Enhanced stochastic averaging of non-integrable nonlinear systems subjected to stochastic excitations

2018 ◽  
Vol 113 ◽  
pp. 256-264 ◽  
Author(s):  
Omar El-Khoury ◽  
Abdollah Shafieezadeh
Keyword(s):  
Nonlinear Systems ◽  
Stochastic Averaging ◽  
Stochastic Excitations

Calculation of Nonlinear Systems Under Narrow Band Excitation Using Equivalent Linearization and Path Continuation

2021 ◽  
Author(s):  
Alwin Förster ◽  
Lars Panning-von Scheidt
Keyword(s):  
Nonlinear Systems ◽  
Narrow Band ◽  
Gaussian White Noise ◽  
Random Excitation ◽  
Mean Value ◽  
Equivalent Linearization ◽  
Efficient Procedure ◽  
Stochastic Excitations ◽  
Wide Range ◽  
The One

Abstract Turbomachines experience a wide range of different types of excitation during operation. On the structural mechanics side, periodic or even harmonic excitations are usually assumed. For this type of excitation there are a variety of methods, both for linear and nonlinear systems. Stochastic excitation, whether in the form of Gaussian white noise or narrow band excitation, is rarely considered. As in the deterministic case, the calculations of the vibrational behavior due to stochastic excitations are even more complicated by nonlinearities, which can either be unintentionally present in the system or can be used intentionally for vibration mitigation. Regardless the origin of the nonlinearity, there are some methods in the literature, which are suitable for the calculation of the vibration response of nonlinear systems under random excitation. In this paper, the method of equivalent linearization is used to determine a linear equivalent system, whose response can be calculated instead of the one of the nonlinear system. The method is applied to different multi-degree of freedom nonlinear systems that experience narrow band random excitation, including an academic turbine blade model. In order to identify multiple and possibly ambiguous solutions, an efficient procedure is shown to integrate the mentioned method into a path continuation scheme. With this approach, it is possible to track jump phenomena or the influence of parameter variations even in case of narrow band excitation. The results of the performed calculations are the stochastic moments, i.e. mean value and variance.

A Stochastic Averaging Approach for Feedback Control Design of Nonlinear Systems Under Random Excitations

2002 ◽  
Vol 124 (4) ◽  
pp. 561-565 ◽  
Author(s):  
O. Elbeyli ◽  
J. Q. Sun
Keyword(s):  
Feedback Control ◽  
Nonlinear Systems ◽  
Control Design ◽  
Stochastic Averaging ◽  
Moment Equation ◽  
Design Criteria ◽  
Numerical Examples ◽  
Rayleigh Approximation ◽  
Stochastic Controls ◽  
The Moment

This paper presents a method for designing and quantifying the performance of feedback stochastic controls for nonlinear systems. The design makes use of the method of stochastic averaging to reduce the dimension of the state space and to derive the Ito^ stochastic differential equation for the response amplitude process. The moment equation of the amplitude process closed by the Rayleigh approximation is used as a means to characterize the transient performance of the feedback control. The steady state and transient response of the amplitude process are used as the design criteria for choosing the feedback control gains. Numerical examples are studied to demonstrate the performance of the control.

scholarly journals The influence of nonlinear terms in mechanical systems having two degrees of freedom

2006 ◽  
Vol 28 (3) ◽  
pp. 155-164
Author(s):  
Nguyen Duc Tinh
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Degrees Of Freedom ◽  
Averaging Method ◽  
Duffing Oscillator ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Higher Order ◽  
White Noise Excitation ◽  
Two Degree Of Freedom

For many years the higher order stochastic averaging method has been widely used for investigating nonlinear systems subject to white and coloured noises to predict approximately the response of the systems. In the paper the method is further developed for two-degree-of-freedom systems subjected to white noise excitation. Application to Duffing oscillator is considered.

Survival Probability Determination of Nonlinear Oscillators Subject to Evolutionary Stochastic Excitation

2014 ◽  
Vol 81 (5) ◽  
Author(s):  
Pol D. Spanos ◽  
Ioannis A. Kougioumtzoglou
Keyword(s):  
Survival Probability ◽  
Computational Cost ◽  
Stochastic Averaging ◽  
Nonlinear Ordinary Differential Equation ◽  
Stochastic Excitation ◽  
Analytical Technique ◽  
Stochastic Excitations ◽  
Response Variance ◽  
Time Varying Frequency

A novel approximate analytical technique for determining the survival probability and first-passage probability density function (PDF) of nonlinear/hysteretic oscillators subject to evolutionary stochastic excitation is developed. Specifically, relying on a stochastic averaging/linearization treatment of the problem, approximate closed form expressions are derived for the oscillator nonstationary marginal, transition, and joint-response amplitude PDFs and, ultimately, for the time-dependent oscillator survival probability. The developed technique exhibits considerable versatility, as it can handle readily cases of oscillators exhibiting complex hysteretic behaviors as well as cases of evolutionary stochastic excitations with time-varying frequency contents. Further, it exhibits notable simplicity since, in essence, it requires only the solution of a first-order nonlinear ordinary differential equation (ODE) for the oscillator nonstationary response variance. Thus, the computational cost involved is kept at a minimum level. The classical hardening Duffing and the versatile Preisach (hysteretic) oscillators are considered in a numerical examples section, in which comparisons with pertinent Monte Carlo simulations data demonstrate the reliability of the proposed technique.

Optimal Time-Delay Control for Multi-Degree-of-Freedom Nonlinear Systems Excited by Harmonic and Wide-Band Noises

2021 ◽  
pp. 2150053
Author(s):  
Rongchun Hu ◽  
Qiangfeng Lü
Keyword(s):  
Time Delay ◽  
Nonlinear Systems ◽  
Control Strategy ◽  
Wide Band ◽  
Stochastic Averaging ◽  
Degree Of Freedom ◽  
Optimal Time ◽  
Time Delay Control ◽  
Strongly Nonlinear ◽  
Delay Control

In this paper, an optimal time-delay control strategy is designed for multi-degree-of-freedom (multi-DOF) strongly nonlinear systems excited by harmonic and wide-band noises. First, by using the generalized harmonic functions, a stochastic averaging method (SAM) is employed for the time-delay controlled strongly nonlinear system under combined harmonic and wide-band noise excitations, by which a set of partially averaged Itô equations are obtained. Then, by solving the dynamical programming equation associated with the partially averaged Itô equations, the optimal control law can be obtained. Finally, by solving the Fokker–Planck–Kolmogorov (FPK) equation, the responses of the optimally time-delay controlled system are predicted. The analytical results are compared with the Monte Carlo simulation to verify the effectiveness and efficiency of the proposed control strategy.

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