The mechanical watch became a vital

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1 Tourbillon Watch Movement Fights Gravity and Wins Almost by David Moline (SC) and John Wagner (SC) The mechanical watch became a vital part of the nineteenth century when train schedules were introduced and needed to be strictly followed to prevent crashes on the tracks. 1 The railroad-grade pocket watch was mandated by the 1893 General Railroad Timepiece Standard for conductors, resulting in booming sales for Ball, Elgin National, Hamilton, Illinois, and Waltham watch companies. Wristwatches evolved into the de facto timepiece in the early 1900s because of their versatility on the World War I battlefield. 2 The underlying manufacturing technology continued to progress and resulted in smaller movements, higher precision, and improved performance. Eventually, fashion trends elevated wristwatches to accessories. The advent of electronics, including quartz watches and mobile phones, has certainly affected the watch industry. Consumers tend to seek highend timepieces for their complex mechanical movements and ensuing status symbols. 3 The tourbillon watch 4 was developed by Abraham-Louis Breguet in 1795 and the watch s escapement escape wheel, pallet fork, balance wheel, and hairspring was placed in a rotating cage. 5 This design attempted to improve timekeeping operation by offsetting the gravitational effects when the movement was placed, or worn, in various positions. 6 Although temperature and friction also influence watch operation, the tourbillon was created to offset the effect of gravity. A brief literature review presents research into escapements, tourbillons, and high-precision timepieces. The research offers insight into the following: Figure 1A. Front view of the Benz & Thomas mechanical watch. Figure 1B. Back view of the Benz & Thomas mechanical watch s movement. Figure 1C. Close-up of the Benz & Thomas mechanical watch s movement that features a tourbillon carousel with fixed brass jewel cup. Operating principles of the escape wheel and pallet fork interaction 7 Mechanical behavior of the prevalent Swiss level escapement found in many watches 8 General framework for escapement analysis 9 Friction and its impact on power consumption in clocks and watches 10 Tourbillon model with a description of the balance wheel s general motion 11 Examination of tourbillon and other watch movements to evaluate the effect of complex motion mechanisms and their impact on performance 12 Double- and triple-axis tourbillons, which rotate about two or three axes once per minute, 13 hour, 14 or both, depending on the given configuration. Two single-axis tourbillon watches that used host pocket watch movements (e.g., ebauche) with added escapement carousels were examined. Figures 1A 314 July August 2017 NAWCC Watch & Clock Bulletin

2 and 1B, respectively, display the front and rear images of the watch designed in 1985 by George Thomas. 15 A close-up of the tourbillon assembly in Figure 1C rotates about the center axis while carrying the oscillating pallet fork, balance wheel, and hairspring. The timekeeping performance of two ebauche tourbillon movements is analyzed in this article and compared with five more common watches operated in six different spatial positions. The remainder of the article discusses the operation of tourbillon watches, including their components and system functionality; introduces a simplified model of a tourbillon watch and describes the dynamic behavior; and analyzes the numerical and experimental results of the movement. Tourbillon Operation A mechanical watch converts the potential energy stored in the spiral mainspring into oscillations of the balance wheel and movement of the hands. 16 Figure 2 shows the three primary subsystems gear train with pinions and arbors, tourbillon, and motion works with the energy flowing from the mainspring to the hairspring. 17 The gear train reduces the mainspring s torque by using a set of wheels or gears while simultaneously increasing the rotational displacement of the 4th wheel that interfaces with the tourbillon structure. The tourbillon contains the escapement, which features an escape wheel-driven pallet fork that impulses the balance wheel. The balance wheel with attached hairspring oscillates, thus establishing the basis for periodic time intervals. The carousel rotates about its own axis while hosting these components. The motion works host the hour, minute, and second hands from various gear ratios, usually driven by the 2nd wheel, to display on the dial. The interfaces between the mechanical parts and the rotating shaft must be clear of debris and lightly lubricated to reduce friction effects. 18 The tourbillon design was proposed to minimize positional errors (e.g., time rate variations due to position and the influence of gravity) by systematically moving the escapement to average out these effects. The operation of a typical Waltham watch movement (Figure 3) is briefly discussed. 19 This pocket watch features a mainspring, a center wheel hosting the Figure 2. Components and energy flow within the tourbillon watch. Figure 3. Typical Waltham 1883 watch movement. Escapement is on left and motion works is on the right. NAWCC Watch & Clock Bulletin July August

