Vintage Watch Straps

Straps for vintage fixed wire lug trench or officer's wristwatches.

I have split the old page about watch movements into two. This page is intended to go into some more technical details about watch movements.

The effects of temperature on the rate of a watch and attempts to compensate for this are such a large topic that I have made a separate page about temperature effects.

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

## The balance spring

The balance spring is the elastic element that gives the oscillating time controller, the balance assembly, its natural frequency. If that sounds a bit abstract it is easier to think of the balance assembly as simply two things; the "balance", the circular flywheel thing that you see swinging backwards and forwards, and the spiral "balance spring" or "hair spring".

The balance spring is attached at one end to the balance cock by the "stud". The spring passes through a hole in the stud and is fixed in place with a pin, and the stud itself is the attached to the balance cock by the stud carrier. At the other end the balance spring is fixed to the balance staff by the collet, again passing through a hole in the collet and being fixed in place by a pin. When the balance turns from its rest position in either direction, the balance spring exerts a twisting force in the opposite direction, trying to return the balance to its rest position. If the balance is swinging freely, at some point the restoring force of the swing arrests its motion and accelerates it in the opposite direction. When the balance reaches its rest position, the spring no longer exerts any twisting force, but now the balance is turning quickly and it momentum carries it past the rest position to an opposite extreme where the force of the spring again arrests it and accelerates it back towards the rest position. In this way the balance is caused to swing backwards and forwards with a natural frequency, which is what determines the timekeeping.

### Isochronism

When he first thought of using a balance spring around 1660, Dr Robert Hooke thought that it would make the watch an even better timekeeper than a pendulum clock. The balance of a watch swings backwards and forwards around a fixed axis and therefore does not suffer from the problem that prevents pendulums from being "isochronous".

In horological terms, "isochronous" means that the regulator that controls the timekeeping, whether balance or pendumum, will oscillate with the same frequency irrespective of the amplitude of its arc of vibration. A simple pendulum can't do this because of circular error. I don't propose to go into this in detail here, but it arises from the pendulum swinging in a circular arc but gravity always pulling vertically. In a perfect oscillator the restoring force is always proportional to the displacement but, because gravity pulls on a pendulum bob vertically rather than tangentially, geometry means that this condition is not fulfilled by a simple pendulum.

Robert Hooke had observed a property of springs from which he formulated "Hooke's law", which says that the force exerted by a spring is proportional to its extension. Hooke had realised that this property of springs could be used in watches and hoped to develop a commercial product so to prove the date at which he had the idea without actually revealing it publicly he recorded his discovery as a Latin cipher in an endnote in a book he wrote about helioscopes: "The true Theory of Elaſticity or Springineſs, ... ceiiinosssttuu." Hooke later revealed that this cipher encoded the phrase "Ut tensio sic vis" or "As extension so force", meaning that the force exerted by a spring is proportional to its extension. Hooke applied this idea to a watch in 1658 and attempted to secure a patent on the invention, but he encountered problems in getting a satisfactory patent and the idea remained on the back burner. In 1675 Christiaan Huygens announced that the timekeeping of watches could be dramatically improved if a spiral spring was attached to the balance. This kicked off a furious row with Hooke about which of them should have the credit for the invention, which has still not been entirely settled.

When a stationery balance with an attached spiral balance spring are rotated away from the rest position, the spring exerts a turning force in opposition to the rotation. When the balance is released at some point away from the rest position, the force of the spring accelerates it back towards the rest position. When the balance reaches the rest position the force from the spring is zero, but the momentum the balance has gained from being accelerated causes it to carry on rotating past the rest position. The spring then starts to exert a turning force in the opposite direction, which slows and eventually stops the balance, and then accelerates it in the opposite direction. If there was no friction the swinging back and forth of the balance and spring would continue for ever, the balance and its spring continually swapping between them the energy that was initially given when they were disturbed from their rest position. The balance and its spring form a simple harmonic oscillator whose regular oscillations can be used to measure time.

