have tested that it worked as expected. The mainspring barrel and arbor would then have been made the correct size to accommodate the spring and the hard work was done.

But springs need to be replaced when they get tired, and you need to know what size to use. For more modern movements it is possible simply to look up the correct spring in tables of data, but for older watches this is often not possible. If there is a spring in the barrel it can be measured to establish its dimensions, but how do you know that it is the correct one? The mainspring will most likely have been replaced at least once over the life of the watch, and repairers sometimes used whatever they had on hand that would fit.

I have seen springs in identical old movements that were significantly different sizes — different width or "height" and different thickness. For instance; a blue coloured carbon steel spring from a Marvin 13 ligne wristwatch movement measured height 1.4mm, thickness 0.18mm and length 320mm. A brown coloured carbon steel spring from an identical movement of similar age measured height 1.5mm, thickness 0.15mm and length 345mm. The 1.5mm high brown spring looked correct for the internal height of the barrel, so perhaps the blue spring had been chosen slightly thicker to compensate for its lower height. But the strength of a spring depends on the product of its height and the cube of its thickness and the slight extra thickness meant that even though it was not as high, the blue spring was 1.6 times stronger than the brown spring.

This means that it is a good idea to determine what is the correct size of mainspring. The manufacturer will have determined the size of the mainspring barrel and arbor to accommodate the mainspring, so we can use the dimensions of these to determine what size the spring should be. The correct height can be determined by measuring the barrel, but determining the length and thickness is not quite so straightforward. There are two different ways of doing this depending on which book you read. The most common description says that the the radius of the barrel should be divided into three equal parts which are occupied by the arbor, the unwound spring and the space between the arbor and the spring. This is not correct. The correct method is based on the spring occupying half of the area between the arbor and the barrel wall.

#### Quick estimate

If you just want a quick answer and don't want to read all of this page, then I suggest that you go to the "Reverse engineering" calculator and enter the internal diameter of your barrel, which will give you a suggested strength (thickness) and length.

The correct height of spring can be determined by measuring the free height within the barrel and allowing a little clearance.

Mainspring materials: white alloy and two carbon steel

If you want a simple rule that you can carry in your head, then note that the often repeated rules of the spring occupying about one third of the barrel diameter and the thickness being the barrel diameter divided by 100 do not give good results. A better rule is to select a spring that when unwound will occupy one quarter of the barrel diameter, and divide the barrel internal diameter by 80 for the strength or thickness.

### Mainspring materials

The image here shows three mainsprings side by side. The blue, almost purple, and brown springs are carbon steel, the white spring on the left is a modern alloy "unbreakable" spring.

The BHI DLC Technician Grade course says *Because of the potential damage from mainspring breakage, you should always replace a broken spring with a modern, 'unbreakable' type.* The reason for only mentioning broken springs is not made clear, and the sentence begs the question of what to do if the old spring is carbon steel but is **not** broken? The clear and logical implication of the BHI statement is that a carbon steel spring should always be replaced with a modern 'unbreakable' type and I don't know why the BHI course doesn't just say that clearly.

Watch mainsprings have not been made from carbon steel since the 1950s so any carbon steel mainspring, whether found in a watch mainspring barrel or"new old stock" (NOS) are at least sixty years old. The most serious problem that affects carbon steel mainsprings is rust. Rather perversely, this is often worse for NOS springs than for springs that have been stuck inside an oily mainspring barrel for sixty or more years.

The old practice of storing replacement mainsprings in paper packets made rusting more likely because paper is hygroscopic will and absorb moisture from the air. NOS carbon steel mainsprings that I have bought have almost all had traces of rust on them, and those that don't appear to have any rust most likely do actually have tiny traces that are just not visible. Even a minute trace of rust can lead to a crack starting and subsequent early fatigue failure of the spring. NOS carbon steel mainsprings are often found shattered into multiple pieces still in their paper packets. Even worse, if an NOS carbon steel mainspring is installed and subsequently breaks, it can have a lot more power stored in it than an old fatigued mainspring and consequently cause far more damage.

There is a section about carbon steel and alloy mainsprings further down on this page. The inherent resistance of alloy mainsprings to all forms of corrosion and rust is quite amazing.

It seems sensible to me that if you have a choice, then a carbon steel mainspring should always be replaced with a modern white alloy spring. Alloy mainsprings have such good resistance to corrosion, and to setting, that I wouldn't hesitate to reuse one, or to use an old NOS alloy spring.

