Dynamics of the verge and foliot clock escapement

TL;DR

This paper analyzes the dynamics of the verge and foliot clock escapement, deriving analytical expressions for its swing angle, period, and amplitude, and exploring how these depend on driving torque and friction.

Contribution

It provides a detailed analytical model of the verge and foliot escapement, a historically significant clock mechanism, which has been little studied in recent horology research.

Findings

Period scales as P ∝ μ⁻¹ for weak driving

Period scales as P ∝ μ^(-1/3) for strong driving

Analytical expressions for swing angle, period, and amplitude are derived

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 $\mu$).