Abstract
In this paper we examine the effect of dissipation on the emergence of resonance in a weakly dissipative Klein-Gordon chain subjected to harmonic forcing applied to the first oscillator. Both asymptotic approximations and numerical simulations prove that weak linear dissipation counteracts resonant oscillations in the entire chain even if a similar undamped array exhibits resonance. Stable resonance may occur either in a short-length chain or in an initial segment of a long-length weakly damped chain but motion of distant oscillators becomes non-resonant. Furthermore, an increase of dissipation diminishes the localization length for resonant oscillations. The conditions of the emergence of resonance as well as an expected length of localization are obtained from the equations for the steady solutions of the system under consideration. The closeness of the approximate solutions to exact (numerical) results is demonstrated.
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
Similar content being viewed by others
References
Vazquez, L., MacKay, R., Zorzano, M.P.: Localization and Energy Transfer in Nonlinear Systems. World Scientific, Singapore (2003)
Dauxois, T., Litvak-Hinenzon, A., MacKay, R., Spanoudaki, A.: Energy Localization and Transfer. World Scientific, Singapore (2004)
Vakakis, A.F., Gendelman, O., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Passive Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems. Springer, Berlin New (2008)
May, V., Kühn, O.: Charge and Energy Transfer Dynamics in Molecular Systems. Wiley, Weinheim, Germany (2011)
Sato, M., Hubbard, B.E., Sievers, A.J.: Nonlinear energy localization and its manipulation in micromechanical oscillator arrays. Rev. Mod. Phys. 78, 137–157 (2006)
Flach, S., Gorbach, A.V.: Discrete breathers with dissipation. Lect. Notes Phys. 751, 289–320 (2008)
Egorov, O., Peschel, U., Lederer, F.: Mobility of discrete cavity solitons. Phys. Rev. E 72, 066603 (2005)
Sato, M., Hubbard, B.E., Sievers, A.J., Ilic, B., Czaplewski, D.A., Craighead, H.G.: Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array. Phys. Rev. Lett. 90, 044102 (2003)
Kimura, M., Hikihara, T.: Coupled cantilever array with tunable on-site nonlinearity and observation of localized oscillations. Phys. Lett. A 373, 1257 (2009)
Dick, A., Balachandran, B., Mote Jr., C.D.: Localization in micro-resonator arrays: influence of natural frequency tuning. J. Comput. Nonlinear Dyn. 5(1), 011002 (2009)
Balachandran, B., Perkins, E., Fitzgerald, T.: Response localization in micro-scale oscillator arrays: influence of cubic coupling nonlinearities. Int. J. Dyn. Control 3, 183 (2015)
Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems. Springer, Berlin (2007)
Kovaleva, A.: Capture into resonance of coupled Duffing oscillators. Phys. Rev. E 92, 022909 (2015)
Manevitch, L.I., Kovaleva, A., Shepelev, D.: Non-smooth approximations of the limiting phase trajectories for the Duffing oscillator near 1:1 resonance. Phys. D 240, 1–12 (2011)
Kovaleva, A., Manevitch, L.I.: Limiting phase trajectories and emergence of autoresonance in nonlinear oscillators. Phys. Rev. E 88, 024901 (2013)
Korn, G.A., Korn, T.M.: Mathematical Handbook for Scientists and Engineers, 2nd edn. Dover, New York (2000)
Acknowledgements
Support for this work from the Russian Foundation for Basic Research (grants 16-02-00400, 17-01-00582) is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Kovaleva, A. (2020). Energy Transport and Localization in Weakly Dissipative Resonant Chains. In: Kovacic, I., Lenci, S. (eds) IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems. ENOLIDES 2018. IUTAM Bookseries, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-030-23692-2_17
Download citation
DOI: https://doi.org/10.1007/978-3-030-23692-2_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-23691-5
Online ISBN: 978-3-030-23692-2
eBook Packages: EngineeringEngineering (R0)