Nonlinear system stochastic response determination via fractional equivalent linearization and Karhunen–Loève expansion

A practical approach is developed for analyzing the spectral response of a nonlinear system subjected to both parametric and external Gaussian white noise excitations. The technique is implemented through the combined methods of equivalent external excitation and equivalent linearization to derive an equivalent linear system under equivalent external noise excitation. The spectral response is then obtained through utilizing the input/output spectral relation and covariance matching condition. A parametric noise excited linear system, Duffing oscillator, and nonlinear system with hysteretic nonlinearity are selected for investigation. The validity of the proposed method for analyzing spectral response is further supported by some analytical solutions and FFT technique through Monte Carlo simulations.

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Stochastic response of nonlinear system in probability domain

Sadhana ◽

10.1007/bf02716780 ◽

2006 ◽

Vol 31 (4) ◽

pp. 325-342

Author(s):

Deepak Kumar ◽

T. K. Datta

Keyword(s):

Nonlinear System ◽

Stochastic Response ◽

Probability Domain

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On using block pulse transform to perform equivalent linearization for a nonlinear Van der Pol oscillator under stochastic excitation

Nonlinear Engineering ◽

10.1515/nleng-2015-0022 ◽

2016 ◽

Vol 0 (0) ◽

Author(s):

Amir Younespour ◽

Hosein Ghaffarzadeh

Keyword(s):

Nonlinear System ◽

Present Method ◽

Gaussian White Noise ◽

Van Der Pol Oscillator ◽

Equivalent Linearization ◽

Van Der Pol ◽

White Noise Excitation ◽

Linearization Methods ◽

The Mean ◽

Methods Numerical

AbstractThis paper applied the idea of block pulse (BP) transform in the equivalent linearization of a nonlinear system. The BP transform gives effective tools to approximate complex problems. The main goal of this work is on using BP transform properties in process of linearization. The accuracy of the proposed method compared with the other equivalent linearization including the stochastic equivalent linearization and the regulation linearization methods. Numerical simulations are applied to the nonlinear Van der Pol oscillator system under Gaussian white noise excitation to demonstrate the feasibility of the present method. Different values of nonlinearity are considered to show the effectiveness of the present method. Besides, by comparing the mean-square responses for divers values of nonlinearity and excitation intensity depicted the present method is able to approximate the behavior of nonlinear system and is in agreement with other methods.

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An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems

Journal of Applied Mechanics ◽

10.1115/1.4030890 ◽

2015 ◽

Vol 82 (10) ◽

Cited By ~ 37

Author(s):

Ioannis A. Kougioumtzoglou ◽

Alberto Di Matteo ◽

Pol D. Spanos ◽

Antonina Pirrotta ◽

Mario Di Paola

Keyword(s):

Path Integral ◽

Computational Cost ◽

High Dimensional ◽

System Response ◽

Stochastic Response ◽

Mdof Systems ◽

Low Dimensional ◽

Response Determination ◽

Efficiency And Reliability

The recently developed approximate Wiener path integral (WPI) technique for determining the stochastic response of nonlinear/hysteretic multi-degree-of-freedom (MDOF) systems has proven to be reliable and significantly more efficient than a Monte Carlo simulation (MCS) treatment of the problem for low-dimensional systems. Nevertheless, the standard implementation of the WPI technique can be computationally cumbersome for relatively high-dimensional MDOF systems. In this paper, a novel WPI technique formulation/implementation is developed by combining the “localization” capabilities of the WPI solution framework with an appropriately chosen expansion for approximating the system response PDF. It is shown that, for the case of relatively high-dimensional systems, the herein proposed implementation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard WPI technique and an MCS approach. Several numerical examples are included, whereas comparisons with pertinent MCS data demonstrate the efficiency and reliability of the technique.

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