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, experimental data on a model soil in a cubical cell are compared with both discrete element (DEM) simulations and continuum analyses and the results show that the generally satisfactory agreement between experimental observations and DEM simulations can be seen as a validation and support the use of DEM to investigate the influence of grain interaction on wave propagation.
Abstract: In this study experimental data on a model soil in a cubical cell are compared with both discrete element (DEM) simulations and continuum analyses. The experiments and simulations used point source transmitters and receivers to evaluate the shear and compression wave velocities of the samples, from which some of the elastic moduli can be deduced. Complex responses to perturbations generated by the bender/extender piezoceramic elements in the experiments were compared to those found by the controlled movement of the particles in the DEM simulations. The generally satisfactory agreement between experimental observations and DEM simulations can be seen as a validation and support the use of DEM to investigate the influence of grain interaction on wave propagation. Frequency domain analyses that considered filtering of the higher frequency components of the inserted signal, the ratio of the input and received signals in the frequency domain and sample resonance provided useful insight into the system response. Frequency domain analysis and analytical continuum solutions for cube vibration show that the testing configuration excited some, but not all, of the system’s resonant frequencies. The particle scale data available from DEM enabled analysis of the energy dissipation during propagation of the wave. Frequency domain analysis at the particle scale revealed that the higher frequency content reduces with increasing distance from the point of excitation.
43 citations
Cites background from "Frequency filtering in disordered g..."
...This confirms that frequency filtering occurs as a function of the wave travelling through a granular system as noted by [49,50] and others....
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TL;DR: This work studies the spreading of initially localized excitations in one-dimensional disordered granular crystals to investigate localization phenomena in strongly nonlinear systems, which it is demonstrated to differ fundamentally from localization in linear and weakly non linear systems.
Abstract: We study the spreading of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to differ fundamentally from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder-an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements)-and for two types of initial conditions (displacement excitations and velocity excitations). We find for strongly precompressed (i.e., weakly nonlinear) chains that the dynamics depend strongly on the type of initial condition. In particular, for displacement excitations, the long-time asymptotic behavior of the second moment m(2) of the energy has oscillations that depend on the type of disorder, with a complex trend that differs markedly from a power law and which is particularly evident for an Anderson-like disorder. By contrast, for velocity excitations, we find that a standard scaling m(2)∼t(γ) (for some constant γ) applies for all three types of disorder. For weakly precompressed (i.e., strongly nonlinear) chains, m(2) and the inverse participation ratio P(-1) satisfy scaling relations m(2)∼t(γ) and P(-1)∼t(-η), and the dynamics is superdiffusive for all of the cases that we consider. Additionally, when precompression is strong, the inverse participation ratio decreases slowly (with η<0.1) for all three types of disorder, and the dynamics leads to a partial localization around the core and the leading edge of a propagating wave packet. For an Anderson-like disorder, displacement perturbations lead to localization of energy primarily in the core, and velocity perturbations cause the energy to be divided between the core and the leading edge. This localization phenomenon does not occur in the sonic-vacuum regime, which yields the surprising result that the energy is no longer contained in strongly nonlinear waves but instead is spread across many sites. In this regime, the exponents are very similar (roughly γ≈1.7 and η≈1) for all three types of disorder and for both types of initial conditions.
40 citations
TL;DR: In this paper, a discrete element method (DEM) model of a face-centred cubic packing of uniform spheres was created to simulate bender element tests to investigate this test from a fundamental perspective.
