Frequency filtering in disordered granular chains
TL;DR: In this article, disorder-induced frequency filtering is studied for one-dimensional systems composed of random, pre-stressed masses interacting through both linear and nonlinear (Hertzian) repulsive forces.
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.
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TL;DR: In this paper, the rectification effect on the propagation of solitary waves in the symmetric Y-shaped granular chain was numerically investigated and it was shown that both nonlinearity and collision effects dominate the rectifying process.
Abstract: The rectification effect on the propagation of solitary waves in the symmetric Y-shaped granular chain is numerically investigated. A heterojunction with mass mismatch occurs at the position of Y-junction by adjusting the branch angle. And the heavy-light heterojunction is more favorable for the solitary wave passing. Based on the characteristics of wave propagation velocity and gap’s opening, we argue that both nonlinearity and collision effects dominate the rectification process. The rectification efficiency can be improved by adjusting the branch angle and the direction of incident solitary wave. The results have particularly practical significance for the potential design of acoustic diode devices.
4 citations
TL;DR: In this paper, a simpler, faster stochastic model for energy propagation is presented, where all frequencies/energies are grouped into bands, and the mean field model is calibrated from the deterministic analytical solutions by ensemble averaging various band-to-band transfer situations for short times, as well as considering the basis energy levels (decaying with the wavenumber increasing) that are not transferred.
Abstract: Energy transfer is one of the essentials of mechanical wave propagation (along with momentum transport) Here, it is studied in disordered one-dimensional model systems mimicking force-chains in real systems The pre-stressed random masses (other types of disorder lead to qualitatively similar behavior) interact through (linearized) Hertzian repulsive forces, which allows solving the deterministic problem analytically The main goal, a simpler, faster stochastic model for energy propagation, is presented in the second part, after the basic equations are re-visited and the phenomenology of pulse propagation in disordered granular chains is reviewed First, the propagation of energy in space is studied With increasing disorder (quantified by the standard deviation of the random mass distribution), the attenuation of pulsed signals increases, transiting from ballistic propagation (in ordered systems) towards diffusive-like characteristics, due to energy localization at the source Second, the evolution of energy in time by transfer across wavenumbers is examined, using the standing wave initial conditions of all wavenumbers Again, the decay of energy (both the rate and amount) increases with disorder, as well as with the wavenumber The dispersive ballistic transport in ordered systems transits to low-pass filtering, due to disorder, where localization of energy occurs at the lowest masses in the chain Instead of dealing with the too many degrees of freedom or only with the lowest of all the many eigenmodes of the system, we propose a stochastic master equation approach with reduced complexity, where all frequencies/energies are grouped into bands The mean field stochastic model, the matrix of energy-transfer probabilities between bands, is calibrated from the deterministic analytical solutions by ensemble averaging various band-to-band transfer situations for short times, as well as considering the basis energy levels (decaying with the wavenumber increasing) that are not transferred Finally, the propagation of energy in the wavenumber space at transient times validates the stochastic model, suggesting applications in wave analysis for non-destructive testing, underground resource exploration, etc
4 citations
01 Sep 2018
TL;DR: In this paper, the influence of disorder on energy and momentum transport during vibration propagation (mechanical/sound wave), the sound wave speed and its low-pass frequency-filtering characteristics is the subject of a study.
Abstract: {What?} Disorder of size (polydispersity) and mass of discrete elements or particles in randomly structured media (e.g., granular matter such as soil) has numerous effects on materials’ sound propagation characteristics. The influence of disorder on energy and momentum transport during vibration propagation (mechanical/sound wave), the sound wave speed and its low-pass frequency-filtering characteristics is the subject of this study. {Why?} The goal is understanding the connection between the particle-microscale disorder and dynamics and the system-macroscale wave propagation, which can be applied to nondestructive testing, seismic exploration of buried objects (oil, mineral, etc.) or to study the internal structure of the Earth. {How?} The mechanical wave/vibration propagating through granular media exhibits a specific signature in time; a coherent pulse or wavefront arrives first with multiply scattered waves (coda) arriving later. The coherent pulse is of low frequency nature and micro-structure independent i.e. it depends only on the bulk properties of the disordered granular sample, the sound wave velocity and hence, bulk and shear moduli. The coda or the multiply scattered waves are of high frequency nature and are micro-structure dependent. Numerical and stochastic techniques for 1-D, 2-D discrete element systems and experiments with 1-D photoelastic particles constituting a granular chain have been employed to isolate and study different modes of the propagating waves (namely, P and S waves), disorder dependent dispersion relations, sound wave velocity and diffusive transport of spectral energy. {Results} Increase in mass disorder (where disorder has been defined such that it is independent of the shape of the probability distribution of masses) decreases the sound wave speed along a granular chain. Averaging over energies associated with the eigenmodes can be used to obtain better quality dispersion relations and can be formulated in a way to give a Master Equation of energy in terms of wavenumber or frequency which identifies the switching and cross-talk of energy between different frequency bands; these dispersion relations confirm the decrease in pass frequency and wave speed with increasing disorder acting opposite to the wave acceleration close to the source. Also, it is observed that an ordered granular chain exhibits ballistic propagation of energy whereas, a disordered granular chain exhibits more diffusive like propagation, which eventually becomes localized at long time periods.