A spiral balance spring that obeys Hooke's law combined with a pivoted balance that is free to rotate about its axis means that the restoring force on the balance towards its rest position is always proportional to its displacement from that position, which would make a perfect harmonic oscillator; one that oscillates with a constant frequency irrespective of the arc of vibration. The frequency of such an oscillator would be immune to the change in amplitude that occurs as the power of the mainspring diminishes between windings, and as manufacturing techniques and balance springs improved, watch escapements approached this ideal.

### Breguet overcoil

The first balance springs were flat spirals, but it was realised that a flat spiral spring in a watch does not exactly obey Hooke's law and produce a restoring force that is exactly proportional to its deflection, or degree of rotation in this case. This is because the spring is pinned in to the collet on the balance staff so that its inner termination is not exactly on the centre line of the balance staff. The ends of the balance spring are also fixed, at one end in the stud, the other in the collet, so they can't change angle as the spring is coiled and uncoiled. When the balance rotates the balance spring, instead of coiling and uncoiling around a fixed centre point at the axis of the balance staff, has its inner end moved around in a circular path by the collet into which it is fixed. As the balance spring "breathes" in and out the fixed angles of the terminations in the collet and stud cause the coils to "develop" asymetrically; they bunch up or spread out on one side of the spring more than the other. This means that the flexure of the spring is not constant along its length, the bunched up sections are bent at a tighter radius than those on the other side where the coils are more speard out. This results in the restoring force exerted by the spring on the balance varying with different amplitudes of oscillation, which causes a lack of isochronism as the mainspring runs down over time and the amplitude of the balance decreases.

This purely mechanical / elastic effect is different to the effect of gravity on the spring, which is called the Grossman effect. It was first studied in detail theoretically by Phillips and Lossier at the end of the nineteenth century, but it had much earlier come to the attention of Abraham Louis Breguet who had realised that by drawing the outer coil of the spring upwards it could be bent inwards and pinned closer to the centre of rotation, which resulted in the balance spring developing more symmetrically and improved isochronism. The coil which rises up and passes over the other coils of the balance spring is called the "overcoil" and a balance spring with this feature is often called a "Breguet balance spring".

### Escapement errors

If the balance and balance spring were set in motion and not interfered with, then the balance would swing back and forth at the natural frequency of the assembly. But it would also slow down and eventually stop as a result of the small but inevitable amount of friction at the pivots of the balance staff, and due to disturbing the air. So there has to be a means of adding energy to the balance to replace the energy lost to friction, and also there has to be a means of recording each swing of the balance so that this can be used to display the time. The escapement performs both functions; as the balance passes through its zero position it unlocks the escapement, which allows the train to advance one tick and move the hands a corresponding amount, and the escapement gives the balance a little push called an impulse that makes up the energy it has lost during its swing.

The actions of unlocking and impulsing the balance affect its period of oscillation. The action of unlocking has to occur before the zero point, and it takes energy from the balance which causes a loss. If the impulse could be delivered instantaneously to the balance exactly as it passed through the zero point, then it would have no effect on the period. But this is not possible and the impulse is delivered over a finite period while the balance moves through a small angle of rotation. Part of the impulse is delivered before the zero point, which causes a gain, but the greater part of the impulse is delivered to the balance after it has passed the zero point and causes a loss.

The changes in period and rate due to unlocking and impulse are traditionally called "escapement errors". In the lever escapement they result in a losing rate compared to the natural frequency of the balance.

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### Effects of position

Balance in Crown Down Position

Although a wristwatch can assume lots of positions, there are two orientations that have a big difference in amplitude, which can affect the rate, and hence the timekeeping, if the balance is not isochronous and because of escapement error.

These two positions can be characterised as "dial up" such as when you are sitting at a desk writing and the dial of you watch is on the top of your wrist, and crown down, such as when you are standing up with your arm hanging down so that the crown is facing the floor and the dial is vertical. The amplitude when dial up is usually 270° or more, when crown down it can fall to 220°. The escapement error takes place over a fixed angle, so as the amplitude falls it has a greater effect.

