Equivalent Nonlinear System Method for Stochastically Excited Hamiltonian Systems

1994 ◽  
Vol 61 (3) ◽  
pp. 618-623 ◽  
Author(s):  
W. Q. Zhu ◽  
T. T. Soong ◽  
Y. Lei
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
White Noise ◽  
Probability Density ◽  
Hamiltonian Systems ◽  
Gaussian White Noise ◽  
Energy Balancing ◽  
Dissipation Energy ◽  
System Method ◽  
Approximate Probability

An equivalent nonlinear system method is presented to obtain the approximate probability density for the stationary response of multi-degree-of-freedom nonlinear Hamiltonian systems to Gaussian white noise parametric and/or external excitations. The equivalent nonlinear systems are obtained on the basis of one of the following three criteria: least mean-squared deficiency of damping forces, dissipation energy balancing, and least mean-squared deficiency of dissipation energies. An example is given to illustrate the application and validity of the method and the differences in the three equivalence criteria.

Equivalent Nonlinear System Method for Stochastically Excited and Dissipated Integrable Hamiltonian Systems

1997 ◽  
Vol 64 (1) ◽  
pp. 209-216 ◽  
Author(s):  
W. Q. Zhu ◽  
Y. Lei
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Hamiltonian Systems ◽  
Energy Balancing ◽  
Integrable Hamiltonian Systems ◽  
Dissipation Energy ◽  
System Method ◽  
Special Cases ◽  
Approximate Probability ◽  
White Noises

An equivalent nonlinear system method is proposed to obtain the approximate probability density for the stationary response of multi-degree-of-freedom integrable Hamiltonian systems with linear and (or) nonlinear dampings and subject to external and (or) parametric excitations of Gaussian white noises. The equivalent nonlinear systems are obtained on the basis of one of the following three criteria: least mean-squared difference in damping forces, dissipation energy balancing, or least mean-squared difference in dissipation energies. Two examples are given to illustrate the application and validity of the method and the differences in the three equivalence criteria. The method is also extended to a more general class of systems which include the stochastically excited and dissipated integrable Hamiltonian systems as special cases.

Exact Stationary Response Solution for Second Order Nonlinear Systems Under Parametric and External White Noise Excitations: Part II

1988 ◽  
Vol 55 (3) ◽  
pp. 702-705 ◽  
Author(s):  
Y. K. Lin ◽  
Guoqiang Cai
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Probability Distribution ◽  
White Noise ◽  
Probability Density ◽  
Detailed Balance ◽  
Stationary Probability ◽  
Probability Flow ◽  
Parametric And External Excitations ◽  
White Noises

A systematic procedure is developed to obtain the stationary probability density for the response of a nonlinear system under parametric and external excitations of Gaussian white noises. The procedure is devised by separating the circulatory portion of the probability flow from the noncirculatory flow, thus obtaining two sets of equations that must be satisfied by the probability potential. It is shown that these equations are identical to two of the conditions established previously under the assumption of detailed balance; therefore, one remaining condition for detailed balance is superfluous. Three examples are given for illustration, one of which is capable of exhibiting limit cycle and bifurcation behaviors, while another is selected to show that two different systems under two differents sets of excitations may result in the same probability distribution for their responses.

Stochastic Averaging of Quasi-Nonintegrable-Hamiltonian Systems

1997 ◽  
Vol 64 (1) ◽  
pp. 157-164 ◽  
Author(s):  
W. Q. Zhu ◽  
Y. Q. Yang
Keyword(s):  
Nonlinear System ◽  
Probability Density ◽  
Hamiltonian Systems ◽  
Present Method ◽  
Transition Probability ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Kolmogorov Equation ◽  
Transition Probability Density ◽  
System Method

A stochastic averaging method is proposed to predict approximately the response of multi-degree-of-freedom quasi-nonintegrable-Hamiltonian systems (nonintegrable Hamiltonian systems with lightly linear and (or) nonlinear dampings and subject to weakly external and (or) parametric excitations of Gaussian white noises). According to the present method, a one-dimensional approximate Fokker-Planck-Kolmogorov equation for the transition probability density of the Hamiltonian can be constructed and the probability density and statistics of the stationary response of the system can be readily obtained. The method is compared with the equivalent nonlinear system method for stochastically excited and dissipated nonintegrable Hamiltonian systems and extended to a more general class of systems. An example is given to illustrate the application and validity of the present method and the consistency of the present method and the equivalent nonlinear system method.

