## Abstract

This paper addresses the nature of noise in machines. As a concrete example, we examine the dynamics of clock escapements from experimental, historical and analytical points of view. Experiments on two escapement mechanisms from the Reuleaux kinematic collection at Cornell University are used to illustrate chaotic-like noise in clocks. These vibrations coexist with the periodic dynamics of the balance wheel or pendulum. A mathematical model is presented that shows how self-generated chaos in clocks can break the dry friction in the gear train. This model is shown to exhibit a strange attractor in the structural vibration of the clock. The internal feedback between the oscillator and the escapement structure is similar to anti-control of chaos models.

## 1. Introduction

All machines exhibit a certain amount of noise. The question arises as to whether a certain level of noise is natural or inevitable in a complex assembly of mechanical devices. For example, in the early history of the steam turbine, Charles Parsons patented a mechanical oscillator in 1895 to break the friction in servo control valves (Conway 1953–1955). Such a designed noise in control systems is sometimes called *dither*. Recently, several papers have examined schemes to maintain the otherwise periodic systems in a state of chaos using anti-control of chaos, such as Christini & Collins (1995) in neuron biology and Moon *et al*. (1996, 2003) in a dry friction oscillator. In those papers, the dynamical system is controlled by an impulsive driver, when the orbit enters a small region of phase space. In the present paper, the clock oscillator is given an impulse when a chaotic orbit reaches a threshold and when the orbit enters a small region of phase space. We propose that complex machines are often designed to admit both periodic motions and chaotic or noisy vibrations, and that these vibrations may be beneficial to the operation of the machine. We examine the nature of regular and noisy dynamics in a specific class of complex multi-body machines, namely the clock escapement.

The mechanical clock is an assemblage of at least half a dozen basic machine elements such as gears, bearings, linkages and impact elements. In nonlinear dynamics of machine elements, there are well-documented examples of chaotic vibrations in bearings, gears, ball bearings and linkages (e.g. Moon 1992). Therefore, it should not be surprising that clock escapements might exhibit unpredictable dynamics as well as predictable oscillations. In fact, there are many physical systems, including fluid convection (e.g. Berge *et al*. 1984) and electrical circuits, in which the dynamics becomes partitioned into sub-regions of mostly steady, regular or periodic motions and sub-regions in which there is a certain level of unpredictable or noisy vibrations.

We have chosen to examine noise in the mechanical clock for several reasons. First, it has a long and documented history of about five centuries. Second, although there have been many books written on the clock, there are very few that describe the complete nonlinear dynamics. Finally, the clock is a machine in which most people would assume the absence of noise. Thus establishing the existence of deterministic noise in clocks might give credence to the broader claim of natural chaos in complex machines. In addition, the mechanical clock has no sources of randomness since it is driven by a falling weight.

In a recent paper by the first author (Moon 2003, 2005), the premise of self-excited chaos in the near periodic machines was examined but only the qualitative experimental data were presented. In this paper, we present more quantitative, experimental evidence for the coexistence of both periodic and noisy vibrations, using probability density functions (PDFs), phase plane plots and return maps. In the clock, the periodic motions reside in the rigid-body motions of the pendulum or balance wheel and the noisy or chaotic-like motions reside in the structural vibrations that support the escapement, gear train and balance wheel or pendulum.

## 2. Historical review

There are claims that the oldest clock escapement, the *verge and foliot*, was invented by Pope Sylvester around 990. However, most historians place this invention between the thirteenth and the fourteenth centuries. For example, the verge and foliot escapement clock at Salisbury Cathedral in England dates back to 1386. The original escapement did not have a pendulum or spring and balance wheel to fix the frequency, and the period depended on the friction in the machine. The verge consists of two paddles fixed to an axel that interacts with the ‘scape’ wheel. The foliot is a bar with weights that acts as an inertial element or angular momentum storage. However, without a spring attached to the foliot, there is no natural frequency except that determined by friction.

