G.Q. Cai

A new path integration procedure based on Gauss-Legendre scheme

A new path integration scheme based on the Gauss-Legendre quadrature integration rule is proposed for calculating the probability density of the response of a dynamical system under Gaussian white-noise excitation. The new scheme is capable of producing accurate results of probability density as it evolves with time, including the tail region where the probability level is very low. This low probability region is important for the system reliability estimation.

Exact and approximate solutions for randomly excited MDOF non-linear systems

An exact solution technique is developed to obtain stationary probability densities for a class of multi-degree-of-freedom (MDOF) non-linear systems under Gaussian white-noise excitations, without the restriction of equipartition of kinetic energies as with the case of previous exact solutions. The conditions under which exact solutions are obtainable are given. When these conditions are not satisfied, approximate solutions are obtained on the basis of minimum weighted residuals. Examples are given for illustration.

Response spectral densities of strongly nonlinear systems under random excitation

A new analytical procedure is developed to evaluate the response spectral densities for nonlinear systems excited by Gaussian white noises or filtered Gaussian white noises. The cumulant-neglect closure scheme is applied to truncate the governing differential equations for statistical moments of the response variables at two different times. The truncated equations in the time domain are transformed to a set of linear algebraic equations in the frequency domain, which include the response spectral densities as unknowns. This new procedure is illustrated in the example of a Duffing oscillator, and analytical results are compared with those obtained from Volterra series method and Monte Carlo simulations.

On exact stationary solutions of stochastically perturbed Hamiltonian systems

Exact stationary solutions are obtained for Hamiltonian systems with linear or nonlinear damping and subject to external or parametric excitations of Gaussian white-noises. Sufficient conditions for the existence and uniqueness of the solutions are given. It is shown that some exact stationary solutions published recently of nonlinear multi-degree-of-freedom stochastic systems are particular cases of the present results.

Application of quasi-conservative averaging to a non-linear system under non-white excitation

The method of quasi-conservative averaging was developed originally to treat the vibration of a system with a strongly non-linear stiffness and under white-noise excitation. The method is extended herein to the case of non-white broad-band excitation, using an energy-dependent white-noise approximation. The results are compared with those of a Fourier expansion technique developed earlier. The relative simplicity, accuracy and applicability of the two methods are compared.

Approximate solution for nonlinear random vibration problems by partial stochastic linearization

Abstract The accuracy of the stochastic linearization methods is improved by the proposed method of partial stochastic linearization, in which only the nonlinear damping force in the original system is replaced by a linear viscous damping, while the nonlinear restoring force remains unchanged. The replacement is based on the criterion of equal mean work, performed by the nonlinear damping force in the original system and its linear counterpart. The resulting nonlinear stochastic differential equation is then solved exactly, keeping the equivalent damping coefficient as a parameter, which can be determined for a specific system by solving a nonlinear algebraic equation.

Some thoughts on averaging techniques in stochastic dynamics

Stochastic averaging and quasi-conservative averaging are used essentially to obtain a one-dimensional Markov approximation describing the response of a randomly excited one-degree-of-freedom system. Difficulties arise when the original physical system is more than one degree of freedom with both low and high damping modes, or/and it has a strongly nonlinear stiffness and the stochastic excitation is non-white. The ways in which such difficulties can be overcome are discussed.

Sliding motion of anchored rigid block under random base excitations

Abstract The sliding motion of an anchored rigid block, subjected to combined horizontal and vertical base excitations is investigated. Modeling the excitations as Gaussian white noises, the stationary probability density of the total energy of the motion is obtained by solving the associated Fokker-Planck equation. The stationary probability density of the displacement amplitude then follows through a simple transformation. Statistical moments for the first-excursion time, at which the total energy exceeds a pre-set critical value, are also obtained by solving the Pontryagin type equations numerically.

Markov process modeling of groundwater contamination problems

The problem of groundwater contamination is investigated by invoking the analogy to the diffusive Markov process in the probability theory. Special emphases are placed on the following issues: (1) the amount of pollutants entering the protected zone; (2) probability distribution and statistical properties of the random time for a tracer particle to reach the protected zone; and (3) the time rate of contamination arriving at the protected zone. It is shown that these are governed by differential equations analogous to the so-called Kolmogorov backward equation, or derivable from such an equation. Application of the new theory is illustrated by simple examples.

IUTAM Symposium on Nonlinear Stochastic Dynamics and Control

One of types of stochastic retarded systems is under consideration. Our scheme of analysis is applicable for investigation of linear and nonlinear differential difference equations with single and multiple constant delays, linear differential equations with variable delays, linear neutral delay differential equations, and separate linear differential difference equations. In addition, a problem of sensitivity estimation for linear dynamic systems described by stochastic differential difference equations can be explored too. All these schemes are based on extensions of phase spaces.