3 Figure 4. Schematic diagram of the tourbillon carousel with interface to the 4th wheel. The top view is on the left and the side view is on the right. motion works, a 3rd wheel, and a 4th wheel, which drives the second hand. The escapement, driven by the 4th wheel, consists of an escape wheel, pallet fork, balance wheel, and a hairspring that regulate the passage of time. The motion works for the minute and hour hands encompass the cannon pinion, minute wheel, and hour wheel, all of which interface to the center wheel. In a tourbillon design that uses a retrofit approach to this watch movement, the design varies after the 4th wheel. A compound 5th wheel is introduced to be driven by the 4th wheel and rotates the carousel that hosts the escapement. The escape wheel interfaces with a 6th wheel fixed to the watch frame. The escape wheel travels around the 6th wheel and drives the pallet fork, balance wheel, and hairspring contained in the tourbillon. The tourbillon movement can be considered a planetary gear train. For instance, the carrier is the rotating 5th wheel that meshes with the driven 4th wheel and indirectly moves the escape wheel s lower gear about the circumference of the 6th wheel. The sun is the fixed 6th wheel and the planet is the escape wheel s lower gear; the upper mechanism engages the ring or pallet fork. The rotation of the whole carriage contributes to the action of the escapement system (Figure 4). Simplified Model A simplified balance wheel model to investigate gravitational sensitivity is shown in Figure 5. The n axis system is a global, fixed reference that does not move; gravity acts in the negative n 3 direction. However, the a axis system holds the watch. The model may be rotated by the pitch angle φ from horizontal to study the effect of watch orientation relative to gravity while the balance wheel rotates in the a 1 - a 2 plane. The balance wheel angle θ measures the position of the balance wheel as it oscillates about the a 3 axis (balance pivots) with angular velocity θ. The angle θˈ is the zero position of the balance wheel, that is, the angular position it would stop at if not driven or locked by a pallet and at which the spring torque is then zero. This balance wheel is not perfect; a small imbalance m exists at some distance R from the axis of rotation with angular position α from the zero position of the balance wheel. Using the described physical model, the balance wheel s equation of motion may be written as (I b + mr 2 ) Ӫ + b θ + kθ + (mrg) sin (θ + θ' + α) sin (ϕ) = rf <wheel inertia> <damping> <hairspring> <gravity imbalance> <impulsive force> The parameters b, g, and k denote the viscous damping, gravity, and hairspring stiffness. A discrete impulse force, F, from the pallet fork acts on the balance wheel at a distance, r, from the center of rotation that creates the excitation torque. However, this model can predict the effect of gravity without the damping and impulse force terms; therefore, they will be neglected. Furthermore, the description does not consider the more subtle effects of gear machining tolerances, changes in clearances at the pivot jewels and end jewels, and the associated friction changes at these points with orientation variations. These influence the impulse torque delivered to the balance wheel and induce amplitude fluctuations in the balance wheel motion, which in the presence of a 316 July August 2017 NAWCC Watch & Clock Bulletin

