Frequency bands of strongly nonlinear homogeneous granular systems

Abstract

Recent numerical studies on an infinite number of identical spherical beads in Hertzian contact showed the presence of frequency bands [Jayaprakash, Starosvetsky, Vakakis, Peeters, and Kerschen, Nonlinear Dyn. 63, 359 (2011)]. These bands, denoted here as propagation and attenuation bands (PBs and ABs), are typically present in linear or weakly nonlinear periodic media; however, their counterparts are not intuitive in essentially nonlinear periodic media where there is a complete lack of classical linear acoustics, i.e., in "sonic vacua." Here, we study the effects of PBs and ABs on the forced dynamics of ordered, uncompressed granular systems. Through numerical and experimental techniques, we find that the dynamics of these systems depends critically on the frequency and amplitude of the applied harmonic excitation. For fixed forcing amplitude, at lower frequencies, the oscillations are large in amplitude and governed by strongly nonlinear and nonsmooth dynamics, indicating PB behavior. At higher frequencies the dynamics is weakly nonlinear and smooth, in the form of compressed low-amplitude oscillations, indicating AB behavior. At the boundary between the PB and the AB large-amplitude oscillations due to resonance occur, giving rise to collisions between beads and chaotic dynamics; this renders the forced dynamics sensitive to initial and forcing conditions, and hence unpredictable. Finally, we study asymptotically the near field standing wave dynamics occurring for high frequencies, well inside the AB.

The study of chain-like materials for use in biomedical ultrasound

“…This leads to the generation of solitary wave impulses, and these would be amplified after multiple reflections between the two ends of the chain. Thus, the approach would be an interaction between the generation of different frequency components [7,8], and the natural normal modes of the chain itself [3,4]. This would need to be considered carefully in the experimental design, so that the conditions might be created for optimal generation of solitary travelling wave impulses.…”

The generation of impulses from narrow bandwidth signals using resonant spherical chains

“…It is clear from these results that the spherical chain naturally selects certain lengths at which a set of resonant peaks occur, each of which are separated by a sub-harmonic of the driving frequency. These resonances of the chain are examples of non-linear normal modes (NNMs) of the system, and the properties of NNMs have been described by other authors [4]. The interesting point is that such behaviour has not been observed previously -other studies were limited to larger spheres or limited drive amplitudes, and tended to be at low frequencies.…”

Generation of Ultrasound Pulses in Water Using Granular Chains with a Finite Matching Layer

“…Wave propagation in homogeneous granular chains and interactions of spherical elastic objects have been well studied both theoretically and experimentally [1][2][3][4][5]. In such a system with Hertzian contact, a nonlinear forcedeformation relation exists as F ∝ δ 1.5 .…”

“…This study, therefore, focuses on maximizing the transferred energy from a chain of spheres into biological tissue and also the harmonic content, which increases the bandwidth of the generated ultrasound waves. After Nesterenko's pioneering work in 1983 [1], a great number of models were developed to simulate wave propagation in infinite and finite chains [2,4,18]. However, predicting the resonance behavior in harmonically driven monoatomic granular chains, modeling the nonlinear and dispersive effects, and minimizing the overall structure are still big challenges [5,19].…”