The first pendulum clock is attributed to Huygens in 1657, although there is some posthumous claim to the invention by Galileo through his son. The Huygens clock is a combination of the verge and the pendulum. The next major improvement was the invention of the *anchor escapement* that replaced the verge with the two-arm device shown in figure 1. This invention is often attributed to Robert Hooke, but other sources give credit to a clockmaker, William Clement, in 1670. The anchor, like its predecessor the verge, served to regulate the amount of energy or torsional impulse imparted to the pendulum from the falling weight in each cycle. One fault of this device was the recoil that occurred when one of the two anchor pallets impacted the escape wheel teeth. This was corrected by the invention of the so-called *deadbeat escapement* invented by the clock and instrument maker George Clement in 1715. This improvement redesigned the shape of the anchor pallet arms as well as the escape wheel to prevent recoil on impact. These design improvements greatly increased the precision of clocks. It is interesting to observe from historical evidence that, contrary to popular belief about the motion of the pendulum, clocks at the beginning of the eighteenth century neither produced accurate nor regular periodic motion (Baillie *et al*. 1956).

The pioneering work by the English clockmaker John Harrison (1693–1776) advanced the design of accurate clocks for marine travel. Without listing all his improvements, clock accuracy went from seconds per day to seconds per month during the Harrison dynasty. Other contributors at this time were Pierre Le Roy and Ferdinand Berthoud of France as well as Arnold and Earnshaw in England (Bruton 1979). In addition, many other escapements were invented, such as the detent, cylinder, duplex, pin wheel and gravity escapement, the last of which was installed in the clock tower in London known as Big Ben in 1859, a period of over four centuries of invention, design and development. Despite this progress, the historical record is replete with evidence and discussion in the literature of the irregularities, inaccuracies and unpredictability in the mechanical clock. (See §12 for more details.)

The major works on the dynamics of clock escapements are those of Airy (1826), Bloxam (1854) and Denison (Lord Grimthorpe, 1868). These works contain a reasonable level of mathematics. Later, important mathematical analyses in the twentieth century were those of the Russians, Andronov *et al*. (*ca* 1940–1966). There have appeared a series of papers on the mathematical analysis of escapement dynamics such as Kauderer (1958), Kesteven (1978), Lepschy *et al*. (1992), Bernstein (2000), Roup *et al*. (2001) and Headrick (2001).

## 3. Multi-body models for clock escapements

Post-Huygens clocks (1658) have either a pendulum or a balance wheel and spring as the basic oscillator (figure 1). Energy is supplied each cycle, using either a falling weight or an elastic spring, through an escapement mechanism. The energy in most escapements is transmitted from the escape wheel to one of the two pallets through impact forces. These impact forces propagate vibrations through the supporting structure (not shown in figure 1) and we posit that these impulse forces play a role in unlocking static friction in the gear train.

The torque from the falling weight {8} is delivered through a train of gears {4–7} that acts in the direction opposite from most gear transmissions. The motion is driven from slow speed (falling weight) to the higher speeds of the clock hands {15–16}. Thus, it generally takes a large torque to deliver a small torque to the pendulum through the pallet arms of the escapement {2}. In many theoretical papers on clock dynamics, the gear train is not treated. However, we will show that the static friction in the gear train may be a key element in the existence of internal, chaotic noise in the clock.

## 4. Dynamics experiments on clock escapements

To investigate the role of noise in clock escapements, dynamic experiments were made on several escapement models from the Reuleaux Collection of Kinematic Mechanisms at Cornell University. This nineteenth century collection of 230 iron and brass models contains 10 clock escapement models (e.g. Reuleaux 1893, pp. 167–169). (Visit the website http://kmoddl.library.cornell.edu to see videos of the motion of these escapements.) Two models are shown in figures 2 and 3. The first model is of a cylinder escapement that drives a balance wheel and torsional spring oscillator. This escapement was once used in pocket watches. The second escapement is a three-tooth device that is very old and drives a pendulum oscillator. In the experiments described in Moon (2003, 2005), the dynamics of the pendulum (or balance wheel) and the structural vibrations were measured with an accelerometer, placed on the half-cylindrical arm on the cylinder escapement and on the pendulum arm on the three-tooth escapement.

