Archetypal oscillator for smooth and discontinuous dynamics

Cao, Qingjie; Wiercigroch, Marian; Pavlovskaia, Ekaterina; Grebogi, Celso; Thompson, J. M. T.

doi:10.1103/physreve.74.046218

Cited by 238 publications

(102 citation statements)

References 25 publications

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“…The motivation of this paper is to study the limit discontinuous case of the smooth and discontinuous (SD) oscillator introduced in [19] and modelled as a piecewise linear system in [20]. The SD oscillator is derived from an arch model which was first proposed by Thompson and Hunt, 1973 in [21], where stability of the arch was investigated by the energy method.…”

Section: Introductionmentioning

confidence: 99%

The limit case response of the archetypal oscillator for smooth and discontinuous dynamics

Cao

Wiercigroch

Pavlovskaia

et al. 2008

International Journal of Non-Linear Mechanics

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In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied. This system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well. The substantial deviation from the standard dynamics is the non-smoothness of the velocity in crossing from one well to another, caused by the loss of local hyperbolicity due to the discontinuity. Without dissipation, the KAM structure on the Poincaré section is constructed with generic KAM curves and a series of fixed points associated with surrounded islands of quasi-periodic orbits and the chaotic connection orbits. It is found that, for a fixed set of parameters, a special chaotic orbit exits there which fills a finite region and connects a series of islands dominated by different chains of fixed points. As one adds weak dissipation, the periodic solutions in this finite region remain unchanged while the quasi-periodic solutions (isolated islands) are converted to the corresponding periodic solutions. The relevant dynamics for the system with weak dissipation under external excitation is shown having period doubling bifurcation leading to chaos, and multi-stable solutions.

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Section: Introductionmentioning

confidence: 99%

The limit case response of the archetypal oscillator for smooth and discontinuous dynamics

Cao

Wiercigroch

Pavlovskaia

et al. 2008

International Journal of Non-Linear Mechanics

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“…[20][21][22][23][24] In a complicated real system, frequency often shifts from one mode to another. It involves a versatile dynamic damper which can lead itself to linear dynamic dampers or nonlinear one to meet reduction demands.…”

Section: Introductionmentioning

confidence: 99%

Vibration reduction in beam bridge under moving loads using nonlinear smooth and discontinuous oscillator

Tian

Yang

Zhang

et al. 2016

Advances in Mechanical Engineering

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The coupled system of smooth and discontinuous absorber and beam bridge under moving loads is constructed in order to detect the effectiveness of smooth and discontinuous absorber. It is worth pointing out that the coupled system contains an irrational restoring force which is a barrier for conventional nonlinear techniques. Hence, the harmonic balance method and Fourier expansion are used to obtain the approximate solutions of the system. The first and the second kind of generalized complete elliptic integrals are introduced. Furthermore, using power flow approach, the performance of smooth and discontinuous absorber in vibration reduction is estimated through the input energy, the dissipated energy, and the damping efficiency. It is interesting that only depending on the value of the smoothness parameter, the efficiency parameter of vibration reduction is optimized. Therefore, smooth and discontinuous absorber can adapt itself to effectively reducing the amplitude of the vibration of the beam bridge, which provides an insight to the understanding of the applications of smooth and discontinuous oscillator in engineering and power flow characteristics in nonlinear system.

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“…The Hamiltonian function of system (3) . It is worth pointing out that the black solid points represent the standard equilibrium and the cycle mean the nonstandard equilibrium, which is named as ``saddle-like" points (Cao et al, 2006). Particularly, the phase trajectories marked by the dotted line and the dashed line correspond to the oscillations and rotations, respectively.…”

Section: Unperturbed Dynamicsmentioning

confidence: 99%

Dynamic Behavior Analysis of a Rotating Smooth and Discontinuous Oscillator with Irrational Nonlinearity

Han¹,

Liu²

2018

MAS

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This paper focuses on a novel rotating mechanical model which provides a cylindrical example of transition from smooth to discontinuous dynamics. The remarkable feature of the proposed system is a cylindrical dynamical system with strongly irrational nonlinearity exhibiting both smooth and discontinuous characteristics due to the geometry configuration. By using nonlinear dynamical technique, the unperturbed dynamics of the proposed system are studied including the irrational restoring force, stability of equilibria, Hamiltonian function and phase portraits. Note that a pair of double heteroclinic-like orbits connecting two non-standard saddle points are proposed in discontinuous case. For the perturbed system, we introduce a cylindrical approximate system for which the analytical solutions can be obtained successfully to reflect the nature of the original system without barrier of the irrationalities. Melnikov method is employed to detect the chaotic thresholds for the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing in smooth regime. Finally, numerical simulations show the efficiency of the proposed method and demonstrate the predicated periodic solution and chaotic attractors. It is found that a good degree of correlation is demonstrated in the bifurcation diagram, the phase portraits of periodic solution, the chaotic attractor' structures and the Lyapunov characteristics between the original system and approximate system.

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Archetypal oscillator for smooth and discontinuous dynamics

Cited by 238 publications

References 25 publications

The limit case response of the archetypal oscillator for smooth and discontinuous dynamics

The limit case response of the archetypal oscillator for smooth and discontinuous dynamics

Vibration reduction in beam bridge under moving loads using nonlinear smooth and discontinuous oscillator

Dynamic Behavior Analysis of a Rotating Smooth and Discontinuous Oscillator with Irrational Nonlinearity

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