Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Dynamic Behaviour of the Platform-vibrator with Soft Impact. Part 2. Interior Crisis. Crisis-Induced Intermittency

Discontinuity, Nonlinearity, and Complexity 11(1) (2022) 107--124 | DOI:10.5890/DNC.2022.03.009

V.A. Bazhenov, O.S. Pogorelova, T.G. Postnikova

Kyiv National University of Construction and Architecture, 31, Povitroflotskiy avenu, Kyiv, Ukraine

Download Full Text PDF

 

Abstract

Platform-vibrator with shock is widely used in the construction industry for compacting and molding large concrete products. Its mathematical model corresponds to a two-body 2-DOF vibro-impact system with a soft impact. A soft impact is simulated with nonlinear Hertzian contact force. When the control parameter (technological mass of mold with concrete) changes, the model exhibits interesting phenomena inherent in a nonlinear non-smooth vibro-impact system, namely: boundary and interior crises, crisis-induced intermittency, transient chaos, and a hysteresis zone. Phase trajectories with Poincar\'{e} maps, graphs of time series and contact forces, Fourier spectra, and wavelet characteristics are used in numerical investigations to determine the realized oscillatory modes. We hope this analysis can help to avoid undesirable platform-vibrator behaviour during design and operation due to inappropriate system parameters.

