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 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.

Zero Parasitic Shift Pivoting Kinematic Structures Based on Coupled n-RRR Planar Parallel Mechanisms for Flexure Pivot Design

“…In this case and if the flexure are used as time base oscillators, their isochronism defect is therefore minimized [14]. Alternatively, 𝐾 " can be set to a specific value to compensate for externally induced isochronism defects (such as the ones introduced by escapements or if the oscillating balance wheel has a varying inertia) [11,28].…”

Gravity-Compensation Design Approaches for Flexure-Pivot Time Bases

“…The co-RCC (Figure 1a) was designed such that each pair of remote-center-of-compliance (RCC) flexures, whose role is to guide the rotation, is connected in series to parallel flexures whose degree of freedom (DOF) is along the main component of the RCC flexures' center shift, that is, their bisector [6,7,23]. As a result, the center shift is drastically reduced in comparison to a standard RCC pivot.…”

“…Flexure time bases have gained significant traction in the watchmaking industry in the past decade thanks to their high quality factor and monolithic design offering the prospect of increased timekeeping accuracy, increased power reserve, reduced maintenance and simplified assembly [1][2][3][4]. They have been the topic of our previous research [4][5][6][7][8][9][10], and recently gave rise to commercial products [11][12][13]. These mechanisms use the deformation of slender elastic elements, i.e., flexures, to guide their motion [14][15][16].…”

Design of a Triple Crossed Flexure Pivot With Minimized Parasitic Shift

“…Alternatively, the "GIFP" pivot [9,15] allows to optimally distribute the stress in the flexures and significantly reduce the parasitic shift in comparison to the CFP, but at the expense of compactness. The "co-RCC" pivot [16,17] presents an interesting way of reducing the parasitic center shift while having a planar design but the remaining center shift is not negligible [9]. Other solutions exist to tackle the issue of parasitic shift while maximizing angular stroke or having a planar design, but they are either overconstrained [18] or have additional elasticity at one extremity [19].…”

“…The "co-RCC" pivot [16,17] presents an interesting way of reducing the parasitic center shift while having a planar design but the remaining center shift is not negligible [9]. Other solutions exist to tackle the issue of parasitic shift while maximizing angular stroke or having a planar design, but they are either overconstrained [18] or have additional elasticity at one extremity [19].…”