Abstract
This paper studies the emergence of autoresonance (AR) in a two-degrees-of-freedom system consisting of an excited nonlinear actuator (the Duffing oscillator) linearly coupled to a passive linear oscillator. Two classes of system are considered: (1) In the system of the first type, a periodic force with constant (resonance) frequency is applied to the nonlinear (Duffing) oscillator with slowly time-decreasing linear stiffness; (2) in the system of the second type, the nonlinear oscillator with constant parameters is excited by a force with slowly increasing frequency. In both cases, the linear oscillator and the linear coupling are time-invariant, and the system is initially engaged in resonance. In the first step, the dynamics of the undamped system is analyzed. It is shown here that in the system of the first type, AR may occur in both oscillators, but in the system of the second type, the amount of energy transferred from the nonlinear oscillator is insufficient to excite high-energy motion in the attached oscillator. Different dynamical behavior arises due to different resonance properties of the systems. In the next step, an effect of viscous damping in the oscillators and the coupling is investigated. As this paper demonstrates, the response enhancement may be observed in the system of the first type on a bounded time interval provided that dissipation is weak enough. Explicit asymptotic approximations of the solutions are obtained. Close proximity of the derived approximations to exact (numerical) results is proved.
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References
Vazquez, L., MacKay, R., Zorzano, M.P.: Localization and Energy Transfer in Nonlinear Systems. World Scientific, Singapore (2003)
Vakakis, A.F., Gendelman, O., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Targeted Energy Transfer in Mechanical and Structural Systems. Springer, Berlin (2008)
May, V., Kühn, O.: Charge and Energy Transfer Dynamics in Molecular Systems. Wiley, Weinheim (2011)
Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillators. Pergamon Press, Oxford (1966)
Astashev, V.K., Babitsky, V.I.: Ultrasonic Processes and Machines. Springer, Berlin (2007)
Veksler, V.I.: A new method of accelerating relativistic particles. Comptes Rendus (Dokaldy) de l’Academie Sciences de l’URSS 43, 329–331 (1944)
McMillan, E.M.: The synchrotron—a proposed high energy particle accelerator. Phys. Rev. 68, 143–146 (1945)
Chacón, R.: Control of Homoclinic Chaos by Weak Periodic Perturbations. World Scientific, Singapore (2005)
Charman, A.E.: Random Aspects of Beam Physics and Laser-Plasma Interactions. University of California, Berkeley (2007)
Sinclair, A.T.: On the origin of commensurabilities amongst satellites of Saturn. Mon. Not. R. Astron. Soc. 160, 169–187 (1972)
Naaman, O., Aumentado, J., Friedland, L., Wurtele, J.S., Siddiqi, I.: Phase-locking transition in a chirped superconducting Josephson resonator. Phys. Rev. Lett. 101, 117005 (2008)
Fajans, J., Friedland, L.: Autoresonant (nonstationary) excitation of pendulums, Plutinos, plasmas, and other nonlinear oscillators. Am. J. Phys. 69, 1096–1102 (2001)
Neishtadt, A.I.: Averaging, passage through resonances, and capture into resonance in two-frequency systems. Russ. Math. Surv. 69, 771–843 (2014)
Zelenyi, L.M., Neishtadt, A.I., Artemyev, A.V., Vainchtein, D.L., Malova, H.V.: Quasiadiabatic dynamics of charged particles in space plasma. Phys. Uspekhi 56, 347–394 (2013)
Klughertz, G., Hervieux, P.-A., Manfredi, G.: Autoresonant control of the magnetization switching in single-domain nanoparticles. J. Phys. D Appl. Phys. 47, 345004 (2014)
Barth, I., Friedland, L.: Multiresonant control of two-dimensional dynamical systems. Phys. Rev. E 76, 016211 (2007)
Gammaitoni, L., Vocca, H., Neri, I., Travasso, F., Orfei, F.: Vibration energy harvesting: linear and nonlinear oscillator approaches. In: Tan, Y.K. (ed.) Sustainable Energy Harvesting Technologies—Past, Present and Future. InTech, Croatia (2011)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, Weinheim (2004)
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)
Manevitch, L.I., Kovaleva, A., Manevitch, E.L., Shepelev, D.: Limiting phase trajectories and non-stationary resonance oscillations of the Duffing oscillator. Part 1. A non-dissipative oscillator. Commun. Nonlinear Sci. Numer. Simul. 16, 1089–1097 (2011)
Manevitch, L.I., Kovaleva, A.: Nonlinear energy transfer in classical and quantum systems. Phys. Rev. E 87, 022904 (2013)
Kovaleva, A., Manevitch, L.I.: Limiting phase trajectories and emergence of autoresonance in nonlinear oscillators. Phys. Rev. E 88, 024901 (2013)
Kovaleva, A., Manevitch, L.I., Kosevich, Y.: Fresnel integrals and irreversible energy transfer in an oscillatory system with time-dependent parameters. Phys. Rev. E 83, 026602 (2011)
Kovaleva, A., Manevitch, L.I.: Classical analog of quasilinear Landau–Zener tunneling. Phys. Rev. E 85, 016202 (2012)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products, 6th edn. Academic Press, San Diego (2000)
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The authors acknowledge support for this work from the Russian Foundation for Basic Research (Grant 14-01-00284).
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Manevitch, L.I., Kovaleva, A. Autoresonant dynamics of weakly coupled oscillators. Nonlinear Dyn 84, 683–695 (2016). https://doi.org/10.1007/s11071-015-2517-z
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DOI: https://doi.org/10.1007/s11071-015-2517-z