Abstract
The analysis of the rotational dynamics of two coupled pendula is presented. The description of the oscillations of the pendulum on the background of the rotation with the average velocity was performed by the asymptotic method for the single pendulum. The source and the significance of the formation of the Limiting Phase Trajectory is clarified. The stability analysis of the rotation of two coupled pendula shows a qualitative difference between in-phase and out-of-phase rotational modes. It is shown that the origin of the in-phase rotation instability is its parametric excitation by the out-of-phase perturbations. The domain of in-phase rotation instability has been determined in the space of the system parameters. The analytic results are confirmed by the numerical simulation data.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Baker, G.L., Blackburn, J.A.: The Pendulum. A Case Study in Physics. Oxford University Press, New York (2005)
Braun, O.M., Kivshar, Yu.S.: The Frenkel–Kontorova Model. Idea. Concept. Methods. Springer, Berlin (2004)
Cuevas-Maraver, J., Kevrekidis, P.G., Williams, G. (eds.): The Sine-Gordon Model and its Applications. From Pendula and Josephson Junctions to Gravity and High-Energy Physics. Springer, Heidelberg (2014)
Scott, A.: Nonlinear Science. Emergence and Dynamics of Coherent Structures. Oxford University Press, New York (2003)
Yakushevich, L.V.: Nonlinear Physics of DNA. Wiley VCH, Weinheim (2004)
Gendelman, O.V., Savin, A.V.: Normal heat conductivity of the one-dimensional lattice with periodic potential of nearest-neighbor interaction. Phys. Rev. Lett. 84, 2381 (2000)
Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1 (2003)
Takeno, S., Homma, S.: A sine-lattice (sine-form discrete sine-gordon) equation. J. Phys. Soc. Jpn. 55, 65 (1986)
Cadoni, M., et al.: Twist solitons in complex macromolecules: from DNA to polyethylene. Int. J. Nonlinear Mech. 43, 1094 (2008)
Homma, S.: Thermodynamic properties of coupled sine-lattice. Phys. D 113, 202 (1998)
Homma, S.: Statistical mechanical theory of DNA denaturation. J. Biol. Phys. 24, 115 (1999)
Takeno, S., Peyrard, M.: Nonlinear rotating modes: green’s-function solution. Phys. Rev. E 55, 1922 (1997)
Mazo, J., Orlando, T.P.: Discrete breathers in Josephson arrays. Chaos 13, 733 (2003)
Manevitch, L.I., Romeo, F.: Non-stationary resonance dynamics of weakly coupled pendula. EPL 112, 30005 (2015)
Manevitch, L.I., Smirnov, V.V., Romeo, F.: Non-stationary resonant dynamics of the harmonically forced pendulum. Cybern. Phys. 5, 91 (2016)
Qian, M., Wang, J.-Z.: Transitions in two sinusoidally coupled Josephson junction rotators. Ann. Phys. 323, 1956 (2008)
Smirnov, L.A., et al.: Bistability of rotational modes in a system of coupled pendulums. Regul. Chaotic Dyn. 21, 849 (2016)
Sagdeev, R.Z., Usikov, D.A., Zaslavsky, G.M.: Nonlinear Physics: From the Pendulum to Turbulence and Chaos. Harwood Academic Publishers, New York (1988)
Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, New York (2005)
Butikov, E.I.: Parametric excitation of a linear oscillator. Eur. J. Phys. 25, 535 (2004)
Acknowledgements
Author is grateful to Prof. L.I. Manevitch for his attention to the work and fruitful discussion.
   This work was supported by the Program of Fundamental Researchers of the Russian State Academies of Sciences 2013–2020 (project No. 0082-2014-0013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Smirnov, V.V. (2019). Revolution of Pendula: Rotational Dynamics of the Coupled Pendula. In: Andrianov, I., Manevich, A., Mikhlin, Y., Gendelman, O. (eds) Problems of Nonlinear Mechanics and Physics of Materials. Advanced Structured Materials, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-92234-8_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-92234-8_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-92233-1
Online ISBN: 978-3-319-92234-8
eBook Packages: EngineeringEngineering (R0)