Flexure-Pivot Oscillator Restoring Torque Nonlinearity and Isochronism Defect

Flexure pivot based oscillators can advantageously replace the hairspring and balance wheel, the time base used in mechanical watches, by drastically reducing friction. However, flexure pivots have drawbacks including gravity sensitivity and restoring torque nonlinearity. In previous work, we introduced a novel gravity insensitive flexure pivot (GIFP) to solve the problem of gravity sensitivty, but no analytical formulation for the restoring torque nonlinearity was found. In this paper, we use numerical simulation to find an empirical expression for restoring torque nonlinearity. We use this expression to find an analytical formula for the rotational stiffness of GIFP. This formula gives an explicit relationship between restoring torque nonlinearity and the isochronism of the corresponding harmonic oscillator. The results also apply to the widely used generalized cross-spring pivot

Gravity insensitive flexure pivots for watch oscillators

Classical pivots have frictional losses leading to the limited quality factor of oscillators used as time bases in mechanical watches. Flexure pivots address these issues by greatly reducing friction. However, they have drawbacks such as gravity sensitivity and limited angular stroke. This paper analyses these problems for the cross-spring flexure pivot and presents an improved version addressing these issues. We first show that the cross spring pivot cannot be both insensitive to gravity and have a long stroke. A 10 ppm sensitivity to gravity acceptable for watchmaking applications occurs only when the leaf springs cross at about 87.3 % of their length, but the stroke is only 30.88 % of the stroke of the symmetrical cross-spring pivot. For the symmetrical pivot, gravity sensitivity is of the order of 104 ppm. Our solution is to introduce the co-differential concept which we show to be gravity insensitive. We then use the co-differential to build a gravity insensitive flexure pivot with long stroke. The design consists of a main rigid body, two co-differentials and a torsional beam. We show that our pivot is gravity insensitive and achieves 100 % of the stroke of symmetrical pivots

Gravity-Insensitive Flexure Pivot Oscillators

Classical mechanical watch plain bearing pivots have frictional losses limiting the quality factor of the hairspring-balance wheel oscillator. Replacement by flexure pivots leads to a drastic reduction in friction and an order of magnitude increase in quality factor. However, flexure pivots have drawbacks including gravity sensitivity, nonlinearity, and limited stroke. This paper analyzes these issues in the case of the cross-spring flexure pivot (CSFP) and presents an improved version addressing them. We first show that the cross-spring pivot cannot be simultaneously linear, insensitive to gravity, and have a long stroke: the 10 ppm accuracy required for mechanical watches holds independently of orientation with respect to gravity only when the leaf springs cross at 12.7% of their length. But in this case, the pivot is nonlinear and the stroke is only 30% of the symmetrical (50% crossing) crossspring pivot’s stroke. The symmetrical pivot is also unsatisfactory as its gravity sensitivity is of order 104 ppm. This paper introduces the codifferential concept which we show is gravity-insensitive. It is used to construct a gravity-insensitive flexure pivot (GIFP) consisting of a main rigid body, two codifferentials, and a torsional beam. We show that this novel pivot achieves linearity or the maximum stroke of symmetrical pivots while retaining gravity insensitivity

Investigation of Size-Dependency in Free-Vibration of Micro-Resonators Based on the Strain Gradient Theory

Abstract This paper investigates the vibration behavior of micro-resonators based on the strain gradient theory, a non-classical continuum theory capable of capturing the size effect appearing in micro-scale structures. The micro-resonator is modeled as a clamped-clamped micro-beam with an attached mass subjected to an axial force. The governing equations of motion and both classical and non-classical sets of boundary conditions are developed based on the strain gradient theory. The normalized natural frequency of the micro-resonator is evaluated and the influences of various parameters are assessed. In addition, the current results are compared to those of the classical and modified couple stress continuum theories

Mechanical oscillator

wherein k1, k3, k5 . . . are constants and θ is an angle of inclination of said third axis (x, P) of said inertial body with respect to a direction of said third axis (xr) when said inertial body is in said resting position

Flexure Pivot Oscillator Insensitive to Gravity

The mechanical oscillator according to the invention comprises an oscillating body (601), at least one rigid intermediate body (602) and a support (600). Each rigid intermediate body is connected to the support by a pair of elements (610, 611) providing rotational guidance. The elements of each pair are elastically substantially identical to each other and extend along respective axes which, in orthogonal projection onto a plane parallel to the oscillation plane of the oscillating body, cross at a point (G) and are symmetric to each other with respect to a line (x) passing between the points of junction of the first pair of elements to the rigid intermediate body. The rigid intermediate body is connected to the oscillating body by at least one further element (604, 605) providing relative guided mobility between the oscillating body and the rigid intermediate body in a direction substantially parallel to the line (x) during regular functioning of the mechanical oscillator. In a variant the pair of elements connect the rigid intermediate body to the oscillating body and the at least one first further element connects the rigid intermediate body to the support