Chaotic dynamics of the vibro-impact system under bounded noise perturbation

Feng, Jinqian

doi:10.1016/j.chaos.2015.01.003

Cited by 13 publications

(4 citation statements)

References 22 publications

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“…At last, the influences of the additive and multiplicative random noises on the stationary PDF of system (1) are discussed in Figure 8. Figure 8(a) gives the stationary PDF of the displacement, while Figure 8(b) plots the PDF of the velocity of system (1). As shown in Figure 8(a), the multiplicative random noise will enlarge the system displacement response, but the influences are not so significant.…”

Section: Example and Discussionmentioning

confidence: 99%

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Stochastic Responses for the Vibro‐Impact System under the Broadband Noise

Guan

Ling

2019

Shock and Vibration

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The stochastic averaging is applied to derive the random responses of the SDOF vibro-impact system with one-side wall under additive and multiplicative broadband noise, where the impact is described by the modified Hertzian contact model. Compared with the popular simplified “classical model” (i.e., x˙−=rx˙+), the whole impact process is captured by the modified Hertzian contact model without any assumptions. One example is given to illustrate the proposed technique. The comparison between the analytical results and those from Monte Carlo simulations manifests the accuracy of the proposed technique. It should be pointed out that the technique proposed in this paper is powerful for the cases of weak excitation and big enough bandwidth of the broadband noise. Finally, the influences of various parameters are also discussed in detail.

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Section: Example and Discussionmentioning

confidence: 99%

“…e main distinguish between the vibro-impact system and ordinary vibration system is the impact. e impact always introduces the complicate nonlinear behaviors, such as homoclinic bifurcations [1]. In view of the wide engineering demands, the impact has been investigated by many engineers and physicists.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Responses for the Vibro‐Impact System under the Broadband Noise

Guan

Ling

2019

Shock and Vibration

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“…Liu and Xu [42] derived in detail the stochastic Melnikov method for abstract stochastic discontinuous systems under bounded noise by measuring the distance between perturbed stable manifolds and unstable manifolds. Feng and Liu [43] extended the Melnikov method of a determined vibro-impact system to the random Melnikov method under bounded noise, and used it to study the global dynamics of a typical random Duffing vibro-impact system. Zhou and Jin [44] used the mean square criterion and phase space flux function theory to study the effect of noise intensity on chaotic dynamics of bistable coupled SD oscillators under Gaussian colored noise and harmonic excitation.…”

Section: Introductionmentioning

confidence: 99%

Homoclinic bifurcations and chaotic dynamics in a bistable vibro-impact SD oscillator under Gaussian white noise

Jia,

Li,

Kou

et al. 2024

Preprint

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This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable vibro-impact SD oscillator. Firstly, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod so that the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistble vibration characteristic, two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable vibro-impact SD oscillator through studying the persistence of the unique unperturbed non-smooth homoclinic structure. Secondly, the general framework of random Melnikov process for a class of bistable vibro-impact systems under Gaussian white noise is simply derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable vibro-impact SD oscillator. Thirdly, the effectiveness of a semi-analytical prediction by the Melnikov function is verified through the lagest Lyapunov exponent, bifurcation series, and 0 − 1 test. In addition, the sensitivity to the initial values of chaos is verified by the fractal of attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show rich dynamical geometric structures.

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“…Thus, vibro-impact systems under random excitations need to draw close attention (Dimentberg and Iourtchenko, 2004). Feng and Liu studied the chaotic behaviors of a vibro-impact system excited by bounded noise by adopting the Melnikov method (Feng and Liu, 2015). The stationary response of vibro-impact systems under random excitations was studied in references (Jing and Sheu, 1990;Nayak, 1972) by using the Hertz contact law.…”

Section: Introductionmentioning

confidence: 99%

Optimal bounded control of stochastically excited strongly nonlinear vibro-impact system

Deng

2020

Journal of Vibration and Control

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An optimal bounded control strategy for strongly nonlinear vibro-impact systems under stochastic excitations with actuator saturation is proposed. First, the impact effect is incorporated in an equivalent equation by using a nonsmooth transformation. Under the assumption of light damping and weak random perturbation, the system energy is a slowly varying process. By using the stochastic averaging of envelope for strongly nonlinear systems, the partially averaged Itô stochastic differential equation for system energy can be derived. The optimal control problem is transformed from the original optimal control problem for the state variables to an equivalent optimal control problem for the system energy, which decreases the dimensions of the optimal control problem. Then, based on stochastic maximum principle, an adjoint equation for the adjoint variable and the maximum condition of partially averaged control problem are established. For infinite time-interval ergodic control, the adjoint variable is assumed to be a stationary process and the adjoint equation can be further simplified. Finally, the probability density function of the system energy and other statistics of the optimally controlled system are derived by calculating the associated Fokker–Plank–Kolmogorov equation. For comparison, the bang–bang control is also investigated and the control results are compared to show the advantages of the developed control strategy.

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Chaotic dynamics of the vibro-impact system under bounded noise perturbation

Cited by 13 publications

References 22 publications

Stochastic Responses for the Vibro‐Impact System under the Broadband Noise

Stochastic Responses for the Vibro‐Impact System under the Broadband Noise

Homoclinic bifurcations and chaotic dynamics in a bistable vibro-impact SD oscillator under Gaussian white noise

Optimal bounded control of stochastically excited strongly nonlinear vibro-impact system

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