### Rule of thirds

Mainspring in barrel from de Carle de Carle, D. (1983)

*Watch and Clock Encyclopedia*, 3

^{rd}edition, Ipswitch: N.A.G press.

An old rule of thumb quoted by Gazeley
Gazeley, W J. (1965) *Watch and Clock Making and Repairing*, 2^{nd} edition, London: Heywood.
and others, including the BHI DLC Technician Grade course, says that the arbor should occupy one third of the space in the barrel, and the spring half the remaining space, dividing the barrel radius into three equal parts. The figure here "Mainspring in barrel" makes this clear, it is copied from de Carle
de Carle, D. (1983) *Watch and Clock Encyclopedia*, 3^{rd} edition, Ipswitch: N.A.G press.
which, in common with the DLC and others, says that the diameter of the arbor should be one third of the internal diameter of the barrel, and that if the radius of the barrel is divided into three parts A, B and C as shown, the mainspring should occupy "a little less" than one third as the drawing shows. Unfortunately they not specific about exactly how much this "little less" should be.

It turns out that this rule is not very good. A much better rule would be to say that when unwound the spring should occupy one quarter of the barrel diameter or radius.

The use of the term "space" in these definitions is confusing. The diagram shows that the arbor, the spring, and the gap between the spring and the arbor are all equal in terms of *radius*, but this is not the same as them all occupying equal *areas*. The area occupied by the arbor in the diagram is much smaller that the area occupied by the spring. Just looking at the dotted circle that represents the arbor, you can see that the half of the arbor shown superimposed on the spring could be duplicated roughly 12 times around the spring, so the area occupied by the spring in this figure is about six times that of the arbor.

De Carle also gives a simple rule for calculating the thickness of mainsprings. He says that the spring thickness should be the arbor diameter divided by 32 for high grade movements, 30 for medium grade and 28 for low grade movements. He also says it should be approximately the barrel internal diameter divided by 100, although this gives a spring a little on the thin side. If the diameter of the arbor were exactly one third of the diameter of the barrel, then the corresponding figures would be the barrel internal diameter divided by 96, 90 and 84 for high, medium and low grade watches. In practice these are not good recommendations.

#### Arbor size

Minimum ratios of arbor size to spring thickness are quoted by various authorities. This is presumably to avoid causing excessive strain in the inner coils of the spring, leading to cracking and ultimately fatigue failure. The range quoted by de Carle of the arbor diameter being 28 to 32 times the spring thickness is typical, although of course he was using it in the opposite direction to determine spring thickness from the arbor diameter.

The book "The Theory of Horology" by the Swiss Federation of Technical colleges in conjunction with WOSTEP says that ratio of the radius of the arbor to spring thickness should be between 10 and 14, i.e. the diameter of the arbor should be 20 to 28 times the spring thickness. The figure of 20 seems on the low side and in practice the ratio seen is around 28 +/- 1.

It has been widely accepted for a long time that an arbor one third of the inner diameter of the barrel gives enough space for the spring to work whilst at the same time not bending it too much at the inner end, which could initiate cracks leading to fatigue failure.

### Rule of equal area

The figure numbered 491 here is from George Daniels
Daniels, G. (2011) *Watchmaking*, Revised edition, London: Philip Wilson Publishers.
book "Watchmaking". It shows the proportions of the wound and unwound spring in the barrel. The diameter of the barrel is 100 units and the arbor occupies one third of the barrel at 33 units. This is consistent with the rule of thirds of de Carle and Gazeley

Daniels Fig. 491 from George Daniels Daniels, G. (2011)

*Watchmaking*, Revised edition, London: Philip Wilson Publishers.

The Theory of Horology gives a similar definition. It says that **the mainspring always occupies 50% of [the space between the arbor and the barrel].** Although this is ambiguous and could mean 50% of the radial "space" as in the rule of thirds, the equation in section 4.2.5 "Calculating the theoretical length of a mainspring" is based on the spring occupying 50% of the *area* between the arbor and the barrel. Interestingly there is a footnote that says experience shows that the spring needs to be 20% longer than the length calculated by this means.

Daniels says that the spring should take up exactly half the **area** between the arbor and the barrel wall. Note that this "equal area" is different from the "equal space" in the rule of thirds. Daniels then works out on an area basis the proportions of the barrel diameter that the spring would take up when it is fully wound tight to the arbor, and fully unwound and lying against the barrel wall, so that in each case the spring takes up exactly half of the area between the arbor and the barrel. These proportions are shown on the figure as 26 and 41, and I have added the 74 in red.

