Does Faster Mean More Accurate?
by Walt Arnstein
Frequency Response of a Watch Escapement
Throughout most of the first half of the Twentieth Century, mechanical watches operated at an essentially standard 5 beats per second, or 18,000 bph (beats per hour). The choice of this beat rate was dictated by a number of factors, such as the large size of a pocket watch’s balance wheel, requiring exceptional mainspring tension; the need for advanced materials and lubricants if faster rates were contemplated; and, not insignificantly, the well-developed ability of watchmakers to design and build watches -- both pocket and wrist -- capable of outstanding accuracy with a 5 bps movement.
The Minerva Pythagore, including TZ’s own limited edition of this watch, has a movement typical of the above approach. Designed in 1943, it is virtually unchanged today, with its manual winding, absence of a calendar window, screwed balance wheel.....and its 5 bps escapement. Its accuracy, it is reported, is nevertheless above average, mainly because of the painstaking work that goes into its assembly and testing. Such personal care is getting increasingly expensive in view of today’s labor costs and the scarcity of qualified watch technicians, so any technical change that could reduce the product’s dependence on individual attention is always welcomed by the manufacturers. One such change is the implementation of a higher beat rate.
In the late ‘40s, a number of watch manufacturers introduced movements with a beat rate of 6 bps, or 21,600 bph. With synthetic lubricants, including dry types in some cases, the frequency race was on. Today, a beat rate of 8 bps (28,800 bph) is fairly common and a few watches beat at 10 bps (36,000 bph). The performance of many of these watches is on the whole noticeably better than that of watches of comparable quality still beating at 5 or 6 bps. Putting this last sentence another way, it is possible to obtain a given level of accuracy cheaper with a fast-beat watch than it is with a slower-beating timepiece.
What is the basis of the superiority claimed for watches of higher beat rates over their slower-beating counterparts of like quality? The answer lies in the basic laws of physics and control theory. This article will explore this question in brief detail.
1.1 The escapement as a damped oscillatory system.
In a mechanical watch, the escapement is in technical terms an impulse-driven lightly damped second order oscillatory system. What this means is that there is a vibrating element -- the balance wheel/hairspring combination -- that oscillates at its natural frequency, losing a small amount of energy with each swing. The energy it loses is exactly restored by an impulse from a tooth of the escape wheel -- forming the end of the mainspring-powered gear train -- striking a jewel on the pallet fork thereby giving the balance wheel a measured increment of angular momentum. The collision between escape wheel and pallet jewel is what gives a mechanical watch its characteristic ticking sound. If the escapement is properly designed, the balance wheel will settle into a stable oscillatory mode in which it will maintain a constant swing amplitude.
Sounds simple and it is -- providing there are no external mechanical disturbances, like extraneous impulses delivered to the system by mechanical shocks, vibrations and rotations about the balance wheel’s pivot axis. In control system science, these constitute forcing functions to which the balance wheel will respond by changes in its momentary position. Such changes can be represented as equivalent momentary changes in position and frequency. In other words, small timing errors. In the long run, such timing errors accumulate into accuracy degradation.
How does an escapement quantitatively respond to a disturbance? That response is a function of the escapement’s natural frequency -- i.e., the beat rate -- and the frequency distribution of the disturbance. These two concepts will be examined in the next section.
1.2 The concepts of frequency distribution and frequency response
If you hold your wristwatch in hour hand with the dial facing up and move the watch sideways back and forth in a straight line in any direction, the watch will probably exhibit no noticeable response, since the balance wheel is dynamically balanced in all planes. The same can be said for any linear movement up or down. The balance wheel, in short, is immune to linear translational accelerations. So far, so good.
But introduce any amount of rotational acceleration about the balance wheel’s axis and you will have an entirely different situation. The rotational disturbance acts about the system’s oscillatory axis, tending to add to or oppose the oscillator’s stable motion. This is a completely different sort of disturbance! Specifically, if this rotational disturbance is of the proper frequency, it can slow down or speed up the oscillatory motion of the balance wheel/hairspring system and thereby change its steady progress. In short, it can generate a timing error that is cumulative in the long run.
