Nonstationary solutions of nonlinear dynamical systems excited by Gaussian white noise

The problem of estimating parameters of nonlinear dynamical systems based on incomplete noisy measurements is considered within the framework of Bayesian filtering using Monte Carlo simulations. The measurement noise and unmodelled dynamics are represented through additive and/or multiplicative Gaussian white noise processes. Truncated Ito–Taylor expansions are used to discretize these equations leading to discrete maps containing a set of multiple stochastic integrals. These integrals, in general, constitute a set of non-Gaussian random variables. The system parameters to be determined are declared as additional state variables. The parameter identification problem is solved through a new sequential importance sampling filter. This involves Ito–Taylor expansions of nonlinear terms in the measurement equation and the development of an ideal proposal density function while accounting for the non-Gaussian terms appearing in the governing equations. Numerical illustrations on parameter identification of a few nonlinear oscillators and a geometrically nonlinear Euler–Bernoulli beam reveal a remarkably improved performance of the proposed methods over one of the best known algorithms, i.e. the unscented particle filter.

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The Response Statistics, Jump and Bifurcation of Nonlinear Dynamical Systems Subjected to White Noise and Combined Sinusoidal and White Noise Excitation

Volume 5: Turbo Expo 2007 ◽

10.1115/gt2007-28102 ◽

2007 ◽

Cited By ~ 1

Author(s):

Pankaj Kumar ◽

S. Narayanan

Keyword(s):

Nonlinear Systems ◽

White Noise ◽

Nonlinear Dynamical Systems ◽

Gaussian White Noise ◽

Nonlinear Dynamical ◽

White Noise Excitation ◽

Bladed Disk Assembly ◽

Noise Frequency ◽

System Nonlinearity ◽

Non Gaussian

The prediction of the response of nonlinear systems subjected to stochastic parametric, narrowband and wideband or coloured external excitation is of importance in the field of structural and rotor dynamics. The transitional probability density function (pdf) for the random response of nonlinear systems under white or coloured noise excitation (delta-correlated) is governed by both the forward Fokker-Planck (FP) and backward Kolmogorov equations. This paper presents efficient numerical solution of the FP equation for the pdf of response for general nonlinear systems subjected to external white noise and combined sinusoidal and white noise excitation. The effect of intensity of white noise, frequency and amplitude of sinusoidal excitation and level of system nonlinearity on the non-Gaussian nature of response caused by the system nonlinearity are investigated. Stochastic behaviours like stability, jump, bifurcation are examined as the system parameters change. The finite element (FE) scheme is used to solve the FP equation and obtain the statistics of a two degree-of-freedom linear system representative of the vibration of gas turbine tip-shrouded bladed disk assembly subjected to Gaussian white noise excitation as an illustrative example.

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