These are all rounded numbers of course, the exact proportions would have decimals, e.g. one third is closer to 33.333 rather than the 33 shown on the figure. The other figures would be more closely approximated by 74.535, 41.202 and 25.465. This sounds like splitting hairs but it does actually make a notable difference to the thickness of spring calculated by this method.

The spring will always take up the same amount of space in the barrel whether it is fully wound, unwound, or part way between, but naturally when the spring is unwound there are fewer coils lying one on top of the other than when it is fully wound, so the outer dimension is smaller than the inner. The widths of the two spaces are shown as the figures 26 over 2 and 41 over 2 (the 2s are there for radial rather than diametral distances; 26 + 41 + 33 = 100). Britten
Britten, F. J. (1978) *Watch & Clockmaker's Handbook Dictionary and Guide*, 16^{th} edition revised by Richard Good, London: Eyre Methuen.
gives the same proportions, 33 for the arbor plus 41 to give 74 as the dividing diameter of the equal space area. This is quite different from the Gazely and de Carle one third / one third / one third rule; 26 is a lot less than 41.

Daniels then goes on to use these results to determine the thickness of a mainspring to fit a barrel of 21.9mm diameter which he has already decided is the correct size for a watch he is making. This struck me as strange. Rather than deciding the strength of spring needed to drive the movement, he first fixes the size of the barrel and then sizes the spring to fit into that. How he could know that the spring that fitted into the barrel would be adequate to drive the movement is not explained. In reality, it turns out that the spring that fits into the barrel is stronger than it needs to be, which Daniels says is an advantage because a longer spring delivers a more constant torque.

If the correct spring size had been worked out first then the barrel could have been made to fit it, in this case smaller in diameter or thinner. But it is very difficult, probably impossible, to determine the required mainspring size from first principles and all watch movement designers have to fall back on experience, and trial and error. Luckily for Daniels he chose a generous size for the barrel and the calculated spring turned out to be too strong; if it had been too weak he would have been in trouble!

To determine the size of spring to fit in the barrel Daniels first works out what the diametral widths of the two equal area spaces would be with the spring wound and unwound. This is done simply by ratio, 21.9 * 41 / 100 and 21.9 * 26 / 100 which he gets to be 8.979mm and 5.694mm respectively.

The number of turns a barrel makes is the difference between the number of coils of the spring when it is fully wound and when it is unwound. Daniels establishes this by taking the difference of the two widths 3.285mm and halves this to turn it into a radial distance 1.642mm. This gives the difference in the overall thickness of the sets of coils when the watch is wound and unwound. He then divides this by the six turns required to arrive at a spring thickness of 0.273mm.

Daniels goes on to say that in practice a spring 0.21mm thick was found to be sufficient, which is pretty much exactly the same as de Carle's "barrel internal diameter divided by 100", which for a 21.9mm barrel would suggest a spring 0.22mm thick. By reversing the previous calculation, a spring 0.21mm thick will fit into the 1.642mm radial difference between the wound and unwound condition 7.8 times, advantageously giving more turns.

Because Daniels uses the proportions of the equal areas method to arrive at the required spring thickness for a given number of turns it is possible to work out the ratio of spring thickness to barrel diameter that this method gives. The equation relating turns and spring thickness to barrel diameter for a barrel with a one thirds diameter arbor and a spring that occupies half the area between arbor and the barrel wall is simply:

$$ thickness = \frac {barrel \, ID}{12.71 \times Turns} $$From this it is easy to see that Daniels target number of six turns gives a ratio of spring thickness to barrel diameter of 76, considerably less than the 96 de Carle suggests for a high grade watch, and I am sure that Daniels wasn't intending to make a low grade watch. With six and a half turns the ratio would be 83, and with the 7.8 turns that Daniels arrives at in practice the ratio is 99, a figure that de Carle would probably have said was eminently suitable for a very high grade watch.

### Mainspring calculations

If we have a mainspring from a watch and measure its thickness, then we can calculate how long a replacement should be by using these rules.

#### Equal areas

Daniels' / Theory of Horology rule says that the spring should take up half the *area* between the arbor and the barrel inner wall. The area between the arbor and the barrel inner wall is determined by subtracting the area taken up by the arbor from the internal area of the barrel. The area to be occupied by the spring is then half of this. The area of a mainspring is given simply by multiplying its length by its thickness.

