## Pendulum Error Part II.

Looking at the same one-second pendulum and adding a suspension spring, we get new graphs. The coefficient of elasticity of a spring deflected by bending remains constant, which is more or less true for small angles of 3° or less. In this example, I assume that the initial force applied by the suspension spring is equal to the initial force applied by gravity to bring the pendulum back to a vertical position, and the two forces are added together. Looking at a pendulum with an arc of only 3°, the graph for gravity looks like this one.

When a suspension spring is added in this simulation, the acceleration increases, the velocity remains the same, the angle decreases, and the period decreases.

When a stronger suspension spring is added, the acceleration increases further, the velocity remains the same, the angle decreases further, and the period decreases further. When an escapement is added to keep a pendulum going, as in a clock, the acceleration increases because there are now three forces acting upon the pendulum: the force of gravity, which is a function of sinθ, the elastic force from the suspension spring, which is a function of θ, and the force from the Graham escapement, as the escape wheel's tooth slides across the pallet's impulse face. In this example, the pallet has an arc of 6° and the pendulum has an arc of 3°, so the peak acceleration from the escapement that reaches the pendulum is multiplied by cos2θ, which has a range of 0.997 and 1 when the arc is 6°. The force of the escapement changes direction when the velocity of the pendulum changes direction. Acceleration gets out of control in this simulation.

Therefore, an energy loss (resistance) needs to be introduced into the equation, such that the energy gains from the escapement are equal to the energy losses from air resistance, the bending of the suspension spring, and so on. The result makes for an interesting graph.

When the energy from the escapement is increased, the acceleration, the velocity, and the angle increase, until an angle is reached when the energy losses become equal to the energy gains from the escapement, and an equilibrium is reached.

The amplitudes of the curves are greater at equilibrium. The period remains unchanged.

However, when the energy losses are also increased in proportion to energy gains in this simulation, the amplitudes and the period remain unchanged. From the point of view of clock repair, the simulation offers insight into the forces that act upon a pendulum and affect timekeeping accuracy.
1. If you use a stronger suspension spring, the period will decrease significantly and the clock will gain time. The angle will also decrease, so a stronger mainspring or a heavier weight may be needed to keep the clock running.
2. The simulation suggests that increasing the energy from the escapement would not affect the period. In practice, I have found that using a stronger mainspring or a heavier weight will decrease the period slightly, so the clock will gain time.

I would like to thank Cecil Mulholland (1913-2001), Master Watchmaker, for everything he taught me.

Usus Magister Optimus Est.

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