182,797 research outputs found
A Fokker-Planck formalism for diffusion with finite increments and absorbing boundaries
Gaussian white noise is frequently used to model fluctuations in physical
systems. In Fokker-Planck theory, this leads to a vanishing probability density
near the absorbing boundary of threshold models. Here we derive the boundary
condition for the stationary density of a first-order stochastic differential
equation for additive finite-grained Poisson noise and show that the response
properties of threshold units are qualitatively altered. Applied to the
integrate-and-fire neuron model, the response turns out to be instantaneous
rather than exhibiting low-pass characteristics, highly non-linear, and
asymmetric for excitation and inhibition. The novel mechanism is exhibited on
the network level and is a generic property of pulse-coupled systems of
threshold units.Comment: Consists of two parts: main article (3 figures) plus supplementary
text (3 extra figures
Linear response theory for quantum Gaussian processes
Fluctuation dissipation theorems connect the linear response of a physical
system to a perturbation to the steady-state correlation functions. Until now,
most of these theorems have been derived for finite-dimensional systems.
However, many relevant physical processes are described by systems of infinite
dimension in the Gaussian regime. In this work, we find a linear response
theory for quantum Gaussian systems subject to time dependent Gaussian
channels. In particular, we establish a fluctuation dissipation theorem for the
covariance matrix that connects its linear response at any time to the steady
state two-time correlations. The theorem covers non-equilibrium scenarios as it
does not require the steady state to be at thermal equilibrium. We further show
how our results simplify the study of Gaussian systems subject to a time
dependent Lindbladian master equation. Finally, we illustrate the usage of our
new scheme through some examples. Due to broad generality of the Gaussian
formalism, we expect our results to find an application in many physical
platforms, such as opto-mechanical systems in the presence of external noise or
driven quantum heat devices.Comment: Close to the published versio
Experiments and numerical results on nonlinear vibrations of an impacting hertzian contact. Part 2: random excitation
Non linear dynamic behaviour of a normally excited preloaded Hertzian contact
(including possible contact losses) is investigated using an experimental test
rig. It consists on a double sphere plane contact loaded by the weight of a
rigid moving mass. Contact vibrations are generated by a external Gaussian
white noise and exhibit vibroimpact responses when the input level is
sufficiently high. Spectral contents and statistics of the stationary
transmitted normal force are analysed. A single-degree-of-freedom non linear
oscillator including loss of contact and Hertzian non linearities is built for
modelling the experimental system. Theoretical responses are obtained by using
the stationary Fokker-Planck equation and also Monte Carlo simulations. When
contact loss occurrence is very occasional, numerical results shown a very good
agreement with experimental ones. When vibroimpacts occur, results remain in
reasonable agreement with experimental ones, that justify the modelling and the
numerical methods described in this paper. The contact loss non linearity
appears to be rather strong compared to the Hertzian non linearity. It actually
induces a large broadening of the spectral contents of the response. This
result is of great importance in noise generation for a lot of systems such as
mechanisms using contacts to transform motions and forces (gears,
ball-bearings, cam systems, to name a few). It is also of great importance for
tribologists preoccupied to prevent surface dammage
Onsagers fluctuation theory and new developments including non-equilibrium Lévy fluctuations
he first part of the paper briefly reviews and explains basic ideas of the theory of Gaussian fluctuations and their relaxation developed in 1931 by Lars Onsager in the context of a general theory of irreversible processes. Motivated by Onsager’s approach, we extend the theory to fluctuations including Lévy processes. We assume that deviations from Gaussian distributions, which are often observed in non-equilibrium systems, may be described by convoluted Gauss–Lévy distributions and their relation to stationary states by generalized Smoluchowski equations. The central part of the distributions we study here is determined by the Gaussian core with the wings decaying according to a power law characteristic for a Lévy-type contribution to statistics. Furthermore, we develop a generalization of Onsager’s theory of linear relaxation processes to those which include statistically independent Gaussian fluctuations and (non-equilibrium) Lévy noises. We apply the generalized version of the fluctuation-dissipation theorem (FDT) to analyze regime of the linear response of the non-equilibrium system driven by Lévy (Cauchy) white noise and subject to thermal (Gaussian) fluctuations. In the last part, applications to non-Maxwellian velocity fluctuations and their relaxations are investigated
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Stochastic response determination and spectral identification of complex dynamic structural systems
Uncertainty propagation in engineering mechanics and dynamics is a highly challenging problem that requires development of analytical/numerical techniques for determining the stochastic response of complex engineering systems. In this regard, although Monte Carlo simulation (MCS) has been the most versatile technique for addressing the above problem, it can become computationally daunting when faced with high-dimensional systems or with computing very low probability events. Thus, there is a demand for pursuing more computationally efficient methodologies. Further, most structural systems are likely to exhibit nonlinear and time-varying behavior when subjected to extreme events such as severe earthquake, wind and sea wave excitations. In such cases, a reliable identification approach is behavior and for assessing its reliability.
