IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v118y2019icp47-57.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077918310221
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2018.11.012?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Sorin Vlad & Paul Pascu & Nicolae Morariu, 2010. "Chaos Models in Economics," Papers 1001.3492, arXiv.org.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. Liang, Xiyin & Qi, Guoyuan, 2017. "Mechanical analysis of Chen chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 173-177.
    3. Yan, Jun-Juh & Lin, Jui-Sheng & Liao, Teh-Lu, 2007. "Robust dynamic compensator for a class of time delay systems containing saturating control input," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1223-1231.
    4. Borys, Przemyslaw & Grzywna, Zbigniew J., 2006. "Diffusion as a result of transition in behavior of deterministic maps," Chaos, Solitons & Fractals, Elsevier, vol. 30(1), pages 156-165.
    5. Han, Ning & Zhang, Hanfang & Lu, Peipei & Liu, Zixuan, 2024. "Resonance response and chaotic analysis for an irrational pendulum system," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    6. 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).
    7. Wang, Jianzhou & Jia, Ruiling & Zhao, Weigang & Wu, Jie & Dong, Yao, 2012. "Application of the largest Lyapunov exponent and non-linear fractal extrapolation algorithm to short-term load forecasting," Chaos, Solitons & Fractals, Elsevier, vol. 45(9), pages 1277-1287.
    8. Anastasia Sofroniou & Steven Bishop, 2014. "Dynamics of a Parametrically Excited System with Two Forcing Terms," Mathematics, MDPI, vol. 2(3), pages 1-24, September.
    9. C. A. Tapia Cortez & J. Coulton & C. Sammut & S. Saydam, 2018. "Determining the chaotic behaviour of copper prices in the long-term using annual price data," Palgrave Communications, Palgrave Macmillan, vol. 4(1), pages 1-13, December.
    10. Dabrowski, Artur, 2009. "Energy–vector method in mechanical oscillations," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1684-1697.
    11. de Souza, Silvio L.T. & Batista, Antonio M. & Caldas, Iberê L. & Iarosz, Kelly C. & Szezech Jr, José D., 2021. "Dynamics of epidemics: Impact of easing restrictions and control of infection spread," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    12. de Souza, S.L.T. & Batista, A.M. & Baptista, M.S. & Caldas, I.L. & Balthazar, J.M., 2017. "Characterization in bi-parameter space of a non-ideal oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 466(C), pages 224-231.
    13. Tapia Cortez, Carlos A. & Hitch, Michael & Sammut, Claude & Coulton, Jeff & Shishko, Robert & Saydam, Serkan, 2018. "Determining the embedding parameters governing long-term dynamics of copper prices," Chaos, Solitons & Fractals, Elsevier, vol. 111(C), pages 186-197.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:118:y:2019:i:c:p:47-57. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.