Impact of reduced drive-weight on a dead-beat escapement
This screen shot shows the impact of going from a weight that is heavy enough to generate over-swing to a weight that is just enough to keep the mechanism ticking. With a 5 pound weight this 30-day mechanism decreased the maximum difference between the time between the tick and tock to around 0.01 seconds. The weight was switched to a 3 pound weight (where shown by the big red arrow on the slide). At that point the over-swing began to decrease, which amplified the difference between the time between the ticks and tocks.
So what you ask? Well, the first big thing is that, with overswing, a pendulum has a lot of excess swing to allow it to power through problems in the time train. When running with no overswing, and right at the very edge of enough weight, any small disturbance can stop the clock – an easy example is if the clock gets shifted just the least little bit when it is being wound.
Overswing, or the amplitude of the pendulums swing, does have an impact on the period of the pendulum. I drew on Wikipedia for an explanation of the impact of amplitude on the period of a pendulum.
The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, called the amplitude. It is independent of the mass of the bob.
For small swings the period of swing is approximately the same for different size swings: that is, the period is independent of amplitude. This property, called isochronism, is the reason pendulums are so useful for timekeeping. Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.
For larger amplitudes, the period increases gradually with amplitude. For example, at an amplitude of 23° it is 1% larger than with a very small amplitude.
Whilst it is possible to calculate a “true period” for a pendulum with small swings, there is an additional “error”, called “Circular Error” that also impacts a pendulum. In the case of a typical grandfather clock whose pendulum has a swing of 6° and thus an amplitude of 3°, the difference between the true period and the small angle approximation amounts to about 15 seconds per day.
Of course, for real pendulums, corrections to the period may be needed to take into account the buoyancy and viscous resistance of the air, the mass of the string or rod, the size and shape of the bob and how it is attached to the string, flexibility and stretching of the string, and motion of the support.
Or, in the real world, one just lets the clock run and adjusts the pendulum to get the clock to keep time – not really caring about the above.