Abstract
An integrated study is presented on the dynamic modelling and experimental testing of a mid-length Foucault pendulum with the aim of confirming insights from the literature on the reliable operation of this device and setting markers for future research in which the pendulum may be used for the measurement of relativistic effects due to terrestrial gravity. A tractable nonlinear mathematical model is derived for the dynamics of a practical laboratory Foucault pendulum and its performance with and without parametric excitation, and with coupling to long-axis torsion is investigated numerically for different geographical locations. An experimental pendulum is also tested, with and without parametric excitation, and it is shown that the model closely predicts the general precessional performance of the pendulum, for the case of applied parametric excitation of the length, when responding to the Newtonian rotation of the Earth. Many of the principal inherent performance limitations of Foucault pendulums from the literature have been confirmed and a general prescription for design is evolved, placing the beneficial effect of principal parametric resonance of this inherently nonlinear system in a central mitigating position, along with other assistive means of response moderation such as excitational phase control through electromagnetic pushing, enclosure, and the minimization of seismic and EMC noise. It is also shown, through a supporting analysis and calculation, that although the terrestrial measurement of the Lense–Thirring (LT) precession by means of a Foucault pendulum is certainly still within the realms of possibility, there remains a very challenging increase in resolution capability required, in the order of 2 × 109 to be sure of reliable detection, notwithstanding the removal of extraneous motions and interferences. This study sets the scene for a further investigation in the very near future in which these challenges are to be met, so that a new assault can be made on the terrestrial measurement of LT precession.
Highlights
Léon Foucault was a notable French physicist, who, apart from measuring the speed of light and discovering eddy currents during his illustrious career, proposed a striking experiment in 1851 to show visually the rotation of the Earth in a direct manner by means of a carefully suspended long pendulum
In order to derive a mathematical model for the Newtonian dynamics of a terrestrial Foucault pendulum, we start by introducing a fundamental global frame of reference with its origin at the centre of the Earth, EXYZ, and we identify a second frame of reference grounded at the location of the pendulum, defined by pxyz, as shown in figure 1
A mathematical model of the dynamics of the Foucault pendulum has been presented which includes aerodynamic damping with turbulent air flow, parametric excitation of the length, and coupling with pure torsional motion along the long axis
Summary
Léon Foucault was a notable French physicist, who, apart from measuring the speed of light and discovering eddy currents during his illustrious career, proposed a striking experiment in 1851 to show visually the rotation of the Earth in a direct manner by means of a carefully suspended long pendulum. We propose a straightforward but representative mathematical model of the Foucault pendulum including parametric excitation of the length in the form of vertical support motion, and we offer a numerical study of the predicted responses for a number of parameter cases and geographical locations. This model clearly complements those available in the literature and provides a useful level of transparency of derivation, which, we believe, provides an additional aid to understanding. Our goal is to develop the design into a more sensitive, and larger-scale instrument capable of resolving relativistic effects within a terrestrial laboratory in a northerly location
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