4 nonisochronous 20 balance wheel spring can contribute to positional error. Numerical and Experimental Results Table 1 summarizes the six different positions considered for the numerical simulations and physical testing of seven watches to evaluate the influence of spatial orientation on the mechanism performance. The simplified model presented earlier offers some insight. First, if there is no imbalance (i.e., m=0), all positions have the same equation of motion and will look no different. However, when there is an imbalance, its effect may be maximized when sin (ϕ) = 1 (i.e., ϕ = ±90 ), corresponding to the watch being held in a vertical plane. Finally, the computer simulation cannot tell the difference between θˈ and α, because the equation of motion always uses the sum of these two constant angles (e.g., θ varies with time as the balance wheel oscillates). Henceforth, they will be referred to collectively as the imbalance zero position, or the location of the imbalance when θ in Figure 5 is zero. This will be observed to have an important practical interpretation. With nominal data, 21 computer simulations 22 determined the effect of orientation in the presence of an imbalance on the oscillation period. The parameter values were selected as g=9.81 (m/s 2 ), I b = 3.4 * 10-9 (kg * m 2 ), k = 1.7 * 10-6 ( Nm ), m = 8.5 rad * 10-3 (g), R = 2.0 * 10-7 (m). The nominal period of the balance wheel and the balance wheel spring (e.g., oriented in the horizontal plane) was 0.28 seconds. Figure 6 displays the deviation of the period from this value as the amplitude of oscillation, θ, and imbalance location, θ + α, are varied. The first result to note is that at small amplitudes of oscillation, which do not normally occur in a watch, the imbalance acts like a pendulum and increases or decreases the period according to the position of the imbalance. When located below the axis of rotation, the action of gravity on the imbalance increases the restoring torque and shortens the period. However, as the imbalance zero position angle is increased, the gravity term loses any restoring action, though the term itself is numerically largest as the imbalance zero position approaches horizontal from the axis. The sign of this term changes and the imbalance, in fact, retards the motion when above the axis of rotation, increasing the period. Numerical investigation shows this effect to be linear to greater than 0.2 Figure 5. Watch balance wheel axis systems with an imbalance, m, on the wheel. Figure 6. Deviation of the period of oscillation of the balance wheel as the amplitude of oscillation and zero position of any imbalance is varied. Figure 7. Deviation of an imperfect balance wheel s oscillation period for amplitudes between 0 and 1,000 and an imbalance zero position of 0 (subset of Figure 6). NAWCC Watch & Clock Bulletin July August

5 Table 1. Summary of watch testing positions Position a b c d e f Description Face up Face Down Edge 12:00 top Edge 9:00 top Edge 6:00 top Edge 3:00 top Pitch, Ø 0º 180º 90º 90º 90º 90º Orientation relative to vertical n/a n/a 0º 90º 180º 270º Diagram Note: The use of n/a refers to not applicable. percent for the imbalance mass at even 100 times the imbalance mass values used in this analysis. The second, and perhaps most significant, result to note is that at large amplitudes of oscillation, the net effect of the imbalance on the period can be nearly eliminated. The gravity term is now highly nonlinear, though easily evaluated by simulation. Figure 7 shows the period deviation for an imbalance at zero position of 0 while the amplitude of oscillation is varied. At a zero to peak amplitude of 219, there is no effect on the period. This also happens at amplitudes of oscillation of 402 and 583, but these are not physically possible for mechanically impulsed balance wheels. Note that at an amplitude of 200, the period error in this case is 1 microsecond. Over the course of 24 hours, the accumulated error is nearly 0.3 seconds. It is important to understand the nature of this result and its source. Balance wheels are precisely made and balanced, but in practice they are installed in pivots that have clearances within the pivot bearing jewel (and some imbalance may yet remain). Operating in the vertical plane as discussed, the balance wheel pivots move slightly within the jewel, and the center of pressure and corresponding center of rotation can shift. More importantly, however, the balance wheel spring is changing shape between the two extreme positions of the balance wheel oscillation. It is well known among watchmakers that this changes the center of gravity of the balance wheel spring and thereby acts as the imbalance modeled above. However, it should be noted that the balance wheel spring center of gravity moves throughout the range of motion and is not fixed as modeled. Fortunately, early research showed that with proper attention to the shape of the balance wheel spring, this effect could be minimized. 23 Although careful attention to workmanship may be given, it is possible that if the balance wheel and spring system are not properly balanced when installed, the result may be an imbalance that acts substantially as modeled. Recall from Figure 6 that the effect of the imbalance itself varies with the initial angular position of the imbalance relative to vertical when the watch operates in a vertical plane. Between two extremes imbalance zero position at 0 and 180 an operating point exists in which the effect is zero as evident by the mean period deviation amplitude, regardless of the amplitude of oscillation. This result can be demonstrated when testing a watch. The watchmaker tests the watch in six positions, four of which correspond to running with the balance wheel rotating in a vertical plane. In each of these four positions, the watch is rotated 90 from the previous position (e.g., winding stem at noon, 3, 6, and 9 o clock) and the rate error is noted. As long as the balance spring has been adjusted to minimize the effect of variations in oscillation amplitude, any imbalance is easily noted and corrected by adjusting the screws on the balance wheel rim (see Figure 1C). Discussion so far noted the effect of the imbalance, but it was also mentioned that when the balance wheel is oscillating with an amplitude of about 219, the imbalance has no effect. Naturally, one might 318 July August 2017 NAWCC Watch & Clock Bulletin

