Moment Equations of Linear It Stochastic Systems with Applications

In this paper, with the Kronecker's product and Kronecker's sum of matrices, the 2nd order moment equations of linear Ito stochastic systems are dervided. Based on the moment equations obtained, a necessary and sufficient condition for the mean-square asymptotic stability of linear Ito stochastic systems is obtained.For the time-invariant stochastic systems,the necessary and sufficient condition is just the same as the Hurwitz property of certain matrices related to the coefficient matrices of the systems. An algorithm STILSS is given for testing the mean-square asymptotic stability of time-invariant linear Ito stochastic systems.

Stability of a delayed predator prey model in a random environment

The stability of the first-order and second-order solution moments for a Harrison-type predator-prey model with parametric Gaussian white noise is analyzed in this paper. The moment equations of the system solution are obtained under Ito interpretations. The delay-independent stable condition of the first-order moment is identical to that of the deterministic delayed system, and the delay-independent stable condition of the second-order moment depends on the noise intensities. The corresponding critical time delays are determined once the stabilities of moments lose. Further, when the time delays are greater than the critical time delays, the system solution becomes unstable with the increase of noise intensities. Finally, some numerical simulations are given to verify the theoretical results.

Laplace-Transform Finite Element Solution of Nonlocal and Localized Stochastic Moment Equations of Transport

Morales-Casique et al.(Adv.Water Res.,29(2006),pp.1238-1255)developed exact first and second nonlocal moment equations for advective-dispersive transport in finite,randomly heterogeneous geologic media.The velocity and concentration in these equations are generally nonstationary due to trends in heterogeneity,conditioning on site data and the influence of forcing terms.Morales-Casique et al.(Adv.Water Res.,29(2006),pp.1399-1418)solved the Laplace transformed versions of these equations recursively to second order in the standard deviationσY of(natural)log hydraulic conductivity,and iteratively to higher-order,by finite elements followed by numerical inversion of the Laplace transform.They did the same for a space-localized version of the mean transport equation.Here we recount briefly their theory and algorithms;compare the numerical performance of the Laplace-transform finite element scheme with that of a high-accuracy ULTIMATE-QUICKEST algorithm coupled with an alternating split operator approach;and review some computational results due to Morales-Casique et al.(Adv.Water Res.,29(2006),pp.1399-1418)to shed light on the accuracy and computational efficiency of their recursive and iterative solutions in comparison to conditional Monte Carlo simulations in two spatial dimensions.

RANDOM RESPONSES OF A TWO-DEGREE-OF-FREEDOM STRUCTURE WITH STRONGLY NONLINEAR COUPLING AND PARAMETRIC INTERACTION

This paper investigates the random responses of a TDOF structure with strongly nonlinear coupling and parametric vibration. With the nonlinear cou- pling of inertia in the equations of motion of the system being removed by successive elimination, the non-Gaussian moment equation method (NGM) is applied and 69 moment equations are integrated with central cumulative truncation technique. The stochastic central difference-cum-statistical linearization method(SCD-SL) and the digital simulation method(DSM) are also used. A comparison of results by different methods are given and the SCD-SL method is the most efficient method.

Friction moment analysis of space gyroscope bearing with ribbon cage under ultra-low oscillatory motion

This paper presents the model of calculating the total friction moment of space gyroscope ball bearings which usually work under ultra-low oscillatory motion and are very sensitive to the friction moment. The aim is to know the proportion of the friction moment caused by each frictional source in the bearing＇s total friction moment, which is helpful to optimize the bearing design to deduce the friction moment. In the model, the cage dynamic equations considering six degree-of-freedom and the balls dynamic equations considering two degree-of-freedom were solved.The good trends with different loads between the measured friction moments and computational results prove that the model under constant rate was validated. The computational results show that when the speed was set at 5 r/min, the bearing＇s maximum total friction moment when oscillation occurred was obviously larger than that occurred at a constant rate. At the onset of each oscillatory motion, the proportion of the friction moment caused by cage in the bearing＇s total friction moment was very high, and it increased with the increasing speed. The analyses of different cage thicknesses and different clearances between cage pocket and ball show that smaller thickness and clearance were preferred.