Parametric Gaussian White Noise Research Articles

Feedback Stabilization of Chain-Like MDOF Nonlinear Structural Systems Under Purely Parametric Gaussian White Noises

A feedback control method is proposed to asymptotically stabilize the chain-like multi-degree-of-freedom (MDOF) nonlinear structural system under purely parametric Gaussian white noises. First, the motion equation of a chain-like nonlinear structural system with [Formula: see text] coupled elements and an undetermined control law is derived. It is reduced to a one-dimensional partially averaged Itô stochastic differential equation of the controlled Hamiltonian by applying the stochastic averaging method, which bypasses the challenge of calculating multiple Lyapunov exponents. Next, based on the dynamical programming principle, the dynamical programming equation of the averaged system with an undetermined cost function is established and addressed to obtain the optimal control law. Then, the Lyapunov exponent of the completely averaged Itô equation is derived. For the given requirement of stabilizing the system, the Lyapunov exponent is entirely fixed and used to analyze the stability of the originally chain-like MDOF nonlinear structural system. A controlled chain-like 8-DOF nonlinear structural system is taken as an example to illustrate the application and effectiveness of the proposed procedure and the effect of control forces on stabilizing the system.

Transient probabilistic analysis of nonlinear systems excited by correlated external and parametric Gaussian white noise

This paper explores using the exponential-polynomial-closure (EPC) method to analyze the transient probabilistic solutions to nonlinear stochastic oscillators excited by correlated external and parametric Gaussian white noise. The probabilistic solution to the transient responses of the nonlinear oscillator is governed by the Fokker–Planck–Kolmogorov (FPK) equation, in which the solution is denoted as an exponentially polynomial function with time-variant coefficients. These coefficients can be solved numerically by introducing a proper weighted function and making the projection of the residual error vanish. Meanwhile, three numerical examples are presented to verify the effectiveness and efficiency of the EPC method for analyzing the nonlinear oscillators through the comparisons with the Monte Carlo simulation (MCS) method. Moreover, the influence of the correlation coefficient between the excitations of external and parametric Gaussian white noise is investigated.

Transient influence of correlation between excitations on system responses

In this paper, transient influence of correlation between external and parametric excitations on system response is investigated. Time variable has been taken into account. This does not seem to have been studied previously. By describing correlation between excitations quantitively with correlated coefficient, general Fokker-Planck-Kolmogorov (FPK) equation is reformulated with two parts. The first part relates to the one of independent excitations, while the other part is caused by the correlated excitations. Then, exponential polynomial closure (EPC) method, by taking time variable into account, is further developed for transient responses of nonlinear oscillators under correlated excitations. With the improved EPC method, typical system of Duffing oscillator under correlated external and parametric Gaussian white noise excitations is investigated. Based on the results, it is found the influence of correlation between excitations on system response depends directly on the magnitude and the sign of the correlation coefficient. Besides, the influence seems stationary, and not affected by the time. In addition, the results obtained from the EQL method are not consistent with the actual ones with unsymmetrical PDFs and nonzero means. It is indicated that the EQL method is not applicable to such case.

Exponential Mean‐Square Stability of Stochastic String Hybrid Systems Under Continuous Non‐Gaussian Excitation

AbstractThe problem of the exponential mean‐square stability for nonlinear stochastic string hybrid system under parametric (multiplicative) Gaussian and external (additive) continuous non‐Gaussian excitations is considered. The string hybrid system is treated as an infinite‐dimensional family of strings (subsystems) with a switching rule that has the form of a right continuous Markov chain. It is described by infinite‐dimension Ito stochastic differential equations. The excitations are assumed to be parametric Gaussian white noises and the continuous non‐Gaussian processes modeled by polynomials of a Gaussian process. The nonlinear strings under external continuous non‐Gaussian excitation are transformed to extended dimensional nonlinear strings with a special structure under a parametric Gaussian excitation. Using the methodology of the stability analysis of nonlinear hybrid systems with Markovian switchings the sufficient conditions of the exponential mean‐square stability of nonlinear stochastic string systems under parametric Gaussian and external continuous non‐Gaussian excitation with a Markovian switching are derived. The detailed calculations are given for linear systems.

Stationary Response Probability Distribution of SDOF Nonlinear Stochastic Systems

There has been no significant progress in developing new techniques for obtaining exact stationary probability density functions (PDFs) of nonlinear stochastic systems since the development of the method of generalized probability potential in 1990s. In this paper, a general technique is proposed for constructing approximate stationary PDF solutions of single degree of freedom (SDOF) nonlinear systems under external and parametric Gaussian white noise excitations. This technique consists of two novel components. The first one is the introduction of new trial solutions for the reduced Fokker–Planck–Kolmogorov (FPK) equation. The second one is the iterative method of weighted residuals to determine the unknown parameters in the trial solution. Numerical results of four challenging examples show that the proposed technique will converge to the exact solutions if they exist, or a highly accurate solution with a relatively low computational effort. Furthermore, the proposed technique can be extended to multi degree of freedom (MDOF) systems.

