Y.K. Lin

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.

Some observations on the stochastic averaging method

Abstract Formulas required for the application of Stratonovichs stochastic averaging method are rederived using a mathematical procedure more appealing to engineers. By so doing the physical implications are made clear, allowing certain variations and relaxation of some restrictions. In particular, the time-averaging portion of the original method is removed so that it becomes applicable to non-stationary excitations as well as cases where deterministic time-varying properties must be preserved. A less restrictive general condition is proposed for this modified stochastic averaging method, and heuristic arguments are given for further relaxation of the condition in stability analyses. Application of this modified method is illustrated by a simple example.

On fatigue crack growth under random loading

Abstract A probabilistic analysis of the fatigue crack growth, fatigue life and reliability of a structural or mechanical component is presented on the basis of fracture mechanics and theory of random processes. The material resistance to fatigue crack growth and the time-history of the stress are assumed to be random. Analytical expressions are obtained for the special case in which the random stress is a stationary narrow-band Gaussian random process, and a randomized Paris-Erdogan law is applicable. As an example, the analytical method is applied to a plate with a central crack, and the results are compared with those obtained from digital Monte Carlo simulations.

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 Stochastic Averaging for Nonlinear Dynamical Systems with High Damping

The asymptotic behavior of coupled nonlinear dynamical systems in the presence of noise is studied using the method of stochastic averaging. It is shown that, for systems with rapidly oscillating and decaying components, the stochastic averaging technique yields a set of equations of considerably smaller dimension, and the resulting equations are simpler. General results of this method are applied to stochastically perturbed nonlinear nonconservative systems in R4. It is shown that in such systems the contribution of the stochastic components in the damped modes to the drift term of the critical mode may be beneficial in terms of stability in certain cases.

On random pulse train and its evolutionary spectral representation

Abstract In this paper, relationship is established between two seemingly unrelated stochastic processes, both of which have been applied widely to engineering problems; namely, the random pulse train and the evolutionary process. In particular, a random pulse train has an evolutionary spectral representation if the pulse shape remains unchanged, and the pulse arrivals are uncorrelated events. This is also true for correlated pulse arrivals, provided that the ensemble average of the random pulse magnitude is zero.

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.

Stochastic stability of wind-excited long-span bridges

Due to its heavy weight, the motion instability of a wind-excited long-span bridge exhibits some unique features different from those of lightweight structures, such as airfoils. These features are explained, and accounts are given of earlier experimental and analytical investigations in the field. Since turbulence is generally present in a natural wind flow, its effects on bridge motion stability are discussed from a stochastic dynamics point of view. Some outstanding problems are indicated.