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Concept# Pendulum

Summary

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing.
The regular motion of pendulums was used for timekeeping and was the world's most accurate timekeeping technology until the 1930s. The pendulum clock invented by Christiaan Huygens in 1656 became the world's standard timekeeper, used in homes and offices for 270 years, and achieved accuracy of about one second per year before it was superseded as a time standard by th

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Christiaan Huygens

Christiaan Huygens, Lord of Zeelhem, (ˈhaɪɡənz , USˈhɔɪɡənz , ˈkrɪstijaːn ˈɦœyɣə(n)s; also spelled Huyghens; Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, ast

Pendulum clock

A pendulum clock is a clock that uses a pendulum, a swinging weight, as its timekeeping element. The advantage of a pendulum for timekeeping is that it is an approximate harmonic oscillator: It swin

Oscillation

Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar example

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PHYS-101(k): General physics : mechanics

Le but du cours de physique générale est de donner à l'étudiant.e les notions de base nécessaires à la compréhension des phénomènes physiques. L'objectif est atteint lorsque l'étudiant.e est capable de prévoir quantitativement les conséquences de ces phénomènes avec des outils théoriques appropriés.

PHYS-100: Advanced physics I (mechanics)

La Physique Générale I (avancée) couvre la mécanique du point et du solide indéformable. Apprendre la mécanique, c'est apprendre à mettre sous forme mathématique un phénomène physique, en modélisant la situation et appliquant les lois de la physique.

PHYS-101(d): General physics : mechanics

Le but du cours de physique générale est de donner à l'étudiant les notions de base nécessaires à la compréhension des phénomènes physiques. L'objectif est atteint lorsque l'étudiant est capable de prévoir quantitativement les conséquences de ces phénomènes avec des outils théoriques appropriés.

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Many real-world systems are intrinsically nonlinear. This thesis proposes various algorithms for designing control laws for input-affine single-input nonlinear systems. These algorithms, which are based on the concept of quotients used in nonlinear control design, can break down a single-input system into cascade of smaller subsystems of reduced dimension. These subsystems are well defined for feedback-linearizable systems. However, approximations are required to handle non-feedback-linearizable systems. The method proceeds iteratively and consists of two stages. During the forward stage, an equivalence relationship is defined to isolate the states that are not directly affected by the input, which reduces the dimension of the system. The resulting system is an input-affine single-input system controlled by a pseudo-input which represents a degree of freedom in the algorithm. The pseudo-input is a complementary state required to complete the diffeomorphism. This procedure is repeated (n − 1) times to give a one-dimensional system, where n is the dimension of the system. The backward stage begins with the one-dimensional system obtained at the end of the forward stage. It iteratively builds the control law required to stabilize the system. At every iteration, a desired profile of the pseudo-input is computed. In this next iteration, this desired profile is used to define an error that is driven asymptotically to zero using an appropriate control law. The quotient method is implemented through two algorithms, with and without diffeomorphism. The algorithm with diffeomorphism clearly depicts the dimension reduction at every iteration and provides a clear insight into the method. In this algorithm, a diffeomorphism is synthesized in order to obtain the normal form of the input vector field. The pseudo-input is the last coordinate of the new coordinate system. A normal projection is used to reduce the dimension of the system. For the algorithm to proceed without any approximation, it is essential that the last coordinate appears linearly in the projection of the transformed drift vector field. Necessary and sufficient conditions to achieve linearity in the last coordinate are given. Having the pseudo-input appearing linearly enables to represent the projected system as an input-affine system. Hence, the whole procedure can be repeated (n−1) times so as to obtain a one-dimensional system. In the second algorithm, a projection function based on the input vector field is defined that imitates both operators, the push forward operater and the normal projection operator of the previous algorithm. Due to the lack of an actual diffeomorphism, there is no apparent dimension reduction. Moreover, it is not directly possible to separate the drift vector field from the input vector field in the projected system. To overcome this obstacle, a bracket is defined that commutes with the projection function. This bracket provides the input vector field of the projected system. This enables the algorithm to proceed by repeating this procedure (n−1) times. As compared with the algorithm with diffeomorphism, the computational effort is reduced. The mathematical tools required to implement this algorithm are presented. A nice feature of these algorithms is the possibility to use the degrees of freedom to overcome singularities. This characteristic is demonstrated through a field-controlled DC motor. Furthermore, the algorithm also provides a way of approximating a non-feedback-linearizable system by a feedback-linearizable one. This has been demonstrated in the cases of the inverted pendulum and the acrobot. On the other hand, the algorithm without diffeomorphism has been demonstrated on the ball-on-a-wheel system. The quotient method can also be implemented whenever a simulation platform is available, that is when the differential equations for the system are not available in standard form. This is accomplished numerically by computing the required diffeomorphism based on the data available from the simulation platform. Two versions of the numerical algorithm are presented. One version leads to faster computations but uses approximation at various steps. The second version has better accuracy but requires considerably more computational time.

Patrick Robert Flückiger, Simon Nessim Henein, Ilan Vardi

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

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.