Design of a Flexure Rotational Time Base With Varying Inertia

Abstract

Flexure oscillators are promising time bases, thanks to their high quality factor and monolithic design compatible with microfabrication. In mechanical watchmaking, they could advantageously replace the traditional balance and hairspring oscillator, leading to improvements in timekeeping accuracy, autonomy, and assembly. As MEMS oscillators, their performance can rival that of the well-established quartz oscillator. However, their inherent nonlinear elastic behavior can introduce a variation of their frequency with amplitude called isochronism defect, a major obstacle to accurate timekeeping in mechanical watches. Previous research has focused on addressing this issue by controlling the elastic properties of flexure oscillators. Yet, these oscillators exhibit other amplitude-related frequency variations caused by changes of inertia with amplitude. In this article, we not only improve existing models by taking into account inertia effects but also present a new way of using them to adjust the isochronism defect. This results in a better understanding of flexure oscillators and an alternative way of tuning isochronism by acting on inertia instead of stiffness. This also opens the door to promising architectures such as the new rotation–dilation coupled oscillator (RDCO) whose symmetry has the advantage of minimizing the influence of linear accelerations on its frequency (the other major limitation of flexure oscillators). We derive analytical models for the isochronism of this oscillator, show a dimensioning with compensating inertia and stiffness variations, and present a practical method for post-fabrication isochronism tuning by displacing masses. The models are validated by finite element method (FEM) and mockups serve as preliminary proof-of-concept.