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Nonlinear Response and Stability Analysis of Vessel Rolling Motion in Random Waves Using Stochastic Dynamical SystemsSu, Zhiyong 2012 August 1900 (has links)
Response and stability of vessel rolling motion with strongly nonlinear softening stiffness will be studied in this dissertation using the methods of stochastic dynamical systems. As one of the most classic stability failure modes of vessel dynamics, large amplitude rolling motion in random beam waves has been studied in the past decades by many different research groups. Due to the strongly nonlinear softening stiffness and the stochastic excitation, there is still no general approach to predict the large amplitude rolling response and capsizing phenomena. We studied the rolling problem respectively using the shaping filter technique, stochastic averaging of the energy envelope and the stochastic Melnikov function. The shaping filter technique introduces some additional Gaussian filter variables to transform Gaussian white noise to colored noise in order to satisfy the Markov properties. In addition, we developed an automatic cumulant neglect tool to predict the response of the high dimensional dynamical system with higher order neglect. However, if the system has any jump phenomena, the cumulant neglect method may fail to predict the true response. The stochastic averaging of the energy envelope and the Melnikov function both have been applied to the rolling problem before, it is our first attempt to apply both approaches to the same vessel and compare their efficiency and capability. The inverse of the mean first passage time based on Markov theory and rate of phase space flux based on the stochastic Melnikov function are defined as two different, but analogous capsizing criteria. The effects of linear and nonlinear damping and wave characteristic frequency are studied to compare these two criteria. Further investigation of the relationship between the Markov and Melnikov based method is needed to explain the difference and similarity between the two capsizing criteria.
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Théorie cinétique et grandes déviations en dynamique des fluides géophysiques / Kinetic theory and large deviations for the dynamics of geophysical flowsTangarife, Tomás 16 November 2015 (has links)
Cette thèse porte sur la dynamique des grandes échelles des écoulements géophysiques turbulents, en particulier sur leur organisation en écoulements parallèles orientés dans la direction est-ouest (jets zonaux). Ces structures ont la particularité d'évoluer sur des périodes beaucoup plus longues que la turbulence qui les entoure. D'autre part, on observe dans certains cas, sur ces échelles de temps longues, des transitions brutales entre différentes configurations des jets zonaux (multistabilité). L'approche proposée dans cette thèse consiste à moyenner l'effet des degrés de liberté turbulents rapides de manière à obtenir une description effective des grandes échelles spatiales de l'écoulement, en utilisant les outils de moyennisation stochastique et la théorie des grandes déviations. Ces outils permettent d'étudier à la fois les attracteurs, les fluctuations typiques et les fluctuations extrêmes de la dynamique des jets. Cela permet d'aller au-delà des approches antérieures, qui ne décrivent que le comportement moyen des jets.Le premier résultat est une équation effective pour la dynamique lente des jets, la validité de cette équation est étudiée d'un point de vue théorique, et les conséquences physiques sont discutées. De manière à décrire la statistique des évènements rares tels que les transitions brutales entre différentes configurations des jets, des outils issus de la théorie des grandes déviations sont employés. Des méthodes originales sont développées pour mettre en œuvre cette théorie, ces méthodes peuvent par exemple être appliquées à des situations de multistabilité. / This thesis deals with the dynamics of geophysical turbulent flows at large scales, more particularly their organization into east-west parallel flows (zonal jets). These structures have the particularity to evolve much slower than the surrounding turbulence. Besides, over long time scales, abrupt transitions between different configurations of zonal jets are observed in some cases (multistability). Our approach consists in averaging the effect of fast turbulent degrees of freedom in order to obtain an effective description of the large scales of the flow, using stochastic averaging and the theory of large deviations. These tools provide theattractors, the typical fluctuations and the large fluctuations of jet dynamics. This allows to go beyond previous studies, which only describe the average jet dynamics. Our first result is an effective equation for the slow dynamics of jets, the validityof this equation is studied from a theoretical point of view, and the physical consequences are discussed. In order to describe the statistics of rare events such as abrupt transitions between different jet configurations, tools from large deviation theory are employed. Original methods are developped in order to implement this theory, those methods can be applied for instance in situations of multistability.
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Peak response of non-linear oscillators under stationary white noiseMuscolino, G., Palmeri, Alessandro January 2007 (has links)
Yes / The use of the Advanced Censored Closure (ACC) technique, recently proposed by the authors for
predicting the peak response of linear structures vibrating under random processes, is extended to
the case of non-linear oscillators driven by stationary white noise. The proposed approach requires
the knowledge of mean upcrossing rate and spectral bandwidth of the response process, which in
this paper are estimated through the Stochastic Averaging method. Numerical applications to
oscillators with non-linear stiffness and damping are included, and the results are compared with
those given by Monte Carlo Simulation and by other approximate formulations available in the literature.
