Stochastic Parametric Excitations Research Articles

On the stability of systems with stochastically variable parameters

The paper analyzes questions related to the construction of dynamic stability boundaries of elastic systems subjected to stochastic parametric excitation. It is supposed that the parametric action is a combination of a deterministic static component and a stochastic fluctuating component. The fluctuating component is taken to be a stationary ergodic process. The stability boundaries are built in the region of combination resonance using the stochastic averaging method and a probabilistic approach due to Khasminskii. In this connection, the stochastic averaging method based on the Stratonovich-Khasminskii theory is used. The probabilistic approach consists in using explicit asymptotic expressions for the largest Lyapunov exponent, from which the asymptotic stability boundaries are determined. As an application, the stability of a simply supported thin-walled bar subjected to a stochastically varying longitudinal load is investigated

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.

Moment Lyapunov Exponents of a Two-Dimensional Viscoelastic System Under Bounded Noise Excitation

The moment Lyapunov exponents of a two-dimensional viscoelastic system under bounded noise excitation are studied in this paper. An example of this system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The stochastic parametric excitation is modeled as a bounded noise process, which is a realistic model of stochastic fluctuation in engineering applications. The moment Lyapunov exponent of the system is given by the eigenvalue of an eigenvalue problem. The method of regular perturbation is applied to obtain weak noise expansions of the moment Lyapunov exponent, Lyapunov exponent, and stability index in terms of the small fluctuation parameter. The results obtained are compared with those for which the effect of viscoelasticity is not considered.

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.

Lyapunov Exponents and Moment Lyapunov Exponents of a Two-Dimensional Near-Nilpotent System

The Lyapunov exponents and moment Lyapunov exponents of a near-nilpotent system under stochastic parametric excitation are studied. The system considered is the linearized system of a two-dimensional nonlinear system exhibiting a pitchfork bifurcation. The effect of stochastic perturbation in the vicinity of static pitchfork bifurcation is investigated. Approximate analytical results of Lyapunov exponent are obtained. The eigenvalue problem for the moment Lyapunov exponent is converted to a two-point boundary value problem, which is solved numerically by the method of relaxation.

Exact stationary solutions of the Fokker-Planck equation for nonlinear oscillators under stochastic parametric and external excitations

A systematic procedure is developed to obtain the stationary probability density function for the response of general single-degree-of-freedom nonlinear oscillators under parametric and external Gaussian white noise excitations. In a previous paper (Wang and Zhang 1998 J. Eng. Mech. ASCE 18 18-23) we expressed a nonlinear function of oscillators using a polynomial formula. The nonlinear system described here has the following form: , where and 1, 2 are Gaussian white noise functions. Thus, this paper is a generalization of the results studied in our previous paper. The stationary Fokker-Planck equation is employed to obtain the governing equation of the probability density function. Based on this procedure, the exact stationary probability densities of many nonlinear stochastic oscillators 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.

Exact Stationary Response Solutions of Six Classes of Nonlinear Stochastic Systems under Stochastic Parametric and External Excitations

A systematic procedure is developed to obtain the stationary probability density function for the response of a general nonlinear system under parametric and external excitations of Gaussian white noises. The nonlinear system described here has the following form: x + 8o(X) + 81(X)X + 82(X).e + 83(X)i 3 = kl~I(t) + k2X~2(t) + k3i~3(t), where ~I(t) = I, 2, 3 are Gaussian white noise. The reduced Fokker-Planck equation is employed to get the governing equation of the probability density function. Based on this procedure, the primary focus of this paper is to find an undetermined function hex, i), which can satisfy the general FPK equation, so that the solution of the FPK equation can be found for each class of the nonlinear stochastic systems. By doing the parameter studies, we get the exact stationary response solutions of six classes of nonlinear stochastic systems. This paper will illustrate that previous certain classes of exact steady-state solutions available up until now have been special cases of exact stationary response solutions under specific conditions of stochastic par­ ametric and external excitations.

A finite-element method for analysis of a non-linear system under stochastic parametric and external excitation

A finite-element method approach is developed to solve the Fokker-Planck-Kol-mogorov equation for the probability density function of stationary response of a general non-linear system subject to both parametric and external Gaussian white noise excitation. N-dimensional shape functions are expanded by a one-dimensional shape function in global coordinate. As the domain of the density function is usually infinite, adaptive grid generation is adopted for the estimation of a finite range of the finite-element mesh. Several examples with existing exact solutions are used to illustrate the validity of the method. Results are also compared with those obtained by the Gaussian closure method, the cumulant-neglect method, equivalent external excitation and Monte Carlo simulation.

Behavior of forced vibrating systems with stochastic parameters introducing parametric excitation effects

The behavior of systems subjected to stochastic parametric excitations is studied in this paper. The method of analysis is based upon integral transforms together with a perturbation-type expansion. The results are illustrated by means of a three-degree-of-freedom system. Different models of stochasticity of parameters and correlation among their elements are studied. It is found that consideration of stochastic parametric excitation helps in establishing a better prediction of system performance. This work is important in mechanical or space structures where operation at different frequencies is involved. Another application has to do with the dynamic modeling of structures.

