Gaussian White Noise Excitation Research Articles

On statistical quasi-linearization

In carrying out the statistical linearization procedure to a non-linear system subjected to an external random excitation, a Gaussian probability distribution is assumed for the system response. If the random excitation is non-Gaussian, however, the procedure may lead to a large error since the response of bother the original non-linear system and the replacement linear system are not Gaussian distributed. It is found that in some cases such a system can be transformed to one under parametric excitations of Gaussian white noises. Then the quasi-linearization procedure, proposed originally for non-linear systems under both external and parametric excitations of Gaussian white noises, can be applied to these cases. In the procedure, exact statistical moments of the replacing quasi-linear system are used to calculate the linearization parameters. Since the assumption of a Gaussian probability distribution is avoided, the accuracy of the approximation method is improved. The approach is applied to non-linear systems under two types of non-Gaussian excitations: randomized sinusoidal process and polynomials of a filtered process. Numerical examples are investigated, and the calculated results show that the proposed method has higher accuracy than the conventional linearization, as compared with the results obtained from Monte Carlo simulations.

Effect of random noise on chaotic motion of a particle in a ϕ6 potential

Abstract The chaotic behaviors of a particle in a triple well ϕ6 potential possessing both homoclinic and heteroclinic orbits under harmonic and Gaussian white noise excitations are discussed in detail. Following Melnikov theory, conditions for the existence of transverse intersection on the surface of homoclinic or heteroclinic orbits for triple potential well case are derived, which are complemented by the numerical simulations from which we show the bifurcation surfaces and the fractality of the basins of attraction. The results reveal that the threshold amplitude of harmonic excitation for onset of chaos will move downwards as the noise intensity increases, which is further verified by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the more possible chaotic domain in parameter space. The effect of noise on Poincare maps is also investigated.

Stochastic stabilization of first-passage failure of Rayleigh oscillator under Gaussian White-Noise parametric excitations

Abstract Stochastic stabilization of first-passage failure of Rayleigh oscillator under Gaussian White-Noise parametric excitation is studied. The equation of motion of the system is first reduced to an averaged Ito stochastic differential equation by using the stochastic averaging method. Then, a backward Kolmogorov equation governing the conditional reliability function of first-passage failure is established. The conditional reliability function, and the conditional probability density are obtained by solving the backward Kolmogorov equation with boundary conditions. Finally, the cost function and optimal control forces are determined by the requirements of stabilizing the system by evaluating the maximal Lyapunov exponent. The numerical results show that the procedure is effective and efficiency.

Stochastic optimal control of first-passage failure for coupled Duffing–van der Pol system under Gaussian white noise excitations

Abstract In this paper, the stochastic averaging method of quasi-non-integrable-Hamiltonian systems is applied to Duffing–van der Pol system to obtain partially averaged Ito stochastic differential equations. On the basis of the stochastic dynamical programming principle and the partially averaged Ito equation, dynamical programming equations for the reliability function and the mean first-passage time of controlled system are established. Then a non-linear stochastic optimal control strategy for coupled Duffing–van der Pol system subject to Gaussian white noise excitation is taken for investigating feedback minimization of first-passage failure. By averaging the terms involving control forces and replacing control forces by the optimal ones, the fully averaged Ito equation is derived. Thus, the feedback minimization for first-passage failure of controlled system can be obtained by solving the final dynamical programming equations. Numerical results for first-passage reliability function and mean first-passage time of the controlled and uncontrolled systems are compared in illustrative figures to show effectiveness and efficiency of the proposed method.