In each model, the torque was generated by the falling weight acting through a single-stage pinion and gear mechanism with the driving torque applied to the large diameter gear wheel. Thus, torque flows from the falling weight through the pinion and is transferred to the oscillator by an impact between the escapement arm that is connected to the pendulum or to the balance wheel. This impact generates vibration in the clock structure that is measured by the accelerometer.

The Reuleaux models were designed for demonstration and the restricted motion of the falling weights limited the running time of the escapements to about 10–20 s, so that only qualitative experiments were reported in Moon (2003, 2005). In the present paper, a new drive system using the falling weight was designed in order to obtain long-time datasets.

Experimental data on these clock escapements show the existence of both periodic dynamics and non-periodic noise as shown in figure 4. The vibrations for the pendulum escapement clearly show both the low-frequency pendulum oscillations that are used to measure the time in seconds and minutes. However, riding on top of these motions are non-periodic vibrations associated with the impact of the escapement wheel with the drive arm.

## 5. Experimental procedure

A goal of the new experiments was to separate the periodic motions from the structural vibrations. This was done by measuring the periodic motions of the rigid-body pendulum or balance wheel with an optical follower camera and using an accelerometer fixed to the structure close to the escapement impact zone to measure the structural noise. In figure 4, the accelerometer was attached to the pendulum and both types of motions were measured simultaneously.

### (a) Pulley device

A pulley device was constructed to transfer the path of the falling weight of each escapement so that the range of falling motion would be greatly increased from Reuleaux's original design. The pulley system is shown in figure 5.

The device was held together by two long strips of plastic that were joined by three threaded rods and tension rods. These rods squeezed the plastic strips tightly against the escapement frame. In this manner, the 120 year old escapements suffered no damage and the device was able to operate on both escapements. The most important parts of the device were the clamped pulley piece, shown in figure 6, and two brass frictionless ball bearing pulleys that are visible in figure 5. The clamped plastic pulley piece was designed with CAD software and printed using a rapid prototyping machine.

The clamped pulley piece was designed in two pieces that could be fastened near the gear train and around the torque bar of each escapement. In addition, the pulley piece was designed to have a large spool area to accommodate up to 50 ft worth of standard size string. Small clamps were used to fasten the plastic pulley onto the escapement torque shaft. These clamps fit easily within the round slots of the top and bottom plastic pieces and were able to be adjusted to ensure a no-slip condition between the pulley and the escapement torque bar.

## 6. Experimental results for cylinder escapement and balance wheel clock

The cylinder escapement was run with the pulley device and measured with an electro-optical biaxial follower sensor. Tests were run in intervals of 300 s and data collected at a rate of 1000 Hz. The escapement was not driven with the original weight and was instead driven by two weights with a combined mass of 2.93 kg. The data were processed with a PC-based program called Labview.

A trend was observed in the behaviour of the balance wheel or flywheel position over large time intervals. While the motion looked periodic in real-time observation, further inspection revealed a quasi-periodic trend. Figure 7 shows 200 s of data and contains vertical marker lines in intervals of 12 times the calculated period. The cylinder escapement had 12 teeth (figure 2) that were affecting the maximum amplitudes of the flywheel position. Figure 7 shows that the flywheel position was dependent on the escape wheel location of the cylinder impact.

## 7. Probability density functions

Probability density functions were used to show the distribution of balance wheel periods and oscillation amplitudes. The amplitudes of the peaks of the flywheel position were divided into equally spaced bins ranging from zero to maximum amplitudes. The plot of figure 8 shows the probability that the maximum amplitude in a cycle fell in a particular bin. Closely packed data or plots with sharp spikes indicate that a region is very probable and implies periodic or predictable motion

### (a) Phase plane plots

A common state space is a plot of velocity versus position. This type of plot was used to analyse the behaviour of the flywheel. Figure 9 is a plot of the flywheel phase plane over a 15 s time-span or approximately 10 periods of data.

