Engineering Applications of Non-smooth Dynamics

Wiercigroch, Marian; Pavlovskaia, Ekaterina

doi:10.1007/978-94-007-2473-0_5

Cited by 7 publications

(5 citation statements)

References 54 publications

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“…Sticking phenomenon on the surfaceẋ = 0 (depicted in dark) of the system (6) equilibrium set are stable since the attracting sliding mode occurs.…”

Section: Fig 11mentioning

confidence: 98%

“…Since U ∩ ε 2 = φ, ε 1 is the largest stationary set in D r , we can easily know that the solutions of the system (6) can not stay in γ \ε 1 , with γ = {(x,ẋ)|ẋ = 0} and γ ⊂ U , by using the equation of motion for system (6). Thus, ∃t > t 0 , such that x(t) / ∈ γ , moreover, every solutions of system (6) is continuous in time and x(t) / ∈ γ for a small time interval (t 0 , t 1 ).…”

Section: Instability Analysis Of the Equilibrium Set εmentioning

confidence: 99%

“…In order to examine the influence of the parameter α and friction coefficient μ on the SD oscillator, we investigated the bifurcations of the equilibrium sets in the system (6), assuming g 1 = 2, ω 0 = 1. Figure 12a shows the body of the equilibrium sets ε for the system (6) in (x, α, μ) space, which is divided into four domains of A, B, C, and D by the planes α = 1 and μ = 0.5. The bifurcation of equilibrium of SD oscillator without dry friction (μ = 0) is shown in Fig.…”

Section: Bifurcation Of the Equilibriummentioning

confidence: 99%

“…Dry friction induced dynamics has attracted the attention of researchers for many years [1][2][3][4][5], which is motivated by engineering applications [6]. The presence of dry friction can affect the behavior and the performance of mechanical systems as it can induce complex phenomena, such as periodic motion [7], stick-slip vibrations [8,9], etc.…”

Section: Introductionmentioning

confidence: 99%

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The equilibrium stability for a smooth and discontinuous oscillator with dry friction

Cao

Léger

2015

Acta Mech. Sin.

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In this paper, we investigate the equilibrium stability of a Filippov-type system having multiple stick regions based upon a smooth and discontinuous (SD) oscillator with dry friction. The sets of equilibrium states of the system are analyzed together with Coulomb friction conditions in both ( f n , f s ) and (x,ẋ) planes. In the stability analysis, Lyapunov functions are constructed to derive the instability for the equilibrium set of the hyperbolic type and LaSalle's invariance principle is employed to obtain the stability of the nonhyperbolic type. Analysis demonstrates the existence of a thick stable manifold and the interior stability of the hyperbolic equilibrium set due to the attractive sliding mode of the Filippov property, and also shows that the unstable manifolds of the hyperbolic-type are that of the endpoints with their saddle property. Numerical calculations show a good agreement with the theoretical analysis and an excellent efficiency of the approach for equilibrium states in this particular Filippov system. Furthermore, the equilibrium bifurcations are presented to demonstrate the transition between the smooth and the discontinuous regimes.

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“…Sticking phenomenon on the surfaceẋ = 0 (depicted in dark) of the system (6) equilibrium set are stable since the attracting sliding mode occurs.…”

Section: Fig 11mentioning

confidence: 98%

Section: Instability Analysis Of the Equilibrium Set εmentioning

confidence: 99%

Section: Bifurcation Of the Equilibriummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The equilibrium stability for a smooth and discontinuous oscillator with dry friction

Cao

Léger

2015

Acta Mech. Sin.

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“…In view of the increasing research on piecewise-smooth systems in the real world, such as mechanics 1 , engineering application 2 , physiology 3 and so on 4,5 . The relevant research methods and theories are constantly improving and developing with the authenticity of system modeling and the complexity of the derived structure.…”

Section: Introductionmentioning

confidence: 99%

Melnikov-type method for a class of planar hybrid piecewise-smooth systems with impulsive effect and noise excitation: Heteroclinic orbits

Wei

Moroz

et al. 2022

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The classical Melnikov method for heteroclinic orbits is extended theoretically to a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation. We assume that the unperturbed system is a piecewise Hamiltonian system with a pair of heteroclinic orbits. The heteroclinic orbit transversally jumps across the first switching manifold by an impulsive effect and crosses the second switching manifold continuously. In effect, the trajectory of the corresponding perturbed system crosses the second switching manifold by applying the reset map describing the impact rule instantaneously. The random Melnikov process of such systems is then derived by measuring the distance of perturbed stable and unstable manifolds, and the criteria for the onset of chaos with or without noise excitation is established. In this derivation process, we overcome the difficulty that the derivation method of the corresponding homoclinic case cannot be directly used due to the difference between the symmetry of the homoclinic orbit and the asymmetry of the heteroclinic orbit. Finally, we investigate the complicated dynamics of a particular piecewise-smooth system with and without noise excitation under the reset maps, impulsive effect, and non-autonomous periodic and damping perturbations by this new extended method and numerical simulations. The numerical results verify the correctness of the theoretical results and demonstrate that this extended method is simple and effective for studying the dynamics of such systems.

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The effect of friction on the vibro-impact locomotion system: modeling and dynamic response

Nguyen

et al. 2021

Meccanica

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Engineering Applications of Non-smooth Dynamics

Cited by 7 publications

References 54 publications

The equilibrium stability for a smooth and discontinuous oscillator with dry friction

The equilibrium stability for a smooth and discontinuous oscillator with dry friction

Melnikov-type method for a class of planar hybrid piecewise-smooth systems with impulsive effect and noise excitation: Heteroclinic orbits

The effect of friction on the vibro-impact locomotion system: modeling and dynamic response

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