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About Pendulum Clocks |
A mechanical pendulum clock is a device that counts the number of times a pendulum has swung.
It must do these things:
Because a pendulum clock counts whole numbers of swings, it is actually a digital device. The face plate may look like an analog display, but in fact it is very digital: the gears leave no doubt that the hands will be in exactly the same position again after an exact and constant number of pendulum swings. One revolution of the hour hand does not correspond to an approximate number of swings, it corresponds to a fixced number which can be derived from the numbers of the teeth of the gears.
Thus, all mechanical clocks and watches are in fact digital. Analog clocks would be water clocks, hourglasses and the like.
A pendulum swings at a steady rate. If it consists of a concentrated mass at the end of a thin rod of length l then the swing time is given by
where g is the acceleration of gravity, normally equal to 9.8m/s2 at an intermediate latitude. The period does not depend on the size of the mass, only on the length. A pendulum that ticks out one second should be of length close to 25cm (one to-and-fro in one second). To compensate for expansion by temperature, real precision pendulums are made of an assembly of rods of different metals combined such that the different expansion rates lead to no increase in length.
Our clock's pendulum is more modest in that it only has an adjustable length so we can tune it to beat a second with some moderate precision.
This is no doubt the most difficult part of the clock.
First, the escape mechanism must give a slight push to the pendulum at each tick, so it compensates for loss of swing through friction.
Second, it must not have a position such that the gear counting the ticks can rotate unimpeded: the gear's teeth must pass one by one.
Third, it must not introduce much friction because it would wear out to fast.
Fourth, the whole must be driven by the gear.
It's quite an accomplishment of the early clock makers that they invented a mechanism that satisfies all these requirements. A tremendous amount of thinking must have gone into the escape mechanism and an even greater amount of experimentation to find a good configuration that would perform well and stand up to wear.
I used a large wheel with six pegs to make the "gear" and then found that the swing needed by the escape hand was too large for the pendulum. This explains why the pendulum drives the escape hand through a rod: the rod can be fixed to the pendulum lower than to the escape hand, thus amplifying the motion.
Once the pendulum is tuned to tick out a second, we know the gear goes round once every six seconds and we have to reduce this to an axle for the minute hand that goes around once every 3600 seconds (one hour), or 600 times slower. 600 = 2 x 2 x 2 x 3 x 5 x 5 and fortunately Lego has these gear ratios. In actual fact I used a different set for other reasons.
Here we face two problems: one is to reduce the minute hand rotation by a factor 12, the other is using a gear set that ends up on the same axle. This was somewhat tricky to build in a compact fashion, especially as I did not have the famous floating gear.
When pendulum clocks were the most precise timekeepers we had, it was very important not to stop timekeeping while winding up or pulling the weights up. Losing a few seconds during the winding up operation was (and is) not important in a home setting, but for astronomical purposes or for measuring longitude at sea this would have been unacceptable. I used a differential gear to combine the motions of the weight and the winding motor in such a fashion that the escape gear is driven while the motor winds the weight back up. I found no Lego clock on the web that had this characteristic.
To get an idea of how important this feature may be, consider this: the earth turns once in 24 hours. At the equator, the earth's circumference is 40'072km (by definition it is 40'000km along a meridian, the extra 72km along the equator are from the slight bulging caused by the rotation). There are 24x60x60=86'400 seconds in a day, hence one second represents 40'072/86'400 km = 0.46km. About half a kilometer. If you go on a trip around the world for several months by sailing ship and want to keep track of your longitude, then your ship's chronometer should not be off by too many seconds! A minute results in an error of almost 30km. Quite clearly you cannot afford to loose ten seconds each day when you wind up your clock.
Finally, once in a while you may need to set the hands, e.g. when changing from summer to winter time. You must be able to disconnect the hands from the rest of the gearing to do this. The hands must remain in their 1/12 gearing though, so this type of "clutch" must be between the main reduction gear and the hands gear. I used a simple 1:1 gear set on a pivoting beam to get this function.