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Fractional Stochastic Dynamics in Structural Stability AnalysisDeng, Jian January 2013 (has links)
The objective of this thesis is to develop a novel methodology of fractional
stochastic dynamics to study stochastic stability of viscoelastic
systems under stochastic loadings.
Numerous structures in civil engineering are driven by dynamic forces, such as
seismic and wind loads, which can be described satisfactorily only by using
probabilistic models, such as white noise processes, real noise processes, or
bounded noise processes. Viscoelastic materials exhibit time-dependent stress
relaxation and creep; it has been shown that fractional calculus provide a
unique and powerful mathematical tool to model such a hereditary property.
Investigation of stochastic stability of viscoelastic systems with fractional
calculus frequently leads to a parametrized family of fractional stochastic
differential equations of motion. Parametric excitation may cause parametric
resonance or instability, which is more dangerous than ordinary resonance as it
is characterized by exponential growth of the response amplitudes even in the
presence of damping.
The Lyapunov exponents and moment Lyapunov exponents provide not only the
information about stability or instability of stochastic systems, but also how
rapidly the response grows or diminishes with time. Lyapunov exponents
characterizes sample stability or instability. However, this sample stability
cannot assure the moment stability. Hence, to obtain a complete picture of the
dynamic stability, it is important to study both the top Lyapunov exponent and
the moment Lyapunov exponent. Unfortunately, it is very difficult to obtain the
accurate values of theses two exponents. One has to resort to numerical and
approximate approaches.
The main contributions of this thesis are: (1) A new numerical simulation
method is proposed to determine moment Lyapunov exponents of fractional
stochastic systems, in which three steps are involved: discretization of
fractional derivatives, numerical solution of the fractional equation, and an
algorithm for calculating Lyapunov exponents from small data sets. (2)
Higher-order stochastic averaging method is developed and applied to
investigate stochastic stability of fractional viscoelastic
single-degree-of-freedom structures under white noise, real noise, or bounded
noise excitation. (3) For two-degree-of-freedom coupled non-gyroscopic and
gyroscopic viscoelastic systems under random excitation, the Stratonovich
equations of motion are set up, and then decoupled into four-dimensional Ito
stochastic differential equations, by making use of the method of stochastic
averaging for the non-viscoelastic terms and the method of Larionov for
viscoelastic terms. An elegant scheme for formulating the eigenvalue problems
is presented by using Khasminskii and Wedig’s mathematical transformations from
the decoupled Ito equations. Moment Lyapunov exponents are approximately
determined by solving the eigenvalue problems through Fourier series expansion.
Stability boundaries, critical excitations, and stability index are obtained.
The effects of various parameters on the stochastic stability of the system are
discussed. Parametric resonances are studied in detail. Approximate analytical
results are confirmed by numerical simulations.
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Nonequilibrium statistical mechanics of a crystal interacting with medium / Mécanique statistique hors d'équilibre d'un cristal interagissant avec un milieu continuDymov, Andrey 17 June 2015 (has links)
Dans cette thèse nous étudions des systèmes hamiltoniens de particules en interaction, où chaque particule est faiblement couplée avec son propre thermostat de type Langevin de température positive arbitraire. Les modèles peuvent être vu comme des cristaux plongés dans un milieu continue et interagissants faiblement avec ce dernier.Nous nous intéressons au transport d'énergie dans les systèmes quand les couplages des particules avec leurs thermostats tendent vers zéro simultanément avec les couplages entre eux.Nous examinons deux situations opposées, quand la mesure de Lebesgue des resonances du système de particules découplées est nulle et quand elle est pleine. Dans le premier cas, en utilisant la méthode de moyennisation stochastique, nous démontrons que dans la limite ci-dessus le comportement de l'énergie locale des particules sur des intervalles de temps longs, et dans le régime stationnaire est donné par une équation autonome stochastique, laquelle predit uniquement le transport d'énergie non hamiltonien.Dans le second cas, en utilisant la méthode de moyennisation resonante stochastique, nous prouvons que la dynamique limite de l'énergie locale est contrôlée par une équation efficace stochastique. La dernière prevoit le transport d'energie hamiltonien entre les particules. Cependant, elle n'est pas autonome pour l'énergie locale. En utilisant cette asymptotique, nous montrons que dans la limite ci-dessus le flux d'énergie hamiltonien du système satisfait des relations qui ressemblent à la loi de Fourier et à la formule de Green-Kubo (cependant, elles ne le sont pas).La plupart des résultats et convergences que nous obtenons dans la thèse sont uniformes par rapport au nombre de particules dans les systèmes, qui rend nos résultats pertinents du point de vue de la physique statistique. / In the present thesis we study Hamiltonian systems of particles with weak nearest-neighbour interaction, where each particle is weakly coupled with its own stochastic Langevin-type thermostat of arbitrary positive temperature.The models can be seen as crystals plugged in some medium and weakly interacting with it.We are interested in the energy transport through the systems when the couplings of the particles with the thermostats go to zero simultaneously with their couplings with each other.We investigate two opposite situations, when resonances of the system of uncoupled particles have Lebesgue measure zero and when they are of full Lebesgue measure.In the first case, using the method of stochastic averaging, we prove that under the limit above behaviour of the local energy of particles on long time intervals and in a stationary regime is given by an autonomous stochastic equation, which does not provide any Hamiltonian energy transport.For the second situation, using the method of resonant stochastic averaging, we show that the limiting dynamics of the local energy is governed by a stochastic effective equation. The latter provides Hamiltonian energy transport between the particles, however, is not an autonomous equation for the local energy. Using this asymptotics, we prove that under the limit above the Hamiltonian energy flow in the system satisfies some relations which resemble the Fourier law and the Green-Kubo formula (however, which are not).Most of results and convergences obtained in the thesis are uniform with respect to the number of particles in the systems, what makes our results relevant from the point of view of statistical physics.
