eesti teaduste
akadeemia kirjastus
Estonian Journal of Engineering

Complete bifurcation analysis of driven damped pendulum systems; pp. 76–87

Full article in PDF format | doi: 10.3176/eng.2011.1.08

Mikhail Zakrzhevsky, Alex Klokov, Vladislav Yevstignejev, Eduard Shilvan

The pendulum systems are widely used in engineering, but their qualitative behaviour has not been investigated enough. Therefore the aim of this work is to study new non-linear effects in three driven damped pendulum systems, which are sufficiently close to the real models used in dynamics of machines and mechanisms. In this paper the existence of new bifurcation groups, rare attractors and chaotic regimes in the driven damped pendulum systems is shown.

  1. Zakrzhevsky, M., Ivanov, Yu. and Frolov, V. NLO: Universal software for global analysis of nonlinear dynamics and chaos. In Proc. 2nd European Nonlinear Oscillations Conference. Prague, 1996, vol. 2, 261–264.

  2. Schukin, I. T. Development of the Methods and Algorithms of Simulation of Nonlinear Dynamics Problems. Bifurcations, Chaos and Rare Attractors. PhD Thesis, Riga–Daugavpils, 2005 (in Russian).

  3. Zakrzhevsky, M. V. The theory of rare phenomena and rare attractors. In XXIX Summer School “Advanced Problems in Mechanics”. St. Petersburg, 2001.

  4. Zakrzhevsky, M. V. Typical bifurcation groups in a nonlinear oscillation theory. In XV Symposium DYVIS-06, RAS. Moscow, 2006, 116–122 (in Russian).

  5. Zakrzhevsky, M. V. Global stable oscillations near unstable equilibrium positions: the hilltop effect. In Proc. IUTAM Symposium on New Applications of Nonlinear and Chaotic Dynamics in Mechanics (Moon, F. C., ed.). Ithaca, USA, 1997. Kluwer Academic Publishers, Dordrecht, 117–124.

  6. Zakrzhevsky, M. V. New concepts of nonlinear dynamics: complete bifurcation groups, protuberances, unstable periodic infinitiums and rare attractors. J. Vibroeng., 2008, 10, 421–441.

  7. Zakrzhevsky, M. V. Global nonlinear dynamics based on the method of complete bifurcation groups and rare attractors. In Proc. ASME 2009 (IDETC/CIE 2009). San Diego, USA, 2009, 8.

  8. Zakrzhevsky, M. V. Rare attractors and the method of complete bifurcation groups in nonlinear dynamics and the theory of catastrophes. In Proc. 10th Conference on Dynamical Systems – Theory and Applications. Łódź, Poland, 2009, vol. 2, 671–678.

  9. Schukin, I. T., Zakrzhevsky, M. V., Ivanov, Yu. M., Kugelevich, V. V., Malgin, V. E. and Frolov, V. Yu. Realization of direct methods and algorithms for global analysis of nonlinear dynamical systems. J. Vibroeng., 2008, 10, 510–518.

10. Stephenson, A. On a new type of dynamic stability. Memoirs Proc. Manchester Literary Philos. Soc., 1908, 52, 1–10.

11. Kapitza, P. L. Dynamic stability of a pendulum with an oscillating point of suspension. J. Exp. Theor. Phys., 1951, 21, 588–597 (in Russian).

12. Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York, Applied Mathematical Sciences, vol. 42, 1983.

13. Blekhman, I. I. Vibrational Mechanics. World Scientific, Singapore, 2000.

14. Thomsen, J. J. Vibrations and Stability. Advanced Theory, Analysis, and Tools, 2nd ed. Springer, 2003.

15. Horton, B., Sieber, J., Thompson, J. M. T. and Wiercigroch, M. Dynamics of the elliptically excited pendulum. Cornell University Library, arXiv:0803.1662 (math.DS), 2008.

16. Klokov, A. and Zakrzhevsky, M. Bifurcation analysis and rare attractors in parametrically excited pendulum systems. In Proc. 10th Conference on Dynamical Systems – Theory and Applications. Łódź, Poland, 2009, vol. 2, 623–628.

17. Zakrzhevsky, M., Klokov, A., Yevstigņejev, V., Shilvan, E. and Kragis, A. Nonlinear dynamics and rare attractors in driven damped pendulum systems. In Proc. 7th International DAAAM Baltic Conference “Industrial Engineering”. Tallinn, Estonia, 2010, vol. 1, 136–141.
Back to Issue