Current work addresses two research themes in the field of stochastic engineering dynamics related to the above challenges.
In the first part of the dissertation, the recently developedWiener Path Integral (WPI) technique for determining the joint response probability density function (PDF) of nonlinear systems subject to Gaussian white noise excitation is generalized herein to account for non-white, non-Gaussian, and non-stationary excitation processes. Specifically, modeling the excitation process as the output of a filter equation with Gaussian white noise as its input, it is possible to define an augmented response vector process to be considered in the WPI solution technique. A significant advantage relates to the fact that the technique is still applicable even for arbitrary excitation power spectrum forms. In such cases, it is shown that the use of a filter approximation facilitates the implementation of the WPI technique in a straightforward manner, without compromising its accuracy necessarily. Further, in addition to dynamical systems subject to stochastic excitation, the technique can also account for a special class of engineering mechanics problems where the media properties are modeled as non-Gaussian and non-homogeneous stochastic fields. Several numerical examples pertaining to both single- and multi-degree-of freedom systems are considered, including a marine structural system exposed to flow-induced non-white excitation, as well as a beam with a non-Gaussian and non-homogeneous Young’s modulus. Comparisons with MCS data demonstrate the accuracy of the technique.
In the second part of the dissertation, a novel multiple-input/single-output (MISO) system identification technique is developed for parameter identification of nonlinear time-variant multi-degree-of-freedom oscillators with fractional derivative terms subject to incomplete non-stationary data. The technique utilizes a representation of the nonlinear restoring forces as a set of parallel linear subsystems. In this regard, the oscillator is transformed into an equivalent MISO system in the wavelet domain. Next, a recently developed L1-norm minimization procedure based on compressive sampling theory is applied for determining the wavelet coefficients of the available incomplete non-stationary input-output (excitation-response) data. Finally, these wavelet coefficients are utilized to determine appropriately defined time- and frequency-dependent wavelet based frequency response functions and related oscillator parameters. A nonlinear time-variant system with fractional derivative elements is used as a numerical example to demonstrate the reliability of the technique even in cases of noise corrupted and incomplete data
Linearization techniques for non-linear dynamical systems
This dissertation is concerned with the application of linearization techniques to the study of the response of non-linear dynamical systems subjected to periodic and random excitations.
A general method for generating an approximate solution of a multi-degree-of-freedom non-linear dynamical system is presented. This method relies on solving an optimum equivalent linear substitute of the original system.
The applicability of the method for determination of the amplitudes and phases of the approximate steady-state solution of a multi-degree-of-freedom non-linear system under harmonic monofrequency excitation is considered. The implementation of the method for several special classes of non-linear functions is discussed in detail. In addition, the manner in which the method may be applied to generate an approximate solution for the covariance matrix of the stationary random response of a multi- degree- of freedom dynamical system subjected to stationary Gaussian excitation is outlined.
The potential of the method to treat transient solutions of non-linear systems is indicated in the context of the non-stationary response of a lightly damped and weakly non-linear oscillator subjected to monofrequency harmonic or to a Gaussian white noise disturbance. For both classes of excitation the method produces a first-order differential equation governing the response amplitude. The results pertinent to the harmonically excited oscillator are compared with existing solutions. A non-stationary solution of the Fokker-Planck equation associated with the stochastic differential equation governing the response amplitude of the randomly excited oscillator is accomplished by perturbation techniques; the stationary solution is determined without making any approximation in the Fokker-Planck equation.
