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Nonlinear oscillators subjected to colored Gaussian/non-Gaussian excitations are modelled through a set of three coupled first-order stochastic differential equations by representing the excitation as a first-order filtered white noise. A 3-D finite element (FE) formulation is developed to solve the corresponding 3-D Fokker Planck (FP) equations. The joint probability density functions of the state variables, obtained as a solution of the FP equation, are typically non-Gaussian and are used for computing the crossing statistics of the response - an essential metric for time variant reliability analysis. The method is illustrated through a noisy Lorenz attractor and a Duffing oscillator subjected to additive colored noise. The increase in state-space dimension when the Duffing oscillator is additionally excited with a parametric Gaussian noise is effectively handled by using stochastic averaging to reduce the state-space dimension. Investigations are carried out to examine the accuracy of the FE method vis-a-vis Monte Carlo simulations. The proposed method is observed to be computationally significantly cheaper for these three problems. � 2014 Elsevier Ltd. All rights reserved.

This paper proposes a derivative-free two-stage extended Kalman filter (2-EKF) especially suited for state and parameter identification of mechanical oscillators under Gaussian white noise. Two sources of modeling uncertainties are considered: (1) errors in linearization, and (2) an inadequate system model. The state vector is presently composed of the original dynamical/parameter states plus the so-called bias states accounting for the unmodeled dynamics. An extended Kalman estimation concept is applied within a framework predicated on explicit and derivative-free local linearizations (DLL) of nonlinear drift terms in the governing stochastic differential equations (SDEs). The original and bias states are estimated by two separate filters; the bias filter improves the estimates of the original states. Measurements are artificially generated by corrupting the numerical solutions of the SDEs with noise through an implicit form of a higher-order linearization. Numerical illustrations are provided for a few single- and multidegree-of-freedom nonlinear oscillators, demonstrating the remarkable promise that 2-EKF holds over its more conventional EKF-based counterparts. DOI: 10.1061/(ASCE)EM.1943-7889.0000255. (C) 2011 American Society of Civil Engineers.

Chaotic motion of a harmonically excited square prism modelled as a Duffing oscillator and kept in fluid flow is considered. The fluid dynamic forces contribute additional non-linear terms to the inherent non-linearity of the system. The harmonically excited oscillator without the fluid flow exhibits three types of steady state periodic motions, dual period 1 motions and period 3 motion without dual. The flow changes the nature of motion of the oscillator. With the flow velocity as bifurcation parameter, the system exhibits period 1 motion without dual, bifurcating into dual period 1 motions which period double into dual period 2, 4, 8 motions, etc., leading to chaos as the flow velocity is increased. Lyapunov exponents for different flow velocities are computed which show the bifurcation points. The harmonic balance method is used to obtain approximate solutions for the periodic motions and to predict the period doubling bifurcations by a stability analysis.

The active control of flow-induced oscillations, specifically, vortex-induced oscillations of circular cylinders and galloping oscillations of circular cylinders and galloping oscillations of a square prism, are considered. In the case of vortex-induced oscillations, the vibrating cylinder is modeled as a single-degree-of-freedom (sdof) linear damped oscillator and the fluid motion as a Rayleigh oscillator. Galloping oscillations are modeled as a sdof limit-cycle system with cubic and higher order non-linearities in the velocity term. In formulating the control method it is assumed that the fluid interaction is an external excitation/disturbance on the structure. A suitable disturbance model is chosen based on the spectral content of the disturbance. For the problems considered here, the disturbance model assumes the form of a harmonic oscillator the frequency of which, in the case of vortex-induced oscillations, is equal to the natural frequency of the structure or the "lock-in" frequency, and in the case of galloping oscillations, is equal to the limit-cycle frequency. A disturbance counteracting control law is synthesized by using a Luenberger observer incorporating the structure and disturbance dynamics. It is shown that using this approach one is able to completely suppress the fluid-induced oscillations of the structure. ? 1993 Academic Press. All rights reserved.