Abstract
We present results of analytical and numerical study of planar dynamics of a membrane consisting of one longitudinal and N transversal strings without a preliminary stretching with uniformly distributed discrete masses. In the previous paper it was shown that in the most significant case of predominantly transversal dynamics an effective non-local axial force is formed in longitudinal string and equations of motion cannot be linearized. Therefore the membrane oscillates in the conditions of “non-local” acoustic vacuum and can be used as an efficient energy trap. Adequate analytical description of non-stationary resonance dynamics at such conditions is achieved in terms of limiting phase trajectories (LPT) corresponding to maximum possible energy exchange between different parts (clusters) of the membrane. We have revealed also in these terms the conditions of energy localization in the initially excited cluster. Analytical results are confirmed by 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
Manevitch, L.I., Vakakis, A.F.: Nonlinear oscillatory acoustic vacuum. SIAM J. 74, 1742–1762 (2014)
Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N., Zevin, A.A.: Normal Modes and Localization in Nonlinear Systems. Wiley, New York (1996)
Rosenberg, R.M.: On nonlinear vibrations of systems with many degrees of freedom. Adv. Appl. Mech. 9, 156–243 (1966)
Manevitch, L., Kovaleva, A., Smirnov, V., Starosvetsky, Y.: Nonstationary Resonant Dynamics of Oscillatory Chains and Nanostructures. Springer Science and Business Media, Berlin (2017)
Nesterenko, V.F.: Nonlinear waves in sonic vacuum. Fizika gorenia i vzryva 28(3), 121–123 (1992). (in russian)
Nesterenko, V.: Propagation of nonlinear compression pulses in granular media. J. Appl. Mech. Tech. Phys. 24, 733–743 (1983)
Kivshar, Y.S.: Intrinsic localized modes as solitons with a compact support. Phys. Rev. E 48(1), 43–45 (1993)
Rosenau, P., Pikovsky, A.: Breathers in strongly anharmonic lattices. Phys. Rev. E 89(022,924) (2014)
Starosvetsky, Y., Ben-Meir, Y.: Nonstationary regimes of homogeneous hamiltonian systems in the state of sonic vacuum. Phys. Rev. E 87(062,919) (2013)
Koroleva(Kikot), I., Manevitch, L., Vakakis, A.F.: Non-stationary resonance dynamics of a nonlinear sonic vacuum with grounding supports. J. Sound Vib. 357, 349–364 (2015)
Kauderer, H.: Nichtlineare Mechanik. Springer, Berlin (1958)
Kirchhoff, G.: Vorlesungen ber Mathematische Physik. Erster Band, Mechanik (1897)
Koroleva(Kikot), I.P., Manevitch, L.I.: Oscillatory chain with grounding support in conditions of acoustic vacuum. Rus. J. Nonlinear Dyn. 11(3), 487–502 (2015). (in russian)
Pilipchuk, V.N.: Nonlinear Dynamics: Between Linear and Impact Limits, vol. 52. Springer Science and Business Media, Berlin (2010)
Acknowledgements
We are grateful to the Russian Foundation for Basic Research (Grants No. 17-01-00582, 16-02-00400) for financial support of this work.
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
Koroleva (Kikot), I.P., Manevitch, L.I. (2019). Is Energy Localization Possible in the Conditions of Non-local Acoustic Vacuum?. 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_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-92234-8_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-92233-1
Online ISBN: 978-3-319-92234-8
eBook Packages: EngineeringEngineering (R0)