# Watch Mainsprings

Copyright © David Boettcher 2005 - 2023 all rights reserved.**NB:** The mainspring calculator spreadsheet and the quick introduction to spreadsheets pdf have both been updated in September 2019. The calculators have also all been updated to display the results more clearly and consistently.

Watch Mainspring in Going Barrel

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 the energy that makes the watch run.

The barrel arbor is made larger in diameter where the spring hooks onto it so that the spring is not bent too tightly. This larger diameter element is usually made as part of the barrel arbor, but it does not have to be; it can be made as a separate part. In English watchmaking, the larger diameter part that the mainspring hooks onto is usually called the arbor. Sometimes it is called the “arbor collet” to distinguish it from the arbor itself. In Swiss French it is called the “bonde”, which translates literally as bung or plug. In engineering terms it would be called a boss or hub.

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 wound around the arbor. In watches with fusees the outside of the barrel is smooth and one end of the fusee chain is hooked onto it. Teeth on the bottom of the fusee drive the centre wheel. As the watch runs and the fusee turns, 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. In watches with "motor barrels" the arbor drives the centre wheel and the barrel is turned to wind the spring.

The image here shows a mainspring in a going barrel. The inner end of the spring has a hole which catches on a hook on the barrel arbor. The middle part of the arbor is increased in diameter so that the inner end of the spring is not overly stressed, and the hook is recessed so that the second coil of the spring can lie on top of the first without the hook causing a bulge. The outer end of the spring has a ‘resilient hook’ which bears against a step formed in the barrel wall. The resilient hook is usually made by riveting a short length of material to the end of the spring. In this case, a watch repairer has formed a hook in the end of the spring by annealing it and bending it over, catching a short piece of spring in it.

## Replacing Mainsprings

Modern white alloy “unbreakable” mainsprings rarely need replacing, but watches that I work on from the early twentieth century have carbon steel mainsprings which need to be replaced because they are tired and “set”, and are also very prone to breaking, which releases a lot of energy that can break pivots and jewels. To replace the spring, you need to know what size to use.

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 section and length of mainspring for it, by calculation or more likely 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.

For more modern movements it is possible simply to look up the correct mainspring in tables of data, but for older watches this is usually 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, and most likely several times, over the life of the watch, and repairers sometimes used whatever they had on hand that would fit.

I ran into this problem when I stripped three identical old movements at the same time and found that the mainsprings were significantly different sizes – different thicknesses and height (widths). This left me with a quandary; which one, if any, was the correct one?

HJ article about Watch Mainsprings. S6040 I3044

Download Spreadsheet: MainspringCalculator.xlsx

Download: Spreadsheets – A Quick Introduction

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 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.

### Mainspring Spreadsheet

An article by me about calculating the dimension of mainsprings was published in the September 2017 edition of the British Horological Institute's Horological Journal as shown in the image here. A second article about mainspring materials was published in the October 2017 Horological Journal.

This page is based on the information contained in the article. A spreadsheet that replicates the function of several of the special purpose mainspring calculators from this page was made available to readers of the Horological Journal. The articles were only available to BHI members who receive the HJ, but the spreadsheet can be downloaded from this link: MainspringCalculator.xlsx. Please let me know if you find it useful or have any comments.

The spreadsheet was updated to version 1.2 to include a calculation of Turns, Fill% and other ratios of the spring to barrel and arbor from the dimensions of the barrel, arbor and spring. This can be used to analyse and evaluate an existing mainspring. If you don't have Version 1.2 of the spreadsheet dated September 2019, you can download the updated version from the links here. The spreadsheet is in Excel format. If you don't have Microsoft Office Excel spreadsheet software, then Libre Office contains an excellent alternative that can open Excel format spreadsheets and is available absolutely free from Download Libre Office

Spreadsheets were created to simplify and automate business models originally created with chalk and blackboards. They are a powerful tool, easy to use and incredibly useful but, like all complicated tools, if you have never been shown how to use one they can be initially daunting. I have created a quick introduction into how they work to get you going. Download it from this link: Spreadsheets – A Quick Introduction. NB: Updated to Rev. 2.1 September 2019.