Noise-induced chaos and basin erosion in softening Duffing oscillator

Abstract It is common for many dynamical systems to have two or more attractors coexist and in such cases the basin boundary is fractal. The purpose of this paper is to study the noise-induced chaos and discuss the effect of noises on erosion of safe basin in the softening Duffing oscillator. The Melnikov approach is used to obtain the necessary condition for the rising of chaos, and the largest Lyapunov exponent is computed to identify the chaotic nature of the sample time series from the system. According to the Melnikov condition, the safe basins are simulated for both the deterministic and the stochastic cases of the system. It is shown that the external Gaussian white noise excitation is robust for inducing the chaos, while the external bounded noise is weak. Moreover, the erosion of the safe basin can be aggravated by both the Gaussian white and the bounded noise excitations, and fractal boundary can appear when the system is only excited by the random processes, which means noise-induced chaotic response is induced.

Stochastic stability and control of coupled Duffing-van der Pol systems

By using the stochastic averaging method of quasi-Hamilton system and stochastic stbilization strategy, the stochastic stability and stabilization control of a coupled Duffing-van der Pol system under Gaussian white noise excitation with in semi-infinite time interval are investigated. The results show that, with suitable controlled parameters, effective stochastic stabilization can be realized.

Solutions of path integration for nonlinear dynamical system under stochastic parametric and external excitations

The numerical path integration based on GaussLegendre scheme is extended to the case of nonlinear dynamical system under stochastic parametric and external excitations. For the purpose of comparison between the numerical solutions and the analytic solution(if the system has) or MonteCarlo simulation, we discuss the system under parametric and external Gaussian white noise excitations. The numerical method is shown to give accurate results. Via the numerical solutions of path integration, we have studied the P bifurcation of the stochastic system.

Stochastic Optimal Semi-Active Control of Nonlinear Systems by Using MR Dampers

A stochastic optimal semi-active control strategy for strongly nonlinear oscillator subjected to external and/or parametric excitations of Gaussian white noises using magneto-rheological (MR) damper is proposed. The dynamic behavior of an MR damper is characterized by using the Bouc-Wen hysteretic model. The control force produced by the MR damper is split into a passive part and a semi-active part. The passive part is incorporated with the uncontrolled system to form a passive control system. Then the system is converted into an equivalent nonlinear non-hysteretic stochastic control system, from which an Itô stochastic differential equation for total energy is derived by using the stochastic averaging method of energy envelope. For the ergodic control problem, a dynamical programming equation for the controlled total energy process is established based on the stochastic dynamical programming principle. The optimal control law is obtained by minimizing the dynamical programming equation and can be implemented by the MR damper without clipping. Then the fully averaged Itô equation for the controlled total energy process is obtained by replacing the control force with the optimal control force and completing the averaging. Finally, the response of semi-actively controlled system is obtained from solving the final dynamical programming equation and the Fokker-Planck-Kolmogorov (FPK) equation associated with the fully averaged Itô equation. The efficacy of the proposed stochastic optimal semi-active control strategy is illustrated by using the numerical results for two examples.

Stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems

A new procedure is developed to study the stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems under the Gaussian white noise excitation. Firstly, the singular boundaries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system's energy levels with respect to the stochastic averaging method. Secondly, the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones. Lastly, a quasi-integrable-Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure. Moreover, simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure. It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system's parameters. Therefore, one can see that the numerical results are consistent with the theoretical predictions.

Stationary motion of active Brownian particles.

The stationary motion of active Brownian particles is studied by using the stochastic averaging method for quasi-integrable Hamiltonian systems. First the stochastic averaging method for quasi-integrable Hamiltonian systems is briefly introduced. Then the stationary solution of the dynamic equations governing an active Brown particle in plane with the Rayleigh velocity-dependent friction model subject to Gaussian white noise excitations is obtained by using the stochastic averaging method. The solution is validated by comparison with the result from Monte Carlo simulation. Finally, two more stationary solutions of the dynamic equations governing active Brownian particle with the Schienbein-Gruler and Erdmann velocity-dependent friction models, respectively, subject to Gaussian white noise excitations are also given.