D : Internal diameter of barrel

d : Diameter of arbor

L : Length of spring

t : Thickness of spring

So we can write that the area of the spring is equal to half the area of the barrel less the area of the arbor:

$$ L\times{t} = \frac 12 \left\{ \left( \frac{\pi}{4} \times D^2 \right) - \left( \frac{\pi}{4} \times d^2 \right) \right\} $$which simplifies to

$$ L = \frac {\pi}{8t} \left( D^2 - d^2 \right) $$This is the same as the formula given in section 4.2.5 of the Theory of Horology book, although the formula given in the book uses radial dimensions rather than diameters so the 8 on the bottom line is 2 in their formula. I prefer to work in diameters because they can be measured directly from the components, and the computer doesn't mind doing the extra division to save me the trouble.

However, the results given by this method are different, shorter, than actual spring lengths, at least for Swiss 12 and 13 ligne movements in the wristwatches that I work on. Mainsprings taken from watches, straightened and measured, are longer than calculated by this method. The principal reason for this is that Swiss manufacturers allowed the spring to take up more than half of the area between the arbor and the barrel. This is reflected in the note in section 4.2.5 of the Theory of Horology book, which says that in practice springs are made 20% longer than the theory suggests. This is not supported by data on actual springs in the Generale Ressorts catalogue, which does show that springs are longer than the equal areas rule suggests but not by as much as 20%, more like 10% on average.

#### Thirds rule

If the spring when unwound takes up one third of the radius of the barrel, then it takes up more than half of the space in area terms between the arbor and the barrel wall. We can work out the area occupied by the spring, L x t as before, by subtracting two thirds of the area of the barrel from its total area as follows:

$$ L\times{t} = \left( \frac{\pi}{4} \times D^2 \right) - \left( \frac{\pi}{4} \times \left( \frac {2}{3} D \right) ^2 \right) $$which simplifies to

$$ L = \frac {5\pi}{36t} D^2 $$The lengths of springs calculated by the thirds rule are longer than those calculated by the equal areas method. This would appear to the the reason that de Carle says that the spring should occupy "a little less" than one third of the barrel diameter.

### Relative dimensions of the two methods

Mainspring relative dimensions

The figure here "Mainspring relative dimensions" illustrates the difference between the "thirds rule" and "equal areas" methods. The barrel inside diameter is taken as 100 units and the outside diameter of the arbor is one third of the barrel internal diameter at 33 units.

The 67 and 74 figures show the areas delineated by the two rules; the thirds rule says that the unwound spring occupies the area outside the circular boundary indicated by 67, with the equal areas rule the unwound spring is outside the boundary indicated by 74.

Because the 74 line is the boundary where half the area between the arbor and the barrel is on one side and the other half on the other, a spring that occupies half the available space sits on the inside of this line when it is wound, and on the outside when it is unwound.

However, with the thirds rule the unwound spring occupies 33 parts of the diameter and so its inner boundary is given by the 67 line. This boundary encompasses more than half of the area between the arbor and the barrel wall. Because of this the outer boundary of the wound spring is greater than 67. In fact, it turns out to be 82 as I have shown on the drawing.

Although this seems at first sight to be a relatively minor difference, apparently only "a little less" as de Carle says, the area occupied by the spring given by the thirds rule is actually 25% greater than the area given by the equal areas rule. I was rather surprised when I first worked this out, if you want to see the proof please email me. This means that for a spring of a defined thickness, the thirds area rule calculates a length that is 25% longer than one calculated by the equal areas rule.

An analysis of data in the Generale Ressorts catalogue of mainsprings indicates that the springs listed fall between the boundaries defined by the two rules. The average unwound inner dimensions for the springs that I analysed are around 72 parts of the barrel diameter, i.e. between the two dimensions given by the equal areas and thirds rules. This indicates that in practice the spring occupies slightly more than half the area between the arbor and the barrel, but not so much as the thirds rule suggests. Actual spring lengths in the catalogue are about 10% longer than lengths calculated by the equal areas method.

On the BHI basic mechanical watch course the tutor John Murphy quoted a rule for calculating the length of a spring from its thickness, \( L = \frac {3 B^2}{8 \, t} \), which works out on an equivalent basis to 69 parts of the barrel diameter, i.e. again between the boundaries given by the thirds and equal areas rule. I am not sure where this equation came from, the absence of any messy fractional numbers or π, which almost inevitably pops up in calculations involving circular areas, makes me think it is a rule of thumb based on experience. It doesn't exactly match the data in the Generale Ressorts catalogue although it is close. It also doesn't help if you don't know the correct spring thickness.