On Randomly Excited Hysteretic Structures

1990 ◽  
Vol 57 (2) ◽  
pp. 442-448 ◽  
Author(s):  
G. Q. Cai ◽  
Y. K. Lin
Keyword(s):  
Energy Dissipation ◽  
Narrow Band ◽  
Probability Distributions ◽  
Gaussian White Noise ◽  
Energy Balancing ◽  
Dissipation Energy ◽  
System Parameters ◽  
Hysteretic Systems ◽  
Approximate Probability ◽  
Simulation Results

Approximate probability distributions of certain response variables are obtained for hysteretic systems under Gaussian white-noise excitations. The approximate method used is a generalization of a dissipation-energy-balancing procedure, developed previously for nonlinear but basically nonhysteretic systems. Some new issues related particularly to hysteresis models are explained and resolved. The method is applicable to either bilinear or smooth-type hysteresis without the restriction that the response be a narrow-band process or the energy dissipation be small. Comparison of computed results with available simulation results indicates that the proposed method is accurate for wide ranges of excitation levels and system parameters.

scholarly journals EMD OF GAUSSIAN WHITE NOISE: EFFECTS OF SIGNAL LENGTH AND SIFTING NUMBER ON THE STATISTICAL PROPERTIES OF INTRINSIC MODE FUNCTIONS

2009 ◽  
Vol 01 (04) ◽  
pp. 517-527 ◽  
Author(s):  
GASTÓN SCHLOTTHAUER ◽  
MARÍA EUGENIA TORRES ◽  
HUGO L. RUFINER ◽  
PATRICK FLANDRIN
Keyword(s):  
White Noise ◽  
Probability Density ◽  
Kernel Smoothing ◽  
Gaussian White Noise ◽  
Large Data ◽  
Intrinsic Mode Functions ◽  
Mode Decomposition ◽  
Non Gaussian ◽  
Data Length ◽  
Number Of Iterations

This work presents a discussion on the probability density function of Intrinsic Mode Functions (IMFs) provided by the Empirical Mode Decomposition of Gaussian white noise, based on experimental simulations. The influence on the probability density functions of the data length and of the maximum allowed number of iterations is analyzed by means of kernel smoothing density estimations. The obtained results are confirmed by statistical normality tests indicating that the IMFs have non-Gaussian distributions. Our study also indicates that large data length and high number of iterations produce multimodal distributions in all modes.

scholarly journals Analysis of exceedance probability of displacement response of randomly nonlinear structures

2000 ◽  
Vol 22 (4) ◽  
pp. 212-224 ◽  
Author(s):  
Luu Xuan Hung
Keyword(s):  
Probability Density Function ◽  
Nonlinear Systems ◽  
White Noise ◽  
Probability Density ◽  
Density Function ◽  
Exceedance Probability ◽  
Nonlinear Structures ◽  
Exact Probability ◽  
Displacement Response ◽  
Linearization Techniques

The paper presents the estimation of the exact exceedance probability (EEP) of stationary responses of some white noise-randomly excited nonlinear systems whose exact probability density function can be known. Consequently, the approximate exceedance probabilities (AEPs) are evaluated based on the analysis of equivalent linearized systems using the traditional Caughey method and the extension technique of LOMSEC. Comparisons of the AEPs versus the EEP are demonstrated. The obtained results indicate important characters of the exceedance probability (EP) as well as the influence of nonlinearity over EP. The evaluation of the applied possibility of the proposed linearization techniques for estimating AEPs are made.

scholarly journals Description of the macroscopic dynamics of populations of phase elements with white non-Gaussian noise based on the circular cumulant approach

2021 ◽  
pp. 05-12
Author(s):  
A. V. Dolmatova ◽  
◽  
I. V. Tiulkina ◽  
D. S. Goldobin ◽  
◽  
...  
Keyword(s):  
White Noise ◽  
Probability Density ◽  
Gaussian Noise ◽  
Gaussian White Noise ◽  
Macroscopic Description ◽  
Reduced Equations ◽  
Non Gaussian ◽  
Dynamics Of Populations ◽  
Stable Noise ◽  
Good Agreement

We use the method of circular cumulants, which allows us to construct a low-mode macroscopic description of the dynamics of populations of phase elements subject to non-Gaussian white noise. In this work, we have obtained two-cumulant reduced equations for alpha-stable noise. The application of the approach is demonstrated for the case of the Kuramoto ensemble with non-Gaussian noise. The results of numerical calculations for the ensemble of N = 1500 elements, the numericalsimulation of the chain of equations for the Kuramoto–Daido order parameters (Fourier modes of the probability density) with 200 terms (in the thermodynamic limit of an infinitely large ensemble) and the theoretical solution on the basis of the two-cumulant approximation are in good agreement with each other.

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