Homoclinic bifurcations and chaotic dynamics for a piecewise linear system under a periodic excitation and a viscous damping

Li, S. B.; Shen, Chao; Zhang, W.; Hao, Yuxin

doi:10.1007/s11071-014-1820-4

Cited by 19 publications

(7 citation statements)

References 29 publications

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“…The Melnikov method for non-smooth systems is an important analytical method [38][39][40][41][42][43][44] used to analyze the global chaos of piecewise-smooth systems. Hence, based on this mechanism, it is necessary to reduce the dimensions of the EI-BNES from four to two and analyse the sufficient conditions to induce transient chaos by means of the Melnikov function of non-smooth systems.…”

Section: Dimension Reduction Of the Motion Equationsmentioning

confidence: 99%

Global Dynamics and Optimal Design of a New Highly Efficient Nonlinear Energy Sink Based on Magnetic-Elastic Impacts with Negative Stiffness

Wang

Chen

2021

Preprint

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A new highly efficient elastic-impact bistable nonlinear energy sink (EI-BNES) based on magnetic-elastic impacts with negative stiffness and bistability is proposed and optimized through global dynamical analysis. The EI-BNES has better robustness and higher energy dissipation rates with nearly more than 96.5\% for broadband impulsive excitations than the traditional cubic NESs and single bistable NESs. The structure of negative stiffness impacts is realized by reasonable layout of permanent ring magnets and springs. A two-degree-of-freedom (two-DOF) elastic-impact system is established to describe the coupled nonlinear interaction between the main structure and the attached EI-BNES. A global Melnikov reduction analysis (GMRA) is proposed to study global dynamics and homoclinic bifurcations of the reduced two-dimensional subsystem, which is used to explain the mechanism of nonlinear targeted energy transfer (TET) and detect the threshold of impulsive amplitudes of EI-BNES for in-well and compound motions between in-well and cross-well resonance responses. A special type of saddle-center equilibrium points is also found in the non-smooth system of the EI-BNES and can be used to effectively increase the energy dissipation rates. The optimal design criterion of the tuned EI-BNES for better dissipation performance is also first discussed based on the GMRA and numerical techniques for calculating the Melnikov function of the non-smooth systems. The effectiveness of the analytical GMRA is also verified by numerical simulations.

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Section: Dimension Reduction Of the Motion Equationsmentioning

confidence: 99%

Global Dynamics and Optimal Design of a New Highly Efficient Nonlinear Energy Sink Based on Magnetic-Elastic Impacts with Negative Stiffness

Wang

Chen

2021

Preprint

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“…Previously, the criteria for chaotic motion have been constructed in some non-smooth systems. [4][5][6][7][8][9][10][11][12][13][14] A lot of effort will be made to extend the Melnikov method to this kind of non-smooth system. Homoclinic bifurcation is detected here and heteroclinic bifurcation will be carried out in a separate paper.…”

Section: Non-smooth Homoclinic Orbits Bifurcationmentioning

confidence: 99%

“…[1][2][3] Melnikov's method, a tool to obtain analytical results, has been extended to piecewise-smooth dynamical systems by geometrical or analytical technique. [4][5][6][7][8][9][10][11][12][13][14] It is important to note that the particular piecewise-smooth dynamical system was investigated in Tian et al [11][12][13] and Niziol and Swiatoniowski. 14 Although the known works have given some good ideas, these results were limited in terms of the theoretical results, which encourage continued research on the practical application.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation and chaos for piecewise nonlinear roll system of rolling mill

Tian

Feng

et al. 2017

Advances in Mechanical Engineering

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A non-smooth cold roll system of rolling mill is studied to reveal the bifurcation of the piecewise-smooth and discontinuous system. To examine the influence of the parameters on the dynamics, the bifurcation diagram is constructed when it is unperturbed. Hamilton phase diagrams of the non-smooth system are detected, which differ significantly from the ones obtained in the smooth system. Non-smooth homoclinic, heteroclinic, and periodic orbits are determined, which depend on the classical heteroclinic periodic orbits, periodic orbits, and a necessary condition. A extended Melnikov function is employed to obtain the criteria for the non-smooth homoclinic bifurcation in this class of piecewise-smooth and discontinuous system, which implies that the existence of homoclinic bifurcation arises from the breaking of homoclinic orbits under the perturbation of damping. The results reveal that this kind of non-smooth factor has little influence on the chaotic threshold apart from calculating piecewise integrals. The efficiency of the criteria for non-smooth homoclinic bifurcation mentioned above is verified by the phase portraits, Poincaré section, and bifurcation diagrams, which laid a theoretical foundation for parameter design, stable operation, and fault diagnosis of rolling mills.

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“…Shaw and Holmes 4 analyzed the harmonic, subharmonic, and chaotic motions existing in a piecewise linear oscillator. Li et al 5 investigated the homoclinic bifurcations and chaotic dynamics of a piecewise linear system with periodic excitation and viscous damping. Gao and Chen 6 studied the resonance and stability for a SDOF system with a piecewise linear/nonlinear stiffness term.…”

Section: Introductionmentioning

confidence: 99%

Dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative

Wang

Shen

Yang

et al. 2019

Journal of Low Frequency Noise, Vibration and Active Control

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In this paper, the dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative is studied. First, a mathematical model of the single-degree-of-freedom system is established, and the approximate steady-state solution associated with the amplitude-frequency equation is obtained based on the averaging method. Then, the amplitude-frequency response equations are used for stability analysis, and the stability condition is founded. To validate the correctness and precision, the approximate solutions determined by the analytical method are compared with the solutions based on the numerical integration method. It is found that the approximate solutions and the numerical solutions are in excellent agreement. Finally, the effects of system parameters, such as fractional-order coefficient, order, clearance, and piecewise stiffness, on the complex dynamical behaviors of the piecewise linear singledegree-of-freedom oscillator are studied. The results show that the system parameters not only influence resonance amplitude and resonance frequency but also affect the size of the unstable region.

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Homoclinic bifurcations and chaotic dynamics for a piecewise linear system under a periodic excitation and a viscous damping

Cited by 19 publications

References 29 publications

Global Dynamics and Optimal Design of a New Highly Efficient Nonlinear Energy Sink Based on Magnetic-Elastic Impacts with Negative Stiffness

Global Dynamics and Optimal Design of a New Highly Efficient Nonlinear Energy Sink Based on Magnetic-Elastic Impacts with Negative Stiffness

Bifurcation and chaos for piecewise nonlinear roll system of rolling mill

Dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative

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