## 8. Structural vibrations and noise

Experiments were performed to identify the non-uniform behaviour of the cylinder clock escapement. The clock was tested with a constant input of torque provided by the pulley device. Vibration data were collected using an accelerometer and an amplifier. The time history the accelerometer results obtained is shown in figure 10.

When the cylinder escapement operated, every tooth impacted the half-cylinder piece twice before the next tooth struck. The second impact was generally less forceful than the first. Figure 10 shows 4 s worth of data and approximately five impacts; therefore, these data correspond to the motion of three teeth of the 12-toothed escapement. Owing to this behaviour, an analysis was performed on the odd and the even peaks since both occur once per tooth impact.

### (a) Probability density functions

Probability density functions were plotted for four cases: minimum odd and even vibration peaks, and maximum odd and even vibration peaks. Figures 11 and 12 show the PDF plots for maximum even-peak vibration amplitudes. The stepped nature of the plots reflected the analogue to digital device in the Labview software used in the experiments.

The PDF plots of maximum vibration amplitudes are approximately Gaussian in shape. The probability function is spread over a wide range of bins.

The highest probabilities were less than 5 and 3.5% for the odd and the even peaks, respectively, for the structural vibrations. When the same plot was generated for the flywheel position using an equal number of bins, the highest probabilities were closer to 10% for the maximum amplitudes. In addition, the vibration PDF plots contain data in a larger percentage of the bins.

Tables 1 and 2 show the results of PDF quantitative comparisons between the flywheel oscillation amplitudes and the escapement impact vibrations. In all the cases, the PDF plots for the flywheel position yielded higher maximum probabilities, which indicated a larger spike in the data and implied that the flywheel was more predictable. Moreover, the fraction of bins occupied was much larger for all the vibration PDF plots that indicated that the vibration peaks were more random and less predictable.

What is more revealing is a comparison for the relative deviation statistics of the balance wheel period compared to the relative deviation of the vibration noise amplitude. The relative deviation of vibration amplitudes noise is 10–15 times that of the zero-crossing periods for the balance wheel that measures the time (table 3). This means that the *function* of the machine, i.e. measuring time, is more predictable than the amplitude of the internal vibrations created by the energy escapement.

### (b) Return maps

To examine whether there is a relationship between the successive structural vibration peaks, return maps were plotted as in figure 13. To separate the variation of impact on the 12th teeth of the escape wheel, the plots use peak amplitudes *X*_{n} versus *X*_{n+12}. Figure 13 shows a wide scatter compared to a similar plot for the balance wheel amplitude positions (not shown.).

The map of maximum vibration peaks contains no obvious structures. Plotting maps of flywheel position via the same technique revealed definitive correlations between peaks. The maps of minimum peak vibration amplitudes looked very similar to those for the maximum peak amplitudes.

Maps were also constructed in three dimensions by plotting three consecutive peaks. The plot of consecutive maximum vibration peaks is shown in figure 14. When colours are used, they illustrate six subsets of consecutive points. Again, there is no obvious relationship here.

## 9. Experimental results for the three-tooth escapement and pendulum clock

The three-tooth escapement with a pendulum is a more primitive clock (figure 2). The energy of the falling weight is fed into the pendulum with a three-tooth impact on the inside of the oval-shaped ‘gear’ built into the pendulum. Although the design is much different from that of the cylinder escapement, Reuleaux used the same gear train and structure as in the cylinder escapement model. This is convenient for testing our hypothesis about clock noise since the basic structural vibrations of each model are the same, as well as the friction in the gear train. The results of these experiments are similar to those for the cylinder escapement-balance wheel clock.

