Abstract
In this paper, a regular perturbation tool is suggested to bridge the gap between weakly and strongly nonlinear dynamics based on exactly solvable oscillators with trigonometric characteristics considered by Nesterov (Proc. Mosc. Inst. Power Eng. 357:68–70, 1978). It is shown that the corresponding action-angle variables linearize the original oscillators with no special functions involved. As a result, linear and strongly nonlinear areas of the dynamics are described within the same perturbation procedure. The developed tool is applied then to analyzing the nonlinear beat and energy localization phenomena in two linearly coupled Duffing oscillators. It is shown that the principal phase variable describing the beat phenomena is governed by the hardening Nesterov oscillator with some perturbation due to qubic nonlinearity and coupling between the oscillators. As a result, the above class of strongly nonlinear oscillators is given clear physical meaning, whereas a closed form analytical solution is obtained for nonlinear beat and localization dynamics. Based on this solution, necessary and sufficient conditions for onset of energy localization are obtained.
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Coppola, V.T., Rand, R.H.: Averaging using elliptic functions: approximation of limit cycles. Acta Mech. 81, 125–142 (1990)
Xu, Z., Cheung, Y.: Averaging method using generalized harmonic functions for strongly non-linear oscillators. J. Sound Vib. 174(4), 563–576 (1994)
Nayfeh, A.H.: Perturbation methods in nonlinear dynamics. In: Nonlinear Dynamics Aspects of Particle Accelerators, Santa Margherita di Pula, 1985, pp. 238–314. Springer, Berlin (1986)
Salenger, G., Vakakis, A.F., Gendelman, O., Manevitch, L., Andrianov, I.: Transitions from strongly to weakly nonlinear motions of damped nonlinear oscillators. Nonlinear Dyn. 20(2), 99–114 (1999)
Mikhlin, Y.V.: Analytical construction of homoclinic orbits of two- and three-dimensional dynamical systems. J. Sound Vib. 230(5), 971–983 (2000)
Andrianov, I.V.: Asymptotics of nonlinear dynamical systems with a high degree of nonlinearity. Dokl. Math. 66(2), 270–273 (2002)
Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)
Ozorio de Almeida, A.M.: Hamiltonian Systems: Chaos and Quantization. Cambridge University Press, Cambridge (1988)
Liao, S.-J., Chwang, A.: Application of homotopy analysis method in nonlinear oscillations. ASME J. Appl. Mech. 65, 914–922 (1998)
Nesterov, S.V.: Examples of nonlinear Kline–Gordon equations, solvable in terms of elementary functions. In: Proceedings of the Moscow Institute of Power Engineering, vol. 357, pp. 68–70 (1978)
Dimentberg, M.F., Bratus, A.S.: Bounded parametric control of random vibrations. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456(2002), 2351–2363 (2000)
Dimentberg, M.F., Iourtchenko, D., Bratus, A.S.: Transition from planar to whirling oscillations in a certain nonlinear system. Nonlinear Dyn. 23, 165–174 (2000)
Pilipchuk, V.N.: Transient mode localization in coupled strongly nonlinear exactly solvable oscillators. Nonlinear Dyn. (2007). http://dx.doi.org/10.1007/s11071-007-9207-4
Timoshenko, S.P., Yang, D., Wiver, U.: Vibration Problems in Engineering. Wiley, New York (1974)
Kosevich, A., Kovalev, A.: Introduction to Nonlinear Physical Mechanics. Naukova Dumka, Kiev (1989) (in Russian)
Holm, D.D., Lynch, P.: Stepwise precession of the resonant swinging spring. SIAM J. Appl. Dyn. Syst. 1(1), 44–64 (2002)
Manevich, A., Manevitch, L.: The Mechanics of Nonlinear Systems with Internal Resonances. Imperial College Press, London (2005)
Uzunov, I., Muschall, R., Golles, M., Kivshar, Y., Malomed, B., Lederer, F.: Pulse switching in nonlinear fiber directional couplers. Phys. Rev. E 51, 2527–2537 (1995)
Kutz, N.: Mode-locked soliton lasers. SIAM Rev. 48(4), 629–678 (2006)
Manevitch, L.I., Musienko, A.: Limiting phase trajectory and beating phenomena in systems of coupled nonlinear oscillators. In: 2nd International Conference on Nonlinear Normal Modes and Localization in Vibrating Systems, Samos, Greece, 19–23 June 2006, pp. 25–26 (2006)
Pierce, J.: Coupling of modes of propagation. J. Appl. Phys. 25(2), 179–183 (1954)
Scott, L.P.S., Eilbeck, J.C.: Between the local-mode and normal-mode limits. Chem. Phys. Lett. 113(1), 29–36 (1985)
Manevitch, L.: New approach to beating phenomenon in coupled nonlinear oscillatory chains. Arch. Appl. Mech. 77(5), 301–312 (2007)
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Pilipchuk, V.N. Transitions from strongly to weakly-nonlinear dynamics in a class of exactly solvable oscillators and nonlinear beat phenomena. Nonlinear Dyn 52, 263–276 (2008). https://doi.org/10.1007/s11071-007-9276-4
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DOI: https://doi.org/10.1007/s11071-007-9276-4