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**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.

### MSC:

70K50 | Bifurcations and instability for nonlinear problems in mechanics |

70J35 | Forced motions in linear vibration theory |

### Citations:

Zbl 1039.70506
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\textit{P. Malatkar} and \textit{A. H. Nayfeh}, Nonlinear Dyn. 47, No. 1--3, 167--179 (2007; Zbl 1180.70037)

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### References:

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