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Publication# Foucault pendulum properties of spherical oscillators

Abstract

In 1851 Léon Foucault created a sensation with his pendulum providing a direct demonstration of the turning of the Earth. This simple device consists of a pendulum which is launched in a purely planar orbit. Following Mach's principle of inertia, the mass will continue to oscillate in the same planar orbit with respect to absolute space. For an observer on Earth, however, the plane of oscillation will turn. Conceptually speaking, Foucault constructed a very precise demonstrator showing that, when put on a rotating table, planar oscillations of an isotropic two degree of freedom oscillator remain planar with respect to an inertial frame of reference. These oscillators have currently been under study in order to construct new horological time bases. A novel concept was a spherical isotropic two degree of freedom oscillator. Theoretical computations indicate that when put on a rotating table, planar oscillations of the spherical oscillator neither remain planar in the inertial frame nor in the rotating frame of reference, but in a frame of reference rotating at exactly half the rotational speed of the rotating table. This intriguing result led to the design, construction and experimental validation of a proof of concept demonstrator placed on a motorized rotating table. The demonstrator consists of a spherical isotropic oscillator, a launcher to place the oscillator on planar orbits, a motorized rotating table and a measurement setup. The experimental data recorded by the lasers validates the physical phenomenon.

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Roland Andreas Bitterli, Nicolas Ferrier, Simon Nessim Henein, Mohammad Hussein Kahrobaiyan, Billy Nussbaumer, Lennart Rubbert, Ilan Vardi

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.

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The Foucault pendulum is a well-known mechanism used to demonstrate the rotation of the Earth. It consists in a pendulum launched on linear orbits and, following Mach’s Principle, this line of oscillation will remain fixed with respect to absolute space but appear to slowly precess for a terrestrial observer due to the turning of the Earth. The theoretical proof of this phenomenon uses the fact that, to first approximation, the Foucault pendulum is a harmonic isotropic two degree of freedom (2-DOF) oscillator. Our interest in this mechanism follows from our research on flexure-based implementations of 2-DOF oscillators for their application as time bases for mechanical timekeeping. The concept of the Foucault pendulum therefore applies directly to 2-DOF flexure based harmonic oscillators. In the Foucault pendulum experiment, the rotation of the Earth is not the only source of precession. The unavoidable defects in the isotropy of the pendulum along with its well-known intrinsic isochronism defect induce additional precession which can easily mask the precession due to Earth rotation. These effects become more prominent as the frequency increases, that is, when the length of the pendulum decreases. For this reason, short Foucault pendulums are difficult to implement, museum Foucault pendulum are typically at least 7 meters long. These effects are also present in our flexure based oscillators and reducing these parasitic effects, requires decreasing their frequency. This paper discusses the design and dimensioning of a new flexure based 2-DOF oscillator which can reach low frequencies of the order of 0.1[Hz]. The motion of this oscillator is approximatelyplanar, like the classical Foucault pendulum, and will have the same Foucault precession rate. The construction of a low frequency demonstrator is underway and will be followed by quantitative measurements which will examine both the Foucault effect as well as parasitic precession.

2020Patrick Robert Flückiger, Simon Nessim Henein, Ilan Vardi

The Foucault pendulum provides a demonstration of the turning of the Earth. The principle at work is that linear oscillations of a two-degree-of-freedom isotropic harmonic oscillator remain unchanged in an inertial frame of reference, so appear to precess in a rotating frame of reference. In recent work, we applied two-degree-of-freedom isotropic oscillators to mechanical timekeeping. In this paper, we note that the spherical oscillators we considered have qualitatively different behavior in a non-inertial frame. We show that when in a rotating frame, linear oscillations precess at one half the rotational speed of the rotating frame. We validate this result experimentally by designing and constructing a proof of concept demonstrator placed on a motorized rotating table. The demonstrator consists of a spherical isotropic oscillator, a launcher to place the oscillator on planar orbits, a motorized rotating table, video recording for qualitative observation, and a laser measurement setup for quantitative results. The experimental data recorded by the lasers strongly validate the physical phenomenon.

2020