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

Lin, Yuqing; Cai, Guoqiang

doi:10.1115/1.3125852

Cited by 85 publications

(29 citation statements)

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“…It is now intended to turn the attention to a class of non-linear SDOF problems with known stationary probability density obtainable through exact solutions of reduced FokkerPlanck equations [29,30]. While relative merits of the HWSNM over EM and SH methods have so far been numerically established in the context of known response statistics of whitenoise-driven linear SDOF oscillators, the latter two methods have considerably lower orders of convergence.…”

Section: Numerical Examplesmentioning

confidence: 99%

A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

Roy

2006

Int. J. Numer. Meth. Engng

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SUMMARYEngineers are often more concerned with the computation of statistical moments (or mathematical expectations) of the response of stochastically driven dynamical systems than with the determination of path-wise response histories. With this in perspective, weak stochastic solutions of dynamical systems, modelled as n degrees-of-freedom (DOF) mechanical oscillators and driven by additive and/or multiplicative white noise (or, filtered white noise) processes, are considered in this study. While weak stochastic solutions are simpler and quicker to compute than strong (sample path-wise) solutions, it must be emphasized that sample realizations of weak solutions have no path-wise similarity with strong solutions. However, the statistical moment of any continuous and sufficiently differentiable deterministic function of the weak stochastic response is 'close' to that of the true response (if it exists) within a certain order of a given time step size. Computation of weak response therefore assumes great significance in the context of simulations of stochastically driven dynamical systems (oscillators) of engineering interest. To efficiently generate such weak responses, a novel class of weak stochastic Newmark methods (WSNMs), based on implicit Ito-Taylor expansions of displacement and velocity fields, is proposed. The resulting multiple stochastic integrals (MSIs) in these expansions are replaced by a set of random variables with considerably simpler and discrete probability densities. In fact, yet another simplifying feature of the present strategy is that there is no need to model and compute some of the higher-level MSIs. Estimates of error orders of these methods in terms of a given time step size are derived and a proof of global convergence provided. Numerical illustrations are provided and compared with exact solutions whenever available to demonstrate the accuracy, simplicity and higher computational speed of WSNMs vis-à-vis a few other popularly used stochastic integration schemes as well as the path-wise versions of the stochastic Newmark scheme.

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Section: Numerical Examplesmentioning

confidence: 99%

A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

Roy

2006

Int. J. Numer. Meth. Engng

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“…(9-1) and (9-2). Much success has been achieved in finding exact solutions for second order nonlinear stochastic systems based on this solution technique [1,6]. Thus, the present paper is focused on higher order systems; in particular, higher order Hamiltonian systems.…”

Section: Generalized Stationary Potentialmentioning

confidence: 99%

“…A more general case was considered by Yong and Un [11] where a systematic procedure ~as given to obtain the exact solutions under an assumption called detailed balance. Further extension was made by Un and Cai [6] and Cai and Un [1] to systems belonging to the class of generalized stationary potential, which may or may not be in detailed balance. Additional exact solutions were obtained by Zhu [12], Soize [9], and Zhu, Cai and Un [13].…”

Section: Introductionmentioning

confidence: 97%

Stochastic Excited Hamiltonian Systems

Zhu

Cai

1992

Nonlinear Stochastic Mechanics

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SummaryThe method of gent;ralized stationary potential is applied to obtain exact probability densities for multi-degree-of-freedom nonlinear Hamiltonian systems excited by Gaussian white noises. The excitations can be additive, or multiplicative, or both, and governing equations are more general than those previously reported in the literature. Further extension is made to a still more general class, which include the Hamiltonian systems as special cases.

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“…Zhu ( )single-degree-of-freedom (SDOF) systems. Even in the case of the SDOF systems, only a few stationary PDF solutions have been obtained in some special cases by solving the associated Fokker-Planck-Kolmogorov (FPK) equation (Caughey and Ma 1982;Dimentberg 1982;Lin and Cai 1988;Vasta 1995;Proppe 2002aProppe , 2003. Most problems need some approximation methods, such as perturbation method (Roberts 1972;Cai and Lin 1992), finite element method (Langley 1985), PetrovGalerkin method (Köylüoglu et al 1994), cell-to-cell mapping (path integration) technique (Köylüoglu et al 1995;Di Paola and Santoro 2008), and finite difference approach (Wojtkiewicz et al 1999).…”

Section: Introductionmentioning

confidence: 98%

Approximate Probability Density Function Solution of Multi-Degree-of-Freedom Coupled Systems Under Poisson Impulses

Zhu

2014

Numerical Methods for Reliability and Safety Assessment

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A solution procedure is proposed to approximate the probability density function (PDF) solution of high-dimensional non-linear systems under Poisson impulses. The PDF solution yields the generalized Fokker-Planck-Kolmogorov (FPK) equation. First a state-space-split method is proposed to reduce the highdimensional generalized FPK equation to a low dimensional equation. After that, the exponential-polynomial closure method is further adopted to solve the reduced FPK equation for the PDF solution. In order to show the effectiveness of the proposed solution procedure, a two-degree-of-freedom coupled pitch-roll ship motion system and a 10-degree-of-freedom mass-spring-damper system are investigated, respectively. Compared to the simulated results, the proposed solution procedure is effective to obtain the PDF solution, especially in the tail region which is very important for reliability analysis.

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

Cited by 85 publications

References 0 publications

A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

Stochastic Excited Hamiltonian Systems

Approximate Probability Density Function Solution of Multi-Degree-of-Freedom Coupled Systems Under Poisson Impulses

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