scholarly journals The Kolmogorov equation for a plane barrier problem

1981 ◽  
Vol 58 (2) ◽  
pp. 283-283
Author(s):  
E. M. Caba�a
Keyword(s):  
Kolmogorov Equation ◽  
Plane Barrier ◽  
Barrier Problem

Stochastic Modelling of Particle Segregation in a Horizontal Drum Mixer

1992 ◽  
Vol 57 (10) ◽  
pp. 2100-2112 ◽  
Author(s):  
Vladimír Kudrna ◽  
Pavel Hasal ◽  
Andrzej Rochowiecki
Keyword(s):  
Differential Equation ◽  
Diffusion Coefficient ◽  
Stochastic Model ◽  
Diffusion Process ◽  
Stochastic Modelling ◽  
Solid Particles ◽  
Kolmogorov Equation ◽  
Particle Segregation ◽  
Drum Mixer ◽  
Horizontal Drum

A process of segregation of two distinct fractions of solid particles in a rotating horizontal drum mixer was described by stochastic model assuming the segregation to be a diffusion process with varying diffusion coefficient. The model is based on description of motion of particles inside the mixer by means of a stochastic differential equation. Results of stochastic modelling were compared to the solution of the corresponding Kolmogorov equation and to results of earlier carried out experiments.

Reliability of nonlinear stochastic controlled systems considering the dynamics of sensors and actuators

2021 ◽  
pp. 107754632110037
Author(s):  
Sun Jiaojiao ◽  
Xia Lei ◽  
Ying Zuguang ◽  
Huan Ronghua ◽  
Zhu Weiqiu
Keyword(s):  
Closed Loop ◽  
First Passage Time ◽  
Hamiltonian Function ◽  
Small Time ◽  
Kolmogorov Equation ◽  
Sensors And Actuators ◽  
First Passage ◽  
Controlled System ◽  
Main Structure ◽  
Controlled Systems

A closed-loop controlled system usually consists of the main structure, sensors, and actuators. The dynamics of sensors and actuators may influence the motion of the main structure. This article presents an analytical study on the first-passage reliability of a nonlinear stochastic controlled system under the consideration of the dynamics of sensors and actuators. The coupled dynamic equations of the controlled systems with sensors and actuators are first given, which are further integrated into a controlled, randomly excited, dissipated Hamiltonian system. By applying the stochastic averaging method for quasi-Hamiltonian systems, a one-dimensional averaged differential equation for the Hamiltonian function is obtained. The backward Kolmogorov equation associated with the averaged equation is then derived for the first-passage reliability analysis, from which the approximate reliability function and probability density of first-passage time are obtained. The accuracy of the proposed procedure is demonstrated by an example. A comparative analysis of the reliability of the system with/without sensors and actuators is carried out, which indicates that ignoring sensors and actuators will make underestimation of the reliability of the closed-loop system with small time. However, when time increases, there appears the opposite trend. Our findings provide a reference for control strategy design.

Power Spectral Density of Nonlinear System Response: The Recursion Method

1993 ◽  
Vol 60 (2) ◽  
pp. 358-365 ◽  
Author(s):  
R. Vale´ry Roy ◽  
P. D. Spanos
Keyword(s):  
Broad Class ◽  
Formal Solution ◽  
Power Series Expansion ◽  
Kolmogorov Equation ◽  
System Response ◽  
Lanczos Algorithm ◽  
Spectral Densities ◽  
Recursion Method ◽  
Approximate Form ◽  
Single Well

Spectral densities of the response of nonlinear systems to white noise excitation are considered. By using a formal solution of the associated Fokker-Planck-Kolmogorov equation, response spectral densities are represented by formal power series expansion for large frequencies. The coefficients of the series, known as the spectral moments, are determined in terms of first-order response statistics. Alternatively, a J-fraction representation of spectral densities can be achieved by using a generalization of the Lanczos algorithm for matrix tridiagonalization, known as the “recursion method.” Sequences of rational approximations of increasing order are obtained. They are used for numerical calculations regarding the single-well and double-well Duffing oscillators, and Van der Pol type oscillators. Digital simulations demonstrate that the proposed approach can be quite reliable over large variations of the system parameters. Further, it is quite versatile as it can be used for the determination of the spectrum of the response of a broad class of randomly excited nonlinear oscillators, with the sole prerequisite being the availability, in exact or approximate form, of the stationary probability density of the response.

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