Missing gears example

A clock restorer had received a replacement antique movement for a single-hand wall clock, and found that one pinion and one wheel were missing. Further, since the case for this clock covered only the face and works, with the pendulum and drive weight exposed below, there was no guidance for the period of the missing pendulum. The train from the great wheel through the third wheel to the escape wheel was complete. Using the convention that tooth counts for gears in mesh are separated by a colon (:) and gears on the same shaft are separated by a comma (,), the counts for this train were: 78:6,72:6,45. The great wheel has 78 teeth, and the escape wheel has 45 teeth. The pinion on the great wheel shaft was missing, as was the wheel on the hour shaft it was to drive. We were asked to determine the period of the pendulum and the tooth counts for the missing gears. These, of course, are not independent of one another.

The age of the clock and proportions of the works suggested that it once had a seconds pendulum, so that was tried. On that basis, analysis indicated that the missing pinion on the great wheel shaft had 13 leaves, and the missing wheel on the hour shaft 80 teeth. Measurement of the allotted space between the great and hour shafts, considering the tooth spacing of the other gears in the train, confirmed that these were, in fact, the counts of the missing gears. So, the train from the hour shaft up to the escape wheel was, and is: 80:13,78:6,72:6,45. With the release of one tooth per second on the escape wheel, it is easily verified that the hour shaft with the 80-tooth wheel turns at the rate of one turn per 12 hours.