### (a) Probability density functions for pendulum motion

The PDF measurements for the pendulum again show a narrow spike illustrative of periodic oscillations as shown in figure 15.

### (b) Clock structure vibrations

Within each period of the pendulum oscillation, complex structural vibration was observed. The impact of the three-toothed rod within a four-toothed groove created a complicated relationship between vibration patterns and pendulum movement. Vibration responses are shown in figure 16; they clearly demonstrate the presence of five impacts per period.

### (c) Evaluating period and amplitude data

The vibration response data included five sets of minimum and maximum vibration peaks per period. The structural vibration periods were highly regular, which perhaps reflects the natural frequencies of the system. However, the vibration amplitudes were more unpredictable as seen in the data in tables 4 and 5.

The relative deviations of the escapement structure vibrations were much larger than those of the pendulum position, as illustrated by comparing data in tables 4 and 5, with table 6. The period of the minimum vibrations also had a significantly higher relative deviation than the period of each pendulum position data set. However, as a machine, the significant output is the period of the pendulum whose statistics are shown in table 7. One can see that the noise amplitude variations of the structure are much greater than the variations of the pendulum period by a factor of 10 or more. Thus the clock, as a time-measuring device, can tolerate a great amount of structural noise so long as the time-measuring periodic dynamics is somehow partitioned in the mechanism.

The ratio of relative deviations for structural vibration amplitude over pendulum amplitude ranged from 12.0 to 17.6 for the minimum vibration sets and from 8.4 to 18.0 for the maximum vibration sets. These data indicates that the pendulum system output is 8.4 to 18.0 times more regular than the clock structure vibrations inside the system.

### (d) Return maps for vibration data

Maps of clock structure vibration data were constructed using the time difference between peak vibrations.

The return map in figure 17 illustrates the time relationship between each escapement impact. The seven clusters of points demonstrate seven combinations of time and impact interaction due to the interplay of the three-toothed rod and four-toothed escapement sprocket.

Maps were also made using consecutive structure vibration amplitude peaks, which are shown in figure 18. While the majority of the points are scattered throughout the map, there exist some chains of structure which can be identified by colour sets. The plot containing maximum peaks exhibits a complicated ordering of data as three to four chains of points are present. Analysis of three-dimensional mapping did not reveal any additional information, as the plots appeared randomly scattered.

## 10. Theoretical clock escapement model

### (a) Friction in gear trains

Gear trains are used in clocks to change the very slow motion of the falling weight into motions associated with hours, minutes and seconds. A clock gear train is driven in the opposite direction than a speed reducer. Small friction torques in the pinion require large torques in the large gear. Thus, it is possible for clock gear trains to lock-up. Because of this friction, large weights are sometimes required to maintain reliable running of a clock. For example, an 8-day long-case clock might require a 12–16 lbf weight. A one- to three-month clock might require a weight of 40 lbf (Bruton 1968).

However, it is known that motion in the gears can reduce the friction loss by up to 80%. Evidence for this can be found in the classic book on gears by Buckingham (1949) in which he reported on experiments by the American Society of Mechanical Engineers (ASME) that showed a dramatic drop in friction loss with gear speed (figure 19). With analogy to the Parsons *dither* mechanism to break valve friction in steam turbines (Conway 1953–1955), we postulate that a similar phenomenon occurs due to vibration in the gear transmission system. In the clock, this vibration is self-induced by the ‘tick-tock’ impact of the escapement. From these observations, we posit a mathematical model with coupling between the escapement and the structure to which it is attached.

### (b) Theoretical model assumptions

In an earlier paper (Moon 2003, 2005), a mathematical model for a class of mechanical clock escapements that involve impact was postulated as a generic model designed to capture the essential features for chaos in clocks. The model incorporates the following assumptions and features

the pendulum is modelled by a linear harmonic oscillator with light damping,

the impact dynamics in the escapement and the propagation of structural dynamics through bearings with gaps is modelled by a cubic oscillator of the Duffing-type coupled linearly to the pendulum equation,

the driving gear train torque and static friction lock-up are modelled by a threshold condition of structural impact velocity as measured by the Duffing oscillator, and

the driving torque from the weight-driven gear train, when released by the Duffing oscillator noise, acts to add energy through the escapement pallet when the pendulum velocity is positive.