### Calculators

The calculators below developed as I studied this topic and realised that it was not as simple as it appeared at first sight. Make sure that you use all consistent units (i.e. all inches, or all millimetres) and the result will be displayed in the same units.

#### Length from known thickness

This was the first calculator I developed. My premise was that it is easy to measure the thickness of a spring and, assuming that this is the correct thickness for the barrel, one then needs just to know the correct length. The calculator returns results for spring length calculated by (1) the equal areas method, (2) a spring 10% longer, and (3) one calculated by the rule of thirds, which is even longer still.

The spring lengths returned by the thirds method are too long and take up too much area in the barrel, restricting the number of turns achieved. I came to the conclusion that the equal areas method is probably the starting point for most springs, with some extra length allowed for the dead areas at each end where the spring is hooked to the barrel and arbor. The Theory of Horology says that in practice springs are 20% longer than calculated by the equal areas rule, but this is contradicted by data in the Generale Ressorts catalogue where springs are on average 10% longer, which is what this calculator shows.

The calculator also gives the relative size of the arbor as a fraction of the barrel diameter, if you leave the arbor size blank it will guess that it is one third of the barrel diameter and use that; the input box remains blank. It also gives the ratio of the spring thickness to the arbor and barrel diameters.

The number of turns for each length of spring is also displayed. If the number of turns given by equal areas is less than 6½ then the spring may be too thick, which will result in a calculated length that is on the short side. You might want to see what the "reverse engineering" calculator suggests for spring thickness, or try the spring dimensions in the evaluator and see what that says.

If you don't have a watch in pieces in front of you but want to try this calculator, then you can use the data that I took from the two identical identical Marvin movements I mentioned above. The barrels are 11.5mm ID and the arbors 3.75mm OD. The springs I measured as follows: blue coloured spring: thickness 0.18mm and length 320mm, brown coloured spring thickness 0.15mm and length 345mm. You will see that both springs look a bit on the thick side in terms of the ratios of their thickness to the barrel and arbor sizes, especially the 0.18mm thick blue spring, and give a low number of turns.

#### Reverse engineering

The calculator below makes a reasonable estimate of mainspring dimensions from just the internal diameter of the barrel, using data gathered from a regression analysis of data in the in the Generale Ressorts catalogue of mainsprings fitted to a large number of movements. It might be thought that basing the spring size on just the barrel dimensions is a bit odd, but the manufacturer of the movement will have made the barrel the correct size to accommodate the correct mainspring for the movement so this is essentially "reverse engineering" the dimensions of the mainspring from the dimensions of the barrel that was designed to accommodate the spring. And as I mentioned above, even that great watchmaker George Daniels started off with the dimensions of the barrel and fitted a spring into it.

When I created charts of the data, removed some outliers and fitted trend lines, I found that forcing a 0,0 intercept did not give the best fit, so this calculator uses coefficients from the trendlines that fit the data better. There is a fair amount of scatter in the data showing that there was no universally agreed way to size a mainspring.

There are a huge number of mainsprings listed in the Generale Ressorts catalogue. For 11mm barrels alone there are around 150 different springs ranging in thickness from 0.095mm to 0.160mm, in length from 300mm to 460mm and fill ratios from 43% to 65%. Why there is such a range is difficult to understand. Many of the springs are close to the mean dimensions of 0.13mm and 360mm, which gives a fill ratio of 55%. It seems likely that a spring of these dimensions and the correct height would probably be perfectly satisfactory for all 11mm barrels.

All the springs in the Generale Ressorts catalogue are made of white alloy, so for watches that would originally have had carbon steel springs the spring sizes quoted in the catalogue might have been adjusted to take account of the slightly greater strength of the "reverse curve" alloy springs. This is difficult to establish without reliable data for the original carbon steel springs. I had the feeling that carbon springs were generally thicker than the alloy replacements but my movements are usually not listed in the Generale Ressorts catalogue. Then I took a carbon steel spring from an IWC cal. 64 movement and it was the same thickness, 0.135mm, as the listed alloy replacement. See the section below about alloy springs for a discussion about carbon and alloy springs.

#### Equal areas calculator

This calculator works out the spring thickness and length from the required number of turns using the equal areas basis, you just need to specify the barrel and arbor dimensions and the number of turns required.

The calculator works with any arbor size, not just one that is exactly one third of the barrel inside diameter. It also uses a more exact equation than the rounded whole numbers ratios 74/26/41 so the results are slightly different from calculations using those approximations. If you leave the arbor diameter blank it will assume a diameter one third of the barrel ID.

