Number Of Nonlinear Oscillators Research Articles

Efficient path integral approach via analytical asymptotic expansion for nonlinear systems under Gaussian white noise

In this paper an efficient formulation of the Path integral (PI) approach is developed for determining the response probability density functions (PDFs) and first-passage statistics of nonlinear oscillators subject to stationary and time-modulated external Gaussian white noise excitations. Specifically, the evolution of the response PDF is obtained in short time steps, by using a discrete version of the Chapman-Kolmogorov equation and assuming a Gaussian form for the conditional response PDF. Next, the technique involves proceeding to treating the problem via an analytical asymptotic expansion procedure, namely the Laplace’s method of integration. In this manner, the repetitive double integrals involved in the standard implementation of the PI approach are evaluated in a closed form, while the response and first-passage PDFs are obtained by mundane step-by-step application of the derived approximate analytical expression. It is shown that the herein proposed formulation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard PI technique and Monte Carlo solution (MCS) approach. A number of nonlinear oscillators are considered in the numerical examples. Notably, for these systems both response PDFs and first-passage probabilities are presented, whereas comparisons with pertinent MCS data demonstrate the efficiency and accuracy of the technique.

The Glassy State as the Lawfully Disarranged State. Vibration Uncertainty and Chaos-Like Movements

Abstract The effect of a non-linear oscillator's vibrations (either in solids or in liquids) is the primary source of the amorphous state of the matter. The number of nonlinear oscillators (N.Os.) is considered to be very small at Tg (approximately 2.5%), and rises with the increase in temperature, reaching approximately 100% at the critical temperature Tc. In the main transition zone (covering the area between Tg and the crossover temperature Tcr), there are high amplitudes of N.Os. restricted by the particle neighbors and also by the internal energy related to the shear modulus. The shear modulus disappears above the Tcr, where the matter exists either in the form of a liquid or in the form of a rubber-like state (which is not considered in this study). In the present study, the liquid is considered as a heterogeneous structure, formed on the one hand by domains, which are responsible for the linear viscoelastic behavior, and on the other hand with the stand alone N.Os., which act as the trembling motion sources of the optically observed aspect of Brownian motion. Under the Tcr and Tg temperatures, the secondary β and tertiary γ processes are studied as well.

Response and First-Passage Statistics of Nonlinear Oscillators via a Numerical Path Integral Approach

AbstractA numerical path integral solution approach is developed for determining the response and first-passage probability density functions (PDFs) of nonlinear oscillators subject to evolutionary broad-band stochastic excitations. Specifically, based on the concepts of statistical linearization and of stochastic averaging, the system response amplitude is modeled as a one-dimensional Markov diffusion process. Further, using a discrete version of the Chapman-Kolmogorov equation and the associated first-order stochastic differential equation, the response amplitude and first-passage PDFs are derived. The main concept of the approach relates to the evolution of the response PDF in short time steps, assuming a Gaussian form for the conditional response PDF. A number of nonlinear oscillators are considered in the numerical examples section including the versatile Preisach hysteretic oscillator. For this oscillator, first-passage PDFs are derived for the first time to the authors’ best knowledge. Comparisons ...

Mechanism of detachment of particles from the surface of high-temperature superconducting targets absorbing laser pulses

A cluster mechanism is considered for the detachment of particles when the surface of a high-temperature superconducting target absorbs laser pulses. By proposing a fractal nature for the surface, one can model it as an assembly of a large number of nonlinear oscillators (clusters). Utilizing the method of sudden perturbations, a calculation was made of the probability of dissociation of these oscillators, i.e., of the detachment of clusters from the surface. The calculated probability decreases rapidly as the power density of the laser radiation is reduced to some limiting value. This is in agreement with experiments.