The first assumption (i) is based on the fact that pendulums in clocks rotate through a very small amplitude such that the nonlinear effects are not important. The linear oscillator assumption is also good for balance wheel clocks. The second assumption (ii) is motivated by research on the propagation of stress waves in structures by groups like Pao *et al*. (1998). Both the experimental and the theoretical research show that a single impact or a step input load on a structure leads to complex wave patterns through reflections and dispersions which excite many modes in the structure. Thus, the escapement impact energy redistribution can propagate into the gear train, break the friction and prevent lock-up.

These assumptions lead to the following equations of motion for the coupled pendulum, structural dynamics and driving train. This fourth-order model employs a vibration-sensitive torque to capture the escapement impact:(10.1a)(10.1b)where

Here *x*_{1}(*t*) represents the motion of the pendulum or balance wheel oscillator; *x*_{3}(*t*) represents the motion of the structural connection between the escapement and the driving train; the cubic term is a nonlinear surrogate for the gaps between bearings and gear teeth. The torque dependence on structural velocity is an attempt to capture the static friction in the drive train and its dependence on the structural vibration.

The frequency *ω*_{1} represents the regular motion of the clock oscillator. The damping constants *β*_{1} and *β*_{2} measure the oscillator and structural dampings, respectively. The escapement torque is only applied when the amplitude is in a given sector of the phase space, e.g. 0<*x*_{1}<*Δ*. Thus, *Δ* should be a small fraction of the limit cycle amplitude. In addition, the noise threshold to release the gears and apply the escapement torque is measured by the constant *δ*.

To ensure that the model does not generate vibrations when the impact torque *tq* is zero, an energy function can be constructed that places restrictions on the constants in the above model. Multiplying the first equation above by *α*_{2}(d*x*_{1}/d*t*), the second equation by *α*_{1}(d*x*_{2}/d*t*) and adding, one comes up with the following energy equation:(10.2)where

Here *T* acts like a kinetic energy function and *V* like the potential energy function.

To ensure energy decay when *tq*=0, both *α*_{1} and *α*_{2} must be positive as well as the nonlinear stiffness constant, *κ*. When the torque is active, the product, *tq*(d*x*_{1}/d*t*), must be also be positive, thus the reason for the sign(d*x*_{1}/d*t*) function in the model.

## 11. Simulation results of clock proto-model

The above coupled oscillator equations of motion were numerically integrated using MATLAB software. A few of the results are shown in figures 20–23. Figure 20 shows a near limit cycle oscillation in the primary clock variable while figure 21 shows a more ‘chaotic-looking’ signal in the coupled structural state variable. A low-dimensional model might neglect the structural ‘noise’, but an analysis of the problem shows that the noise is self-generated and is essential to provide the trigger for the escapement torque in order to drive the nearly periodic oscillator. The trigger in this model is included as a mechanism for the friction-breaking *dither* in real machines.

Typical values for the parameters in equations (10.1*a*) and (10.1*b*), are: *α*_{1}=0.005, *α*_{2}=0.1, *β*_{1}=0.3, *β*_{2}=0.07, *κ*=1.0, *τ*=9, *δ*=0.004, *Δ*=0.2, , . No attempt was made to match the parameters of the experimental clock escapements. While some of these parameters are somewhat arbitrary, others are not. The structural damping *β*_{2}=0.07 is usually small and this is built into this model. In addition, the nonlinear parameter, *κ*=1.0, was chosen to emphasize the nonlinear nature of gaps between the gear teeth. The impact duration parameter, *Δ*=0.2, was chosen to be much less than the pendulum vibration amplitude, which from figure 20 is of the order of 3 units. The torque threshold parameter, *δ*=0.004, corresponds to a structure velocity amplitude of 0.063, which is about 9% of the maximum structural vibration velocity shown in figure 21.

