Physics of the Automatic Watch's
Winding System

by Walt Arnstein

Introduction

Owners of automatic watches generally take for granted the ability of their watches' self-winding mechanism to keep the mainsprings wound without the need for manual intervention. Many of the owners don't really know how this task is accomplished beyond what they have been told when they bought their timepieces: That the "natural" motions of their wrists in the course of daily wear wind the mainspring and that the watches need to be worn daily if they are expected to keep running.

Imagine the surprise and dismay of the hapless wearer, then, who finds his Rolex, say, stopped or 2-3 hours behind the correct time and discovers that the important client he was to meet in the lobby of the Waldorf at 3 PM left a couple of hours ago and is at this moment probably meeting with a competitor's representative to discuss the same business deal.

The above situation is, fortunately, fictitious and a bit exaggerated, but the factors that cause an automatic watch to fall short of expectations are potentially present in everyone's daily life. For this reason, it is an advantage to know how an automatic winding mechanism works and what keeps it happy.

This discussion will examine the physical principles that govern the operation of the automatic watch's winding system.

The Watch Winding System as a Black Box

In science and engineering it is often convenient to represent a system under study as an opaque enclosure whose contents we may not know or care about but whose inputs and outputs we can very clearly define and analyze. This is called a black box model. The inputs can be energy, physical forces, or matter and so can the outputs. Without needing to know what is inside the box, we can apply the laws of physics to account for all processes taking place at its boundaries. For example, if mechanical energy is seen to be flowing into the box at one rate and going out of the box at a lower rate, we can assume that there is some form of energy storage inside the box. If there is measured amount of heat going into the box and a lesser amount coming out, some of the heat must be being consumed inside the box in the form of physical work or thermal storage. Whatever the case, all inputs and outputs of the black box must balance quantitatively if the model is valid.

In the case of an automatic watch's winding system, our black box model's boundaries will enclose:

The black box model above has three input sources:

It has only two output sources:

Among input sources, inertial forces represent the major portion of the watch's energy supply by an overwhelming margin. Gravity plays only a minor role on Earth and none in space. The only exception to this rule is the electrical watch winder, which depends totally on gravity to keep the rotor at the bottom as it turns the watch in a vertical plane about the axis of its hands.

The laws of Newtonian mechanics and the First Law of Thermodynamics tell us that every turn of the crown, every turn of the rotor, and every joule of energy stored in the mainspring can be completely accounted for and will ultimately end up as energy in the form of turns of the mainspring barrel (driving the watch's gear train and escapement) plus a varying amount of heat generated by friction.

With a watch, we have a very convenient reference about which to perform our analysis of energy balance: The output rate of our black box is known and essentially constant. That is, the mainspring barrel gear turns at a steady rate with reasonably constant torque as it drives the watch's main gear train and escapement. So, to keep the watch running dependably, we need to provide enough input energy, from inertial and gravitational sources -- and the crown if necessary -- to offset losses due to the rotation of the mainspring barrel and to a small amount of heat dissipation. Of course, the mainspring barrel turns at a constant rate while our rotor's movement is sporadic and uneven. But as long as the mainspring has some capacity left to absorb and store this uneven motion, our system will work reliably providing the long term average input rate matches or exceeds the output rate.

And it is this last proviso that leads to grief in the case of a quite a few owners!

We will examine the way a rotor mechanism works in the following section.

The Kinematics and Dynamics of the Rotor and Mainspring.

The rotor of a modern automatic watch is an unbalanced pivoted mass, free to rotate a full 360 degrees about a bearing located away from the its center of mass. (We will mention and briefly discuss the older "bumper" type rotors later in this discussion). The mass is distributed in a particular way so as to maximize its moment of inertia about the pivot. The moment of inertia is a physical quantity that may be thought of as the mass's tendency to keep rotating once it is spun up and hence its ability to store kinetic energy. In the case of the rotor, we definitely wish to get this parameter as high as possible, given its physical restrictions.

[For the technically minded, the moment of inertia of a distributed mass about a pivot is given by the integral:

 

I=çr,m@r2dm

 

where r is the distance of a mass element dm from the pivot.]

Since the moment of inertia is sensitive to the square of each of its elements' distance from the pivot, it is obviously advantageous to concentrate as much of a rotor's mass away from the pivot as possible. This might explain the typical shape of a fine watch's rotor: A thin web-like structure near the center of the watch and a thick, heavy ring of metal -- frequently dense, like gold, platinum, etc. -- at the periphery of the movement. The same total mass, if located closer to the pivot, would drastically reduce the rotor's moment of inertia.