Balance in Dial Up Position

In the dial up position, as shown in the first drawing, the balance staff is vertical so all the weight of the balance is taken on the point where the rounded end of the staff sits on the cap jewel, indicated by the red arrow. In this position there is virtually no frictional torque acting on the balance staff. Frictional torque, the amount of friction that acts to retard the turning of the balance staff, is determined by the amount of friction multiplied by the length of the torque arm over which it acts. In the dial up position, although all the weight of the balance is taken on the end of the balance staff, so the friction is considerable, because the end of the balance staff is rounded where it contacts the cap jewel, the length of the torque arm is virtually zero, so the frictional torque is also virtually zero.

In the crown down position, as shown in the second picture, the weight of the balance is taken where the balance staff sit on the sides of the two jewel holes, as shown by the two red arrows. There is a lot more frictional torque in this position than in the dial up position because the torque arm over which the friction operates on the balance staff is the radius of the balance staff itself. Although this is small, it is still a lot greater than the length of the torque arm acting in the dial up position, which as we saw above is virtually zero.

The loss of amplitude cannot be prevented, but careful engineering of the balance and jewels can minimise the effect. This is the principal reason why the pivots of the balance staff are made so thin and the jewel holes are olives: the thinner the pivot the shorter the torque arm, and the olive shaped jewel has a smaller area of contact with the pivot than a jewel with a cylindrical hole. The less friction there is in the crown down position, the less is the effect of changing attitude.

So there is more friction at the balance staff pivots when the watch is vertical than when it is in the dial up or down position. Does this mean that the watch will run slower? Well, it shouldn't. The balance will have a much smaller amplitude in the crown down position than in the dial up position but if the oscillator, the balance and its spring, are truly isochronous, then their frequency of vibration should be the same whatever the amplitude. However, perfect isochronism is difficult to achieve and there are always the escapement errors, and a watch that is keeping time in the dial up position can have a different rate in the crown down position.

It is interesting to note that the tourbillon, which is supposed to counter the effects of position, has absolutely no effect on this. The tourbillon turns the whole escapement to iron out the effects of a balance that is out of poise or a balance spring that does not expand and contract perfectly concentrically. But it only turns it in one plane and on this much greater positional effect it makes no improvement whatsoever. When pocket watches were the norm, which usually sit vertically in a pocket, then the tourbillon may have had some use, but for wristwatches that are constantly moving, it has no noticeable effect.

As the two positional effects that the tourbillon does counter can also be eliminated by correctly poising the balance and using an overcoil balance spring, one is left wondering if the tourbillon is anything more than an impressive optical effect — if it is visible, that is!

Tourbillons with inclined cages and two and three axes of rotation have been made.

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## Magnetism

Magnetism can have an effect on the timekeeping of a watch if the balance or balance spring become magnetised. If the balance becomes magnetised the cross bar can act like a North/South magnet and try to align itself with the earth's magnetic field. This will cause the watch to run fast or slow depending on the direction it is pointing.

This was thought to be a particular concern for chronometers on ships. The earth's magnetic field is too weak to act on an unmagnetised balance, and before the age of electricity it was not easy for a balance to become accidentally magnetised. In 1840 the Astronomer Royal Sir George Airy reported one chronometer, Brockbanks No. 425, that had become magnetised and altered in its rate depending in which direction it was oriented. Evidently the balance bar had become magnetised so that when the neutral position was aligned with the earth's magnetic field it was faster than when it was at 90°.

How the chronometer had become magnetised is not mentioned in Airy's report, it may have been due to a lightning strike on the ship in which it was being carried. Before electricity was generated and distributed, lightning strikes were the only sources of large electrical currents and magnetic fields that could magnetise a chronometer. Naturally occurring lodestones that are magnetic were most likely themselves magnetised by lightning strikes. At the time there was no easy way to demagnetise the chronometer but Airy showed how t could be corrected by placing a compass beneath it. The field of the compass needle opposed the earth's field and the two effects cancelled each other.