Abstract: While bender element testing is now well-established as a laboratory technique to determine soil stiffness, a robust technique to interpret the data remains elusive. A discrete element method (DEM) model of a face-centred cubic packing of uniform spheres was created to simulate bender element tests to investigate this test from a fundamental perspective. During the DEM simulations transmitter and receiver signals were recorded, analogous to the data available in laboratory tests, and these macro-scale data were supplemented with particle scale measurements (forces, stresses and displacements). A range of approaches previously applied in experimental and numerical studies were used to analyse the resulting data in both the time and frequency domains. The shortcomings in these approaches are clear from the differences in the resultant shear stiffness values and the frequency-dependent nature of the values. The particle-scale data enabled visualization of the passage of the wave through the sample, and it was found not to be possible to precisely link the arrival of the shear wave at the receiver and any of the previously proposed characteristic points along the signal recorded at the receiver. The most reliable determination of the shear wave velocity was obtained by applying a two-dimensional fast Fourier transform (2D FFT) to the data describing the velocity of the particles lying between the transmitter and receiver elements. Use of the DEM model and this 2D FFT approach facilitated the sensitivity of the system response to small variations in the interparticle force–displacement law (the contact model) to be established.
38 citations
01 Jan 1972
32 citations
TL;DR: In this article, discrete element method (DEM) simulations of planar compression wave propagation were performed to generate the data for the study, and the assembly stiffness and mass matrices were extracted from the DEM model and these data were used in an eigenmode analysis that provided significant insight into the observed overall dynamic response.
Abstract: Laboratory geophysics tests including bender elements and acoustic emission measure the speed of propagation of stress or sound waves in granular materials to derive elastic stiffness parameters. This contribution builds on earlier studies to assess whether the received signal characteristics can provide additional information about either the material’s behaviour or the nature of the material itself. Specifically it considers the maximum frequency that the material can transmit; it also assesses whether there is a simple link between the spectrum of the received signal and the natural frequencies of the sample. Discrete element method (DEM) simulations of planar compression wave propagation were performed to generate the data for the study. Restricting consideration to uniform (monodisperse) spheres, the material fabric was varied by considering face-centred cubic lattice packings as well as random configurations with different packing densities. Supplemental analyses, in addition to the DEM simulations, were used to develop a more comprehensive understanding of the system dynamics. The assembly stiffness and mass matrices were extracted from the DEM model and these data were used in an eigenmode analysis that provided significant insight into the observed overall dynamic response. The close agreement of the wave velocities estimated using eigenmode analysis with the DEM results confirms that DEM wave propagation simulations can reliably be used to extract material stiffness data. The data show that increasing either stress or density allows higher frequencies to propagate through the media, but the low-pass wavelength is a function of packing density rather than stress level. Prior research which had hypothesised that there is a simple link between the spectrum of the received signal and the natural sample frequencies was not substantiated.
24 citations
Cites background from "Frequency filtering in disordered g..."
...The FCC sample consisted of 3200 particles (4 × 4 × 200 layers) and so is equivalent to that considered by Mouraille et al. [19] and Mouraille and Luding [12]; it was created by considering the lattice geometry of the packing....
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...Lawney and Luding [15] examined a 1-...
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...Santamarina and Aloufi [10] and Santamarina et al. [11] related the maximum transmitted frequency ( flow−pass) and the associated wavelength (λlow−pass) to particle size, while Mouraille and Luding [12] related λlow−pass to the layer spacing....
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...Santamarina et al. [11] and Santamarina and Aloufi [10] assumed the particle diameter to be an internal scale ( α) of granular materials where λlow−pass = 2α, while Mouraille and Luding [12] took α to be the layer distance for a FCC sample, i.e. α = √2R....
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...D chain model, Lawney and Luding [15] showed that the low-frequency eigenmodes are not affected by small random variations in particle mass. Somfai et al. [30] considered a 2-...
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References
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TL;DR: In this article, a mathematical method was developed which gives fairly generally the density of eigenstates for one-dimensional disordered systems, applied first to a disordered linear chain of elastically coupled masses.
Abstract: A mathematical method is developed which gives fairly generally the density of eigenstates for one-dimensional disordered systems. The method is applied first to a disordered linear chain of elastically coupled masses. The results for the energy spectrum are closely related to those obtained by Dyson.
230 citations
TL;DR: The experimental findings support the appearance of a fluidized regime at low dissipation and of a Feigenbaum-type bifurcation scenario at high dissipation.
