Dynamics of the verge and foliot clock escapement

The verge and foliot escapement has received relatively little attention in horology, despite the fact that it has been used in clocks for ages. We analyse the operation of a verge and foliot escapement in stationary swing. It is driven by a torque $m=\pm\mu$, switching sign at fixed swing angles $\pm\phi_0$, and $\mu$ is taken to be constant. Friction is assumed to exert a torque proportional to the angular speed. We determine the shape of the swing angle $\varphi(t)$, and compute the period and the swing amplitude of the foliot analytically, as a function of the model parameters. We find that the period of the foliot scales as $P\propto\mu^{-1}$ for weak driving, gradually changing into $P\propto\mu^{-1/3}$ for strong driving (large $\mu$).