Abstract
The study of disorder-induced frequency filtering is presented for one-dimensional systems composed of random, pre-stressed masses interacting through both linear and nonlinear (Hertzian) repulsive forces. An ensemble of such systems is driven at a specified frequency, and the spectral content of the propagated disturbance is examined as a function of distance from the source. It is shown that the transmitted signal contains only low-frequency components, and the attenuation is dependent on the magnitude of disorder, the input frequency, and the contact model. It is found that increased disorder leads to a narrower bandwidth of transmitted frequencies at a given distance from the source and that lower input frequencies exhibit less sensitivity to the arrangement of the masses. Comparison of the nonlinear and linear contact models reveals qualitatively similar filtering behavior; however, it is observed that the nonlinear chain produces transmission spectrums with a greater density at the lowest frequencies. In addition, it is shown that random masses sampled from normal, uniform, and binary distributions produce quantitatively indistinguishable filtering behavior, suggesting that knowledge of only the distribution’s first two moments is sufficient to characterize the bulk signal transmission behavior. Finally, we examine the wave number evolution of random chains constrained to move between fixed end-particles and present a transfer matrix theory in wave number space, and an argument for the observed filtering based on the spatial localization of the higher-frequency normal modes.
Highlights
One-dimensional analogs of electronic, magnetic, and mechanical systems are often employed for their use as simple models which have the potential to reveal the physics of more general, higher-dimensional systems [25]
Sen et al [43] provide a detailed account of prior studies concerning solitary waves in granular chains
We study the effect of disorder and nonlinearity on the transmission of signals in one-dimensional systems
Summary
One-dimensional analogs of electronic, magnetic, and mechanical systems are often employed for their use as simple models which have the potential to reveal the physics of more general, higher-dimensional systems [25]. These treatments are typically limited to infinitely repeatable unit cells containing one or two particles/atoms for which dispersion equations relating the oscillation frequency and wavelength are analytically accessible It is from these periodic, linear systems that more recent studies on inhomogeneous, disordered, and nonlinear chains originate. Ponson et al [39] employ a nonlinear chain of two-particle unit cells which are randomly oriented, as in a spin system, and studies the effect of their disorder parameter on the spatial decay of the force transmitted by such systems. Tournat et al [48] observe the propagation of these low-frequency signals in nonlinear chains, terming itself demodulation Such behavior is due to nonlinear interaction and is not a mass disorder-induced effect.
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