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.

Strongly Nonlinear Spatially Periodic Traveling Waves in Granular Dimer Chains

2012 ◽  
Author(s):  
K. R. Jayaprakash ◽  
Alexander F. Vakakis ◽  
Yuli Starosvetsky
Keyword(s):  
Traveling Waves ◽  
Periodic Boundary Conditions ◽  
Mass Ratio ◽  
Efficient Energy ◽  
Periodic Traveling Waves ◽  
Strongly Nonlinear ◽  
Spatial Periodicity ◽  
Spatially Periodic ◽  
New Formulation ◽  
Primary Pulse

In the present work we study the dynamics of spatially periodic traveling waves in granular 1:1 (each bead is followed and preceded by a bead of different mass and/or stiffness) dimer chain with no pre-compression. The dynamics of a 1:1 dimer chain is governed by a single parameter, the mass ratio of the two beads forming each dimer pair of the chain. In particular, we demonstrate numerically the formation of special families of traveling waves with spatially periodic waveforms that are realized in semi-infinite dimer chains with the application of an arbitrary impulse. These traveling waves were first observed in the form of oscillatory tails in the trail of the propagating primary pulse. The energy radiated by the propagating primary pulse manifests in the form of traveling waves of varying spatial periodicity depending on the mass ratio. These traveling waves depend only on the mass ratio and are rescalable with respect to any arbitrary applied energy. The dynamics of these families of traveling waves is systematically studied by considering finite dimer chains (termed the ‘reduced systems’) subject to periodic boundary conditions. We demonstrate that these waves may exhibit interesting bifurcations or loss of stability as the system parameter varies. In turn, these bifurcations and stability exchanges in infinite dimer chains are correlated to previous studies of pulse attenuation in finite dimer chains through efficient energy radiation from the propagating pulse to the far field, mainly in the form of traveling waves. Based on these results a new formulation of attenuation and propagation zones (stop and pass bands) in semi-infinite granular dimer chains is proposed.

Traveling and solitary waves in monodisperse and dimer granular chains

2017 ◽  
Vol 31 (10) ◽  
pp. 1742001 ◽  
Author(s):  
Yuli Starosvetsky ◽  
K. R. Jayaprakash ◽  
Alexander F. Vakakis
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Traveling Waves ◽  
Granular Media ◽  
Equations Of Motion ◽  
Continuum Approximation ◽  
Granular Chains ◽  
Solitary Pulse ◽  
Countably Infinite ◽  
Dimer Granular Chains

We provide a review of propagating traveling waves and solitary pulses in uncompressed one-dimensional ([Formula: see text]) ordered granular media. The first such solution in homogeneous granular media was discovered by Nesterenko in the form of a single-hump solitary pulse with energy-dependent profile and velocity. Considering directly the discrete, strongly nonlinear governing equations of motion of these media (i.e., without resorting to continuum approximation or homogenization), we show the existence of countably infinite families of stable multi-hump propagating traveling waves with arbitrary wavelengths. A semi-analytical approach is used to study the dependence of these waves on spatial periodicity (wavenumber) and energy, and to show that in a certain asymptotic limit, these families converge to the single-hump Nesterenko solitary wave. Then the study is extended in dimer granular chains composed of alternating “heavy” and “light” beads. For a set of specific mass ratios between the light and heavy beads, we show the existence of multi-hump solitary waves that propagate faster than the Nesterenko solitary wave in the corresponding homogeneous granular chain composed of only heavy beads. The existence of these waves has interesting implications in energy transmission in ordered granular chains.

scholarly journals Demonstration of accelerating and decelerating nonlinear impulse waves in functionally graded granular chains

2018 ◽  
Vol 376 (2127) ◽  
pp. 20170136 ◽  
Author(s):  
Rajesh Chaunsali ◽  
Eunho Kim ◽  
Jinkyu Yang
Keyword(s):  
Functionally Graded ◽  
Contact Forces ◽  
Granular System ◽  
Nonlinear Energy Transfer ◽  
Granular Chains ◽  
Highly Nonlinear ◽  
Impulse Excitation ◽  
Impulse Waves ◽  
Nonlinear Energy ◽  
Stiffness Distribution

We propose a tunable cylinder-based granular system that is functionally graded in its stiffness distribution in space. With no initial compression given to the system, it supports highly nonlinear waves propagating under an impulse excitation. We investigate analytically, numerically and experimentally the ability to accelerate and decelerate the impulse wave without a significant scattering in the space domain. Moreover, the gradient in stiffness results in the scaling of contact forces along the chain. We envision that such tunable systems can be used for manipulating highly nonlinear impulse waves for novel sensing and impact mitigation purposes. This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

scholarly journals Nonlinear Pulse Equipartition in Weakly Coupled Ordered Granular Chains With No Precompression

