scholarly journals Quasiperiodic granular chains and Hofstadter butterflies

2018 ◽  
Vol 376 (2127) ◽  
pp. 20170139 ◽  
Author(s):  
Alejandro J. Martínez ◽  
Mason A. Porter ◽  
P. G. Kevrekidis
Keyword(s):  
Fractal Dimension ◽  
Contact Angles ◽  
Spherical Particles ◽  
Localization Transition ◽  
Strongly Nonlinear ◽  
Nonlinear Energy Transfer ◽  
Nonlinear Lattices ◽  
Granular Chains ◽  
A Chain ◽  
Cylindrical Particles

We study quasiperiodicity-induced localization of waves in strongly precompressed granular chains. We propose three different set-ups, inspired by the Aubry–André (AA) model, of quasiperiodic chains; and we use these models to compare the effects of on-site and off-site quasiperiodicity in nonlinear lattices. When there is purely on-site quasiperiodicity, which we implement in two different ways, we show for a chain of spherical particles that there is a localization transition (as in the original AA model). However, we observe no localization transition in a chain of cylindrical particles in which we incorporate quasiperiodicity in the distribution of contact angles between adjacent cylinders by making the angle periodicity incommensurate with that of the chain. For each of our three models, we compute the Hofstadter spectrum and the associated Minkowski–Bouligand fractal dimension, and we demonstrate that the fractal dimension decreases as one approaches the localization transition (when it exists). We also show, using the chain of cylinders as an example, how to recover the Hofstadter spectrum from the system dynamics. Finally, in a suite of numerical computations, we demonstrate localization and also that there exist regimes of ballistic, superdiffusive, diffusive and subdiffusive transport. Our models provide a flexible set of systems to study quasiperiodicity-induced analogues of Anderson phenomena in granular chains that one can tune controllably from weakly to strongly nonlinear regimes. This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

scholarly journals Propagation of short stress pulses in discrete strongly nonlinear tunable metamaterials

2014 ◽  
Vol 372 (2023) ◽  
pp. 20130186 ◽  
Author(s):  
Yichao Xu ◽  
Vitali F. Nesterenko
Keyword(s):  
Power Law ◽  
Elastic Moduli ◽  
Dynamic Behaviour ◽  
Spherical Particles ◽  
Short Pulses ◽  
One Dimensional ◽  
Linear Elastic ◽  
Strongly Nonlinear ◽  
Granular Chains ◽  
Different Levels

The propagation of short pulses with wavelength comparable to the size of a unit cell has been studied in a one-dimensional discrete metamaterial composed of steel discs alternating with toroidal nitrile O-rings under different levels of precompression using experiments, numerical simulations and theoretical analysis. This strongly nonlinear metamaterial is more tunable than granular chains composed of linear elastic spherical particles and has better potential for attenuation of dynamic loads. A double power-law relationship for compressed O-rings was found to describe adequately their quasi-static and dynamic behaviour with significantly different elastic moduli. It is demonstrated that the double power-law metamaterial investigated allows a dramatic increase in sound speed and acoustic impedance of three to four times using a moderate force.

Nonlinear Dynamics of Granular Chains

2010 ◽  
Author(s):  
Yuli Starosvetsky ◽  
Alexander F. Vakakis
Keyword(s):  
Nonlinear Dynamics ◽  
Traveling Waves ◽  
Equations Of Motion ◽  
Traveling Wave Solutions ◽  
Strongly Nonlinear ◽  
Nonlinear Energy Transfer ◽  
Granular Chains ◽  
Spatial Periodicity ◽  
Analytical Approaches ◽  
Nonlinear Energy

We study strongly nonlinear traveling waves in one-dimensional granular chains with no pre-compression. We directly study the discrete, strongly nonlinear governing equations of motion of these media without resorting to continuum approximations or homogenization, which enables us to compute families of stable multi-hump traveling wave solutions with arbitrary wavelengths. We develop systematic semi–analytical approaches for computing different families of nonlinear traveling waves parametrized by spatial periodicity (wavenumber) and energy. Our findings indicate that homogeneous granular chains possess complex nonlinear dynamics, including the capacity for intrinsic nonlinear energy transfer.

scholarly journals Ergodic and nonergodic many-body dynamics in strongly nonlinear lattices

