  We analyze the dynamics of a driven, damped pendulum as used in mechanical
clocks. We derive equations for the amplitude and phase of the oscillation, on
time scales longer than the pendulum period. The equations are first order ODEs
and permit fast simulations of the joint effects of circular and escapement
errors, friction, and other disturbances for long times. The equations contain
two averages of the driving torque over a period, so that the results are not
very sensitive to the fine structure of the driving. We adopt a constant-torque
escapement and study the stationary pendulum rate as a function of driving
torque and friction. We also study the reaction of the pendulum to a sudden
change in the driving torque, and to stationary noisy driving. The equations
for the amplitude and phase are shown to describe the pendulum dynamics quite
well on time scales of one period and longer. Our emphasis is on a clear
exposition of the physics.