Abstract: As a toy model for dissipative granular materials, we investigate a one-dimensional column of beads undergoing external vibrations. The analysis is both experimental and numerical. We display the crossover from a condensed to a fluidized state of the column; the parameters are the agitation, the number of beads, and the momentum restitution coefficient. We find clustered states for high dissipation and/or a large number of beads. Our experimental findings support the appearance of a fluidized regime at low dissipation and of a Feigenbaum-type bifurcation scenario at high dissipation.
154 citations
"Frequency filtering in disordered g..." refers methods in this paper
...The Hertz and linear models are given by β = 1/2 and β = 0, respectively [22,26,37]....
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TL;DR: In this paper, the acoustical behavior of a 1D model of granular medium, which is a chain of identical spherical beads, is discussed, and the authors compare the predictions of the different models to experimental results that concern linear sound wave propagation in the chain submitted to a static force, and nonlinear solitary wave propagating in an unconstrained chain.
Abstract: We discuss the acoustical behavior of a 1D model of granular medium, which is a chain of identical spherical beads. In this geometry, we are able to test quantitatively alternative models to the Hertz theory of contact between elastic solids. We compare the predictions of the different models to experimental results that concern linear sound wave propagation in the chain submitted to a static force, and nonlinear solitary wave propagation in an unconstrained chain. We use elastic, elastic-plastic and brittle materials, the beads roughness extends on one order of magnitude, and we also use oxidized metallic beads. We demonstrate experimentally that at low static forces, for all types of beads, the linear acoustic waves propagate in the system as predicted by Hertz's theory. At larger forces, after onset of permanent plastic deformation at the contacts, the brass beads exhibit non Hertzian behavior, and hysteresis. Except in the case of brass beads, the nonlinear waves follow the predictions of Hertz theory.
153 citations
TL;DR: In this article, the relevance of the acoustic band gap on the transformation of single and multiple pulses in linear, nonlinear and strongly nonlinear regimes is investigated with numerical calculations and experiments.
Abstract: One-dimensional nonlinear phononic crystals have been assembled from periodic diatomic chains of stainless steel cylinders alternated with Polytetrafluoroethylene spheres. This system allows dramatic changes of behavior (from linear to strongly nonlinear) by application of compressive forces practically without changes of geometry of the system. The relevance of classical acoustic band-gap, characteristic for chain with linear interaction forces and derived from the dispersion relation of the linearized system, on the transformation of single and multiple pulses in linear, nonlinear and strongly nonlinear regimes are investigated with numerical calculations and experiments. The limiting frequencies of the acoustic band-gap for investigated system with given precompression force are within the audible frequency range (20–20,000 Hz) and can be tuned by varying the particle’s material properties, mass and initial compression. In the linear elastic chain the presence of the acoustic band-gap was apparent through fast transformation of incoming pulses within very short distances from the chain entrance. It is interesting that pulses with relatively large amplitude (nonlinear elastic chain) exhibit qualitatively similar behavior indicating relevance of the acoustic band gap also for transformation of nonlinear signals. The effects of an in situ band-gap created by a mean dynamic compression are observed in the strongly nonlinear wave regime.
150 citations
"Frequency filtering in disordered g..." refers background in this paper
..., deliberate insertion of different sized masses) [10,13,14,27] tapering [5,32,47,49] and controlled variation of the particle material [3,15]....
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TL;DR: In this paper, it was shown that the unloading of compression force at the chain edge has a nearly exponential decrease, and the characteristic time is mainly a function involving the grains' masses and the striker mass.
Abstract: The features of solitary waves observed in horizontal monodisperse chain of barely touching beads not only depend on geometrical and material properties of the beads but also on the initial perturbation provided at the edge of the chain. An impact of a large striker on a monodisperse chain, and similarly a sharp decrease of bead radius in a stepped chain, generates a solitary wave train containing many single solitary waves ordered by decreasing amplitudes. We find, by simple analytical arguments, that the unloading of compression force at the chain edge has a nearly exponential decrease. The characteristic time is mainly a function involving the grains’ masses and the striker mass. Numerical calculations and experiments corroborate these findings.
144 citations