The first figure reported called "Diff" is the radial difference between the space occupied by the spring when wound and unwound in millimetres. I use a simple equation to calculate Diff as follows:

$$ Diff = \frac{ \sqrt{ 2 \left( D^2 + d^2 \right)} - \left( D + d \right) }{2} $$This equation returns the correct Diff for a spring that occupies half the area, whatever the ratio of arbor diameter to the barrel ID. If you want to see how it is derived please email me. If the arbor is not one third of the barrel diameter Diff will not be the same as the dimension calculated by the proportional method of the ratios 26/41 used by George Daniels because the arbor will occupy more or less area than that method assumes.

The spring thickness is calculated from Diff and the required number of turns. The length is then calculated using the resulting spring thickness so that the spring will fill 50% of the area between the arbor and the barrel wall.

If you use less than about 6.4 turns you might find that the evaluator below grumbles about the spring dimensions. This is because the evaluator is based on data from Swiss going barrels that use longer springs than are strictly necessary to get the minimum number of turns required for a 36 or 40 hour run. A longer spring gives a more equal torque output over the 24 hours or so that the watch is running, and good quality movements are capable of running with the thinner springs that a higher number of turns requires. If you are trying to work out the correct size for a watch with a fusee then six turns or fewer may be appropriate, see the discussion below about turns.

It is interesting compare the output of this calculator with the results that George Daniels obtained for a barrel with an internal diameter of 21.9mm and arbor 7.3mm using the rounded approximations 41 and 26 for the proportions of the barrel occupied by the spring when wound and unwound. The 41 is rounded down from 41.202 and the 26 is rounded up from 25.465, which has the effect of reducing the difference between the two measurements. Daniels got 1.642mm for the radial difference of these dimension, which for six turns gave a spring thickness of 0.273mm; he doesn't mention the spring length.

### Spring evaluator

This calculator is an experiment in evaluating a mainspring of known dimensions for suitability in a particular application where the barrel and arbor dimensions are also known. Although it gives the ratio of the spring thickness to the barrel ID it evaluates the thickness and length based on the coefficients derived from the Generale Ressorts data as described above. I am still working on this so if it gives you funny results or you have any comments I would be glad to hear about them.

## Turns

The number of turns is greater for going barrels than for fusees. This seems to be because it is difficult to make a fusee "cone" with a large number of turns, and anyway the fusee inherently copes with the large torque variations from a shorter, thicker, spring. A longer thinner spring is desirable in a going barrel to make the torque delivery more even during the normal run of the watch.

Britten says that going barrels should make 6½ turns, whereas for a fusee 5¼ turns is sufficient for the most extreme case and that often only 4½ or 5 turns were used.

Turns v spring fill% for given thickness

If the spring occupies **exactly** half the area between the arbor and the barrel wall, the number of turns can be calculated from its thickness by this equation:

If the exact area occupied by the spring is not 50% we need a more general equation that can calculate the number of turns for a spring of any length and thickness as follows:

$$ Turns = \frac{ \sqrt{ D^2 - \frac{4LT}{\pi} } + \sqrt{ \frac{4LT}{\pi} + d^2 } - \left( D + d \right) }{2 \, t} $$I haven't shown the full derivations of either of these equations because they would take up a lot of space, if you would like to see them please email me.

The figure here uses results calculated by the second equation to show the number of turns for a spring of a certain thickness as its length is increased and it takes up more area in the barrel. The scale across the bottom of the plot is the percentage of the area between the arbor and the barrel wall occupied by the spring.

The figure makes it clear that the maximum number of turns is reached when the spring occupies exactly 50% or half of the available area. Once the length that gives a 50% fill is reached, further lengthening the spring reduces the number of turns.

The maximum number of turns can be varied by altering the thickness of the spring, a thinner spring will give more turns than a thicker spring, but will of course be weaker. A thinner spring will also need to be longer than a thicker spring in order to fill 50% of the barrel area and achieve its maximum number of turns.

In practice mainsprings are made slightly longer than the length that gives exactly 50% fill to allow for the dead areas at each end of the spring where it is attached to the barrel wall and the arbor. Around the 50% fill length an extra 10% or so on the length of the spring makes only a small difference to the number of turns because the curve is fairly flat at this point. For the spring illustrated by the chart, 10% above the optimum length would reduce the maximum number of turns from 6.52 to 6.49. Further away from the 50% fill length, varying the length of the spring has a proportionately greater effect as the slope of the curve gets steeper.