#### Mainsprings and Pendulums

The methods described on this page are valid for mainsprings for pendulum clocks provided that the correct number of theoretical turns are specified. For a watch going barrel, the number of turns is easily established from the required length of run, e.g. a watch that is manually wound every day is usually given a run of 30 hours or so, which gives a 25% margin in reserve. This also works for clocks with fusees, and with platform escapements. However, the number of turns required for a pendulum clock with a going barrel cannot be established in this way.

Circular error means that a pendulum is not isochronous. To reduce the effect of circular error, the pendulum amplitude must be kept as near constant as possible. The pendulum amplitude depends on the impulse, which therefore must also be as constant as possible. A pendulum suspended from a leaf spring has low friction and requires only a small impulse to keep it swinging in a narrow arc. A pendulum timekeeper with a going barrel therefore needs a long thin spring which will produce a small but sensibly constant impulse during the period from when it is fully wound until it is wound up again. This results in mainsprings that have many more theoretical turns than are required to run for the period between windings. E.g., an the mainspring of an eight day going barrel clock may be sized to have enough turns to run the clock for 24 days or more.

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## Quick Estimate

If you just want to quickly size a mainspring 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 the barrel and diameter of the arbor. This will give you a suggested mainspring thickness and length. The Reverse Engineering calculator is based on a regression analysis of the mainsprings of over 1,000 real calibres, which includes manufacturers' experience of what works in practice.

If you want a simple rule to keep in your head, note that the often repeated spring should occupy one third of the barrel diameter and thickness be the barrel diameter divided by 100 do not give good results. When in the unwound state, the spring should occupy about one quarter of the barrel diameter and a better rule is:

Divide the barrel internal diameter by 87 for spring thickness, multiply it by 30 for length.

A spring calculated by these simple rules will give 6.85 theoretical turns and fill 49% of the area between the barrel and the arbor.

The correct height (width) of spring can be determined by measuring the free height within the barrel and allowing a little clearance, a tenth of a millimetre.

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## Mainspring Materials

Mainspring materials: white alloy and two carbon steel

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 logical implication of the BHI statement about the potential damage that a broken mainspring can cause is that if the spring hasn't already broken you are lucky, and that old carbon steel springs should **always** be replaced with a modern 'unbreakable' type. That is my view 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. My view is if at all possible don't take a chance: **don't use old carbon steel mainsprings**.

There is a section about carbon steel and alloy mainsprings in the section about Mainspring Materials 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 NOS alloy spring. I would not use an NOS carbon steel spring. A NOS carbon steel spring is worse than a used one because while the used one is fatigued, the 60 plus years old unused spring will almost certainly have minute traces of rust and is likely to shatter at full power within a short period of time with catastrophic results.

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## 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.

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 Diameter

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, although the coils of the spring that lay against the arbor will have been stressed past their elastic limit when first wound and deformed plastically into a permanent coil.

With an arbor diameter of one third that of the barrel and a ratio of spring thickness to arbor diameter of 28, the ratio of the spring thickness to the barrel diameter would be 84.

## Rule of Half Area

The "Theory of Horology" 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, which shows unambiguously that the statement means the spring should occupy half of the area.

Interestingly, there is a footnote saying that experience shows that the spring needs to be 20% longer than the length calculated by this method.

### George Daniels

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

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

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 diameter at 33 units. This is consistent with the arbor size given in the rule of thirds of de Carle and Gazeley.

Daniels says that the spring should take up exactly half the **area** between the arbor and the barrel wall. He 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 laying 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 from the book as 26 and 41.

I have added the 74 in red, this is the proportional diameter that evenly divides the space between the arbor and the barrel into two equal areas.