When iron hulled ships were introduced there was a concern that this might have an effect on the chronometers, but this fear wasn't realised. Iron in the hull or structure of a ship become magnetised during construction and this does have an effect on compasses carried on board, which are sensitive to magnetic fields because they have magnetised needles. But so long as the balance of a chronometer was not itself magnetised, the earth's and ship's magnetic fields have no noticeable effect on it.

The rate of a chronometer can be affected by bring a magnet close to it, so it might be throughout that the magnetism of a ship would also have an effect. This is not the case for the following reasons. Compasses, which are sensitive to magnetic fields because their needle is magnetised, continue to work on iron ships. They do need to be checked and corrected for the ship's magnetic fields, but this shows that the earth's magnetic field is stronger than the ship's — if it wasn't the compass wouldn't work at all. The rate of a chronometer can be affected by a magnet brought close to it. This is because the magnet induces temporary magnetism into the steel parts of the chronometer. When the magnet is removed the temporary magnetism disappears and the rate returns to normal. A strong magnet can induce permanent magnetism into steel, but then the effect on the chronometer's rate is permanent and noticeable. The earth's magnetic field is not strong enough to induce either permanent or temporary magnetism into chronometers, if it were then they would all be useless. And therefore, a ship's magnetic field, being weaker than the earth's, can have no effect on a chronometer.

With the age of electricity it was much easier for watches to get exposed to strong magnetic fields and non-magnetic watches were produced. These either had balances and springs that were made from non-magnetic material, or the movement was encapsulated in a "soft iron" enclosure — the iron is magnetically soft, not physically soft; it remains "as hard as iron".

Fortunately, the age of electricity also brought with it an easy means of demagnetising a watch that has become magnetised. The watch is placed in the alternating magnetic field of a demagnetiser and slowly removed, which removes the magnetism. But this needs a trip to a watchmaker who has a demagnetiser, which can be avoided if you take care not to expose an old watch to a magnetic field. Modern watches are usually resistant to magnetism.

There is more about the general topic of magnetism on a page on my personal web site at DavidBoettcher.uk/magnetism.

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## 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 the energy that makes the watch run.

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.

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

An old rule of thumb quoted by 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 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 spring that occupied one third of the barrel when wound and unwound is physically impossible, which is why some versions say "a little less" when unwound. 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 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

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). 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 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.

Blank uses ⅓ D.

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.

Blank uses ⅓ B.

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.

Blank uses ⅓ D.

### 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.

1. 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 results returned by the calculator.
2. 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.

## 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:

$$Turns = \frac{ \sqrt{ 2 \left( D^2 + d^2 \right)} - \left( D + d \right) }{2 \, t}$$

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 105 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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### Vibrations

Each swing of the balance from the centre to the point where it reverses and returns to the centre is called a vibration or beat. The Swiss use the term vibration, Americans use beat. Escapements are often characterised by their "vibrations per hour" (vph), sometimes shortened to simply "vibrations", or beats per hour (bph).

At the start of each vibration, as the balance swings through the centre, the impulse pin moves the lever and a tooth of the escape wheel is unlocked by a pallet. The shape of the pallet face and escape wheel tooth are designed so that after unlocking an impulse is given to the balance, replacing the energy lost by friction and keeping it turning. After the escape wheel has turned slightly, the other pallet locks a tooth which generates an audible tick.

Very old pocket watches ran at 14,400, 15,400 or 16,200 vibrations per hour. The 18,000 vph escapement eventually superseded all of these and for many decades this was standard, giving 5 vibrations or ticks per second. With 3,600 seconds in an hour, 5 ticks per second = 18,000 vph.

Higher frequency escapements were introduced to give mechanical watches greater resolution, which was useful for timing events accurately. These operated at 21,600 vph then 28,800 vph, giving 6 or 8 vibrations per second. When electric watches came along higher frequencies were introduced for mechanical watches such as the Girard Perregaux 36,000 vph movement of 1966 and the Zenith El Primero 36,000 vph movement of 1969.

