Abstract
Triple physical pendulum in a form of three connected rods with the first link subjected to an action of constant torque and with a horizontal barrier is used as an example of plane mechanical system with rigid limiters of motion. Special transition rules for solutions of linearized equations at impact instances (Aizerman-Gantmakher theory) are used in order to apply classical tools for Lyapunov exponents computation as well as for stability analysis of periodic orbits (used in seeking for stable and unstable periodic orbits and bifurcations of periodic solutions analysis). Few examples of extremely rich bifurcational dynamics of triple pendulum are presented.
Keywords
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Baker, G.L., Blackburn, J.A.: The Pendulum. A Case Study in Physics. Oxford University Press, Oxford (2005)
Zhu, Q., Ishitobi, M.: Experimental study of chaos in a driven triple pendulum. J. Sound Vib. 227(1), 230–238 (1999)
Awrejcewicz, J., Supeł, K.G., Wasilewski, G., Olejnik, P.: Numerical and experimental study of regular and chaotic motion of triple physical pendulum. Int. J. Bifurcation Chaos 18(10), 2883–2915 (2008)
Awrejcewicz, J., Kudra, G.: The piston - connecting rod - crankshaft system as a triple physical pendulum with impacts. Int. J. Bifurcation Chaos 15(7), 2207–2226 (2005)
Brogliato, B.: Non-smooth Mechanics, Springer, London (1999)
Leine, R.L., van Campen, D.H., van de Vrande, B.L.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23, 105–164 (2000)
Awrejcewicz, J., Kudra, G.: Stability analysis and Lyapunov exponents of a multi-body mechanical system with rigid unilateral constraints. Nonlinear Anal. Theor. Methods Appl. 63(5–7) (2005)
Aizerman, M.A., Gantmakher, F.R.: On the stability of periodic motions. J. Appl. Math. Mech. 22, 1065–1078 (1958)
Müller, P.C.: Calculation of Lyapunov exponents for dynamic systems with discontinuities. Chaos Solitons Fractals 5, 1671–1691 (1995)
Acknowledgments
The work has been supported by the Ministry of Science and Higher Education under the grant no. 0040/B/T02/2010/38. J. Awrejcewicz acknowledges support of the Alexander von Humboldt Award.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this paper
Cite this paper
Awrejcewicz, J., Kudra, G. (2011). Bifurcation and Chaos of Multi-body Dynamical Systems. In: Náprstek, J., Horáček, J., Okrouhlík, M., Marvalová, B., Verhulst, F., Sawicki, J. (eds) Vibration Problems ICOVP 2011. Springer Proceedings in Physics, vol 139. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2069-5_1
Download citation
DOI: https://doi.org/10.1007/978-94-007-2069-5_1
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-2068-8
Online ISBN: 978-94-007-2069-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)