Abstract
Intrinsic Localized Modes (ILMs) are defined as localizations due to strong intrinsic nonlinearity within an array of perfect, periodically repeating oscillators. Such nonlinear phenomena have been studied for a number of years in the solid-state physics literature. Energy can become localized at a specific location in a discrete system as a result of the nonlinearity of the system and not due to any defects or impurities within the considered systems. Here, such mode localization is studied in the context of microcantilever arrays and microresonator arrays, and it is explored if an ILM can be realized as a forced nonlinear normal mode or nonlinear vibration mode. The method of multiple scales and methods to construct nonlinear normal modes are used to study nonlinear vibrations of microresonator arrays. Investigations reported in this article suggest that it is possible to realize an ILM as a forced nonlinear vibration mode. These results are believed to be important for future designs of microresonator arrays intended for signal processing, communication, and sensor applications.
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References
Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958)
Sievers, A.J., Takeno, S.: Intrinsic localized modes in anharmonic crystals. Phys. Rev. Lett. 61(8), 970–973 (1988)
Campbell, D.K., Flach, S., Kivshar, V.S.: Localizing energy through nonlinearity and discreteness. Phys. Today 57, 43–49 (2004)
Ustinov, A.V.: Imaging of discrete breathers. Chaos 13, 716–724 (2004)
Fleischer, J.W., Segev, M., Efremidis, N.K., Christodoulides, D.N.: Observation of two-dimensional discrete solutions in optically induced nonlinear photonic lattices. Nature 4224, 137–150 (2003)
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), 1–4 (2003)
Sato, M., Hubbard, B.E., English, L.Q., Sievers, A.J., Ilic, B., Czaplewski, D.A., Craighead, H.G.: Study of intrinsic localized vibrational modes in micromechanical oscillator arrays. Chaos 13, 702–715 (2003)
Dauxois, T., Peyrard, M., Willis, C.R.: Discrete effects on the formation and propagation of breathers in nonlinear Klein–Gordon equations. Phys. Rev. E 48, 4768–4778 (1993)
Balachandran, B., Li, H.: Nonlinear phenomena in microelectromechanical resonators. In: Proceedings of the IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, pp. 97–106 (2003)
Dick, A.J., Balachandran, B.: Parametric identification of piezoelectric microscale resonators. In: Proceedings of the Fifth ENOC, No. 19-239, pp. 1–10 (2005)
Li, H., Preidikman, S., Balachandran, B., Mote, C.D. Jr.: Nonlinear free and forced oscillations of piezoelectric microresonators. J. Micromech. Microeng. 16(2), 356–367 (2006)
Dick, A.J., Balachandran, B., DeVoe, D.L., Mote, C.D. Jr.: Parametric identification of piezoelectric microscale resonators. J. Micromech. Microeng. 16(8), 1593–1601 (2006)
Lifshitz, R., Cross, M.C.: Response of parametrically driven nonlinear couple oscillators with applications to micromechanical and nanomechanical resonator arrays. Phys. Rev. B 67, 134302 (2003)
Sato, M., Hubbard, B.E., Sievers, A.J., Ilic, B., Craighead, H.G.: Optical manipulation of intrinsic localized vibrational energy in cantilever arrays. Europhys. Lett. 66, 318–323 (2004)
Sato, M., Hubbard, B.E., Sievers, A.J.: Colloquium: Nonlinear energy localization and its manipulation in micromechanical oscillator arrays. Rev. Mod. Phys. 78(1), 137–157 (2006)
Nayfeh, A.H.: Nonlinear Interactions: Analytical, Computational, and Experimental Methods. Wiley, New York (2000)
Nayfeh, A.H.: Perturbation Methods. Wiley, New York (2003)
Pak, C.H.: Nonlinear Normal Modes Dynamics: For Two Degree-of-Freedom Systems. Inha University Press, Seoul (1999)
Andrianov, I., Awrejcewicz, J., Manevitch, L.I.: Asymptotical Mechanics of Thin-Walled Structures. Springer, Berlin (2004)
Dick, A.J., Balachandran, B., Mote, C.D., Jr.: Intrinsic localized modes and nonlinear normal modes in micro-resonator arrays. In: Proceedings of the ASME IMECE, No. 80255 (2005)
Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley, New York (1995)
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Dick, A.J., Balachandran, B. & Mote, C.D. Intrinsic localized modes in microresonator arrays and their relationship to nonlinear vibration modes. Nonlinear Dyn 54, 13–29 (2008). https://doi.org/10.1007/s11071-007-9288-0
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DOI: https://doi.org/10.1007/s11071-007-9288-0