6 ask how this operation point is achieved. In practice it is not. However, through the tuning process, the sources of error within the system are adjusted to reduce the variation and approach this point as a mean. For example, spring design seeks to minimize the variation in drive torque as the watch runs down and delivers an input torque that results in oscillations of this amplitude. It is unavoidable without features, such as a fusee, but in a carefully designed automatic winding watch, the watch spring is wound whenever there is sufficient movement; a clutch prevents overwinding. Temperature compensation seeks to minimize variation in the balance spring stiffness and dimensions of the balance wheel. The precision of the gears and wheels in the drive train causes variations in the torque, which affects the amplitude of oscillation. In extreme cases this source of error is easily identified in the beat-to-beat variation of the period of the watch. 24 Meticulous manufacturing also minimizes the effect of changes in time and geometry with orientation and end play in the pivots. Another source of variation in drive torque are changes in the friction at the pivots and wheel-to-wheel contact. Naturally, good lubrication is in order, but the change from operating in a horizontal plane to a vertical one is significant: the pivots change from running against the end jewels to running against the pivot jewels. This is significant enough that watchmakers have developed means to tune the friction against the end jewels to work in their favor. 25 Two tourbillon watches, created by George Thomas, were tested in addition to five other watches. First, a 1985 tourbillon watch, labeled Benz & Thomas, is based on a Waltham movement (see Figure 1) that was modified to host a rotating carousel. Next, a 1983 tourbillon watch, again based on a Waltham Model 83 pocket watch but this time featuring an Omega 450 lever escapement, was evaluated (Figure 8). The third watch tested was a 1943 Hamilton chronometer AF/45-D-1118 size 35 (Figure 9). The testing for these three movements occurred at the NAWCC Library and Research Center in Columbia, PA. The four other watches considered from our collection are a Girard-Perregaux, a 1960s ladies pendant, a modern Pacific Rim manufactured wristwatch, and a 1910 Waltham 37-size movement used in a wall clock (Figures 10 13). Table 1 summarizes the tests of the watches in six different positions (A F) to evaluate the influence of spatial orientation on the accuracy of the mechanism. Specifically, the rotations of each watch are given for the pitch, ϕ, and orientation relative to vertical. An implicit assumption is that this approach approximates the positions a watch experiences when worn by the owner. An electronic Microset Timer and clamping acoustic watch sensor were used to gather the operating data over a 4-minute interval for each position. The measured beat period, over a defined window, established the beats per hour (BPH) to determine whether a given watch was operating fast or slow. The target rate for the two tourbillons, Hamilton chronometer, Girard-Perregaux, and Waltham movements were similar at 18,000 BPH. In contrast, the ladies pendant and modern Pacific Rim movements had a Figure 8. Tourbillon watch movement with design based on Waltham Model 83 pocket watch. Figure 9. Hamilton chronometer AF/45-D-1118 investigated for operating accuracy. Figure 10. Girard-Perregaux S J movement. NAWCC Watch & Clock Bulletin July August

7 higher target rate of 21,600 BPH. By comparing the observed average rates and their standard deviation of the tourbillon watch or watches against the other timepieces, the perceived, actual advantages, or both were readily determined. Table 2 summarizes the average BPH values, rate deviations, and accompanying statistics. Note that the smallest daily error belonged to the Benz & Thomas tourbillon at 4.3 seconds, whereas the largest was the Waltham 37-size movement at seconds. Figures 14A 14D show a representative sample of the tests and offers insight into the timekeeping performance of select movements based on the design of experiments listed in Table 1. The experimentally measured BPH (4-minute interval) are displayed for the Benz & Thomas tourbillon (position A-face up), Hamilton chronometer (position D-edge, 9 o clock top), Girard-Perregaux (position F-edge, 3 o clock top), and Pacific Rim (position E-edge, 6 o clock top) movements. The Benz & Thomas tourbillon behaved quite well in all position settings with the beat rate clustered tightly about 18,000 BPH while displaying small spikes every 60 seconds. In contrast, the Hamilton chronometer showed large beat rate variations each minute, which led to significant timekeeping problems. The Girard-Perregaux watch beat rate fluctuated continually below the ideal beat rate, which indicates a 79.7 seconds per day maximum error. Finally, the Pacific Figure 11. Ladies pendant movement. Figure 12. Pacific Rim manufactured wristwatch. Figure 13. A 1910 Waltham wall clock, 8-day, 37S, 7J movement. Table 2. Average beat-per-hour rate for a 4-minute period in each position Position Benz & Thomas tourbillon Waltham 83 tourbillon Hamilton chronometer Watches Girard- Perregaux Ladies pendant Pacific Rim watch Waltham 37S a 18002, , , , , , ,3.7 b 18003, , , , , , ,6.2 c 18002, , , , , , ,-36.7 d 18002, , , , , , ,3.3 e 18002, , , , , , ,26.7 f 18002, , , , , , ,-3.2 Standard deviation Average rate Maximum daily error (second) July August 2017 NAWCC Watch & Clock Bulletin