Consider now what happens as a rotor is made to move. For convenience, let's imagine the wearer is standing at rest with his arms at his side. The crown is pointing downward and because of gravity, the center of mass of the rotor is also at the bottom, i.e., underneath the 3 o'clock mark. Now assume the owner suddenly jerks his left arm forward and continues to move forward. The watch and rotor bearing, being firmly attached to the arm, move along, but the rotor's mass does not, since it is free. Instead, it accelerates backward relative to the pivot and as a consequence is imparted an instantaneous linear momentum M equal to mv, where m is the mass of the rotor and v is its new velocity relative to the pivot. However, since the rotor is restrained by the pivot from continuing in a straight line, its motion is transformed into a rotation about the pivot -- counterclockwise, viewing the dial, in this case. The linear impulse of the wearer's arm has imparted a rotational motion to the pivot! Note that even though gravity was stipulated at the beginning to use as a convenient starting point, it was not necessary for this result to ensue. The rotor would have started rotating in outer space under the same circumstances.

Depending on the abruptness and speed of the arm motion, we have now in effect imparted a certain amount of angular momentum to the rotor. The expression for the angular momentum A is given by:

A=l‚—

where I is the moment of inertia of the rotor and w is the angular velocity of the rotor, in radians per second.

The rotational kinetic energy we have imparted to the rotor is given by:

E=1/2 ‚Œ‚—2

 

Immediately, the rotation of the rotor starts being transferred via the winding gears to the mainspring. The rotational kinetic energy of the rotor is gradually transformed to potential energy as it is stored in the mainspring for later use as needed. The mainspring, with its tension growing, resists the rotor, slowing it down. Eventually, the rotor will stop and the mainspring will have gained the energy previously contained in the rotating mass of the rotor -- minus a small amount lost to friction.

Described above was just a single incident of angular momentum being applied to a rotor by a mechanical impulse. (By the way, an impulse is defined in physics as an abrupt change in momentum due to a force acting on a mass over a very short period). In actual situations, the impulses delivered by a watch wearer's wrist are frequent and of varying directions and intensities. To some degree, the process is statistical. If the impulse is applied in the direction of the instantaneous motion of the rotor's center of mass, it will add to the existing momentum. (In fact, it can produce remarkable rotation rates if the impulses happen to be applied at the proper frequency and directions. Most people can generate this kind of motion instinctively by holding the watch face-up and swirling it at the proper rate, making a motion as if trying to dissolve a cube of sugar in a cup of tea). If it is in the opposite sense, it may reduce the present momentum. But in general, frequent impulsive motions typical of everyday activities of most watch wearers, will generate lots of momentum transfer and consequently, lots of energy to transfer to the mainspring.

It should now be clear that, as implied above, any automatic watch would work in the absence of gravity, as on a shuttle or spaceship. The "urban legend" keeps resurfacing about NASA choosing a manual wind chronograph, the Omega Speedmaster Professional, because its engineers thought that an automatic wouldn't wind in the absence of gravity in outer space. In actuality, it is almost certain that these notable scientists were well aware of the minor role of gravity in keeping an automatic watch wound. The real reason a manual watch was chosen is that in 1962, there were relatively few, if any, automatic chronographs on the market. Indeed, if there is any situation where the rotor's winding efficiency is reduced or impaired, it would be under water, particularly with the owner wearing a diving suit. The hydrodynamic drag of the water inhibits the sort of abrupt accelerations of the arm that are ideal for transferring large amounts of energy to the rotor. Luckily, the typical dive represents only a small part of the user's daily wear time so any rotor turns lost underwater are probably made up for once the wearer is back on dry land.
>
> Speaking of momentum transfer, a special case of the action described earlier prevailed in the old 'bumper' watches, common before about 1955. These automatics, including all Omegas, Tissots, and a number of other brands (not including Rolex), had rotors that had the freedom to move only about 270 degrees, their further travel being restricted by tiny coil springs that they struck with amazing resilience. Wearers of these watches could feel distinct bumps as they moved their wrists -- a pleasant feeling, according to some owners. For purposes of our discussion, these rotors received angular momentum much like today's 360 degree rotors. The only difference in their operation was that they transferred their rotational energy to the mainspring in a series of 270 degree swings punctuated by elastic direction reversals and accompanied by satisfying (we are told) bumps. The energy lost to heat dissipation in this way was very low.

So Why Do Watches Stop?

Considering the efficiency with which momentum, hence energy, can be transferred from the wrist to the rotor and thence to the mainspring, it is still a matter of puzzlement to many people that their watches keep stopping at unexpected and often inconvenient times. Setting aside the obvious possibility that the watch is defective in some way or simply needs cleaning and lubrication, the most obvious cause of the inadequate power reserve is one or more of the following:

The first of these can be elusive even to people who essentially wear the same automatic watch every day. They may, for example not put it on until they are ready to leave home for the working day. Then, at work, they may remove the watch when going to some location where strong magnets are present. They may also remove them simply to wash their hands (particularly if the watch is not water-resistant), to play a sport or engage in some other activity that may put the watch at risk, or may simply prop the watch up on the desktop for better full-time visibility. Back at home, there may be an equally logical set of reasons to take the watch off. On weekends, the watch may hardly be worn at all. So, in sum, it is not difficult to develop wear habits that promote impromptu stopping, while at the same time the owner may be convinced that he wears his watch all day, every day.

