Cell Renormalized FPK Equation for Stochastic Non-linear Systems

In this paper, the probabilistic solutions of the multi-degree-of-freedom (MDOF) or large-scale stochastic dynamic systems with polynomial type of nonlinearity and excited by Gaussian white noise excitations are obtained and investigated with the subspace method proposed recently by the authors. The space of the state variables of large-scale nonlinear stochastic dynamic (NSD) system excited by white noises is separated into two subspaces. Both sides of the Fokker–Planck–Kolmogorov (FPK) equation corresponding to the NSD system is then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in another subspace is formulated. Therefore, the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of large-scale NSD systems solvable with the exponential polynomial closure (EPC) method. A simple flexural beam on nonlinear elastic springs is analyzed with the subspace method to show the effectiveness of the subspace-EPC method in this case.

In this paper, the method proposed recently by the author for the solution of probability density function (PDF) of nonlinear stochastic systems is presented in detail and extended for more general problems of stochastic differential equations (SDE), therefore the Fokker-Planck-Kolmogorov (FPK) equation is expressed in general form with no limitation on the degree of nonlinearity of the SDE, the type of δ-correlated excitations, the existence of multiplicative excitations, and the dimension of SDE or FPK equation. Examples are given and numerical results are provided for comparing with known exact solution to show the effectiveness of the method.

Probabilistic solutions of nonlinear oscillators excited by combined colored and white noise excitations

Using reservoir computing to solve FPK equations for stochastic dynamical systems under Gaussian or Non-Gaussian excitation

Random vibration of pretensioned rectangular membrane structures under heavy rainfall excitation

An efficient method for solving high-dimension stationary FPK equation of strongly nonlinear systems under additive and/or multiplicative white noise

Vehicular network is an important application scenario of the fifth generation (5G) mobile communications. Due to the increasing number of vehicles and the users' various requirements, resource allocation problem in vehicular networks becomes more serious and has attracted researchers' attention. In this paper, we investigate the resource allocation in software defined fog vehicular networks where we formulate the problem as a mean-field game (MFG). We present a state space function by considering both interference factors and energy availability. Then the problem can be modeled as a cost minimization with the constraint of state space. Through the mean-field approximation method, we derive the corresponding Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck-Kolmogorov (FPK) equations. A joint finite difference algorithm is proposed to solve the coupled HJB and FPK equations. The numerical results are presented to show the effectiveness of the proposed algorithm.

Stationary response of Duffing oscillator with hardening stiffness and fractional derivative

In the design of gas turbine engines, the analysis of nonlinear vibrations of mistuned and frictionally damped blade-disk assembly subjected to random excitation is highly complex. The transitional probability density function (PDF) for the random response of nonlinear systems under white or coloured noise excitation (delta-correlated) is governed by both the forward Fokker-Planck (FP) and backward Kolmogorov equations. This paper presents important improvement and extensions to a computationally efficient higher order, finite difference (FD) technique for the solution of higher dimensional FP equation corresponding to a two degree of freedom nonlinear system representative of vibration of tip shrouded frictionally damped bladed disk assembly subjected to Gaussian white noise excitation. Effects of friction damping on the mean square response of a blade are investigated. The friction coefficient of the damper is assumed to be a function of the sliding velocity of the contact surface. The effects of stiffness and damping mistuning on the forced response of frictionally damped bladed disk are investigated. Numerical studies are presented for a pair of mistuned blades of cyclic assemblies. The response and reliability of a blade subjected to random excitation is also obtained. With time averaged probability density as an invariant measure, the probability of large excursion in case of damping mistuning is also presented. The results of the FD method are validated by comparing with Monte Carlo Simulation (MCS) results.

Reliability assessment of stochastic dynamical systems using physics informed neural network based PDEM

Modelling of the grain size probability distribution in polycrystalline nanomaterials

Wave propagation in semi-infinite bar with random imperfections of density and elasticity module

We obtain broad sufficient conditions for reconstructing the coefficients of a Kolmogorov operator by means of a solution to the Cauchy problem for the corresponding Fokker–Planck–Kolmogorov equation.

The parameter perturbation makes it difficult to analyze the dynamic mechanism of the brake chatter with the help of an ideal deterministic model. Hence, a stochastic dynamic model of brake chatter with the uncertainty of the torsional stiffness of the brake disc is established, the Stribeck model is used to describe the friction characteristic between the brake pad and brake disc. Then, the Itô stochastic differential equation is solved by means of the stochastic averaging method, the boundary type of the one-dimensional energy process is identified to discuss the stochastic stability of the brake system. Furthermore, the Fokker Planck Kolmogorov equation is derived, the dynamic bifurcation and phenomenological bifurcation are analytically proven, the influences of parameters, such as noise intensity, brake pressure, and friction coefficient, on bifurcation characteristics are revealed. The results show that the noise intensity, brake pressure, and friction coefficient difference need to be reduced to improve the system stability. Finally, the probability density function of the brake system is used to validate the analytical analysis. The relevant results provide a theoretical basis for better suppressing the brake chatter.

The semiclassical approximation and the Fokker–Planck–Kolmogorov equations are used to derive expressions for the density matrix of matter interacting with nonmonochromatic radiation. An allowance is made for correlation between various orders of random processes describing a radiation field.