The histogram plot in figure 22 shows the distribution spread of the model structural vibration amplitudes similar to that for the experimental structural vibration in the Reuleaux model X-2, figures 11 and 12.

A Poincaré map (Moon 1992) is used to show the existence of a fractal strange attractor as shown in figure 23 and provides evidence for chaotic vibrations in the clock. In the simulation, the Poincaré map is generated when the clock variable *x*_{1}(*t*) crosses a certain level. The map is plotted in the phase space variables of the structural oscillator, [*x*_{3}, *x*_{4}]. The resulting Poincaré map has a fractal-like structure. This is rather remarkable since the attractor lies in a four-dimensional state space. However, it shows the dynamical decoupling that occurs in this model between the near periodic clock oscillator and its structural linkage.

The structure of the Poincaré map in figure 23 has the appearance of a fractal. Fractal patterns in a fourth-order ordinary differential equation system are difficult to find since the resulting mapping is third order and the projection onto the plane can obliterate visual evidence of the multi-scale structure of the points. The point distribution in figure 23 does lend itself to a description of ‘fractal-like’ and hence the likelihood of the dynamical system having a strange attractor in the fourth-order space.

In these simulations, other sets of values were found to lead to similar fractal structure. In addition, the tolerance in the MATLAB ode45 solver was changed and the same general pattern was exhibited by the structural vibrations.

This model also exhibits ‘friction’ lock-up as in a real escapement, if the value of the torque threshold trigger *δ* is raised high enough. Another parameter that was studied was the torque duration length, *Δ*. We looked for the minimum distance, *Δ*_{min}, that the torque had to operate in order to sustain the pendulum oscillation for different values of the torque. This resulted in a criterion of the form *tqΔ*_{min}=constant. This represents a measure of the work or energy necessary to balance the loss of kinetic energy due to damping in each cycle.

## 12. General discussion

### (a) Historical evidence for irregular dynamics in clocks

Evidence for irregular and unpredictable dynamics of clock motions can be found in the historical literature on clocks. The quotes below are anecdotal and not considered controlled experiments as regards irregularity in clocks. However, they do represent observations of actual complex machines, in this case, mechanical clocks. Lacking controlled experiments, the historical record from the respected clock designers contains data, that, although coarse, are better than no data at all.

The following quotations are taken from the paper by Bloxam (1854), published by the Royal Astronomical Society. This work was cited by the later paper of Lord Grimthorpe (1868). Bloxam had worked on the gravity escapement and the design for the clock in the tower of Westminster (Houses of Parliament), first operation in 1859. Owing to the size of tower clocks and their exposure to the elements, they were less accurate than marine chronometers at the time. In this unique paper, Bloxam blends mathematical analysis with practical experience to address the sources of irregularities in clocks. The mathematics here is essentially sensitivity analysis of the clock period for various physical parameters in the clock. The quotes here are from the more practical observations of the dynamic behaviour of clocks. The paper starts on page 103 and quotations are in page order.

The real question to be investigated is, the extent to which the rate of oscillation of the pendulum is rendered more or less constant by its being connected to the clock, in consequence of the mechanical imperfections of the latter, …

(Bloxam 1854, p. 108)

It is usual with theorists to consider this force [of the impulse which the train transmits from the going weight to the pendulum], as constant during each impulse, but this assumption is too incorrect to be admitted, at least without examination, …

(Bloxam 1854, p. 109)

It is well known that the force transmitted by clock trains is far from constant. Small defects in the forms of the teeth of the wheels, and of the leaves of the pinions, and also in the depths to which they are set into each other, cause considerable irregularity in the force transmitted from each wheel to the next; and the accidental combinations of these irregularities in a train of four or five wheels makes the force transmitted from each to the last exceedingly variable.