The numbers are all rounded 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 a 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 laying 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 occupied when wound and unwound are shown in the figure as the 26 over 2 and 41 over 2 (the 2s are there to give radial rather than diametral proportions; 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 uses the ratios to determine the thickness of a mainspring to fit a barrel for a watch he is making, which he has already decided will be 21.9mm internal diameter. This initially struck me as strange, that he first decides on the size of the barrel and then calculates a spring to fit into it, rather than trying to work out the torque needed to drive the movement and work out the dimensions of the mainspring from that. However, Daniels explains that it is impossible to determine what torque a movement will need until it is finished, so the course of action he adopts is to make the mainspring barrel as large as possible, and travel in hope.

Daniels calculates the length and thickness of a mainspring to give the number of turns of the barrel required to run the watch for 30 hours. He 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 run the watch for 30 hours to arrive at a spring thickness of 0.273mm.

It turns out that the calculated spring is stiffer than it needs to be, which is fortunate because the barrel is already sized and as large as it can be - if the spring had been too weak he would have been in trouble! A thinner longer spring was substituted in the real watch, which delivers a more constant torque over the first 30 hours.

Daniels says 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. Six turns is too few because no spring will deliver all of its theoretical 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, which applies to any barrel. 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.

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## Method Comparison

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.

#### Half Area

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 here is a 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 division to save me the trouble.

If the arbor is one third of the diameter of the barrel, i.e. d = ⅓D, then this simplifies further to:

$$ L = \frac {\pi D^2}{9t} $$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 figure 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 is the reason that de Carle says that the spring should occupy "a little less" than one third of the barrel diameter, although in practice it is quite a bit less.

### 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.

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## Turns

The mainspring barrel must exert enough force while turning enough times to drive the watch for a minimum period. This period is usually 30 hours for a manually wound watch, to give the wearer some leeway in winding it up every day without spending too long.

The number of turns is fewer for a fusee than for a going barrel. This is because it is difficult to make a fusee "cone" with a large number of turns that will fit between the plates of a movement, which would also require an impractically thin chain. The fusee can inherently cope with the larger 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. However, an extra coil and a half is added to allow for the ends of the spring not being effective. For a going barrel, twelve coils in the unwound coil stack is said to be sufficient, for a fusee its 10.

### Actual Turns

When calculating a mainspring, allowance must be made for two real life effects; "ineffective turns", due to the loss of spring force towards the end of the run, and "lost turns", the difference between the theoretical number of turns a spring will provide and what actually happens in practice.

**Ineffective Turns:**The force produced by a spring in a going barrel drops off sharply during the last few turns of unwinding so these should not be counted on.**Lost Turns:**It is found in practice that mainsprings do not provide the full number of theoretically calculated turns due the effects of hooking at the arbor and barrel, and due to permanent set in the inner coils which mean they do not fully lie against the outer coils when wound off the arbor.

#### Example

A typical wristwatch mainspring barrel has 75 teeth and the centre wheel pinion has 10 leaves. One rotation of the barrel will therefore turn the centre wheel 7½ times. Since the hour hand is mounted on the centre wheel arbor, this equates to seven and a half hours. Therefore to get a 30 hour run, the barrel must make four turns. Allowing for ineffective and lost turns, an additional 2½ turns are added to the theoretical four turns required for a 30 hour run. This makes a total of 6½ turns that the spring needs to be calculated to make, which would give a theoretical 48 hours run.

An alternative way of looking at this says that a spring with enough turns to, in theory, drive a watch for 48 hours inherently takes into account the ineffective and lost turns, although in practice the watch wouldn't actually run for this long, or after a number of hours would run inaccurately due to low balance amplitude. For the example above, 6½ turns would in theory drive the watch for 48 hours although in reality it would not run accurately for this long due to the ineffective and lost turns.

