Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

Han, Yanwei; Cao, Qingjie; Chen, Yushu; Wiercigroch, Marian

doi:10.1016/j.ijnonlinmec.2014.09.007

Cited by 27 publications

(5 citation statements)

References 17 publications

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“…This oscillator bears significant similarities to Duffing oscillator for α > 0 but discontinuous characteristics as α drops to 0. Some other similar nonlinear geometrical oscillators have also been proposed and analyzed, such as an oscillator with two geometrical parameters, seen in [30,31], and an oscillator with three geometrical parameters, seen in [32]. Some nonlinear geometrical oscillators with characteristics of cylindrical system are also studied, such as a novel oscillator consisting of two oblique springs and a rotating disk, seen in [33], and a rotating SD oscillator, seen in [34].…”

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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Section: Introductionmentioning

confidence: 99%

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

Huang

Cao

2023

Preprint

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“…On this basis, Yang and Zhou [24] proposed several feedback-control approaches to control its chaotic vibrations and numerically evaluated the effect of the control. Han and his coworkers proposed another nonlinear oscillator with strong irrational nonlinearities, i.e., the coupled SD oscillator, employed Melnikov method to obtain the analytical criteria of chaotic thresholds, and presented chaos via numerical simulation [25]. Based on SD oscillator, Yang et al [26] designed a multidirectional multistable device (MMD) including displacement sensors, hydraulic shakers, and oscilloscopes to observe multistability of the oscillator which may be adopted for energy harvesting from ultra-low-frequency vibration sources.…”

Section: Introductionmentioning

confidence: 99%

Initial-Sensitive Dynamical Behaviors of a Class of Geometrically Nonlinear Oscillators

Qin

Shang

Huimin

2022

Shock and Vibration

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The vibrating system of a class of linkage-slider structure is considered, and its initial-sensitive dynamical behaviors such as safe jump, locking instability, and chaos are studied. First, static bifurcation of the dynamical system is discussed. Then, via analyzing the effect of the external excitation on the periodic solutions under primary resonance, it is found that the change of the excitation frequency may lead to bistability and safe jump. Furthermore, it follows from the investigation of the heteroclinic bifurcation that the increase of the external excitation amplitude may lead to locking instability, chaos, and static locking. The results have some potential values in the design of geometrically nonlinear oscillators.

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“…Unlike the previous studies, the present multi-stable energy harvester uses the elastic steel slice and spring, rather than the usual magnetic attraction and repulsion eects. This harvester was developed from a simple smooth and discontinuous oscillator proposed by Cao et al [31,32], which has attracted a great interest from the trans ition dynamics between smooth and discontinuous systems [31][32][33][34], to the snapthrough energy harvesting [9,10,17,18,30], as well as complex nonlinear dynamics of vibration isolation [35,36]. The simplicity of the multi-stable energy harvester not only enables a low-cost manufacturing process, but also produces a large output power, which means that it is a viable design for harvesting energy from low and ultra-low level vibration resources.…”

Section: Introductionmentioning

confidence: 99%

Novel multi-stable energy harvester by exploring the benefits of geometric nonlinearity

Yang

Cao

2019

J. Stat. Mech.

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This paper proposes the theoretical model and dynamical investigations of a novel multi-stable energy harvester using geometric nonlinearity technique. Among present energy harvesters one of them uses the elastic steel slices and springs, rather than the usual magnetic attraction and repulsion effects. Such a harvester is shown to outperform conventional linear harvesters and to overcome some of the most severe limitations of present approaches. By adjusting the geometric nonlinearity parameters, the energy harvester can possess multi-stable potential energy functions, ranging from mono-stable to quad-stable. The results show that the combined geometric nonlinearity can not only make jumping between potential wells easier, but also improve the efficiency of energy harvesting under low intensity excitations.

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Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

Cited by 27 publications

References 17 publications

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

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

Initial-Sensitive Dynamical Behaviors of a Class of Geometrically Nonlinear Oscillators

Novel multi-stable energy harvester by exploring the benefits of geometric nonlinearity

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