Do we normally subject our watches to a disturbance such as described above? You bet! The motion of our arms in the course of our normal activity is a complex combination of translational and rotational accelerations. Moreover, it has a constantly varying frequency distribution, all of which can be resolved into specific components by the method of Fourier Analysis. In other words, your wrist’s motion can be described in terms of infinitesimal frequency components of differing magnitudes, with those associated with rotational motion affecting the balance wheel’s motion in a predictable way. This tabulation is the frequency distribution, or frequency spectrum, of your arm’s rotational motion.
Let us now focus our attention on the watch escapement. If we held it as before, face up, and very slowlyand gently -- with a period of 3 seconds, say -- rotated it back and forth about the axis of its hands (and hence those of its balance wheel, of course), there would probably be no noticeable reaction on the part of the escapement. But if we now gradually began to increase the frequency of this tiny back and forth rotation, the watch would start to react, the same as would a heavy steel washer hanging at the end of a rubber band as we moved the band’s opposite end up and down in a steady rhythm. The magnitude of this reaction would grow as we approached the escapement’s natural frequency -- 2.5 Hz for a 5 bps balance wheel, 3 Hz for a 6 bps wheel, etc. -- until at or near that frequency, the escapement would seem to go crazy. The amplitude of the oscillation would grow until the balance wheel began to strike its "crash stops" at the extremes of its travel. We would also note that with this disturbance in the vicinity of the escapement’s normal beat frequency, the actual beat frequency would shift very slightly in the direction of the disturbing function.
If we continued rotating the watch back and forth at increasing frequency beyond its movement’s natural rate, the response of the escapement would gradually diminish and return to more or less normal operation. So if we drew a plot of response amplitude as a function of the disturbance frequency, we would get a curve with a strong and fairly broad peak at the normal frequency of the escapement (1/2 the beat rate) and sharply falling skirts at both sides of the peak. There might also be one or more secondary peaks at some of the multiples of the fundamental frequency, but these would be relatively small. This kind of curve is called the frequency response of the escapement and is understood to mean the response of the escapement to external periodic rotational mechanical disturbances. For the frequency relation just described to hold validity, keep in mind that our external disturbance must be kept very small in amplitude compared to the amplitude of the escapement’s normal oscillation.
From the above plot, we know that the watch will tick happily away unless and until it is subjected to a disturbance at or near its fundamental oscillating frequency. Then we will have troubles.
1.3 Frequency Distribution of the Wearer’s Wrist Motions
A watch worn on the wrist, being somewhat cushioned by flesh and located as it is at the end of a rather massive human arm, is generally constrained to move without exceedingly strong accelerations. In fact, unless the watch itself contacts a brick wall or tile floor, say, it is well protected from heavy jars. To be sure, the complexity of the motions we perform in the normal course of our daily lives includes a lot of small rotations, some of them periodic. As we mentioned before, all these motions can be resolved into superimposed translational and rotational components and further, all of these components can be characterized by their frequency content.
The translational components can be disregarded for our purposes because they don’t affect a well-designed escapement. But the rotational components and their frequency content are very important. What is that frequency content? Studies have established that the rotational physical activity of our wrists is concentrated at frequencies below 3 Hz and the bulk well below 2 Hz. However, the total spectrum does extend into higher frequencies with rapidly falling amplitude. Hence, a watch balance wheel operating at 2.5 Hz (5 bps) will be subjected to a tiny amount of disturbance from our wrist motions -- the components at or near 2.5 Hz -- and will react accordingly.
What about an escapement operating at 3 Hz (6 bps)? Clearly it will react to the smaller amplitude of the disturbance spectrum centered at this frequency. A movement of recent vintage operating at 4 Hz (8 bps) will be far less sensitive to the concentration of low-frequency components delivered by the wrist and the tiny residual components at or near 4 Hz. The 5 Hz (10 bps) movements should do even better. Conclusion? The higher the beat frequency of the escapement the lower its sensitivity to external mechanical disturbances. Hence, the better its achievable accuracy.