In physics terms a cycle is a complete swing of the balance one way and then the other way, returning to where it started, so an 18,000 vph, 5 vibrations per second, balance cycles at 2.5 cycles per second, called Hertz (Hz).

The sketch shows the flow of power from the great wheel through the train to the escapement. The great wheel, which might be on a going barrel or a fusee, drives the pinion of the centre wheel. The minute hand is mounted on the arbor of the centre wheel so it must turn once an hour. The centre wheel turns the pinion of the third wheel, and the third wheel turns the pinion of the fourth wheel. The fourth wheel drives the pinion of the escape wheel, which is locked by the pallets on the lever until a swing of the balance unlocks a pallet and the escape wheel can advance by one tooth. From number of teeth on each wheel and pinion in the train you can work out how fast each wheel is turning and how many vibrations the escapement makes in an hour.

Typical trains might have the following tooth counts.

Vph18,00014,400
Wheel nameWheel teethPinion leavesWheel teethPinion leaves
Centre wheel8064
Third wheel6010608
Fourth wheel708568
Escape wheel157157

The relative speed of two wheels is given by the ratio of the teeth on the wheel and the teeth on the pinion it is driven by or driving. For example, in this train, the third wheel will turn 8 times as fast as the centre wheel, because 80 teeth on the centre wheel engage with 10 teeth on the third wheel pinion. The third wheel pinion will make one revolution when 10 teeth of the centre wheel have turned with its 10 teeth, and as the centre wheel has 80 teeth it will rotate once for 8 turns of the third wheel.

The escape wheel has 15 teeth, and for the wheel to make one complete revolution each tooth has to pass both the entry and the exit pallet in getting back to where it started from, so each tooth on the escape wheel needs two ticks to get back to where it started. For an 18,000 vph train, which ticks 5 times per second, a 15 tooth escape wheel will take 30 ticks to make one revolution. Each tick takes 1/5 seconds, so a complete revolution of the escape wheel takes 30 x 1/5 = 6 seconds.

The fourth wheel engages with the escape wheel pinion, which has 7 teeth. The fourth wheel has 70 teeth, so it will turn once for every 10 revolutions of the escape wheel, that is once every 60 seconds. That's good, because the fourth wheel arbor also carries the seconds hand, so it is useful that it turns once a minute.

Knowing that the centre wheel rotates one an hour allows us to work out the vibrations and frequency of the escapement using the following formula, where w is the number of teeth on a wheel and p is the number of teeth on a pinion, so tw is the tooth count of the third wheel and tp is the number of leaves on its pinion. Wheels have teeth but pinions have leaves. Why? I have no idea. The 2 on the top is there because each tooth of the escape wheel has to escape twice during one revolution, the 3600 on the bottom is there because the centre wheel revolves once in one hour, which is 3,600 seconds.

$\mathrm{Vps}=\frac{\mathrm{cw}.\mathrm{tw}.\mathrm{fw}.\mathrm{ew}.2}{\mathrm{tp}.\mathrm{fp}.\mathrm{ep}.3600}$

The calculator below lets you enter all of these wheel and pinion counts and calculates the escapement's vibrations for you.

#### Watch with seconds

It isn't always necessary to count all the wheel and pinion teeth if you just need to work out how many vibrations a movement does. This is because the ratio of the fourth and centre arbors is usually fixed - if the watch has a seconds hand the fourth arbor turns once a minute to drive the seconds hand, the centre arbor turns once an hour and drives the minute hand. So the ratio between the centre and fourth arbors must be 60, no matter what the counts are of the centre wheel, third wheel, third pinion and fourth pinion are. We can therefore simplify our calculator and just enter the counts of the fourth wheel, escape pinion and escape wheel, like this. The 60 on the bottom is because the seconds hand takes 60 seconds to revolve.

$\mathrm{Vps}=\frac{\mathrm{fw}.\mathrm{ew}.2}{\mathrm{ep}.60}$

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