Non-Gaussian Response of Nonlinear Oscillators with Fourth-Order Internal Resonance

1988 ◽  
pp. 335-350
Author(s):  
R. A. Ibrahim ◽  
A. Soundararajan
Keyword(s):  
Fourth Order ◽  
Internal Resonance ◽  
Nonlinear Oscillators ◽  
Non Gaussian

scholarly journals DETERMINING SOURCE CUMULANTS IN FEMTOSCOPY WITH GRAM–CHARLIER AND EDGEWORTH SERIES

2011 ◽  
Vol 26 (24) ◽  
pp. 1771-1782 ◽  
Author(s):  
H. C. EGGERS ◽  
M. B. DE KOCK ◽  
J. SCHMIEGEL
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Fourth Order ◽  
Momentum Space ◽  
Crucial Role ◽  
Gaussian Distributions ◽  
Edgeworth Series ◽  
The Central Limit Theorem ◽  
Non Gaussian

Lowest-order cumulants provide important information on the shape of the emission source in femtoscopy. For the simple case of noninteracting identical particles, we show how the fourth-order source cumulant can be determined from measured cumulants in momentum space. The textbook Gram–Charlier series is found to be highly inaccurate, while the related Edgeworth series provides increasingly accurate estimates. Ordering of terms compatible with the Central Limit Theorem appears to play a crucial role even for non-Gaussian distributions.

Some measurements of multi-point time correlations in grid turbulence

1970 ◽  
Vol 41 (1) ◽  
pp. 169-178 ◽  
Author(s):  
C. W. Van Atta ◽  
T. T. Yeh
Keyword(s):  
Fourth Order ◽  
High Speed ◽  
Longitudinal Velocity ◽  
Joint Probability ◽  
Grid Turbulence ◽  
Odd Order ◽  
Probability Densities ◽  
Even Order ◽  
Point Correlation ◽  
Non Gaussian

Three-point odd-order correlations and four-point even-order correlations of the longitudinal velocity fluctuations in grid-generated turbulence have been measured using linearized hot-wire anemometry, digital sampling, and a high-speed digital computer. The measured correlations are compared with relations between higher-order correlations corresponding to non-Gaussian Gram-Charlier joint probability densities for three and four variables. The fourth-order, three-point Gram-Charlier distribution accurately describes the relation between measured odd-order three-point correlations. The measured fourth-order even-order correlations may be accurately predicted from the two-point correlation using Millionshtchikov's joint-Gaussian hypothesis, except for small values of the separations. The disagreement at small separations cannot be reduced through use of the Gram-Charlier approximation.

A method for estimating rainflow fatigue damage of narrowband non-Gaussian random loadings

2014 ◽  
Vol 228 (14) ◽  
pp. 2459-2468 ◽  
Author(s):  
HW Cheng ◽  
JY Tao ◽  
X Chen ◽  
Y Jiang
Keyword(s):  
Fatigue Damage ◽  
Fourth Order ◽  
Analytical Solutions ◽  
Estimation Methods ◽  
Accurate Estimation ◽  
Damage Estimation ◽  
Random Loading ◽  
New Approach ◽  
Computational Errors ◽  
Non Gaussian

We describe efforts to improve the accuracy of fatigue damage estimation methods of narrowband non-Gaussian random loading. The available analytical solutions are reviewed and briefly summarized, and the reasons for the occurrence of computational errors during nonlinear transformation-based methods are determined. The computational errors are mainly due to inconsistencies in the statistical moments above fourth order. A new approach is proposed for the evaluation of rainflow fatigue damage. This approach avoids the problem of transformation-based methods and provides accurate estimation for fatigue damage of narrowband leptokurtic non-Gaussian random loading. Additionally, the applicability of the proposed method to Gaussian random loading is investigated. Finally, two examples are carried out and comparisons are made to more commonly used methods to demonstrate the capabilities and brevity of the proposed algorithm.

Methods and Gaussian Criterion for Statistical Linearization of Stochastic Parametrically and Externally Excited Nonlinear Systems

1989 ◽  
Vol 56 (1) ◽  
pp. 179-185 ◽  
Author(s):  
R. J. Chang ◽  
G. E. Young
Keyword(s):  
Nonlinear Systems ◽  
Goodness Of Fit ◽  
Duffing Oscillator ◽  
A Priori ◽  
Nonlinear Oscillators ◽  
Statistical Linearization ◽  
Goodness Of Fit Test ◽  
Chi Square ◽  
Non Gaussian ◽  
Parametric And External Excitations

