Theory and design of spherical oscillator mechanisms

In previous work, we showed that two degree of freedom oscillators can be advantageously applied to horological time bases since they can be used to eliminate the escapement mechanism. We subsequently examined planar two degree of freedom oscillators based on parallel flexure stages. We noted that these oscillators are strongly affected by the orientation of gravity so are not directly suitable for portable timekeepers such as wristwatches. In this paper we examine the design and performance of two degree of freedom spherical oscillators. By spherical oscillator, we mean a spherical mass having purely rotational kinematics and subject to elastic restoring torque. As opposed to our previously examined oscillators, the oscillation period of spherical oscillators is relatively insensitive to the effect of tilting the mechanism in the presence of gravity. In order to restrict spherical rotation to two degrees of freedom, we restrict the kinematics to obey Listing's law, a well–known constraint occurring in human eye movement. We show that a particular central restoring force we call the scissors law is best suited for chronometric performance and propose a number of theoretical mechanisms producing it. We then design an actual spherical oscillator based on our theoretical results. The design uses flexure springs to restrict kinematics to Listing's Law, produce the scissors law and provide the necessary suspension. Finally, we present experimental data based on a physical realization indicating promising chronometric performance.