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

2007 ◽  
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.

A sequential importance sampling filter with a new proposal distribution for state and parameter estimation of nonlinear dynamical systems

2007 ◽  
Vol 464 (2089) ◽  
pp. 25-47 ◽  
Author(s):  
Shuva J Ghosh ◽  
C.S Manohar ◽  
D Roy
Keyword(s):  
Dynamical Systems ◽  
Parameter Identification ◽  
Importance Sampling ◽  
Nonlinear Dynamical Systems ◽  
Gaussian White Noise ◽  
Sequential Importance Sampling ◽  
State Variables ◽  
Nonlinear Dynamical ◽  
Taylor Expansions ◽  
Non Gaussian

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.

Response Statistics and Reliability Analysis of a Mistuned and Frictionally Damped Bladed Disk Assembly Subjected to White Noise Excitation

2010 ◽  
Author(s):  
Pankaj Kumar ◽  
S. Narayanan
Keyword(s):  
White Noise ◽  
Probability Density ◽  
Gaussian White Noise ◽  
Random Excitation ◽  
Forced Response ◽  
Computationally Efficient ◽  
Bladed Disk ◽  
White Noise Excitation ◽  
Bladed Disk Assembly ◽  
Mean Square Response

In the design of gas turbine engines, the analysis of nonlinear vibrations of mistuned and frictionally damped blade-disk assembly subjected to random excitation is highly complex. 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 important improvement and extensions to a computationally efficient higher order, finite difference (FD) technique for the solution of higher dimensional FP equation corresponding to a two degree of freedom nonlinear system representative of vibration of tip shrouded frictionally damped bladed disk assembly subjected to Gaussian white noise excitation. Effects of friction damping on the mean square response of a blade are investigated. The friction coefficient of the damper is assumed to be a function of the sliding velocity of the contact surface. The effects of stiffness and damping mistuning on the forced response of frictionally damped bladed disk are investigated. Numerical studies are presented for a pair of mistuned blades of cyclic assemblies. The response and reliability of a blade subjected to random excitation is also obtained. With time averaged probability density as an invariant measure, the probability of large excursion in case of damping mistuning is also presented. The results of the FD method are validated by comparing with Monte Carlo Simulation (MCS) results.

scholarly journals Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

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