(Bloxam 1854, p. 121)

the theory of this escapement [the dead beat escapement], however perfect it may be by itself, must be rendered practically imperfect by the mechanical imperfections which we cannot estimate.

(Bloxam 1854, p. 128)

One of the greatest defects of a pendulum is to

*wabble*, i.e. to oscillate about its own vertical axis at the same time it oscillates in or near the proper vertical plane. If a pendulum does this to a considerable extent, its rate is usually so irregular as to render it useless.(Bloxam 1854, p. 135)

How it happens that disturbing causes which are not very variable in their nature occasion so much

*irregularity* in the rate of the clock is not immediately obvious.(Bloxam 1854, p. 136)

Denison (1868) (Lord Grimthorpe) later, in a book about clocks, also wrote about the unpredictability in clocks especially the gravity escapement.

there is one position of the lever in which it jams against the teeth and stops the clock for good, …. Sometimes too, the click sticks and it sometimes slips, even if made rightly.

(Bloxam 1854, p. 153)

Thus, there has been a lot of anecdotal evidence for unpredictable dynamics in clocks from many clock experts, although the popular image of the clock is of a highly regular machine. However, as machines, clocks are multi-body devices with many nonlinear components and mechanisms, so that from the point of view of nonlinear dynamics, it is remarkable that clocks are as reliable as many became in the eighteenth and nineteenth centuries.

### (b) Comparison of numerical dynamics and experiments in clock escapements

The theoretical model clearly shows that near-periodic dynamics in one state variable and chaotic-looking dynamics in another state variable can coexist. This phenomenon is not that surprising when one looks at the coupled equations of motion in certain limits. By assumption, the coupling of the clock oscillator with the structure is strong but the effect of the structural vibrations on the oscillator is weak. If we assume that the oscillator-state variable dynamics is periodic, i.e. *x*_{1}(*t*)=*x*_{1}(*t*+*T*), and neglect the structural vibration coupling, then the motion is uncoupled and the structure dynamics has the form of a Duffing oscillator and the Poincaré map in figure 23 has the same form as the so-called Japanese strange attractor, studied by Ueda of Kyoto University in the 1970s. (See Moon 1992 for a discussion of the Japanese Attractor.) In fact, some might say that the parameters in the model were chosen to be close to the Japanese chaotic attractor. Indeed, it is that although the clock escapement is a complicated machine, its behaviour can be understood in terms of simpler nonlinear dynamics models. For the physical clock escapement itself, we have constructed a case for our thesis that complex machines may exhibit both periodic and chaotic motions in the same system provided there is unidirectional coupling between sub-systems.

## 13. Conclusions

New experimental evidence has been presented here for coexisting periodic and noisy or chaotic-looking vibrations in autonomous dynamics of clock escapements. Both the balance wheel and pendulum escapements contain common multi-body elements and impact induced structural vibrations. Both the clocks use a gear train that contains friction between the gear teeth and we postulate that this friction is broken by this structure-borne noise. In both the cases, the scatter in structure vibration amplitudes is much greater than that in the time-measuring oscillator period. In addition, there is no experimental evidence for a simple low-order attractor in the noise. We call the noise chaotic-looking because there is no source of random energy input and the structural dynamics does not show evidence of simple structural modes.

We have also presented a clock dynamics model with gear friction released by structural vibration as an example of ‘good chaos’ in machines. Similar models of good chaotic noise in other machine systems like low-level chaos in ball bearings are also possible. A similar example for hydrodynamic bearings has been published by Boedo (1999). Experimental evidence of good chaos in the cutting of metals has been documented by Johnson & Moon (2001). It is our belief that an examination of real machine systems will reveal low levels of self-generated noise that in many cases may have beneficial effects on the dynamic machine performance.

## Footnotes

One contribution of 15 to a Theme Issue ‘Exploiting chaotic properties of dynamical systems for their control’.

- © 2006 The Royal Society