2021 ◽  
Vol 103 (5) ◽  
Author(s):  
Dominik Hahn ◽  
Juan-Diego Urbina ◽  
Klaus Richter ◽  
Rémy Dubertrand ◽  
S. L. Sondhi
Keyword(s):  
Many Body ◽  
Strongly Nonlinear ◽  
Nonlinear Lattices ◽  
Body Dynamics

Local structure at a ferrofluid surface: chain-like aggregates revealed by non-specular X-ray reflection

2003 ◽  
Vol 36 (2) ◽  
pp. 244-248 ◽  
Author(s):  
I. Takahashi ◽  
N. Tanaka ◽  
S. Doi
Keyword(s):  
Surface Tension ◽  
Fractal Dimension ◽  
Local Structure ◽  
Fine Particles ◽  
Capillary Waves ◽  
Aggregation Kinetics ◽  
X Ray ◽  
Surface Fluctuations ◽  
Irreversible Aggregation ◽  
A Chain

The surface structure of a ferrofluid was investigated by means of non-specular X-ray reflection. Strong intensity that is impossible to explain by surface fluctuations due to capillary waves was observed. It can be related to lateral correlation within aggregates of super-paramagnetic fine particles in the vicinity of the specimen surface. The fractal dimension of these surface-induced aggregates and the surface-tension coefficient of the ferrofluid were simultaneously determined. The fractal dimension was found to be around 1.1, indicating a chain-like character of the aggregates that have few branches. Strong and anisotropic interaction among the particles, as well as irreversible aggregation kinetics must be the origin of such a high-density and low-fractal-dimension system of dipolar 10 nm sized particles. The temperature variation of the fractal dimension indicated that the fractal aggregates stabilize themselves by losing their branches at increasing temperatures.

Transport of Non-Spherical Particles in Turbulence: Application of the LBM

2006 ◽  
Author(s):  
Andreas Ho¨lzer ◽  
Martin Sommerfeld
Keyword(s):  
Angular Velocity ◽  
Turbulent Flow ◽  
Radial Direction ◽  
Stokes Number ◽  
Spherical Particles ◽  
Axis Ratio ◽  
Angular Velocities ◽  
Boltzmann Method ◽  
Short Time ◽  
Cylindrical Particles

Direct numerical simulations of the motion of volume equivalent single cylindrical particles with axis ratios of 1, 2, 3, and 4 and Stokes numbers of 1, 2, 4, and 40 in a homogeneous isotropic turbulent flow field are presented. The forced turbulent flow is simulated using the Lattice Boltzmann Method (LBM). It is observed that the rms velocity and the rms angular velocity in longitudinal and in radial direction are identical for every particle, even though the rms forces can differ more than 100% and the rms torque more than 1000% in both directions. However, these differences in force and torque result in a different short-time behaviour of the particle in longitudinal and in radial direction. The rms particle velocity is found to decrease with increasing axis ratio and the rms particle angular velocity to have a maximum at an axis ratio of about 2.5. The ratio of the rms velocity of the particle to that of the fluid decreases with increasing Stokes number as well as the ratio between the rms angular velocities, as one could expect.

scholarly journals Tunable, synchronized frequency down-conversion in magnetic lattices with defects

2018 ◽  
Vol 376 (2127) ◽  
pp. 20170137 ◽  
Author(s):  
Marc Serra-Garcia ◽  
Miguel Molerón ◽  
Chiara Daraio
Keyword(s):  
Energy Transfer ◽  
Frequency Conversion ◽  
Low Frequency ◽  
Multiple Sources ◽  
Localized Modes ◽  
Nonlinear Energy Transfer ◽  
Nonlinear Mechanical ◽  
Mechanical Analogue ◽  
A Chain

We study frequency conversion in nonlinear mechanical lattices, focusing on a chain of magnets as a model system. We show that, by inserting mass defects at suitable locations, we can introduce localized vibrational modes that nonlinearly couple to extended lattice modes. The nonlinear interaction introduces an energy transfer from the high-frequency localized modes to a low-frequency extended mode. This system is capable of autonomously converting energy between highly tunable input and output frequencies, which need not be related by integer harmonic or subharmonic ratios. It is also capable of obtaining energy from multiple sources at different frequencies with a tunable output phase, due to the defect synchronization provided by the extended mode. Our lattice is a purely mechanical analogue of an opto-mechanical system, where the localized modes play the role of the electromagnetic field and the extended mode plays the role of the mechanical degree of freedom. This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

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