References

  1. [1]  Borschevsky, A.A. and Ilyin A.S. (2009), The Mechanical equipment for manufacture of building materials and products. The textbook for high schools on ``Pr-in builds. And designs''. M: The Publishing house the Alliance. (in Russian).
  2. [2]  Gusev, B.V. and Zazimko, V.G. (1991), Vibration technology of concrete, Budivelnik, Kiev. (in Russian).
  3. [3]  Gusev, B.V., Deminov, A.D., and Kryukov, B.I. (1982), Impact-Vibrational Technology of Compaction of Concrete Mixtures. Stroiizdat, Moscow. (in Russian).
  4. [4]  Bazhenov, V.A., Pogorelova, O.S., Postnikova, T.G, and Otrashevska, V.V. (2020), Dynamic Behaviour of the Platform-vibrator with Soft Impact. Part 1. Dependence on Exciting Frequency, Discontinuity, Nonlinearity, and Complexity. (In press).
  5. [5]  Macau, E.E. (Ed.). (2019), A mathematical modeling approach from nonlinear dynamics to complex systems. Springer International Publishing.
  6. [6]  Lai, Y.C. and T{e}l, T. (2011), Transient chaos: complex dynamics on finite time scales~(Vol. 173). Springer Science & Business Media.
  7. [7]  Mishra, A., Leo Kingston, S., Hens, C., Kapitaniak, T., Feudel, U., and Dana, S.K. (2020), Routes to extreme events in dynamical systems: Dynamical and statistical characteristics, Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(6), 063114.
  8. [8]  Elaskar, S. (2018), Studies on Chaotic Intermittency (Doctoral dissertation, Doctoral Thesis, Universidad Polit{e}cnica de Madrid. Madrid).
  9. [9]  Wang, G., Lai, Y.C., and Grebogi, C. (2016), Transient chaos-a resolution of breakdown of quantum-classical correspondence in optomechanics, Scientific reports, 6, 35381.
  10. [10]  Elaskar, S. and Del R{\i}o, E. (2017), New advances on chaotic intermittency and its applications~(pp. 35-38). New York: Springer.
  11. [11]  Astaf'ev, G.B., Koronovskii, A.A., and Khramov, A.E. (2003), Behavior of dynamical systems in the regime of transient chaos, Technical Physics Letters, 29(11), 923-926.
  12. [12]  Hilborn, R.C. (2000), Chaos and nonlinear dynamics: an introduction for scientists and engineers. Oxford University Press on Demand.
  13. [13]  Grebogi, C., Ott, E., and Yorke, J.A. (1982), Chaotic attractors in crisis, Physical Review Letters, 48(22), 1507.
  14. [14]  Grebogi, C., Ott, E., and Yorke, J.A. (1983), Crises, sudden changes in chaotic attractors, and transient chaos, Physica D: Nonlinear Phenomena, 7(1-3), 181-200.
  15. [15]  Grebogi, C., Ott, E., Romeiras, F., and Yorke, J.A. (1987), Critical exponents for crisis-induced intermittency, Physical Review A, 36(11), 5365.
  16. [16]  Grebogi, C., Ott, E., and Yorke, J.A. (1986), Critical exponent of chaotic transients in nonlinear dynamical systems, Physical Review Letters, 57(11), 1284.
  17. [17]  Ott, E. (2002), Chaos in dynamical systems. Cambridge university press.
  18. [18]  Bazhenov, V.A., Pogorelova, O.S., and Postnikova, T.G. (2017), Stability and Discontinious Bifurcations in Vibroimpact System: Numerical investigations. LAP LAMBERT Academic Publ. GmbH and Co. KG Dudweiler, Germany.
  19. [19]  Bazhenov, V.A., Pogorelova, O.S., and Postnikova, T.G. (2019), Intermittent and quasi-periodic routes to chaos in Vibroimpact System: Numerical simulations, LAP LAMBERT Academic Publ, GmbH and Co. KG Dudweiler, Germany.
  20. [20]  Bazhenov, V.A., Pogorelova, O.S., and Postnikova, T.G. (2020), Quasiperiodic Route to Transient Chaos in Vibroimpact System. Chapter 3 in Book: ``Nonlinear Dynamics, Chaos, and Complexity: In Memory of Professor Valentin Afraimovich (1945-2018)'', Vol. 1, Volchenkov D., (Ed.) Springer.
  21. [21]  Bazhenov, V.A., Pogorelova, O.S., and Postnikova, T.G. (2020), Analysis of Intermittent and Quasi-periodic Transitions to Chaos in Vibro-impact System with Continuous Wavelet Transform. Chapter into Book: ``Discontinuity, Ninlinearity and Complexity. In memory of Valentin Afraimovich (1945-2018)'', Vol 3, Volchenkov D., (Ed.) Spinger. (DNC Vol 11 (issue) in press).
  22. [22]  Danca, M.F. (2016), Hidden transient chaotic attractors of Rabinovich--Fabrikant system, Nonlinear Dynamics, 86(2), 1263-1270.
  23. [23]  Goldsmith, W. (1960), Impact, the theory and physical behaviour of colliding solids. Edward Arnold Ltd.
  24. [24]  Johnson, K.L. (1985), Contact Mechanics. Cambridge: Cambridge University Press.
  25. [25]  Kapitaniak, T. and Bishop, S.R. (1999), The illustrated dictionary of nonlinear dynamics and chaos. Wiley.
  26. [26]  Lamarque, C.H. and Janin, O. (2000), Modal analysis of mechanical systems with impact non-linearities: limitations to a modal superposition, Journal of Sound and Vibration, 235(4), 567-609.
  27. [27]  Moon, F.C. (1987), Chaotic vibrations: an introduction for applied scientists and engineers. --- New York : Wiley -- C. 219.
  28. [28]  Schuster, H.G. and Just, W. (2006), Deterministic chaos: an introduction. John Wiley & Sons.
  29. [29]  Bhalekar, S., Daftardar-Gejji, V., Baleanu, D., and Magin, R. (2012), Transient chaos in fractional Bloch equations, Computers $\&$ Mathematics with Applications, 64(10), 3367-3376
  30. [30]  J{a}nosi, I.M. and T{e}l, T. (1994), Time-series analysis of transient chaos, Physical Review E, 49(4), 2756.
  31. [31]  T{e}l, T. (2015), The joy of transient chaos, Chaos: An Interdisciplinary, Journal of Nonlinear Science, 25(9), 097619.