Dynamics of the verge and foliot clock escapement

Abstract: 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 $\mu$).