Synchronization of two coupled escapement-driven pendulum clocks

The synchronization of two coupled, similarly sized, escapement-driven pendulum clocks is studied. These clocks are coupled by having their extended cases suspended from adjacent stiff beams that can move together horizontally. This setup models the system that Huygens studied in 1665, using clocks that he had designed for determining a ship's longitude. Huygens observed that the two clocks soon ran at a common rate, with the pendulums moving in opposition to each other. A quantitative approximate theory of this synchronization is developed herein. This theory explicitly includes the essential nonlinear elements of Huygens’ system, which are the escapements, as well as the suspended-clock-case and non-identical-clock features of his setup.