Energy Pumping in Nonlinear Mechanical Oscillators: Part II—Resonance Capture

[+] Author and Article Information
A. F. Vakakis

Department of Mechanical and Industrial Engineering, University of Illinois, 1206 W. Green Street, Urbana, IL 61801 e-mail: avakakis@uiuc.edu

O. Gendelman

Institute of Chemical Physics, Russian Academy of Sciences, Kosygin Street 4, 117977 Moscow, Russia e-mail: ovgend@center.chph.ras.ru

J. Appl. Mech 68(1), 42-48 (May 02, 2000) (7 pages) doi:10.1115/1.1345525 History: Received September 29, 1999; Revised May 02, 2000
Copyright © 2001 by ASME
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Gendelman,  O., Manevitch,  L. I., Vakakis,  A. F., and M’Closkey,  R., 2001, “Energy ‘Pumping’ in Coupled Mechanical Oscillators I: Dynamics of the Underlying Hamiltonian Systems,” ASME J. Appl. Mech., Vol. 68, pp. 34–41.
Arnold, V. I., ed., 1988, Dynamical Systems III (Encyclopaedia of Mathematical Sciences), Vol. 3, Springer-Verlag, Berlin.
Lochak, P., and Meunier, C., 1988, Multiphase Averaging for Classical Systems (Series on Applied Mathematical Sciences), Vol. 72, Springer-Verlag, Berlin.
Morozov, A. D., 1998, Quasi-conservative Systems, Cycles, Resonances and Chaos (Series on Nonlinear Science, Series A), Vol. 30, World Scientific, Singapore.
Neishtadt,  A. I., 1975, “Passage Through a Resonance in the Two-Frequency Problem,” Dokl. Akad. Nauk, 221, pp. 301–304.
Neishtadt,  A. I., 1976, “Averaging in Multifrequency Systems II,” Dokl. Akad. Nauk SSSR, 226, pp. 1295–1298.
Neishtadt,  A. I., 1975, “Passage Through a Separatrix in a Resonance Problem With a Slowly-Varying Parameter,” Prikl. Mat. Meck. (PMM), 39, No. 4, pp. 621–632.
Haberman,  P., 1983, “Energy Bounds for the Slow Capture by a Center in Sustained Resonance,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 43, No. 2, pp. 244–256.
Morozov,  A. D., and Shil’nikov,  L. P., 1984, “On Nonconservative Periodic Systems Close to Two-Dimensional Hamiltonian,” Prikl. Mat. Meck. (PMM), 47, No. 3, pp. 327–334.
Lewin,  L., and Kevorkian,  J., 1978, “On the Problem of Sustained Resonance,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 35, No. 4, pp. 738–754.
Kath,  W. L., 1983, “Necessary Conditions for Sustained Reentry Roll Resonance,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 43, No. 2, pp. 314–324.
Kath,  W. L., 1983, “Conditions for Sustained Resonance II,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 43, No. 3, pp. 579–583.
Quinn,  D., and Rand,  R., 1995, “The Dynamics of Resonance Capture,” Nonlinear Dyn., 8, pp. 1–20.
Rand,  R., and Quinn,  D., 1995, “Resonant Capture in a System of Two Coupled Homoclinic Oscillators,” J. Vib. Control, 1, pp. 41–56.
Bosley,  D. L., and Kevorkian,  J., 1995, “On the Asymptotic Solution of Non-Hamiltonian Systems Exhibiting Sustained Resonance,” Stud. Appl. Math., 94, pp. 83–130.
Bosley,  D. L., 1996, “An Improved Matching Procedure for Transient Resonance Layers in Weakly Nonlinear Oscillatory Systems,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 56, No. 2, pp. 420–445.
Percival, I., and Richards, D., 1982, Introduction to Dynamics, Cambridge University Press, Cambridge, UK.
Vakakis,  A. F., 1994, “Exponentially Small Splittings of Manifolds in a Rapidly Forced Duffing System,” J. Sound Vib., 170, No. 1, pp. 119–129.


Grahic Jump Location
Transient responses I1(t) and I2(t) of system (1) for, (a) h=0.5, (b) h=0.8, (c) h=1.125; – oscillator 1, [[dashed_line]] oscillator 2
Grahic Jump Location
Phase portraits of system (14) for, (a) μ>ν, and (b) μ≤ν
Grahic Jump Location
Numerical solutions of system (30): (a) no resonance capture (M=2.8), (b, c, d) resonance capture (M=4.0, 10.0, 15.0)
Grahic Jump Location
Transient response y1(t) of system (1), (a) when no energy pumping occurs (h=0.5), and (b,c) when energy pumping takes place (h=0.8, 1.125). ⋄⋄⋄⋄⋄⋄⋄⋄ Analytical approximations based on (30), – Numerical simulations.




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