Response probability estimation for randomly excited quasi-linear systems using a neural network approach

A quasi-linear system is referred to as a system linear in properties and subjected to multiplicative random excitations appearing also in the linear terms. It is known that exact solutions for the stationary moments can be obtained analytically for such a quasi-linear system if the excitations are Gaussian white noises. However, the exact response probability, which is non-Gaussian, is not obtainable analytically. In this paper, a neural network approach is proposed to evaluate the stationary response probability for quasi-linear systems under both additive and multiplicative excitations of Gaussian white noises based on the obtained exact statistical moments. Numerical examples show that the procedure yields accurate results if an appropriate form is assumed for the probability density function. The accuracy of the results is substantiated by comparing them with those obtained from Monte Carlo simulations.

An original approximate method for estimating the invariant probability distribution of a large class of multi-dimensional nonlinear stochastic oscillators

This paper proposes an original method for obtaining analytical approximations of the invariant probability density function of multi-dimensional Hamiltonian dissipative dynamic systems under Gaussian white noise excitations, with linear non-conservative parts and nonlinear conservative parts. The method is based on an exact result and a heuristic argument. Its pertinence is attested by numerical tests.

Equivalent linearization of MDOF systems under external Poisson white noise excitation

Equivalent linearization (EQL) techniques are developed and evaluated for multidimensional systems under external Poisson white noise excitation. Especially, a simulation strategy for the calculation of the linearization coefficients is proposed. The methods are illustrated by several examples that have been treated under Gaussian white noise excitation in the literature. It is shown that EQL for MDOF systems under Poisson white noise excitation is able to deal with problems of nearly the same dimension as under Gaussian white noise excitation.

Approximate solution of the Fokker–Planck–Kolmogorov equation

The aim of this paper is to present a thorough investigation of approximate techniques for estimating the stationary and non-stationary probability density function (PDF) of the response of nonlinear systems subjected to (additive and/or multiplicative) Gaussian white noise excitations. Attention is focused on the general scheme of weighted residuals for the approximate solution of the Fokker–Planck–Kolmogorov (FPK) equation. It is shown that the main drawbacks of closure schemes, such as negative values of the PDF in some regions, may be overcome by rewriting the FPK equation in terms of log-probability density function (log-PDF). The criteria for selecting the set of weighting functions in order to obtain improved estimates of the response PDF are discussed in detail. Finally, a simple and very effective iterative solution procedure is proposed.

Some properties of multi-degree-of-freedom potential systems and application to statistical equivalent non-linearization

This paper presents some properties of two restricted classes of multi-degree-of-freedom potential systems subjected to Gaussian white-noise excitations. Specifically, potential systems which exhibit damping terms with energy-dependent polynomial form are referred to. In this context, first systems with coupled stiffness terms and damping terms depending on the total energy are investigated. Then, systems with uncoupled stiffness terms and damping terms depending on the total energy in each degree-of-freedom are considered. For these two classes, it is found that algebraic relations among the stationary statistical moments of the energy functions can be derived by applying standard tools of Itô calculus. Further, it is noted that these relations are very useful within the framework of an equivalent statistical non-linearization technique to build approximate solutions for arbitrary non-linear systems.

A technique note of exact stationary solutions on nonlinear systems under stochastic parametric excitations

In this paper, a systematic procedure is developed to obtain the stationary probability density function for the response of a general nonlinear system under parametric and external Gaussian white noise excitations. In Ref. [11], nonlinear function of system was expressed to the polynomial formula. The nonlinear system described here has the following form: x¨ + g(x, x˙) = k1ξ1(t)+ k2xξ2(t), where g(x,x˙) = ∑ i = 0∞g i (x)x˙ i and ξ1,ξ2 are Gaussian white noises. Thus, this paper is a generalization for the results studied in Ref. [11]. The reduced Fokker–Planck (FP) equation is employed to get the governing equation of the probability density function. Based on this procedure, the exact stationary probability densities of many nonlinear stochastic systems are obtained, and it is shown that some of the exact stationary solutions described in the literature are only particular cases of the presented generalized results.

