White Noise Parametric Excitations Research Articles

Comparison of stochastic stability boundaries for parametrically forced systems with application to ship rolling motion

Numerous accidents caused by parametric rolling have been reported on container ships and pure car carriers (PCCs). Considering this dangerous phenomenon, the parametric rolling in irregular seas is examined in this paper based on the stability of the systems origin, which corresponds to the upright condition of the vessel. It provides a novel theoretical explanation of the instability mechanism for two cases: white-noise parametric excitation and colored-noise parametric excitation. Moreover, the authors confirm the usefulness of the previously provided formulae by Roberts and Dostal by means of numerical examples.

MOMENT LYAPUNOV EXPONENTS AND STOCHASTIC STABILITY OF A THIN-WALLED BEAM SUBJECTED TO AXIAL LOADS AND END MOMENTS

In this paper, the Lyapunov exponent and moment Lyapunov exponents of two degrees-of-freedom linear systems subjected to white noise parametric excitation are investigated. The method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises. The Lyapunov exponent and moment Lyapunov exponents are important characteristics for determining both the almost-sure and the moment stability of a stochastic dynamic system. As an example, we study the almost-sure and moment stability of a thin-walled beam subjected to stochastic axial load and stochastically fluctuating end moments. The validity of the approximate results for moment Lyapunov exponents is checked by numerical Monte Carlo simulation method for this stochastic system.

Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and Poisson white noise parametric excitations

The stochastic stability of a Duffing oscillator with fractional derivative damping under combined harmonic and Poisson white noise parametric excitations is investigated. A stochastic averaging method and the Khasminskii’s procedure are applied to evaluate the largest Lyapunov exponent. Then the asymptotic Lyapunov stability with probability one of the original system is determined approximately by using the largest Lyapunov exponent. Finally, the analytical results are verified by those from the Monte Carlo simulation of the original system.

Stochastic stability of Duffing–Mathieu system with delayed feedback control under white noise excitation

The asymptotic Lyapunov stability with probability one of Duffing–Mathieu system with time-delayed feedback control under white-noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. Meanwhile, the stability conditions for the system with different time delays are also obtained. The theoretical results are well verified through digital simulation.

Moment Lyapunov exponents and stochastic stability for two coupled oscillators

CThe Lyapunov exponent and moment Lyapunov exponent of two degree-of-freedom linear systems subjected to white noise parametric excitation are investigated. Through a perturbation method we obtain the explicit asymptotic expressions for these exponents in the presence of low intensity noise. The Lyapunov exponent and moment Lyapunov exponents are important characteristics for determining the almostsure and moment stability of a stochastic dynamical system. As an example, we study the almost-sure and moment stability of the flexural-torsion stability of a thin elastic beam subjected to a stochastically fluctuating follower force. The validity of the approximate results for moment Lyapunov exponents is checked by a numerical Monte Carlo simulation for these stochastic systems.

Moment Lyapunov exponents and stochastic stability of a double-beam system under compressive axial loading

The Lyapunov exponent and moment Lyapunov exponents of two degrees-of-freedom linear systems subjected to white noise parametric excitation are investigated. The method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises. The Lyapunov exponent and moment Lyapunov exponents are important characteristics for determining the almost-sure and moment stability of a stochastic dynamical system. As an example, we study the almost-sure and moment stability of the double-beam system under stochastic compressive axial loading. The validity of the approximate results for moment Lyapunov exponents is checked by the numerical Monte Carlo simulation method for this stochastic system.

Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and white noise parametric excitations

The stochastic stability of a Duffing oscillator with fractional derivative damping of order α (0 < α < 1) under parametric excitation of both harmonic and white noise is studied. First, the averaged Ito equations are derived by using the stochastic averaging method for an SDOF strongly nonlinear stochastic system with fractional derivative damping under combined harmonic and white noise excitations. Then, the expression for the largest Lyapunov exponent of the linearized averaged Ito equations is obtained and the asymptotic Lyapunov stability with probability one of the original system is determined approximately by using the largest Lyapunov exponent. Finally, the analytical results are confirmed by using those from a Monte Carlo simulation of the original system.

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.

Moment Lyapunov exponents of a two-dimensional system under both harmonic and white noise parametric excitations

The moment Lyapunov exponents and the Lyapunov exponents of a 2D system under both harmonic and white noise excitations are studied. The moment Lyapunov exponents and the Lyapunov exponents are important characteristics determining the moment and almost-sure stability of a stochastic dynamical system. The eigenvalue problem governing the moment Lyapunov exponent is established. A singular perturbation method is applied to solve the eigenvalue problem to obtain second-order, weak noise expansions of the moment Lyapunov exponents. The influence of the white noise excitation on the parametric resonance due to the harmonic excitation is investigated.

Dynamic Stability of Fluttered Systems Subjected to Parametric Random Excitations

A general solution for dynamic stability of the fluttered mode of damped, fluttered systems subjected to parametric random excitations is presented in this paper. First, the system equations are pairwisely uncoupled by a modal analysis based on normal modes of the system at the onset of fluttering. The stochastic averaging method is then applied to obtain Ito's equation governing the amplitude of the fluttered mode. Finally, the Lyapunov exponent of the fluttered mode is derived, from which the criterion for asymptotic sample stability of the mode is determined. A cantilevered beam acted upon by a static follower force and a white noise parametric excitation at the free end, and a skew panel subjected to an aerodynamic force in the chordwise direction and a white noise excitation in the spanwise direction are demonstrated as examples. Numerical results show that, although the static follower force or the aerodynamic force exceeds the flutter load, the fluttered mode of the beam or the panel may remain stable in the sense of asymptotic sample stability due to the presence of the white noise excitation.

Parametric Random Vibrations under Non-Gaussian Delta-Correlated Processes

This paper addresses parametric random vibrations subjected to non-Gaussian delta-correlated processes as well as combinations of delta-correlated Gaussian and non-Gaussian processes. The scheme employs a response-moment method, in which response moments are solved through a series of simultaneous equations. This approach requires that a general response moment equation suitable for non-Gaussian/Gaussian, parametric/external excitations be developed. Two problems related to second-order linear systems are explored and their closed-form solutions for response moments up to fourth order are obtained. The first problem considers systems subjected to “physical” Gaussian white-noise parametric excitations together with non-Gaussian delta-correlated external excitations. The second problem considers both parametric and external excitations that are non-Gaussian and delta-correlated.

Probability distributions in simple neutronic systems via the fokker-planck equation

The Fokker-Planck equation for the neutron density probability distribution in a multiplying system, possessing random source and random parametric excitation, is deduced. It is shown that white noise parametric excitation and white noise source excitation both have the same effect on the functional form of the correlation but that the parametric excitation increases its amplitude. A method for obtaining the correlation function directly from the conditional probability distribution is described and applied to a simple point model equation with one group of delayed neutrons.