8 Figure 14A. Experimental data, beats per hour versus sample number, for Benz & Thomas tourbillon watches, position A. Figure 14B. Experimental data, beats per hour versus sample number, for Hamilton chronometer, position D. NAWCC Watch & Clock Bulletin July August

9 Rim watch illustrates a sinusoidal waveform discrepancy, with a period of 1 minute, about a small bias above the nominal beat rate. Conclusions are reached when holistically considering the numerical and test results in the context of whether the tourbillon watch design offers performance improvements. The computer simulations illustrate the positional error problem that the tourbillon attempts to average out through its rotating carousel-based escapement. Watchmakers have recognized for centuries that the balance wheel imbalance produces an error that must be addressed to improve watch operation. The electronic timing (e.g., beat time measured to within a millionth of a second) of seven watches in six positions shows variations in timekeeping performance. Three types of operational errors can be identified: positional, manufacturing, and mechanism fouling. First, positional errors can be attributed to imbalance as evident in the Waltham 37-size watch in positions c e and to a lesser extent in the Waltham 83 and Girard-Perregaux. Second, manufacturing flaws were observed in the Pacific Rim watch with periodic variations that were a function of the drive train wheel tolerances and an off-center wheel. Similarly, the Girard-Perregaux watch revealed uncertainty issues as evident in a large BPH error band. Third, several watches displayed impulse-type spikes that represented temporary interruptions of the drive torque acting on the carousel or escapement likely due to mechanism microparticle contamination. All the watches were not serviced before testing. The Girard-Perregaux had dirty pallets, which may have contributed to the uncertainty in that watch. Overall, the Benz & Thomas tourbillon had, by far, the best performance of the tested watches with minimal positional errors. Tourbillon watches can indeed improve timekeeping performance by minimizing the positional errors through the integration of innovative features that embody the continual horology contributions. Conclusion The tourbillon watch continues to fascinate and capture the attention of consumers and horologists around the world because of its inherent beauty and complexity. Does the tourbillon movement offer an improvement in timekeeping performance by negating the influence of gravity through the rotating escapement carousel? A simple model and test results show the theoretical benefits of the tourbillon action. Gravity can affect the balance wheel s motion whenever operation occurs in a nonhorizontal plane, causing changes in the period of oscillation. The tourbillon can minimize this effect by systematically moving the escapement through a complete change in orientation so that the error is averaged out. A definitive difference exists with the design of mechanical watches, which were examined as evident in the experimentally observed positional error. However, the advent of miniature electronics has established new timekeeping standards that relegate the tourbillon to primarily a vivid example of humanity s quest for functional artistic mechanical movements. Acknowledgments We thank Library and Archives Supervisor Sara Dockery with the NAWCC Library and Research Center and Museum Director Noel Poirier and Carter Harris with the National Watch and Clock Museum for use of select timepieces at the Columbia, PA, campus. We especially thank George Benz for the insightful discussions about the tourbillon watches he created and their functionality. Finally, we express appreciation to Xin She and Tyler Wagner for the initial research contributions and the graphics support, respectively. About the Authors David Moline is an engineer in the Department of Electrical and Computer Engineering at Clemson University in South Carolina. His horology interests include high-precision timepieces and instrumentation of mechanical systems to investigate their experimental behavior. John Wagner is a professor in the Department of Mechanical Engineering at Clemson University. He fondly remembers the cuckoo and Westminster chime wall clocks that sounded throughout the day and night at his grandmother s home, which kindled his fascination with timepieces. References 1. David Landes. Revolution in Time. Cambridge, MA: Harvard University Press, J. E. Barnett. Time s Pendulum: From Sundials to Atomic Clocks, the Fascinating History of Timekeeping and How Our Discoveries Changed the World. New York, NY: Harcourt Books, July August 2017 NAWCC Watch & Clock Bulletin

