Transient solution of stochastic oscillators with both odd and even nonlinearity under modulated Gaussian white noise.

This study presents an improved solution procedure for determining the transient probability density function (PDF) solutions of the nonlinear oscillators with both odd and even nonlinearity under modulated random stimulation, which is an extension of the exponential-polynomial-closure approach. An evolutionary exponential-polynomial function with time-varying undetermined variables is considered as the transient probabilistic solution. By selecting a set of independent evolutionary base functions spanning a R^{n} space as weight functions, a set of ordinary differential equations can be formulated by integrating the weighted residual error. The undetermined variables can be determined numerically by solving those ordinary differential equations. Three numerical examples illustrate that the improved solution procedure can acquire the transient probabilistic responses of the stochastic dynamic systems effectively and efficiently even in the PDF solution tails when compared with Monte Carlo simulation. Moreover, the results indicate that the PDF solutions of the oscillators are asymmetrical at their nonzero means due to the influence of even nonlinearity. The nonstationary behaviors of the system responses are also investigated along with the behavior of modulated Gaussian white noise.

Weight Function Method for Stress Intensity Factors of Semi-Elliptical Surface Cracks on Functionally Graded Plates Subjected to Non-Uniform Stresses.

In this paper, a weight function method based on the first four terms of a Taylor's series expansion is proposed to determine the stress intensity factors of functionally graded plates with semi-elliptical surface cracks. Cracked surfaces that are subjected to constant, linear, parabolic and cubic stress fields are considered. The weight functions for the surface, deepest and general points on the crack faces of long and deep cracked functionally graded plates are derived, which has never been done before in the literature. The accuracy of the method in this study is then validated by comparing the results with those of finite element modeling. The numerical results indicate that the derived weight functions are highly accurate and robust enough to predict the stress intensity factors for cracked functionally graded plates subjected to non-uniform stress distributions. The weight function method is therefore a time-saving technique and suitable for handling non-uniform stress fields.