Gear train example
Consider the problem of representing the phases of the Moon on a watch or clock dial. The synodic month, the period between two successive full Moons, has a length of 29.5305888531 days as of 2000 January 1 noon UTC, so this period is what the designer tries to incorporate into the display. The most common version of such a train is a ratchet wheel containing 59 teeth which contains the two-disk lunar phase display, and which is advanced one tooth per day. This mechanism represents the synodic month as having 29.5 days exactly, leading to an error of 1 day in 2.64 years. Another train which could be driven off the hour shaft of a watch or clock represents the synodic month as having 29.555… days, leading to an error of 1 day in 3.24 years. We are not aware of any other practicable, simple train driven by the hour shaft with about this error. In the following, we will represent a gear train using the following convention: gears in mesh are separated by a colon (:), gears on a common shaft are separated by a comma (,). For a lunar display driven off the hour shaft of a watch or clock, which turns once every 12 hours, to drive the conventional two-disk lunar phase display, which should turn once every two synodic months, the "traditional" train would be 6:56,6:76, where the first 6-leaf pinion is on the hour shaft and the 76-tooth wheel is on the shaft with the two-disk display. The required reduction is, of course, 118.12…; this train produces a reduction of 118.22…
IWC, in designing the perpetual calendar mechanism for their Da Vinci watch, devised a far more accurate, and very compact, gear train for driving a conventional two-disk lunar phase display off the day-of-the-week shaft of the calendar. This train, 30:90,32:90, approximates the synodic month as 29.53125 days. The 30-tooth wheel is on the day-of-the-week shaft which is incremented once a day at local midnight by a ratchet, thus turning at 1/7th of a turn per day. The second 90-tooth wheel is on the edge of the lunar display disk, and turns at 1/59.0625th of a turn per day, producing an error of 1 day in 122 years. This accuracy permits the lunar display to be geared to the perpetual calendar without requiring a stud in the case edge for the user to adjust the display; the watch would have to visit the watchmaker every 122 years or so to have the lunar display wheel adjusted by 1 day. (A similar visit is required on March 1 in the century years 2100, 2200, and 2300 because the watch mechanism treats these as leap years, which they are not in the Gregorian calendar now in use; perhaps three of the four required adjustments to the lunar display over the planned 500-year lifetime of the watch could be accomplished at the same time.)
We have discovered 19 independent candidate gear train ratios which can be implemented with the same number of gears IWC used, with similar, feasible tooth counts for each gear, and with accuracies varying from 1 day in about 520 years to 1 day in over 10,000 years. At least one of these should be suitable for the IWC application (after a bit of reengineering of the movement), where it would eliminate any requirement to adjust the Moon phase display over the planned 500 year lifetime of this watch. The process we used to find these train ratios suggests that our list is exhaustive for four-gear trains with tooth counts low enough on all gears to be practicable for this application.
Audemars Piquet uses this same reduction ratio in some of their watches, including the Equation of Time [follow this link by selecting “Equation of time” from the list of watch models], but mechanizes the train differently. They use 14,16:32:135, where the 32-tooth gear is an idler between the first shaft and the 135-tooth wheel on the edge of the two-disk lunar phase display. The 14-tooth gear is a ratchet which is advanced one tooth twice per day, thus turning at 1/7th of a turn per day like the IWC first shaft. Examination of an Equation of Time watch movement showed that there appears to be sufficient offset between the planes of the 16-tooth wheel on the first shaft and the 135-tooth display wheel so that the rather thick 32-tooth idler could be split into two stacked wheels on a common intermediate shaft. The lower wheel on the intermediate shaft would be dirven by the upper wheel on the first shaft, and the upper wheel on the intermdiate shaft would drive the lunar phase display disk.
We have discovered eight independent candidate gear train ratios which can be implemented with the same number of gears Audemars Piguet used, treating the 32-tooth idler they used as a stacked pair, and with similar, feasible tooth counts for each wheel. The accuracies of these candidates varies from 1 day in about 560 years to 1 day in over 14,000 years. Again, at least one of these should be suitable for the Audemars Piguet application (after a bit of reengineering of the movement).
We know of other manufacturers using this same basic reduction ratio, but do not know the details of their mechanizations.
We have also discovered a lunar phase display gear train suitable for use in a watch about the size of the IWC Da Vinci, driven off the hour shaft at two turns per day, with an accuracy of 1 day in about 293 years. There appear to be no more-accurate four-gear trains with tooth counts low enough on all gears to be practicable for this application.
To the level of accuracy achieved by any of the trains discussed so far, the variation in the length of the synodic month over time will have no practical effect on their accuracy over the period 2000 – 2500. Obtaining a synodic month gear train with an accuracy of the order of a second per year, rather than a day per year, requires a longer train with three intermediate shafts (or more). These are probably unsuitable for a watch, but would fit many clock mechanisms. An example of such a train is 30:92,7:97,59:82,15:30. The first 30-tooth wheel is on the hour shaft the second is on the shaft with a standard two-disk lunar phase display. The two 30-tooth wheels could be replaced with a matched pair of any other tooth count, of course. The error in this train is 1 second in 2.82 years at present. Its error averaged over the period 2000 – 2500 is 1 second in 4.54 years when taking into account the small increase in the length of the synodic month over that interval. This train would have an error of under two minutes over the next 500 years, which probably couldn’t be discerned on the display, and so would be suitable to be in mesh with a perpetual calendar mechanism which accounts for the "missing" leap years for century years not divisible by 400 in the Gregorian calendar. We have also found two practicable trains with errors of only 1 second in 17+ years and 1 second in 24+ years at present, respectively. However, over the interval 2000 – 2500 their averaged errors increases to 1 second in 1.93 years and 1 second in 1.62 years, respectively.
[updated 4 February 2003]