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Concept# Oscillation

Summary

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 examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms.
Oscillations occur not only in mechanical systems but also in dynamic systems in virtually every area of science: for example the beating of the human heart (for circulation), business cycles in economics, predator–prey population cycles in ecology, geothermal geysers in geology, vibration of strings in guitar and other string instruments, periodic firing of nerve cells in the brain, and the periodic swelling of Cepheid variable stars in astronomy. The term vibration is precisely used to describe a mechanical oscillation.
Oscillation, especially rapid oscillation, may be an undesirable phenomenon in process control an

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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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In this thesis we describe a strategy to control robotic knees and ankles. A dynamical system is used to generate a position trajectory to control a servo motor replacing the missing joint. The dynamical system consists in a pool of coupled oscillators modeling a central pattern generator (CPG). As a first step, anthropometric trajectories of the knee and ankle are learned by the system through the convergence of the oscillators to the specific frequencies, corresponding amplitudes and phase relations. The same system is then used to play back these trajectories. As a sensory feedback to trigger the playback we use one adaptive frequency oscillator to synchronized with the acceleration from the thigh. We use a bipedal model in a physics-based robot simulation environment to test the proposed system. Finally we present a simple hardware implementation of this system on the Agonist-Antagonist Active Knee prototype.

2010This thesis was carried out within the framework of a scientific cooperation project entitled “Application of High Power Electromagnetics to Human Safety” developed by the EPFL, the National University of Colombia and Los Andes University, Colombia. The project was funded by the Swiss Agency for Development and Cooperation (SDC) through the EPFL Centre Coopéation & Développement (CODEV). The Scientific Cooperation aimed at the study and development of techniques for the generation of high power electromagnetic signals for the disruption or preemptive activation of Improvised Explosive Devices (IEDs) during humanitarian clearance activities. The results and conclusions of the thesis will be applied to the construction of a resonant radiator, which can be used for securing humanitarian demining operations in Colombia. The thesis is devoted to the analysis of a specific type of resonant radiators known as Switched Oscillators (SWO). An SWO is a radiator constituted by a high voltage charging circuit that drives a quarter-wave transmission line resonator connected to an antenna. An SWO can produce high-amplitude, short duration, electromagnetic fields, with a moderate bandwidth, when compared to the main resonance frequency. The outcome of the thesis can be also be used in electromagnetic compatibility applications, for the production of resonant, high power electromagnetic fields, with the aim of testing the immunity of electronic systems against Intentional Electromagnetic Interference (IEMI) attacks. The thesis is divided in three parts. The first part deals with the electrostatic design of an SWO. A method for producing an optimized design of the electrodes forming the spark gap of the SWO is presented. The method is based on the generation of a curvilinear coordinate space on which the electrodes are conformal to one of the coordinate axis of the space. Laplace equation is solved in the interelectrodic space, obtaining an analytical solution for the electrostatic distribution. Furthermore, using appropriate mathematical manipulations, we derive an analytical expression for the impedance of the transmission line formed by the proposed electrodes. The second part of the thesis is devoted to the analysis of SWOs in the frequency domain. An original analysis approach, based on the chain-parameter technique, is proposed in which the SWO and the connected antenna are described using a two-port network using which a transfer function between the input voltage and the radiated field is established. A closed form expression of the resonance frequency of the SWO is also obtained. The developed technique makes it possible to study the response of an SWO when connected to an arbitrary antenna with a frequency-dependent input impedance. The final part of the thesis presents the construction and test of an SWO prototype. The prototype design is based on the theoretical developments presented in the first two parts of the thesis. The realized SWO is experimentally characterized using different antennas. It is characterized by an input voltage of 30 kV and a resonance frequency of 433 MHz. Radiated electric fields using monopole antennas were in the order of 10 kV/m at a distance of 1.5 m. The prototype is used for testing the validity of the electrodynamic model for the analysis of SWOs connected to frequency dependent antennas. Different monopole antennas connected to the SWO are considered and the radiated fields are measured and compared with theoretical calculations. It is shown that the developed theoretical model is able to reproduce with a good accuracy the behavior of the SWO connected to a frequency dependent antenna.

Coupled dynamical systems are omnipresent in everyday life. In general, interactions between
individual elements composing the system are captured by complex networks. The latter
greatly impact the way coupled systems are functioning and evolving in time. An important
task in such a context, is to identify the most fragile components of a system in a fast and
efficient manner. It is also highly desirable to have bounds on the amplitude and duration
of perturbations that could potentially drive the system through a transition from one equi-
librium to another. A paradigmatic model of coupled dynamical system is that of oscillatory
networks. In these systems, a phenomenon known as synchronization where the individual
elements start to behave coherently may occur if couplings are strong enough. We propose
frameworks to assess vulnerabilities of such synchronous states to external perturbations. We
consider transient excursions for both small-signal response and larger perturbations that can
potentially drive the system out of its initial basin of attraction.
In the first part of this thesis, we investigate the robustness of complex network-coupled
oscillators. We consider transient excursions following external perturbations. For ensemble
averaged perturbations, quite remarkably we find that robustness of a network is given by
a family of network descriptors that we called generalized Kirchhoff indices and which are
defined from extensions of the resistance distance to arbitrary powers of the Laplacian matrix
of the system. These indices allow an efficient and accurate assessment of the overall vulnera-
bility of an oscillatory network and can be used to compare robustness of different networks.
Moreover, a network can be made more robust by minimizing its Kirchhoff indices. Then for
specific local perturbations, we show that local vulnerabilities are captured by generalized
resistance centralities also defined from extensions of the resistance distance. Most fragile
nodes are therefore identified as the least central according to resistance centralities. Based on
the latter, rankings of the nodes from most to least vulnerable can be established. In summary,
we find that both local vulnerabilities and global robustness are accurately evaluated with
resistance centralities and Kirchhoff indices. Moreover, the framework that we define is rather
general and may be useful to analyze other coupled dynamical systems.
In the second part, we focus on the effect of larger perturbations that eventually lead the sys-
tem to an escape from its initial basin of attraction. We consider coupled oscillators subjected
to noise with various amplitudes and correlation in time. To predict desynchronization and
transitions between synchronous states, we propose a simple heuristic criterion based on the
distance between the initial stable fixed point and the closest saddle point. Surprisingly, we
find numerically that our criterion leads to rather accurate estimates for the survival probability and first escape time. Our criterion is general and may be applied to other dynamical
systems.