Theoretical and experimental studies of global dynamics for a class of bistable nonlinear impact oscillators with bilateral rigid constraints

Li, Shuangbao; Wu, Honglei; Zhou, Xinxing; Wang, Tingting; Zhang, Wei

doi:10.1016/j.ijnonlinmec.2021.103720

Cited by 33 publications

(8 citation statements)

References 23 publications

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“…Melnikov method [10,11] is an analytical method to study chaos in nonlinear systems. It has been widely used in smooth and non-smooth systems [12], for example, bimetallic circular plates subjected to periodic heat load [13], Duffing-type system with friction [14], hysteretic models [15], self-excited two-DOF hysteretic systems with friction [16], four-dimensional self-excited systems with Coulomb-like friction [17], coupled oscillators with friction [18], double self-excited Duffing-type oscillator [19], rotated Froude pendulum [20], bistable nonlinear impact oscillators with bilateral rigid constraints [21], and so on. In this section, chaotic motions of system (10) are investigated with the Melnikov method.…”

Section: Chaotic Motions Of the Systemmentioning

confidence: 99%

Chaotic dynamics and subharmonic bifurcations of current‐carrying conductors subjected to harmonic excitation

Zhou

Chen

2022

Z Angew Math Mech

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With both analytical and numerical methods, global dynamics including chaotic motions and subharmonic bifurcations of current‐carrying conductors subjected to harmonic excitation are investigated in this paper. The system parameter conditions for chaos are obtained with the Melnikov method. The monotonicity of the critical value for chaos on the damping, alternating current, and excitation amplitude is studied in detail for three cases. Some interesting dynamic phenomena such as “uncontrollable frequency interval” and “controllable frequency” are presented analytically. Subharmonic bifurcations of odd order or even order are also investigated with the subharmonic Melnikov method. The evolution of subharmonic bifurcation to chaos is studied. It is proved rigorously that the system may undergo chaotic motions through finite or infinite subharmonic bifurcations. Numerical simulations are given to verify the chaos threshold obtained with the analytical method.

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Section: Chaotic Motions Of the Systemmentioning

confidence: 99%

Chaotic dynamics and subharmonic bifurcations of current‐carrying conductors subjected to harmonic excitation

Zhou

Chen

2022

Z Angew Math Mech

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“…For a set of system parameters, the bistable oscillator may experience different dynamic behaviors over time, and its specific dynamic response is closely related to the input initial conditions of the system. From the perspective of engineering applications, some scholars have paid attention to vibro-impact bistable oscillators by considering the collision constraint, such as the vibro-impact inverted pendulum [3,4,5], the vibroimpact SD oscillator [6,7,8], etc. The basic idea of global vibro-impact dynamics is to extend the non-smooth Melnikov method, which was first proposed by Melnikov [9] and developed in Guckenheimer & Holmes [10] and Wiggins [11] for smooth dynamical systems and extended by Kunze [12], Shi et al [13], Battelli and Feckan [14,15,16,17], Tian et al [18,19,20], and Li et al [21,22,23,24,25,26,27] to non-smooth dynamical systems.…”

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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“…In recent years, Dou, and et al studied dynamical behaviours of single-degree-of-freedom and two-degree-of-freedom systems with bilateral rigid constraints and one-side rigid obstacle, considering the nonlinear friction and external force, seen in [39][40][41]. With the help of Melnikov method, numerical calculation and experiment, Li et al [42] investigated the rich dynamics of SD oscillator with a pair of extra bilateral rigid constraints, verifying the thresholds of parameters for the global bifurcations and chaotic oscillations.…”

Section: Introductionmentioning

confidence: 99%

The complicated dynamical behaviours of a geometrical oscillator with a mass parameter

Huang

Cao

2023

Preprint

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In this paper, we consider a special kind of geometrical nonlinear oscillator with a mass parameter admitting two different dynamical states leading to a double-valued potential energy. A cylindrical manifold is introduced to formulate the equation of motion to describe the distinguished dynamical behaviours. With the help of Hamiltonian, the complex bifurcations are demonstrated with the varying of parameters including periodic solutions, the steady states and the blowing up phenomenon near θ = ± π/2 to infinity. A toroidal manifold is introduced to map the infinities into (0, ±2, 0) on the torus exhibiting saddle-node-like behaviour, where the uniqueness of solution is failed, for which a special ‘collision’ parameter is introduced to define the possible motion leaving from the infinities. A numerical method which is proposed to get solution near the infinity where Runge-Kutta method fails, is employed to get the bifurcation diagrams using Poincaré sections for the perturbed system to exhibit the complex dynamics including the co-existence of periodic solutions, the chaos from the coexisted periodic doubling and also the instant chaos from the coexisted periodic solutions. The results demonstrated herein this paper provide a brand new insight into the understanding of enriched nonlinear dynamics and an essential explanation about ‘collision’ of mechanical system with both the geometrical and mass parameters.

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Theoretical and experimental studies of global dynamics for a class of bistable nonlinear impact oscillators with bilateral rigid constraints

Cited by 33 publications

References 23 publications

Chaotic dynamics and subharmonic bifurcations of current‐carrying conductors subjected to harmonic excitation

Chaotic dynamics and subharmonic bifurcations of current‐carrying conductors subjected to harmonic excitation

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

The complicated dynamical behaviours of a geometrical oscillator with a mass parameter

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