Random Vibrations with Impacts: A Review

Dimentberg, M. F.; Iourtchenko, D. V.

doi:10.1023/b:nody.0000045510.93602.ca

Cited by 153 publications

(67 citation statements)

References 26 publications

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“…We next consider the response of a SDOF oscillator constrained by two rigid barriers; this type of system is often referred to in the literature as a vibro-impact system [7]. Let X denote the stationary response of a vibro-impact system to a stationary Gaussian white noise with zero mean and one-sided spectral density 1/π; the driving noise is denoted by W .…”

Section: Vibro-impact Systemmentioning

confidence: 99%

Reliability of dynamic systems under limited information

Field

Grigoriu

2009

Probabilistic Engineering Mechanics

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A method is developed for reliability analysis of dynamic systems under limited information. The available information includes one or more samples of the system output; any known information on features of the output can be used if available. The method is based on the theory of non-Gaussian translation processes and is shown to be particularly suitable for problems of practical interest. For illustration, we apply the proposed method to a series of simple example problems and compare with results given by traditional statistical estimators in order to establish the accuracy of the method. It is demonstrated that the method delivers accurate results for the case of linear and nonlinear dynamic systems, and can be applied to analyze experimental data and/or mathematical model outputs. Two complex applications of direct interest to Sandia are also considered. First, we apply the proposed method to assess design reliability of a MEMS inertial switch. Second, we consider re-entry body (RB) component vibration response during normal re-entry, where the objective is to estimate the time-dependent probability of component failure. This last application is directly relevant to re-entry random vibration analysis at Sandia, and may provide insights on test-based and/or model-based qualification of weapon components for random vibration environments.

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Section: Vibro-impact Systemmentioning

confidence: 99%

Reliability of dynamic systems under limited information

Field

Grigoriu

2009

Probabilistic Engineering Mechanics

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“…Many effective methods have been developed, for example, linearization method used by Metrikyn [2], quasistatic approach method used by Stratonovich [3], exponential polynomial fitting method proposed by Zhu [4], Markov process method used by Jing and Young [5], stochastic averaging method used by Xu et al [6,7], variable transformation method used by Zhuravlev [8], energy balance method used by Iourtchenko and Dimentberg [9], mean impact Poincaré map method used by Feng and He [10], path integration method used by Iourtchenko and Song [11], and numerical simulation method used by Dimentberg et al [12]. In [13], the authors tried to review and summarize the existing methods, results, and literature available for solving problems of stochastic vibroimpact systems. However, most researches focused on responses statistics, such as statistic moment and probability density function of the vibroimpact oscillator, and few are focused on the bifurcations and chaos of the stochastic vibroimpact dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation of Safe Basins and Chaos in Nonlinear Vibroimpact Oscillator under Harmonic and Bounded Noise Excitations

Huang

Wang

et al. 2014

Journal of Applied Mathematics

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The erosion of the safe basins and chaotic motions of a nonlinear vibroimpact oscillator under both harmonic and bounded random noise is studied. Using the Melnikov method, the system's Melnikov integral is computed and the parametric threshold for chaotic motions is obtained. Using the Monte-Carlo and Runge-Kutta methods, the erosion of the safe basins is also discussed. The sudden change in the character of the stochastic safe basins when the bifurcation parameter of the system passes through a critical value may be defined as an alternative stochastic bifurcation. It is founded that random noise may destroy the integrity of the safe basins, bring forward the occurrence of the stochastic bifurcation, and make the parametric threshold for motions vary in a larger region, hence making the system become more unsafely and chaotic motions may occur more easily.

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“…Here stochastic averaging [31,32,33] leads to a simple stochastic differential equation for the energy. The work [29] has been extended to compliant impacts [34], Hertzian contacts [35], and nonlinear oscillators [36]. The presence of forcing adds significant difficulty to the problem but has recently been investigated numerically and via stochastic averaging [35,37] and analyzed with series expansions and mean Poincaré maps in the context of gear rattling [38,39].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Regular Grazing Bifurcations

Simpson¹,

Hogan²,

Kuske³

2013

SIAM J. Appl. Dyn. Syst.

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A grazing bifurcation corresponds to the collision of a periodic orbit with a switching manifold in a piecewise-smooth ODE system and often generates complicated dynamics. The lowest order terms of the induced Poincaré map expanded about a regular grazing bifurcation constitute a Nordmark map. In this paper we study a normal form of the Nordmark map in two dimensions with additive Gaussian noise of amplitude, ε. We show that this particular noise formulation arises in a general setting and consider a harmonically forced linear oscillator subject to compliant impacts to illustrate the accuracy of the map. Numerically computed invariant densities of the stochastic Nordmark map can take highly irregular forms, or, if there exists an attracting period-n solution when ε = 0, be well approximated by the sum of n Gaussian densities centred about each point of the deterministic solution, and scaled by 1 n , for sufficiently small ε > 0. We explain the irregular forms and calculate the covariance matrices associated with the Gaussian approximations in terms of the parameters of the map. Close to the grazing bifurcation the size of the invariant density may be proportional to √ ε, as a consequence of a square-root singularity in the map. Sequences of transitions between different dynamical regimes that occur as the primary bifurcation parameter is varied have not been described previously.

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Random Vibrations with Impacts: A Review

Cited by 153 publications

References 26 publications

Reliability of dynamic systems under limited information

Reliability of dynamic systems under limited information

Bifurcation of Safe Basins and Chaos in Nonlinear Vibroimpact Oscillator under Harmonic and Bounded Noise Excitations

Stochastic Regular Grazing Bifurcations

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