Non-Gaussian Linearization Method for Stochastic Parametrically and Externally Excited Nonlinear Systems

A new practical non-Gaussian linearization method is developed for the problem of the dynamic response of a stable nonlinear system under both stochastic parametric and external excitations. The non-Gaussian linearization system is derived through a non-Gaussian density that is constructed as the weighted sum of undetermined Gaussian densities. The undetermined Gaussian parameters are then derived through solving a set of nonlinear algebraic moment relations. The method is illustrated by a Duffing-type stochastic system with/without parametric noise excited term. The accuracy in predicting the stationary and nonstationary variances by the present approach is compared with some exact solutions and Monte Carlo simulations.

Exact and approximate response of non-linear stochastic systems

ABSTRACT The method of finding the exact solution and a new approximation method for a class of non-linear stochastic systems is presented. In both methods a non-linear transformation of the original system is applied. The idea of this method is presented for a non-linear system with stochastic parametric excitations. If the transformed system is a linear one then one can find an exact solution, otherwise the cumulant neglect closure technique is applied to this new system. The resuts obtained are illustrated by a numerical example.

Stationary Response of States-Constrained Nonlinear Systems Under Stochastic Parametric and External Excitations

A Fourier-series closure scheme is developed for the prediction of the stationary stochastic response of a stochastic parametrically and externally excited oscillator with a nonpolynomial type nonlinearity and under states constraint. The technique is implemented by deriving the moment relations and employing the Fourier series as the expansion of a non-Gaussian density for constructing and solving a set of algebraic equations with unknown Fourier coefficients. A single-arm robot manipulator operated in a constrained working space and subjected to parametric and/ or external noise excitations is selected to illustrate the present approach. The validity of the present scheme is further supported by some exact solutions and Monte Carlo simulations.

Stochastically Excited Linear Nonconservative Systems∗

ABSTRACT Dynamic stability of linear nonconservative systems under stochastic parametric excitation of small intensity is examined. It is assumed that a finite number of modes of a general, multi-degree-of-freedom system undergo flutter instability (nonresonance case). A modified stochastic averaging method is employed to obtain the contribution from the stochastic components of the stable modes to that of the critical modes. The approximate lto equations for the critical modes are then examined. Conditions for mean square stability of dynamic response are obtained. Results are shown to depend only on those values of the excitation spectral density near twice the natural frequencies, the difference and combination frequencies of the system. The results are applied to the problem of a cantilever column subjected to stochastic follower force.

Methods and Gaussian Criterion for Statistical Linearization of Stochastic Parametrically and Externally Excited Nonlinear Systems

The methods of Gaussian linearization along with a new Gaussian Criterion used in the prediction of the stationary output variances of stable nonlinear oscillators subjected to both stochastic parametric and external excitations are presented. The techniques of Gaussian linearization are first derived and the accuracy in the prediction of the stationary output variances is illustrated. The justification of using Gaussian linearization a priori is further investigated by establishing a Gaussian Criterion. The non-Gaussian effects due to system nonlinearities and/or large noise intensities in a Duffing oscillator are also illustrated. The validity of employing the Gaussian Criterion test for assuring accuracy of Gaussian linearization is supported by performing the Chi-square Gaussian goodness-of-fit test.

Optimal Control of Stochastic Parametrically and Externally Excited Nonlinear Control Systems

A sub-optimal nonlinear controller which is synthesized by using the external linearization approach is applied to the optimal control of stochastic parametrically and externally excited nonlinear systems with complete state information. Algebraic necessary conditions are derived for the minimization of the quadratic cost function through the concepts of equivalent external excitation. The concepts and applications of the statistical linearization approach for the externally excited nonlinear systems are extended to the nonlinear systems subjected to both stochastic parametric and external excitations. Two examples which include a first-order nonlinear and a second-order Duffing type stochastic system are used to illustrate the performance of the present design. The applications of the statistical linearization approach to the optimal control of a stochastic parametrically and externally excited Duffing type system is illustrated and compared with the present approach by using Monte Carlo simulation.

Prediction of the response of non-linear oscillators under stochastic parametric and external excitations

The method of equivalent external excitation is derived to predict the stationary variances of the states of non-linear oscillators subjected to both stochastic parametric and external excitations. The oscillator is interpreted as one which is excited solely by an external zero-mean stochastic process. The Fokker-Planck-Kolmogorov equation is then applied to solve for the density functions and match the stationary variances of the states. Four examples which include polynomial, non-polynomial, and Duffing type non-linear oscillators are used to illustrate this approach. The validity of the present approach is compared with some exact solutions and with Monte Carlo simulations.

Stochastically Perturbed Linear Gyroscopic Systems

ABSTRACT Dynamic stability of linear conservative gyroscopic systems under stochastic parametric excitations of small intensity is examined. Conditions for mean square stability of dynamic response are obtained. Results are shown to depend only on those values of the excitation spectral density near twice the natural frequencies and the combination frequencies of the system. These results are applied to the problem of flow induced vibration in a supported pipe conveying fluid with pulsating velocity. The effects of mean flow velocity and virtual mass on the extent of parametric instability regions are then discussed.

Stochastic stability of coupled linear systems: a survey of methods and results

The analysis of dynamic systems subject to stochastic parametric excitation is important in a variety of branches of engineering and physics. For example, models of this type frequently occur in the analysis of linear continuous systems using modal decomposition. The random coupling or parametric excitation can, for example, model the influence of externally applied loads on the system parameters. In this paper we investigate the almost–sure stability properties of the sample trajectories of linear stochastic systems with parametric excitation.