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This model clearly complements those available in the literature and also provides a useful level of transparency of derivation, which, we believe, provides an additional aid to understanding. Firstly, 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.
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The aim of the work discussed within this paper is twofold. Since its inception, the Foucault pendulum has received a large amount of attention and many of the potential design problems, which are now known to be inherent to this system, have been investigated in detail and a multitude of mitigating solutions have been proposed during more than a century of international research since the time of Foucault. The installation of a Foucault Pendulum requires great care and precision if one wants to observe the real precession generated by the rotation of the Earth beneath the laboratory, without other spurious effects intruding and ultimately dominating the motion of the pendulum. 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. 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. 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 × 10 9 to be sure of reliable detection, notwithstanding the removal of extraneous motions and interferences. 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. 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. 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 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.