10 Figure 14C. Experimental data, beats per hour versus sample number, for GirardPerregaux watches, position F. Figure 14D. Experimental data, beats per hour versus sample number, for Pacific Rim watches, position E. NAWCC Watch & Clock Bulletin July August

11 3. Ariel Adams. Are Elitists the Target Market of High-End Watches? Forbes (July 19, 2012). Accessed December 7, com/sites/arieladams/2012/07/19/are-elitiststhe-target-market-of-high-end-watches/. 4. The tourbillon watch was referred to as the whirlwind watch in France. 5. Brevet Du Sept Messidor An IX. Brequet Depuis Accessed December 16, Mark Denny. The Tourbillon and How It Works. IEEE Control System Magazine, Vol. 30, No. 3 (June 2010): L. C. Tam, Y. Fu, and R. Du. Virtual Library of Mechanical Watch Movements. Computer- Aided Design and Application, Vol. 4, Nos. 1 4 (January 2007): Y. Fu and R. Du. A Study on the Swiss Lever Mechanism. Paper presented at International Mechanism and Machine Science Conference, Yinchuan, China, August 14, M. Kesteven. On the Mathematical Theory of Clock Escapements. American Journal of Physics, Vol. 46, No. 2 (February 1978): S. Patel, David Moline, and John Wagner. Thermodynamic Analysis of an Atmospheric Driven Clock with Mechanical Escapement Controller Theory and Test. Paper presented at European Control Conference, Zurich, Switzerland, July Denny, The Tourbillion and How It Works. 12. G. Xu, P. Ko, and R. Du. A Study on the Precision of Mechanical Watch Movement with Tourbillon. Journal of Sound and Vibration, Vol. 330, No. 155 (August 2011): Patricia Tomes. NAWCC Museum and Library. A Visit with a Craftsman. NAWCC Bulletin, No. 314 (June 1998): Bernstein, Exquisite Coupling. 17. George Daniels. Watchmaking, 2nd ed. London, UK: Philip Wilson Publishers, A. L. Rawlings. The Science of Clocks and Watches, 3rd ed. Upton, UK: British Horological Institute, Henry B. Fried. The Watch Repairer s Manual, 4th ed. Cincinnati, OH: American Watchmakers Institute Press, M. Phillips. Memoire sur le spiral reglant des chronometres et des montres. [Memory on the spiral resolving stop watches and watches]. Annales des mines, Vol. 5, No. 20 (Paris, 1861): L. Xie, P. Ko, and R. Du. The Mechanics of Spiral Springs and Its Application in Timekeeping. Journal of Applied Mechanics, Vol. 81, No. 3 (March 2014): : 1 7. Ruxu Du and Longhan Xie. The Mechanics of Mechanical Watches and Clocks, Vol. 21 of History of Mechanism and Machine Science. Berlin, GER: Springer-Verlag, Simulations were performed using Matlab / Simulink, developed by MathWorks. MathWorks. Accessed February 16, www. mathworks.com. 23. Phillips, Memoire sur le spiral reglant des chronometres et des montres. 24. C. Reymondin, G. Monnier, D. Jeanneret, and U. Pelaratti. The Theory of Horology. Neufchatel, CHE: The Swiss Federation of Technical Colleges, Charles Edgar Fritts. A Practical Treatise on the Balance Spring, Including Making, Fitting, Adjusting to Isochronism and Positions and Rating; Also the Adjustment for Heat and Cold. New York, NY: D. H. Hopkinson Office of The Jewelers Circular, Dennis Bernstein. Exquisite Coupling. IEEE Control System Magazine, Vol. 31, No. 2 (April 2011): Tomes, NAWCC Museum and Library. 324 July August 2017 NAWCC Watch & Clock Bulletin