The new method for transient response is applied to the random response of a Duffing Oscillator and a Hysteretic System. The solution for the Duffing Oscillator is compared with data obtained by a Monte Carlo study
Fluctuation-Dissipation relations in Driven Granular Gases
We study the dynamics of a 2d driven inelastic gas, by means of Direct
Simulation Monte Carlo (DSMC) techniques, i.e. under the assumption of
Molecular Chaos. Under the effect of a uniform stochastic driving in the form
of a white noise plus a friction term, the gas is kept in a non-equilibrium
Steady State characterized by fractal density correlations and non-Gaussian
distributions of velocities; the mean squared velocity, that is the so-called
{\em granular temperature}, is lower than the bath temperature. We observe that
a modified form of the Kubo relation, which relates the autocorrelation and the
linear response for the dynamics of a system {\em at equilibrium}, still holds
for the off-equilibrium, though stationary, dynamics of the systems under
investigation. Interestingly, the only needed modification to the equilibrium
Kubo relation is the replacement of the equilibrium temperature with an
effective temperature, which results equal to the global granular temperature.
We present two independent numerical experiment, i.e. two different observables
are studied: (a) the staggered density current, whose response to an impulsive
shear is proportional to its autocorrelation in the unperturbed system and (b)
the response of a tracer to a small constant force, switched on at time ,
which is proportional to the mean-square displacement in the unperturbed
system. Both measures confirm the validity of Kubo's formula, provided that the
granular temperature is used as the proportionality factor between response and
autocorrelation, at least for not too large inelasticities.Comment: 11 pages, 7 figures, submitted for publicatio
Exact closed-form fractional spectral moments for linear fractional oscillators excited by a white noise
In the last decades the research community has shown an increasing interest in the engineering applications of fractional calculus, which allows to accurately characterize the static and dynamic behaviour of many complex mechanical
systems, e.g. the non-local or non-viscous constitutive law. In particular, fractional calculus has gained considerable importance in the random vibration analysis of engineering structures provided with viscoelastic damping. In this case, the evaluation of the dynamic response in the frequency domain presents significant advantages, once a probabilistic characterization of the input is provided. On the other hand, closed-form expressions
for the response statistics of dynamical fractional systems are not available even for the simplest cases. Taking advantage of the Residue Theorem, in this paper the exact expressions of the spectral moments of integer and complex orders (i.e. fractional spectral moments) of linear fractional oscillators driven by acceleration time histories obtained as samples of stationary Gaussian white noise processes are determined
Digital implementation and parameter tuning of adaptive nonlinear differential limiters
Master of ScienceDepartment of Electrical and Computer EngineeringAlexei NikitinBalasubramaniam NatarajanIt has been shown that the performance of communications systems can be severely limited by non-Gaussian and impulsive interference from a variety of sources. The non-Gaussian nature of this interference provides an opportunity for its effective mitigation by nonlinear filtering. In this thesis, we describe blind adaptive analog nonlinear filters, referred to as Adaptive Nonlinear Differential Limiters (ANDLs), that are characterized by several methodological distinctions from the existing digital solutions. When ANDLs are incorporated into a communications receiver, these methodological differences can translate into significant practical advantages, improving the receiver performance in the presence of non-Gaussian interference. A Nonlinear Differential Limiter (NDL) is obtained from a linear analog filter by introducing an appropriately chosen feedback-based nonlinearity into the response of the filter, and the degree of nonlinearity is controlled by a single parameter. ANDLs are similarly controlled by a single parameter, and are suitable for improving quality of non-stationary signals under time-varying noise conditions. ANDLs are designed to be fully compatible with existing linear devices and systems (i.e., ANDLs’ behavior is linear in the absence of impulsive interference), and to be used as an enhancement, or as a simple low-cost alternative, to state-of-the-art interference mitigation methods. We provide an introduction to the NDLs and illustrate their potential use for noise mitigation in communications systems. We also develop a digital implementation of an ANDL. This allows for rapid prototyping and performance analysis of various ANDL configurations and use cases
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