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.