4 citations
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TL;DR: In this article, the rectification effect on the propagation of solitary waves in the symmetric Y-shaped granular chain is numerically investigated, and the results have particularly practical significance for the potential design of acoustic diode devices.
Abstract: The rectification effect on the propagation of solitary waves in the symmetric Y-shaped granular chain is numerically investigated in this Letter. A heterojunction with mass mismatch occurs at the position of Y-junction by adjusting the branch angle. And the heavy-light heterojunction is more favorable for the solitary wave passing. The energy rectification efficiency can be improved by adjusting the branch angle and the direction of incident solitary wave. The results have particularly practical significance for the potential design of acoustic diode devices.
3 citations
TL;DR: In this article, the authors studied non-reciprocal wave transmission across the interface of two dissimilar granular media separated by an elastic solid medium, and the highly discontinuous and nonlinear interaction forces coupling the granular medium to the elastic solid were accurately computed through an algorithm with interrelated iteration and interpolation at successive adaptive time steps.
Abstract: We study nonreciprocal wave transmission across the interface of two dissimilar granular media separated by an elastic solid medium. Specifically, a left, larger-scale and a right smaller-scale granular media composed of two-dimensional, initially uncompressed hexagonally packed granules are interfacing with an intermediate linearly elastic solid, modeled either as a thin elastic plate or a linear Euler-Bernoulli beam. The granular media are modeled by discrete elements and the elastic solid by finite elements assuming a plane stress approximation for the thin plate. Accounting for the combined effects of Hertzian, frictional and rotational interactions in the granular media, as well as the highly discontinuous interfacial effects between the (discrete) granular media and the (continuous) intermediate elastic solid, the nonlinear acoustics of the integrated system is computationally studied subject to a half-sine shock excitation applied to a boundary granule of either the left or right granular medium. The highly discontinuous and nonlinear interaction forces coupling the granular media to the elastic solid are accurately computed through an algorithm with interrelated iteration and interpolation at successive adaptive time steps. Numerical convergence is ensured by monitoring the (linearized) eigenvalues of a nonlinear map of interface forces at each (variable) time step. Due to the strong nonlinearity and hierarchical asymmetry of the left and right granular media, time scale disparity occurs in the response of the interface which breaks acoustic reciprocity. Specifically, depending on the location and intensity of the applied shock, propagating wavefronts are excited in the granular media, which, in turn, excite either (slow) low-frequency vibrations or (fast) high-frequency acoustics in the intermediate elastic medium. This scale disparity is due to the size disparity of the left and right granular media, which yields drastically different wave speeds in the resulting propagating wavefronts. As a result, the continuum part of the interface responds with either low-frequency vibrations---when the shock is applied to the larger-scale granular medium, or high-frequency waves---when the shock is applied to the smaller-scale granular medium. This provides the fundamental mechanism for breaking reciprocity in the interface. The nonreciprocal interfacial acoustics studied here apply to a broad class of asymmetric hybrid (discrete-continuum) nonlinear systems and can inform predictive designs of highly effective granular shock protectors or granular acoustic diodes.
3 citations
References
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TL;DR: In this article, a simple model for spin diffusion or conduction in the "impurity band" is presented, which involves transport in a lattice which is in some sense random, and in them diffusion is expected to take place via quantum jumps between localized sites.
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