But, of course, we all knew that. Now we know why.
2.0 The General Case
The above rule is not limited to mechanical watches with rotating balance wheels. It applies to grandfather clocks and Atmos clocks, whose sensitivity to platform movements is strongly influenced by the oscillation frequency of its escapement and the frequency distribution of the floor or mantle movements. The briefly popular tuning fork watches of the 1960s, like the Bulova Accutron", oscillating at 400 Hz, were admirably insensitive to normal wrist motion -- unless you dropped the watch, in which case you were very likely to have to take the watch to a repair shop.
Quartz watches are also not exceptions from the "faster is better" rule. Despite their electrical trappings, quartz crystals are basically mechanical oscillators. Their oscillatory frequency in wrist watches is nominally 32,768 Hz, almost without exception. As a consequence, their oscillation is essentially immune from mechanical disturbances originating in the wearer’s motions. However, they are not immune from temperature changes. A few movements incorporate temperture compensation in their implementation and therefore have outstanding accuracy compared to most mechanical watches. Other movements vary by tenths of seconds per day over temperature ranges of tens of degrees typical for their environments.
OK, so faster is better, up to a point. So why doesn’t the mechanical watch industry boost the beat rate of all watches to at least 10 bps? Or, for that matter, why don’t we keep raising the beat rate, to 16 bps, say, or try for 20? It turns out that a fast beat is not without its technological problems. The most significant of these is lubrication. To keep a layer of lubricant between critical moving surfaces at that rate takes some very sophisticated materials. Fast-beat Rolexes, for example, use dry lubricants (made with bismuth compounds) in the escapement. The method of applying them is very exacting.
Then there is the problem of supplying the proper power to run these escapements. The energy needed to operate a movement varies approximately as the square of the beat rate (approximately because the losses due to friction and air resistance are nonlinear), so the mainspring tension and forces acting on the gear train’s components grow accordingly. Most important, the consequences of loss of lubrication are far more serious in a fast-beat watch than they are for a 5 bps movement. Walt Odets, for example, wrote about a friend’s watch that had gone without an overhaul for 15 years or so, I seem to recall. It was probably a 5 bps or 6 bps watch. A watch designed to operate at 10 bps would probably have suffered a serious amount of permanent damage if it had been allowed to run that long without an overhaul. In short, given 10 bps watches as presents, a large portion of our consumer population would through neglect allow these fine instruments to literally beat themselves to death.
Also, there are other considerations in comparing various watches. For example, the phenomenally accurate pocket watches designed for railroad use by Hamilton, Elgin, Waltham, Ball, etc. (and now, sadly, no longer made) all had frequencies of 2.5 Hz (5 bps). Why, then, were they so accurate? To begin with, they were adjusted to an unusually high degree of precision. Also, they were generally kept in the wearer’s pocket in a 12-up position most of the time and carefully protected from external disturbances. Equally important is that fact that their balance wheels were very massive compared to the tiny components of a typical wrist watch. It therefore took a strong disturbance indeed to cause a noticeable reaction on the part of the movement.
Finally, we should keep in mind that response to mechanical disturbances is not the only source of error in a watch. There can be poor adjustment for isochronism, sensitivity to temperature, friction in the gear train, etc. How else to explain the fact that Rolexes, Omegas, and other fine watches of the 1930s and 1940s routinely passed COSC tests with flying colors despite their low (5 bps) beat rate? Or the fact that a sizable number of modern watches (notably models by PP, IWC, Minerva, etc.) still continue to exhibit outstanding performance with a beat rate of 5 bps? The answer lies in precision and individual attention to detail.
No, beat rate is not the only factor affecting potential for outstanding performance. But given everything equal, the properly designed fast-beat watch does have a leg up on its slower cousin.