Steady-state dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator. (English) Zbl 1180.70037

Summary: We present a theoretical study of the dynamics of the coupled system of Jiang, McFarland, Bergman, and Vakakis. It comprises a harmonically excited linear subsystem weakly coupled to an essentially nonlinear oscillator. We explored the rich dynamics exhibited by this coupled system by determining its periodic responses and their bifurcations. Not surprisingly, we found a lot of interesting dynamics over a broad frequency range: cyclic-fold, Hopf, symmetry-breaking, and period-doubling bifurcations; phase-locked motions; regions with multiple coexisting solutions; hysteresis; and chaos. We did not find any occurrence of energy transfer via modulation (also known as zero-to-one internal resonance); theoretically, the possibility of its occurrence was ruled out for systems with weak nonlinearity and damping. Finally, we investigated the effectiveness of the so-called nonlinear energy sink (NES) in vibration attenuation of forced linear structures. We found that the NES results in an increase in the vibration amplitude of the linear subsystem, especially when the damping is low, contrary to the claim made by X. Jiang et al. [Nonlinear Dyn. 33, No. 1, 87–102 (2003; Zbl 1039.70506)]. Also, we did not find any indication of nonlinear energy pumping or localization of energy in the NES, away from the directly forced linear subsystem, indicating that the NES is not effective for controlling the vibrations of forced linear structures.


70K50 Bifurcations and instability for nonlinear problems in mechanics
70J35 Forced motions in linear vibration theory


Zbl 1039.70506
Full Text: DOI


[1] Vakakis, A.F., McFarland, D.M., Bergman, L.A., Manevitch, L., Gendelman, O.: Passive vibration control through nonlinear energy pumping. In: Proceedings of the ASME Design Engineering Technical Conferences, Chicago, IL (2003)
[2] Georgiadis, F., Vakakis, A.F., McFarland, D.M., Bergman, L. A.: Shock isolation through passive energy pumping in a system with piecewise linear stiffnesses. In: Proceedings of the ASME Design Engineering Technical Conferences, Chicago, IL (2003)
[3] Bergman, L.A., McFarland, D.M., Vakakis, A.F.: Flutter suppression by means of nonlinear energy sinks. In: Proceedings of AFOSR/AFRL Workshop on Nonlinear Aspects of Aeroelasticity and Related Structural Dynamics, Shalimar, FL (2003)
[4] Gendelman, O.: Transition of energy to a nonlinear localized mode in a highly asymmetric system of two oscillators. Nonlinear Dyn. 25, 237–253 (2001) · Zbl 0999.70019
[5] Gendelman, O., Manevitch, L.I., Vakakis, A.F., M’Closkey, R.: Energy pumping in nonlinear mechanical oscillators: part I – Dynamics of the underlying Hamiltonian systems. J. Appl. Mech. 68, 34–41 (2001) · Zbl 1110.74452
[6] Vakakis, A.F., Gendelman, O.: Energy pumping in nonlinear mechanical oscillators: part II – Resonance capture. J. Appl. Mech. 68, 42–48 (2001) · Zbl 1110.74725
[7] Vakakis, A.F.: Inducing passive nonlinear energy sinks in vibrating systems. Journal of Vibration and Acoustics 123, 324–332 (2001)
[8] Gourdon, E., Lamarque, C.H.: Energy pumping with various nonlinear structures: numerical evidences. Nonlinear Dynamics 40, 281–307 (2005) · Zbl 1101.70015
[9] Jiang, X., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Steady state passive nonlinear energy pumping in coupled oscillators: theoretical and experimental results. Nonlinear Dynamics 33, 87–102 (2003) · Zbl 1039.70506
[10] Anderson, T.J., Balachandran, B., Nayfeh, A.H.: Observations of nonlinear interactions in a flexible cantilever beam. In: Proceedings of the 33rd AIAA Structures, Structural Dynamics, and Materials Conference, Dallas, TX (1992)
[11] Nayfeh, S.A., Nayfeh, A.H.: Nonlinear interactions between two widely spaced modes – external excitation. International Journal of Bifurcation and Chaos 3, 417–427 (1993) · Zbl 0900.70305
[12] Nayfeh, A.H., Mook, D.T.: Energy transfer from high-frequency to low-frequency modes in structures. Journal of Vibration and Acoustics 117, 186–195 (1995)
[13] Nayfeh, A.H.: Nonlinear Interactions, Wiley, New York (2000)
[14] Malatkar, P., Nayfeh, A.H.: On the transfer of energy between widely spaced modes in structures. Nonlinear Dynamics 31, 225–242 (2003) · Zbl 1027.74501
[15] Nayfeh, A.H., Mook, D.T., Nonlinear Oscillations. Wiley, New York (1979)
[16] Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics, Wiley, New York (1995) · Zbl 0848.34001
[17] Hayashi, C.: Nonlinear Oscillations in Physical Systems, McGraw-Hill, New York (1964) · Zbl 0192.50605
[18] Hassan, A.: On the local stability analysis of the approximate harmonic balance solutions. Nonlinear Dynamics 10, 105–133 (1996)
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