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Optimal Performance-Based Control of Structures against Earthquakes Considering Excitation Stochasticity and System NonlinearityEl Khoury, Omar, Mr. 10 August 2017 (has links)
No description available.
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Stochastic description of rare events for complex dynamics in the Solar System / Modélisation stochastique d'événements rares dans des systèmes dynamiques complexes de notre système solaireWoillez, Éric 21 September 2018 (has links)
Cette thèse considère quatre systèmes physiques complexes pour lesquels il est exceptionnellement possible d’identifier des variables lentes qui contrôlent l'évolution à temps long du système complet. La séparation d'échelle de temps entre ces variables lentes et les autres variables permet d'utiliser la technique de moyennisation stochastique pour obtenir une dynamique effective pour les variables lentes. Cette thèse considère la possibilité de prédire les événements rares dans le système solaire. Nous avons étudié deux types d’événements rares. Le premier est un renversement possible de l'axe de rotation de la Terre en l'absence des effets de marée de la Lune. Le second est la désintégration de l'ensemble du système solaire interne suite à une instabilité dans l'orbite de Mercure. Pour chacun des deux problèmes, il existe des variables lentes non triviales, qui ne sont pas données par des variables physiques naturelles. La moyennisation stochastique a permis de découvrir le mécanisme physique qui conduit à ces événements rares et de donner, par une approche purement théorique, l'ordre de grandeur de la probabilité de ces phénomènes. Nous avons également montré que la déstabilisation de Mercure sur un temps inférieur à l'âge du système solaire obéit à un mécanisme d'instanton bien décrit par la théorie des grandes déviations. Le travail effectué dans cette thèse ouvre donc un nouveau champ d'action pour l'utilisation d'algorithmes de calcul d'événements rares. Nous avons utilisé pour la première fois les théorèmes de moyennisation stochastique dans le cadre de la mécanique céleste pour quantifier l'effet stochastique des astéroïdes sur la trajectoire des planètes. Enfin, une partie du travail porte sur un problème de turbulence géophysique: dans l'atmosphère de Jupiter, on peut observer des structures zonales (jets) à grande échelles évoluant beaucoup plus lentement que les tourbillons environnants. Nous montrons qu'il est pour la première fois possible d'obtenir explicitement le profil de ces jets par moyennisation des degrés de liberté turbulents rapides. / The present thesis describes four complex dynamical systems. In each system, the long-term behavior is controlled by a few number of slow variables that can be clearly identified. We show that in the limit of a large timescale separation between the slow variables and the other variables, stochastic averaging can be performed and leads to an effective dynamics for the set of slow variables. This thesis also deals with rare events predictions in the solar system. We consider two possible rare events. The first one is a very large variation of the spin axis orientation of a Moonless Earth. The second one is the disintegration of the inner solar system because of an instability in Mercury’s orbit. Both systems are controlled by non-trivial slow variables that are not given by simple physical quantities. Stochastic averaging has led to the discovery of the mechanism leading to those rare events and gives theoretical bases to compute the rare events probabilities. We also show that Mercury’s short-term destabilizations (compared to the age of the solar system) follow an instanton mechanism, and can be predicted using large deviation theory. The special algorithms devoted to the computation of rare event probabilities can thus find surprising applications in the field of celestial mechanics. We have used for the first time stochastic averaging in the field of celestial mechanics to give a relevant orders of magnitude for the long-term perturbation of planetary orbits by asteroids. A part of the work is about geophysical fluid mechanics. In Jupiter atmosphere, large scale structures (jets) can be observed, the typical time of evolution of which is much larger than that of the surrounding turbulence. We show for the first time that the mean wind velocity can be obtained explicitly by averaging the fast turbulent degrees of freedom.
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