### Theoretical Turns

**NB:** The equations below calculate the theoretical number of turns that an ideal spring should produce. In practice, springs are not ideal and the actual number of turns usefully delivered will be less than this theoretical number. Sometimes this distinction is not made clear; for example, Dr Daniels says that his watch requires six turns, and he calculated a spring to give a 50% fill and exactly six turns, which would have resulted in too few real turns. Fortunately, the calculated spring turned out to be too strong and a thinner one that gave more turns was substituted.

A pendulum clock with a going barrel would require much more than just a few additional turns. This is because the principal requirement for a going barrel pendulum clock is to minimise the torque variation between the mainspring fully wound and unwound. A pendulum clock requires only a small impulse to keep it ticking, so a long thin spring can be used. If the barrel is required to turn four times during the run between windings of an eight day clock, the number of theoretical turns needs to be 12 or more.

Turns v spring fill% for given thickness

If the spring occupies **exactly** half the area between the arbor and the barrel wall, i.e. a 50% fill, the number of turns can be calculated from its thickness by this equation:

T = Turns

D = Barrel ID

d = Arbor OD

L = Spring length

t = Spring thickness

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

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

Fortunately, although most springs occupy more than 50% fill, you don't need to use the second equation to do manual calculations because the mainspring calculator spread sheet mentioned in the introductory section above, which can be freely downloaded, does them for you.

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.

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## 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.

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### Length from Thickness

This is a simple calculator which takes the barrel and arbor dimensions and a spring thickness, and returns a spring length that will take up exactly 50% of the area between the arbor and the barrel, which gives the maximum number of turns.

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 will remain blank. The calculator also returns the number of theoretical turns, the relative size of the arbor as a fraction of the barrel diameter the ratio of the spring thickness to the arbor and barrel diameters.

If the number of turns is less than 6½ for a going barrel, this implies that the spring that was fitted is too thick, resulting 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 out 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 appear to be 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, which also gives a low number of turns.

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### 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. It is impossible to work out exactly the required spring from first principles. The Reverse Engineering calculator takes data from a regression analysis of the mainsprings of over 1,000 calibres to make its estimation, which includes experience of what works in practice.

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%, which seems strange. 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 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 any difference. This is difficult to establish without reliable data for the original carbon steel springs. 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.

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## Required Turns

This calculator works out the spring thickness and length from the required number of turns using the equal areas basis so that the spring occupies 50% of the area between the barrel and the arbor, you just need to specify the barrel and arbor dimensions and the number of theoretical turns required. Don't forget to make an allowance for the lost and ineffective turns, which increases the number of theoretical turns required.

The calculator works with any arbor size, not just one that is exactly one third of the barrel inside diameter. If you leave the arbor diameter blank it will assume a diameter one third of the barrel ID.

If you use less than about 6.4 turns you might find that the evaluator calculator will grumble 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 30 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.

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.

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## 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. If it gives you funny results or you have any comments I would be glad to hear about them.

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## Length from Coils

Length from Coils Measurements

Spring in Carrier

This method works with a mainspring either in a carrier, as shown in the image here, or in a mainspring barrel.

The length of a mainspring can be estimated from the number of coils "N" laying against the carrier or barrel wall when the spring is unwound, the Internal Diameter "D" of the barrel, and the thickness of the spring "t". The equation is:

$$ length = \pi (DN - N^2t) $$Before removing the spring from the carrier or barrel, count the number of coils laying against the wall. Then remove the spring and measure the inside diameter of the carrier or barrel and the thickness of the spring. Pop these measurements into the calculator below and it will give you an estimate of the length of the spring.

If the spring is in a carrier such as the one shown in the second image, measure D as the outside diameter of the spring, then measure the thickness of the coil stack on the side opposite where the bridle and end coils lie. Count the number of coils at this point, the spring thickness is then simply the thickness of the coil stack divided by the number of coils.

