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Evolution of the geometric structure of strange attractors of a quasi-zero stiffness vibration isolator

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  • Margielewicz, Jerzy
  • Gąska, Damian
  • Litak, Grzegorz

Abstract

The aim of this work is to evaluate the dynamics of the quasi-zero stiffness vibration isolator, with particular emphasis on the density distribution of Poincaré cross-section points for different conditions of the system's excitation. The research was mainly focused on model tests carried out to examine the structures representing the geometry of strange attractors of a quasi-zero stiffness vibration isolator. The analysed mechanical system consists of one main spring and two compensation springs. Energy losses are caused by friction at the connection points of the compensation springs and vibroisolated mass. On the basis of a formulated non-linear mathematical model, the ranges of variation of physical parameters of an external dynamic input for which the system motion is chaotic are identified. Taking into account three selected values of the dynamic input amplitude, a bifurcation diagram and a diagram defining the number of phase stream intersections (NPSI) with the abscissa of the phase plane are generated. Based on the diagrams, a solution is selected. The diagram proposed by us (NPSI) is an alternative method of identifying areas in which the system moves chaotically. The classic Poincaré cross-section combined by us with information about the density of point distribution on the trajectory intersection with the control plane serve as a basis for the assessment of evolution of the geometric structure of strange attractors. The evaluation of the density distribution of Poincaré cross-section points provides important information regarding the evolution of geometrical structures of strange attractors. It has been shown that in relation to large ranges of changes in the control parameter, the geometric structure of the strange attractor is stretched and curved. However, in the area of small changes in the control parameter, the evolution of the attractor’ geometric structure can only be observed by analysing the density of point distribution on the Poincaré cross-section by the probability density function (PDF). The areas with the highest density of the Poincaré cross-section are usually located in places where the strange attractor is curved. Taking into account the practical application of research results, operational guidelines are formulated.

Suggested Citation

  • Margielewicz, Jerzy & Gąska, Damian & Litak, Grzegorz, 2019. "Evolution of the geometric structure of strange attractors of a quasi-zero stiffness vibration isolator," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 47-57.
    • Handle: RePEc:eee:chsofr:v:118:y:2019:i:c:p:47-57
    DOI: 10.1016/j.chaos.2018.11.012
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    References listed on IDEAS

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    1. Xu, Xu & Wiercigroch, M. & Cartmell, M.P., 2005. "Rotating orbits of a parametrically-excited pendulum," Chaos, Solitons & Fractals, Elsevier, vol. 23(5), pages 1537-1548.
    2. Stefanski, Andrzej & Dabrowski, Artur & Kapitaniak, Tomasz, 2005. "Evaluation of the largest Lyapunov exponent in dynamical systems with time delay," Chaos, Solitons & Fractals, Elsevier, vol. 23(5), pages 1651-1659.
    3. Sorin Vlad & Paul Pascu & Nicolae Morariu, 2010. "Chaos Models in Economics," Papers 1001.3492, arXiv.org.
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    1. Margielewicz, Jerzy & Gąska, Damian & Litak, Grzegorz & Wolszczak, Piotr & Yurchenko, Daniil, 2022. "Nonlinear dynamics of a new energy harvesting system with quasi-zero stiffness," Applied Energy, Elsevier, vol. 307(C).
    2. Margielewicz, Jerzy & Gąska, Damian & Litak, Grzegorz & Yurchenko, Daniil & Wolszczak, Piotr & Dymarek, Andrzej & Dzitkowski, Tomasz, 2023. "Influence of the configuration of elastic and dissipative elements on the energy harvesting efficiency of a tunnel effect energy harvester," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    3. Zhou, Biliu & Jin, Yanfei & Xu, Huidong, 2022. "Global dynamics for a class of tristable system with negative stiffness," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    4. Grzegorz Litak & Jerzy Margielewicz & Damian Gąska & Piotr Wolszczak & Shengxi Zhou, 2021. "Multiple Solutions of the Tristable Energy Harvester," Energies, MDPI, vol. 14(5), pages 1-17, February.

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