Abstract
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.
Highlights
Unidirectional rotational motion is the key to eliminating the escapement, and this desirable property is achieved by passing from the one degree of freedom oscillators found in classical time keepers: pendulum, balance wheel, tuning fork, to two degree of freedom mechanical oscillators having a central force, that is, a restoring force towards a unique stable position
Since the sphere with fixed center has three degrees of freedom, condition 1 requires a restriction to two degrees of freedom. This is accomplished by specifying that the kinematics of our spherical oscillator respect Listing's law, which is well-known to apply to human eye movement
We show that Listing's law can be realized by a constant velocity joint and our restoring force by a scissors mechanism
Summary
The biggest improvement in timekeeper accuracy was due to the introduction of the oscillator as a time base, first the pendulum by Christiaan Huygens in 1656 [18], the balance wheel-spiral spring by Huygens and Hooke in about 1675, and the tuning fork by N. In [14], we presented new time bases for mechanical timekeepers which, in their simplest form, were based on a harmonic oscillator first described in 1687 by Isaac Newton in Principia Mathematica [25,Book I,Proposition X] This oscillator is the isotropic harmonic oscillator, where a mass m at position r is subject to a central linear restoring force (Hooke's law), and as Newton showed, the resulting trajectories are elliptical (Fig. 2). Unidirectional rotational motion is the key to eliminating the escapement, and this desirable property is achieved by passing from the one degree of freedom oscillators found in classical time keepers: pendulum, balance wheel, tuning fork, to two degree of freedom mechanical oscillators having a central force, that is, a restoring force towards a unique stable position. ☆ Second revision submitted to Precision Engineering, September 20, 2017. ⁎ Corresponding author
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