Controllable wave propagation in a weakly nonlinear monoatomic lattice chain with nonlocal interaction and active control

doi:10.1007/s10483-018-2360-6

Abstract

Abstract: The one-dimensional monoatomic lattice chain connected by nonlinear springs is investigated, and the asymptotic solution is obtained through the Lindstedt-Poincaré perturbation method. The dispersion relation is derived with the consideration of both the nonlocal and the active control effects. The numerical results show that the nonlocal effect can effectively enhance the frequency in the middle part of the dispersion curve. When the nonlocal effect is strong enough, zero and negative group velocities will be evoked at different points along the dispersion curve, which will provide different ways of transporting energy including the forward-propagation, localization, and backwardpropagation of wavepackets related to the phase velocity. Both the nonlinear effect and the active control can enhance the frequency, but neither of them is able to produce zero or negative group velocities. Specifically, the active control enhances the frequency of the dispersion curve including the point at which the reduced wave number equals zero, and therefore gives birth to a nonzero cutoff frequency and a band gap in the low frequency range. With a combinational adjustment of all these effects, the wave propagation behaviors can be comprehensively controlled, and energy transferring can be readily manipulated in various ways.

Key words: Robust control, bilinear system, uncertain, stability analysis, Liapunov function, nonlocal effect, monoatomic lattice chain, band gap, negative group velocity, active control

2010 MSC Number:

74B20

Jiao WANG, Weijian ZHOU, Yang HUANG, Chaofeng LYU, Weiqiu CHEN, Weiqiu ZHU. Controllable wave propagation in a weakly nonlinear monoatomic lattice chain with nonlocal interaction and active control. Applied Mathematics and Mechanics (English Edition), 2018, 39(8): 1059-1070.

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