### Effects of length

If you can't find a spring that is close in length to the one that you want, can a longer or shorter spring be used? The theoretical strength of a spring is inversely proportional to its length, but this supposes that the spring is free. In a watch barrel, many of the coils of the spring rest on top of one another and friction between them must come into play. But if it didn't, then a spring that was say 380mm instead of the 360mm you really wanted would be in theory 5% weaker. By the time friction has also had its say this is negligible. Even a 400mm spring would be only 10% weaker in theory and less in practice. This is less than the difference between fully wound and 24 hours later. So there is quite a scope for using springs of lengths that are different from the ideal. It is the thickness of the spring that is the most important factor, which is why it is often called the "strength" of the spring.

## Carbon steel mainsprings

Portable timepieces with mainsprings in barrels and with fusees to equalise the torque from the spring as it wound down were known in the fifteenth century. The Almanus manuscript has an illustration of a spring driven clock dated 1480. Flat springs were known to locksmiths and weapons makers, the arbalest variation of the crossbow came into use in Europe during the 12th century with a steel bow that required a windlass to pull it, but who had the idea of making a long flat spring into a coil and using it instead of a weight to power a clock is not known.

The first springs would have been of uncertain quality because making steel of uniform composition and mechanical properties was not possible with the techniques that were available at the time. Over the centuries methods of melting steel and forming it into strips of uniform quality improved, but the basic material remained the same. Springs were made from a high carbon steel, an alloy of iron with around 1% carbon. This was formed into a strip or ribbon that was hardened and tempered, and then coiled.

High carbon steel heated to a glowing red, often called cherry red, and then cooled very quickly by being plunged into water or another coolant takes on an angular needle-like crystal structure called martensite, which is very hard and brittle; steel in this condition is often called "glass hard" because it is not only very hard but also likely to break in a brittle manner like glass. In this form it is too brittle to be useful so the hardness is reduced or drawn by tempering, heating to a temperature when some of the martensite transforms into a softer and tougher crystal structure. If the steel is allowed to cool slowly from the tempering temperature the new crystal structure is preserved. If the tempering temperature exceeds 400°C a different crystal structure is formed that has less strength but much greater ductility. It is not used in horology.

Tempering temperatures are often judged by observing changes in the colour of steel as it is heated. In normal air an oxide film forms and as this increases in thickness the observed colour of the steel changes from at first a very pale yellow at about 220°C through straw, suitable for scribers, to shades of brown, then purple and finally blue at about 300°C. A blue temper gives the steel an elastic quality that is ideal for springs. The colour of a commercially produced mainspring might be brown due to the atmosphere in which the tempering has been done.

Carbon steel was the only material used for mainsprings for over 450 years. It had two problems that were never overcome by improvements in the material or technology. Mainsprings made from hardened and tempered carbon steel are prone to breaking, and this often occurred when they were near to fully wound. The sudden energy release could bend teeth or strip them from wheels, or crack jewel bearings. The other problem is that carbon steels springs are not perfectly elastic and over time they "set", that is they take up a greater permanent curvature and can store less energy.

## Alloy mainsprings

Alloy steel mainsprings were introduced in the 1940s by the Elgin National Watch Company of America. Dr. Oscar E. Harder and Simon A. Roberts working under a company fellowship at the Battelle Memorial Institute at Columbus, Ohio, tested hundreds of alloys before finding a successful formula in 1942. The Elgin company then spent the next four years developing methods of making mainsprings from the tough new alloy. In 1946 mainsprings made from the alloy were installed in 100,000 watches, not a single one of which was returned with mainspring trouble. Previously about 30% of the watches returned to the factory under warranty had a problem with their mainspring.

The new material was announced to the news media at a press conference in March 1947 at the company's factory in Elgin by George G. Ensign, director of research. The press were told that mainsprings of the new alloy had also been exposed to a salt spray for 500 hours without any rust developing whereas ordinary steel rusted in an hour under the same test.

Ensign then carried out in front of the reporters an astonishing demonstration of the toughness of the new alloy. A watch movement equipped with one of the new type mainsprings was immersed in a bath of aqua regia, a highly corrosive mixture of nitric acid and hydrochloric acid. Aqua regia is peculiar because it is the only acid that will dissolve gold and platinum, which neither nitric acid or hydrochloric acid alone, nor any other acid, can do. In the acid all the brass and steel parts of the movement dissolved, leaving only the alloy mainspring and the jewel bearings. The reaction between the acid and the metals was exothermic and the mixture got hot. After the acid had cooled the spring and jewels were removed — the spring was apparently completely sound and untouched by the acid.

