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Nonlinear resonances leading to strong pulse attenuation in granular dimer chains. (English) Zbl 1319.70023

Summary: We study nonlinear resonances in granular periodic one-dimensional chains. Specifically, we consider a diatomic (“dimer”) chain composed of alternating “heavy” and “light” spherical beads with no precompression. In a previous work [K. R. Jayaprakash et al., Phys. Rev. E 83, No. 3, 036606 (2011)] we discussed the existence of families of solitary waves in these systems that propagate without distortion of their waveforms. We attributed this dynamical feature to “antiresonance” in the dimer that led to the complete elimination of radiating waves in the trail of the propagating solitary wave. Antiresonances were associated with certain symmetries of the velocity waveforms of the dimer beads. In this work we report on the opposite phenomenon: the break of waveform symmetries, leading to drastic attenuation of traveling pulses due to radiation of traveling waves to the far field. We use the connotation of “resonance” to describe this dynamical phenomenon resulting in maximum amplification of the amplitudes of radiated waves that emanate from the propagating pulse. Each antiresonance can be related to a corresponding resonance in the appropriate parameter plane. We study the nonlinear resonance mechanism numerically and analytically and show that it can lead to drastic attenuation of pulses propagating in the dimer. Furthermore, we estimate the discrete values of the normalized mass ratio between the light and heavy beads of the dimer for which resonances are realized. Finally, we show that by adding precompression the resonance mechanism gradually degrades, as does the capacity of the dimer to passively attenuate propagating pulses.

MSC:

70K30 Nonlinear resonances for nonlinear problems in mechanics
70K60 General perturbation schemes for nonlinear problems in mechanics
70K70 Systems with slow and fast motions for nonlinear problems in mechanics
74J30 Nonlinear waves in solid mechanics
74J35 Solitary waves in solid mechanics
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