Similarly, some owners have occupations that involve long hours at a computer keyboard or similar activities featuring very little motion of the kind that sends rotors spinning.

Finally, we can often place the blame on the watch or, rather, the poor match between the watch and the wearer. Small size ladies' watches are often a major problem in this regard. Simply put, their rotors are just too small to be good grabbers of momentum. Recall that the momentum and energy of a spinning rotor is directly proportional to its moment of inertia and that the moment of inertia is very sensitive to the square of the distance of its component elements from its pivot. Because of this, the physical size of a watch -- hence its rotor -- has a profound effect on the efficiency of the winding system. How profound? Consider this: If we scale down a rotor from a 40 mm size by a factor of 2, i.e., make it a perfect 20 mm replica of the larger rotor -- materials and all -- the smaller rotor's moment of inertia will be 1/32 that of the larger one! Yet the energy needed to operate the smaller watch is not significantly smaller than that needed to operate the large one. At best, it is 1/2 the energy. Small wonder, then, that so many small automatics stop, particularly on the wrists of elderly ladies.

Some complicated watches have unusually high energy requirements, to be able to run the various subsystems built into them. As a result, the winding gear set may have an unusually high transfer ratio in order to wind the watch's stiff mainspring. In other words, it may require a lot of turns of the rotor to keep the watch wound.

Finally, there may be simply a weakness in the design of the watch's winding mechanism. One common in relatively inexpensive watches (e.g., the Swatch Automatic) is a rotor with very low moment of inertia, meaning that it requires a lot of wrist motion to get it to make the necessary number of turns per day. Also, watches with bidirectional winding do lose some transfer efficiency due to the direction sensing components that provide gear reversal with every change of rotor direction. Some watch experts actually suspect that unidirectionally winding watches may in some cases be more efficient in winding their mainsprings even if their rotors do a lot of freewheeling.

It should be added in conclusion that power reserve problems often have very long time constants, sometimes measured in weeks, making the problem exceedingly hard to pinpoint. That is, a fully wound watch falling victim to one of the problems discussed above may lose just a small part of its power reserve in 24 hours. As a result, it may take weeks before the watch stops for the first time. An unfortunate consequence is that the direct cause-effect relationship is obscured as the watch owner searches for some recent event to explain the mysterious stoppage of the watch.

For the above problems, a winder is the obvious solution. Winders are now available in a very wide variety of types and price ranges.

What Happens to Excess Energy?

Until now, we have been discussing the energy interchange process between the rotor and the mainspring, the kinetic energy of the rotor being transformed into the potential energy stored in the mainspring's spiral. Well, there comes a point in a mainspring's power curve when it can no longer store up any energy. It becomes fully wound, in our terminology. But the rotor doesn't know this! So it keeps on trying to put more and more energy into the mainspring. The designers of automatic watches have wisely foreseen this problem -- some of them perhaps learned it the hard way -- and have provided automatic watches with an overwind protection mechanism. Unlike the case of the mainspring in a manually wound watch, the outer end of an automatic watch is not rigidly connected to the inner wall of the mainspring barrel. Instead, it is connected to a bridle, which is in turn interfaced by controlled viscous friction to the barrel. The controlled friction is provided by a very special grease, whose properties and thickness of application are critical for proper functioning of the overwind protection system. The bridle must begin to slip -- smoothly -- against the inner wall of the barrel when the tension in the mainspring reaches a value just a bit below the "brick wall" tightness that signifies the spring's ability to store energy is at an end. If there is too much grease, or the grease is too slippery, or the spring is too weak, slippage will occur too soon and the watch will exhibit inadequate power reserve at all times. If there is not enough grease, or the grease has dried out, or is of the wrong type, or the spring is too strong, the slippage will occur too late or not at all, resulting in overbanking of the balance wheel and possibly eventual damage to parts of the escapement.

But for our purposes, the question that arises regarding the above conditions is, "What happens to the energy from the rotor when the mainspring has begun to slip?" The answer is that it all goes into heat -- an irreversible loss of energy. If this were allowed to continue without restraint, the result would be pointless wear on the mainspring barrel was and the bridle. But luckily, It turns out that on the wrist, the impulsive nature of the wrist's motion tends to be self regulating for the winding mechanism. Rotors working with fully wound watches encounter a strong resistance and bounce off it -- without significantly moving the mainspring -- going back and forth in ping-pong fashion in watches that wind in both directions or freewheeling in the nonwinding direction for watches that wind in only one direction. In this way, energy doesn't have a chance to build up. But in an electric winder, which rotates the watch steadily using gravity, the slippage of the mainspring's bridle against the barrel wall becomes steady with continuous dissipation of heat. For this reason, it is important to limit a watch's number of turns per day it spends on the winder. Most manufacturers of winders either provide a timing mechanism for this purpose or provide tables of requirements for a wide variety of watches, based on the operating cycles of their winders.