The closed-form solution of the reduced Fokker–Planck–Kolmogorov equation for nonlinear systems

In this paper, we propose a new method to obtain the closed-form solution of the reduced Fokker–Planck–Kolmogorov equation for single degree of freedom nonlinear systems under external and parametric Gaussian white noise excitations. The assumed stationary probability density function consists of an exponential polynomial with a logarithmic term to account for parametric excitations. The undetermined coefficients in the assumed solution are computed with the help of an iterative method of weighted residue. We have found that the iterative process generates a sequence of solutions that converge to the exact solutions if they exist. Three examples with known exact steady-state probability density functions are used to demonstrate the convergence of the proposed method.

Response analysis for a vibroimpact Duffing system with bilateral barriers under external and parametric Gaussian white noises

In this paper, a vibroimpact Duffing oscillator with two barriers that are symmetrical with respect to the equilibrium point of the system is considered for the cases of external and parametric Gaussian white noise random excitations. According to the levels of the system energy, the motions of the unperturbed vibroimpact system are divided into two types: oscillations without impacts and oscillations with alternate impacts on both sides. Then, under the assumption that the vibroimpact Duffing system is quasi-conservative, the stochastic averaging method for energy envelope is applied to obtain the averaged drift and diffusion coefficients for the two types of motions, respectively. The Probability Density Functions (PDFs) of stationary responses are derived by solving the corresponding Fokker-Plank-Kolmogorov (FPK) equation. Lastly, results obtained from the proposed procedure are validated by directly numerical simulation. Meanwhile, effects of the position of bilateral barriers and the random excitations on the PDFs of the stationary responses are also discussed.

Stability of a delayed predator—prey model in a random environment**Project supported by the National Natural Science Foundation of China (Grant Nos. 11272051 and 11302172).

The stability of the first-order and second-order solution moments for a Harrison-type predator–prey model with parametric Gaussian white noise is analyzed in this paper. The moment equations of the system solution are obtained under Itô interpretations. The delay-independent stable condition of the first-order moment is identical to that of the deterministic delayed system, and the delay-independent stable condition of the second-order moment depends on the noise intensities. The corresponding critical time delays are determined once the stabilities of moments lose. Further, when the time delays are greater than the critical time delays, the system solution becomes unstable with the increase of noise intensities. Finally, some numerical simulations are given to verify the theoretical results.

Response analysis of Rayleigh–Van der Pol vibroimpact system with inelastic impact under two parametric white-noise excitations

In the present paper, the random vibration problem of Rayleigh–Van der Pol oscillator is considered by means of a single-degree-of-freedom system with a unilateral rigid barrier at its equilibrium position. The random excitations are two parametric Gaussian white noises. The nonsmooth transformation method is applied to convert the vibroimpact system to an equivalent system without velocity jump. The modified stochastic averaging technique for energy envelope is used to deal with the transformed system, and the averaging Ito stochastic differential equation is obtained. The analytical solution of the response of the original vibroimpact system is derived by solving the corresponding Fokker–Planck equation and using the inverse transformation of nonsmooth transformation. The validity of the analytical results is verified by those from Monte Carlo simulations based on original system. Effects of different system parameters and parametric white noises on the response of the system are examined. In addition, the critical condition of stochastic bifurcations is also explored.

Delay-independent stability of moments of a linear oscillator with delayed state feedback and parametric white noise

The stability of a linear oscillator with delayed state feedback driven by parametric Gaussian white noise is studied in this paper. The first and second order moment equations of the system response are derived by using moment method and Itô differential rule. Based on the moment equations, the delay-independent stable conditions of both moments are proposed: For the first order moment, the sufficient and necessary condition that guarantee delay-independent stability is identified to that of the deterministic system; for the second order moment, the sufficient condition that ensure delay-independent stability depends on noise intensity. The theoretical results are also illustrated with numerical simulations.

Stochastic response of a parametrically excited vibro-impact system with a nonzero offset constraint

In this paper, a recently proposed solution procedure is extended in a straightforward manner to obtain the probability density function (PDF) of the stochastic response of a vibro-impact Duffing system with a nonzero offset constraint. The Duffing system is simultaneously excited by external and parametric Gaussian white noises and it undergoes repetitive instantaneous impacts against the constraint. First, in terms of the Zhuravlev non-smooth coordinate transformation, the original equation of motion and the impact condition are combined into a new equation without any constraints by adding an additional impulsive damping term. Second, the PDF of the new system is governed by the associated Fokker–Planck equation which is solved by the exponential-polynomial closure method. Last, the PDF of the original vibro-impact Duffing system is formulated in terms of the methodology on seeking the PDF distribution of a function of random variables. In order to evaluate the effectiveness of the proposed solution procedure, four illustrative examples are studied considering different values of the parameters, namely nonzero offset, nonlinear stiffness and parametric excitation intensity. A comparison with the direct energy balance method is also presented. Comparison with the simulation result shows that the proposed solution procedure can provide a satisfactory PDF solution, even in the tail region for the examined examples. In addition, the nonzero offset constraint significantly affects the PDF distribution of the response.

Stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises

Abstract This study presents a solution procedure for the stationary probability density function (PDF) of the response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises. First the Zhuravlev non-smooth coordinate transformation is adopted to convert a vibro-impact oscillator into an oscillator without barriers. The stationary PDF of the converted oscillator is governed by the Fokker–Planck (FP) equation. The FP equation is solved by the exponential-polynomial closure (EPC) method. Illustrative examples are presented with vibro-impact Duffing oscillators under external and parametric Gaussian white noises to show the effectiveness of the solution procedure. The parametric excitation is acting in displacement and the constraint is a unilateral zero-offset barrier. The restitution coefficient of impacts is taken as 0.90. Comparison with the simulated results shows that the proposed solution procedure can provide good approximate PDFs for displacement and velocity although a little difference exists in the tail of these PDFs. This difference may be due to the weak approximation on the response of the vibro-impact oscillators using a continuous Markov process when the restitution coefficient is not very close to unity.

First-passage failure of MDOF nonlinear oscillator

First-passage failure of multiple-degree-of-freedom nonlinear oscillators with lightly nonlinear dampings and strongly nonlinear stiffness subject to additive and/or parametric Gaussian white noise excitations is studied. First, by using the stochastic averaging method based on the generalized harmonic functions, the averaged Ito stochastic differential equation for the amplitudes of the nonlinear oscillators can be derived. Then the associated backward Kolmogorov equation of the conditional reliability function is established, and the conditional reliability is approximately expressed as a series expansion in terms of Kummer functions with time-dependent coefficients. By using the Galerkin method, the time dependent coefficients of the associated conditional reliability function can be solved by a set of differential equations. Finally, the proposed procedure is applied to Duffing-Van der Pol systems under external and/or parametric excitations of Gaussian white noises. The results are also verified by those obtained from Monte Carlo simulation of the original system. The effects of system parameters on first-passage failure are discussed briefly.

The True Linearization Coefficients for Nonlinear Systems Under Parametric White Noise Excitations

In this paper some properties of the stochastic linearization method applied to nonlinear systems excited by parametric Gaussian white noises are discussed. In particular, it is shown that the linearized quantities, obtained by the author in another paper by linearizing the coefficients of the Ito differential rule related to the original system, show the same properties found by Kozin with reference to nonlinear system excited by external white noises. The first property is that these coefficients are the true linearized quantities, in the sense that their exact values are able to give the first two statistical moments of the true response. The second property is that, in the stationary case and in the field of the parameter estimation theory, they represent the maximum likelihood estimates of the linear model quantities fitting the original nonlinear response.

The Generalized Cell Mapping Method in Nonlinear Random Vibration Based Upon Short-Time Gaussian Approximation

A short-time Gaussian approximation scheme is proposed in the paper. This scheme provides a very efficient and accurate way of computing the one-step transition probability matrix of the previously developed generalized cell mapping (GCM) method in nonlinear random vibration. The GCM method based upon this scheme is applied to some very challenging nonlinear systems under external and parametric Gaussian white noise excitations in order to show its power and efficiency. Certain transient and steady-state solutions such as the first-passage time probability, steady-state mean square response, and the steady-state probability density function have been obtained. Some of the solutions are compared with either the simulation results or the available exact solutions, and are found to be very accurate. The computed steady-state mean square response values are found to be of error less than 1 percent when compared with the available exact solutions. The efficiency of the GCM method based upon the short-time Gaussian approximation is also examined. The short-time Gaussian approximation renders the overhead of computing the one-step transition probability matrix to be very small. It is found that in a comprehensive study of nonlinear stochastic systems, in which various transient and steady-state solutions are obtained in one computer program execution, the GCM method can have very large computational advantages over Monte Carlo simulation.

Theory of fluctuations and parametric noise in a point nuclear reactor model

We present a joint description of internal fluctuations and parametric noise in a point nuclear reactor model in which delayed neutrons and a detector are considered. We obtain kinetic equations for the first moments and define effective kinetic parameters which take into account the effect of parametric Gaussian white noise. We comment on the validity of Langevin approximations for this problem. We propose a general method to deal with weak but otherwise arbitrary non-white parametric noise. Exact kinetic equations are derived for Gaussian non-white noise.