A study on the precision of mechanical watch movement with Tourbillon

Despite its long history, detailed studies on the dynamics of mechanical watch movement are still lacking. In this paper, extending our previous mechanics model, the dynamical wave motion of the hairspring—balance wheel assembly, the core of the timekeeping mechanism called the escapement, is analyzed by partial differential equations. It is found that the corresponding eigenfunctions may be formulated into a Fourier series, through which the relative amplitudes of high order harmonics can be obtained. The relative error incurred owing to the high order harmonics is estimated to be around 10⁻³, or 10² s/day, much higher than the needed accuracy. This is why manual adjustment is needed in practice. Moreover, addition of the pricy Tourbillon mechanism, which is designed to counteract the gravity effect, is found to trigger complexity in the motion and hence, may not be worth much.