The new alloy material was called "Elgiloy" metal after the name of the company, mainsprings made from it were called "DuraPower"

The Elgin company said that the new alloy mainsprings would give the American watch industry a distinct advantage over foreign competitors, particularly the Swiss who supplied about 75 per cent of the watches sold in the United States at the time. They expected to eventually license other *American* manufacturers to use the new alloy, which was said to be protected by patent. Elgin adverts in the 1950s continued to say "patent pending" even though a patent was granted in 1950.

The use of alloy mainsprings was rapidly taken up across the watch industry worldwide to the extent that manufacture of carbon steel mainsprings ceased in the 1950s or early 1960s.

### Carbon vs. Alloy Springs

Old and new springs

Given that blue or brown carbon steel mainspring have not been made since the 1960s, any carbon steel spring, either in a watch or "new old stock" (NOS), will be at least forty years old, should using or re-using one even be considered? Carbon steel springs are extremely sensitive to moisture. Rust on the surface causes stress corrosion cracking, and the cracks will grow each time the spring is wound until eventually it breaks, a condition called "fatigue failure". The rust that initiates this process might not even be visible. It is not uncommon to find NOS carbon steel mainsprings still in their packets already broken into many short lengths because of this. Mainspring failure in a watch, which often happens at or near full wind, can break pivots, teeth and jewels. This is something to be avoided if at all possible, especially in an old watch with rubbed in jewels.

The benefits of alloy springs are just the same today as they ever were. They don't rust and have an extremely low likelihood of breaking, they resist setting or getting tired, and are non-magnetic.

The image here shows an old carbon mainspring that I have removed from a watch and the alloy spring that I replaced it with. The red ring is the carrier that the alloy spring came in, which is slightly smaller than the mainspring barrel. Normally the new spring is just pushed straight out of the carrier into the barrel, but this one wouldn't go in easily so I removed it from the carrier and used a mainspring winder to coil it up and eject it into the barrel. The old and new springs are the same length, so you can see how much more "wind" is going to go onto the new spring to get it in the barrel. The reverse curve of the alloy spring has been retained after being squashed into the red carrier ring. If you put a reverse curved onto a carbon spring and wind it into a barrel, the curve disappears and it becomes a spiral just like the old one here. I know because I have tried it. This shows just how much higher is the elastic limit of the alloy material compared to carbon steel.

I have heard it said that modern alloy springs are stronger than old blue or brown carbon steel mainsprings, and that therefore you should fit a a weaker mainspring. I think that although alloy springs are a bit stronger for the same dimensions the difference is not significant.

The Young's modulus of alloy springs is not much different from hardened and tempered carbon spring steel. The Theory of Horology describes mainsprings as being a structurally hardened alloy of cobalt, nickel and chrome, so these are modern white alloy springs, not blue or brown hardened steel springs. It quotes a value for their Young's modulus of 2.2 to 2.3 x 10^{5} MPa. This is 220 to 230 GPa, not much different to the 210 GPa usually accepted for carbon spring steel.

Two identical springs made of the different materials (white alloy or blue/brown carbon steel) but of identical thickness will produce nearly enough the same torque, certainly closer than the different thicknesses of springs that I find in old watches. When I tried to work out how much thinner than a carbon steel spring an alloy steel spring would have to be to be the same strength I found that the difference was insignificant. The thickness of the spring has a strong bearing on the strength of the spring because it varies as the third power (cube) of thickness. Old books call the thickness of the spring the "strength" because of this.

One of the benefits of an alloy spring is that the material has a higher elastic limit than steel and so does not "set" or become tired in the same was as a steel spring. This allows them to be made with a reverse curve that keeps the torque produced more constant as the spring runs down. If you tried to make a mainspring with a reverse curve out of hardened and blue tempered steel, the first time it was fully wound in the barrel so that it was tight around the arbor, the resulting stress would exceed the elastic limit of the material in the reverse curved part and remove or reverse the curve.

The reverse curve spring is bent more than a spiral spring when it is coiled into the barrel and therefore produces more torque. Exactly how much is difficult to calculate from first principles but I have seen figures suggesting that a reverse curve spring produces 30% to 40% more torque when fully wound than a conventional spiral spring. This doesn't appear to be enough to cause a problem - which would be excessive amplitude and knocking of the impulse jewel against the outside of the lever. Knocking can be caused by applying force to the crown of a manually wound watch when the spring is already fully wound, but this would produce much more excess force than a 30% or 40% increase in the spring force.

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