The methods of Gaussian linearization along with a new Gaussian Criterion used in the prediction of the stationary output variances of stable nonlinear oscillators subjected to both stochastic parametric and external excitations are presented. The techniques of Gaussian linearization are first derived and the accuracy in the prediction of the stationary output variances is illustrated. The justification of using Gaussian linearization a priori is further investigated by establishing a Gaussian Criterion. The non-Gaussian effects due to system nonlinearities and/or large noise intensities in a Duffing oscillator are also illustrated. The validity of employing the Gaussian Criterion test for assuring accuracy of Gaussian linearization is supported by performing the Chi-square Gaussian goodness-of-fit test.

scholarly journals Backbone curves, Neimark-Sacker boundaries and appearance of quasi-periodicity in nonlinear oscillators: application to 1:2 internal resonance and frequency combs in MEMS

Meccanica ◽  
2021 ◽  
Author(s):  
Giorgio Gobat ◽  
Louis Guillot ◽  
Attilio Frangi ◽  
Bruno Cochelin ◽  
Cyril Touzé
Keyword(s):  
Mechanical System ◽  
Internal Resonance ◽  
Boundary Curve ◽  
Nonlinear Oscillators ◽  
Conservative System ◽  
Micro Electro Mechanical System ◽  
Frequency Combs ◽  
Limit Value ◽  
Families Of Periodic Orbits ◽  
Backbone Curves

AbstractQuasi-periodic solutions can arise in assemblies of nonlinear oscillators as a consequence of Neimark-Sacker bifurcations. In this work, the appearance of Neimark-Sacker bifurcations is investigated analytically and numerically in the specific case of a system of two coupled oscillators featuring a 1:2 internal resonance. More specifically, the locus of Neimark-Sacker points is analytically derived and its evolution with respect to the system parameters is highlighted. The backbone curves, solution of the conservative system, are first investigated, showing in particular the existence of two families of periodic orbits, denoted as parabolic modes. The behaviour of these modes, when the detuning between the eigenfrequencies of the system is varied, is underlined. The non-vanishing limit value, at the origin of one solution family, allows explaining the appearance of isolated solutions for the damped-forced system. The results are then applied to a Micro-Electro-Mechanical System-like shallow arch structure, to show how the analytical expression of the Neimark-Sacker boundary curve can be used for rapid prediction of the appearance of quasiperiodic regime, and thus frequency combs, in Micro-Electro-Mechanical System dynamics.

Phase-Locking and Chaotic Phenomena of Circular Mesh Antenna With 1:3 Internal Resonance

2019 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yan Zheng ◽  
Xiangying Guo
Keyword(s):  
Fourth Order ◽  
Internal Resonance ◽  
Chaotic Dynamics ◽  
Circular Cylindrical Shell ◽  
Phase Locking ◽  
State Parameter ◽  
The Finite Element Method ◽  
First Order ◽  
Intermittent Chaos ◽  
Locking Phenomena

Abstract We study chaotic dynamics and the phase-locking phenomenon of the circular mesh antenna with 1:3 internal resonance subjected to the temperature excitation in this paper. Firstly, the frequencies and modes of the circular mesh antenna are analyzed by the finite element method, it is found that there is an approximate threefold relationship between the first-order and the fourth-order vibrations of the circular mesh antenna. Considering a composite laminated circular cylindrical shell clamped along a generatrix and with the radial pre-stretched membranes at both ends subjected to the temperature excitation, we study the nonlinear dynamic behaviors of the equivalent circular mesh antenna model based on the fourth-order Runge-Kutta algorithm, which are described by the bifurcation diagrams, waveforms, phase plots and Poincaré maps in the state-parameter space. It is found that there appear the Pomeau-Manneville type intermittent chaos. According to the topology evolution of phase trajectories, the phase-locking phenomena are found.

Monte Carlo Simulation of Coupled Nonlinear Oscillators Under Random Excitations

1990 ◽  
Vol 57 (4) ◽  
pp. 1097-1099 ◽  
Author(s):  
Wenlung Li ◽  
R. A. Ibrahim
Keyword(s):  
Numerical Simulation ◽  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Nonlinear Interaction ◽  
Internal Resonance ◽  
Initial Conditions ◽  
Non Gaussian ◽  
Energy Absorbing ◽  
Three Degree Of Freedom ◽  
Gaussian Closure

The main objectives of this note are to examine the random response of nonlinear three degree-of-freedom systems in the neighborhood of combination internal resonance by using Monte Carlo simulation and to compare the results with those obtained by first-order non-Gaussian closure. The numerical simulation is found to support the main features of the nonlinear interaction in the neighborhood of internal resonance conditions. For example, the nonlinear interaction takes place in the form of a randomly continuous energy exchange between the modes involved. In addition, the results verify the existence of energy absorbing effect as predicted by the non-Gaussian closure method. While the non-Gaussian closure exhibits regions of multiple solutions in the neighborhood of exact internal resonance, the numerical simulation gives only one solution depending on the assigned initial conditions. This observation requires further investigation to establish the domains of attraction in stochastic nonlinear dynamics.

Sign in / Sign up

Export Citation Format

Share Document