Noise-induced chaotic motions in Hamiltonian systems with slow-varying parameters

This paper studies chaotic motions in quasi-integrable Hamiltonian systems with slow-varying parameters under both harmonic and noise excitations. Based on the dynamic theory and some assumptions of excited noises, an extended form of the stochastic Melnikov method is presented. Using this extended method, the homoclinic bifurcations and chaotic behavior of a nonlinear Hamiltonian system with weak feed-back control under both harmonic and Gaussian white noise excitations are analyzed in detail. It is shown that the addition of stochastic excitations can make the parameter threshold value for the occurrence of chaotic motions vary in a wider region. Therefore, chaotic motions may arise easily in the system. By the Monte-Carlo method, the numerical results for the time-history and the maximum Lyapunov exponents of an example system are finally given to illustrate that the presented method is effective.

First excursion probabilities for linear systems by very efficient importance sampling

An analytical study of the failure region of the first excursion reliability problem for linear dynamical systems subjected to Gaussian white noise excitation is carried out with a view to constructing a suitable importance sampling density for computing the first excursion failure probability. Central to the study are ‘elementary failure regions’, which are defined as the failure region in the load space corresponding to the failure of a particular output response at a particular instant. Each elementary failure region is completely characterized by its design point, which can be computed readily using impulse response functions of the system. It is noted that the complexity of the first excursion problem stems from the structure of the union of the elementary failure regions. One important consequence of this union structure is that, in addition to the global design point, a large number of neighboring design points are important in accounting for the failure probability. Using information from the analytical study, an importance sampling density is proposed. Numerical examples are presented, which demonstrate that the efficiency of using the proposed importance sampling density to calculate system reliability is remarkable.

Optimal displacement feedback control law for active tuned mass damper

AbstractOptimal displacement feedback control law is derived for a vibration control of a single‐degree‐of‐freedom structure with an active tuned mass damper (ATMD). Analytical expressions of the linear quadratic regulator (LQR) feedback gains for the ATMD are derived by solving the Ricatti equation straightforwardly. Based on these solutions, it is found that if the stiffness of the tuned mass damper (TMD) is calibrated to satisfy a certain condition, the control law is simplified to be composed of the feedback gains only for the displacement of the structure and the velocity of the auxiliary mass stroke, which is referred to as ‘optimal displacement feedback control’. The mean‐square responses of the structure as well as the auxiliary mass against Gaussian white noise excitations are evaluated by solving the Lyapunov equation analytically based on the stochastic optimal control theory. Using these analytical solutions, the optimal damping parameter for the auxiliary mass is also derived. Finally, the optimal displacement feedback control law is presented. Copyright © 2001 John Wiley & Sons, Ltd.

Nonlinear bearing stiffness parameter estimation in flexible rotor–bearing systems using Volterra and Wiener approach

Higher order frequency response functions based on Volterra and Wiener series are explored for the inverse problem of stiffness estimation of a flexible rotor supported in nonlinear bearings. The Volterra series has been employed by researchers earlier for the identification of higher order kernels of nonlinear systems through a non-parametric approach. The present study investigates the possibility of employing these kernels for parameter estimation of the system. Numerical simulation has been carried out for a system with three degrees of freedom and with cubic nonlinearity in stiffness. A frequency domain has been adopted for the identification of higher order kernels. The procedure involves extraction of Wiener kernels from the response of the system to a Gaussian white noise excitation. Volterra kernels are in turn synthesised from the Wiener kernels. In addition to direct kernels, the system under consideration, requires definitions of cross-kernels and their estimation. Expressions for the cross and direct kernels are constructed in the frequency domain. A set of third-order kernel factors are algebraically and graphically synthesised from the measured first-order kernels. These third-order kernel factors are then processed with the measured third-order kernels for nonlinear parameter estimation. Damping is taken to be linear in the analysis. The procedure is illustrated through numerical simulation. The assumptions involved and the approximations are discussed.