The spring in the red carrier measured 13.9mm outside diameter and the coil stack was 1.0mm. There were eight coils, so the spring thickness was an estimated 0.125mm. The calculated length from coils is 324mm - try the figures in the calculator and see. When I straightened out and measured the spring it was 330mm long. The length of the spring is the least sensitive measurement as far as its stiffness is concerned, so this is an acceptable, and surprisingly accurate, estimate.

Please remember that the estimated length is only as accurate as the input data allows.

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### Length from Weight

If you have a spring it is easy to measure its width and height, but not so easy to measure its length without straightening it out, risking damage. This calculator estimates the length of a mainspring from its thickness, height and weight. It returns two results, one for a carbon steel spring, the other for an alloy spring made of cobalt, chrome, nickel, etc. such as Elgiloy or Nivaflex.

The input dimensions are the thickness and height in millimetres and the weight in grams. The accuracy of the result depends on the accuracy of the input figures so to get the best result these need to be as accurate as possible.

- It is quite easy to measure the thickness and height, but the result from the calculator is very sensitive to the spring thickness because it is such a small number. I recommend that you measure the thickness in several places, e.g. at least each end and the middle, and with several instruments if possible. Most vernier calipers and micrometers measure to 0.01mm, which means that the reading could be .005 either side of what is displayed, i.e. a spring that gives a reading of 0.15mm could be anywhere between 0.145 and 0.155, which makes a difference to the result - try it in the calculator and see, it will give you a better feel for the accuracy of the results.
- Watch mainsprings don't weigh very much, usually only a fraction of a gram, which is outside the capability of the average domestic scales. Electronic scales with a resolution of 0.001 gram are used by the jewellery and gem trade and are available on eBay for a few pounds, search for "0.001g scales". I recommend the use of a device with this resolution for the same reasons as are explained about spring thickness.

#### Length estimating calculator

Please remember that the estimated length is only as accurate as the input data allows.

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## 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 torque generated by 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.

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## Mainspring Materials

For the first several hundred years, the mainsprings of portable timepieces were made from hardened and tempered high carbon steel. In the twentieth century alloy materials were developed that were more resistant to setting and breakage, the so called "unbreakable" mainsprings.

### 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 wound into the red carrier ring.

I have heard it said that modern alloy mainsprings are stiffer than old blue or brown carbon steel springs, and that therefore you should fit a thinner one. This is not correct. When spring manufacturers make alloy springs for the watch replacement market they make sure that they are interchangeable with carbon steel springs. Some modern watches are fitted with very high yield point Nivaflex mainsprings; in this case it is important that the correct reference spring is used.

Sometimes it is said that modern alloy mainsprings have a greater modulus of elasticity than carbon steel springs. Even if this is the case, it has little effect on the required spring thickness. To understand why this is, consider the bending moment, which determines the torque that the spring will supply to the movement wheel train:

\begin{equation*} \text{Bending Moment } \beta = \frac{Eht^3\theta}{12l} \text{ therefore } \beta \propto Et^3 \end{equation*} Two springs with the same height \(h\) and length \(l\) but with different elastic moduli \( E_1, E_2 \) and thickness \( t_1, t_2 \) will produce the same bending moment if: \begin{align*} E_1t_1^3 &= E_2t_2^3 \end{align*} The thickness \( t_2 \) must therefore be: \begin{align*} t_2 = \sqrt[3]{\frac{E_1}{E_2}} \ t_1 \end{align*} If \( E_1=210 \) GPa and \( E_2=220 \) GPa, then \( t_2 = 0.985 t_1 \). If \( t_1=0.13 \) , then \( t_2= 0.128 \).This shows that although the stiffness of the spring is strongly dependent on its thickness, because the thickness is cubed in the expression for bending moment, for a given bending moment the thickness of the spring is not strongly dependent on the magnitude of the modulus of elasticity.

If you have any comments or questions, please don't hesitate to get in touch via my Contact Me page.

Copyright © David Boettcher 2005 - 2023 all rights reserved. This page updated May 2023. W3CMVS. Back to the top of the page.