Integer Ratio Self-Synchronization in Pairs of Limit Cycle Oscillators

The existence of multiple stable states of higher order m:n locking in coupled limit cycle oscillators has been studied by prior authors in the context of injection-locking in systems driven by an external periodic force. The current work builds on this concept to study the higher order locking characteristics of pairs of limit cycle oscillators self-synchronizing under coupling forces. To this end we analyze three oscillator systems: Van der Pol oscillators using numerical analysis, a simplified model for MEMS oscillators using numerical analysis as well as perturbation theory, and a full model of thermo-optically driven MEMS oscillators using numerical analysis. For the Van der Pol system, higher order locking is observed for the strongly nonlinear case corresponding to relaxation oscillations and the transition from weak to strong nonlinearity is studied using a parameter sweep. Additionally, coupling of a different nature such as quadratic coupling is also capable of inducing higher order coupling in Van der Pol oscillators. For the MEMS systems with linear coupling, higher order locking is observed when a strong cubic stiffness nonlinearity exists. Devil’s staircase-like structures are obtained for the coupling strength-frequency ratio parameter space which suggest overlapping Arnold locking regions for m:n locks corresponding to different integers.

Design of a Flexure Rotational Time Base with Varying Inertia

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 new 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. The models are validated by FEM and mock-ups serve as preliminary proof-of-concept.