Circular Error

The Seth Thomas tower clock shows a prominent correlation between the amplitude of swing and the rate. When the amplitude is wider, the clock runs slower. When the amplitude declines, the clock speeds up. The graph below from March of 2011 illustrates this.

The top line (light blue) represents the amplitude of the pendulum. The higher the line, the wider the amplitude. It's actually a measurement of how long the pendulum blocks the optical sensor each time it passes through center. The duration of this time is shown on the right side of the graph (about .023 seconds). The wider it swings, the faster it swings through the center.

The bottom line (brick red) is the rate of the pendulum, near 1.25000 seconds per beat. The lower the line, the slower the clock is.

The reason for the interaction of these two parameters is "circular error". In a pendulum clock, the pendulum swings in a circular arc because the pivot point is fixed. (If you want to get picky, there is debate about whether or not the pivot point moves up and down as the suspension spring flexes. But generally speaking, a pendulum bob swings in a nearly circular arc.)

But a circular arc is not isochronous. This means that, when the arc is wider, the pendulum takes longer to swing from one side to the other. Christiaan Huygens recognized this problem with amplitude in the 17th century, and determined that a pendulum needed to swing in a "cycloidal" arc to be isochronous. Many people have tried to make pendulum clocks that swing in a cycloidal arc, but it ain't easy. Instead, precision clocks are made to swing with small amplitudes. With a small amplitude, the deviation of a circular arc from a cycloidal arc is small, and clock rate is less effected by inevitable changes in amplitude. But as the amplitude increases, the effect on timekeeping increases.

Tower clocks are made to run in harsh environments. The dials are out in the weather with wind, snow, rain, birds, and all manner of disturbances. It helps such a clock to keep going if the amplitude is wide. The Seth Thomas clock in the Santa Barbara Courthouse was made to have a wide amplitude of swing. As we restored it, the amplitude was measured near 7 degrees of semi-arc (14 degrees total swing). There is no adjustment for the amplitude, it is what it is. And our clock appears to have a large amount of circular error .... the rate is less consistent because it is altered by changes in amplitude. The graph above makes this association clear.

As an experiment, I wanted to see if we could make the clock run better by reducing the amplitude. Since there's no adjustment for this, I temporarily hung lead weights on the gravity arms. You might expect heavier gravity arms to increase the impulse and drive the amplitude higher. But the weights have a stronger effect on limiting the overswing of the pendulum and reducing the amplitude. I have now hung a little over 100 grams of lead on each arm and reduced the amplitude only a modest amount, from 7 degrees of semi-arc to 6 degrees. So far, the results are inconclusive and I don't know if we've gained anything or not.


Bryan Mumford
April 16, 2011



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