2013 ◽  
Vol 8 (3) ◽  
Author(s):  
Yuli Starosvetsky ◽  
M. Arif Hasan ◽  
Alexander F. Vakakis
Keyword(s):  
Nonlinear Dynamics ◽  
Solitary Waves ◽  
Practical Significance ◽  
Drastic Reduction ◽  
Strongly Nonlinear ◽  
Localized State ◽  
Weakly Coupled ◽  
Granular Chains ◽  
Initial Impulse ◽  
Primary Pulse

We report on the strongly nonlinear dynamics of an array of weakly coupled, noncompressed, parallel granular chains subject to a local initial impulse. The motion of the granules in each chain is constrained to be in one direction that coincides with the orientation of the chain. We show that in spite of the fact that the applied impulse is applied to one of the granular chains, the resulting pulse that initially propagates only in the excited chain gets gradually equipartitioned between its neighboring chains and eventually in all chains of the array. In particular, the initially strongly localized state of energy distribution evolves towards a final stationary state of formation of identical solitary waves that propagate in each one of the chains. These solitary waves are synchronized and have identical speeds. We show that the phenomenon of primary pulse equipartition between the weakly coupled granular chains can be fully reproduced in coupled binary models that constitute a significantly simpler model that captures the main qualitative features of the dynamics of the granular array. The results reported herein are of major practical significance since it indicates that the weakly coupled array of granular chains is a medium in which an initially localized excitation gets gradually defocused, resulting in drastic reduction of propagating pulses as they are equipartitioned among all chains.

scholarly journals Transient dynamics in strongly nonlinear systems: optimization of initial conditions on the resonant manifold

2018 ◽  
Vol 376 (2127) ◽  
pp. 20170131 ◽  
Author(s):  
Nathan Perchikov ◽  
O. V. Gendelman
Keyword(s):  
Coupled Oscillators ◽  
Energy Exchange ◽  
Initial Conditions ◽  
Complete Analysis ◽  
Slow Flow ◽  
Coupling Energy ◽  
Strongly Nonlinear ◽  
Nonlinear Energy Transfer ◽  
Exchange Regime ◽  
Nonlinear Energy

We consider a system of two linear and linearly coupled oscillators with ideal impact constraints. Primary resonant energy exchange is investigated by analysis of the slow flow using the action–angle (AA) formalism. Exact inversion of the action-energy dependence for the linear oscillator with impact constraints is not possible. This difficulty, typical for many models of nonlinear oscillators, is circumvented by matching the asymptotic expansions for the linear and impact limits. The obtained energy–action relation enables the complete analysis of the slow flow and the accurate description of the critical delocalization transition. The transition from the localization regime to the energy-exchange regime is captured by prediction of the critical coupling value. Accurate prediction of the delocalization transition requires a detailed account of the coupling energy with appropriate redefinition and optimization of the limiting phase trajectory on the resonant manifold. This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

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’.

The effects of nonlinear energy sink and piezoelectric energy harvester on aeroelastic instability of an airfoil

2021 ◽  
pp. 107754632199358
Author(s):  
Ali Fasihi ◽  
Majid Shahgholi ◽  
Saeed Ghahremani
Keyword(s):  
Energy Harvester ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Dynamical Behavior ◽  
Harmonic Balance Method ◽  
Nonlinear Energy Sink ◽  
Piezoelectric Energy ◽  
Energy Sink ◽  
Two Degree Of Freedom ◽  
Nonlinear Energy

The potential of absorbing and harvesting energy from a two-degree-of-freedom airfoil using an attachment of a nonlinear energy sink and a piezoelectric energy harvester is investigated. The equations of motion of the airfoil coupled with the attachment are solved using the harmonic balance method. Solutions obtained by this method are compared to the numerical ones of the pseudo-arclength continuation method. The effects of parameters of the integrated nonlinear energy sink-piezoelectric attachment, namely, the attachment location, nonlinear energy sink mass, nonlinear energy sink damping, and nonlinear energy sink stiffness on the dynamical behavior of the airfoil system are studied for both subcritical and supercritical Hopf bifurcation cases. Analyses demonstrate that absorbing vibration and harvesting energy are profoundly affected by the nonlinear energy sink parameters and the location of the attachment.

STABILITY OF STEADY STATE AND TRAVELING WAVES SOLUTIONS IN COUPLED MAP LATTICES

2008 ◽  
Vol 18 (01) ◽  
pp. 219-225 ◽  
Author(s):  
DANIEL TURZÍK ◽  
MIROSLAVA DUBCOVÁ
Keyword(s):  
Steady State ◽  
Essential Spectrum ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Linear Operators ◽  
Coupled Map Lattices ◽  
Basic Tool ◽  
Wave Solutions ◽  
The Stability

We determine the essential spectrum of certain types of linear operators which arise in the study of the stability of steady state or traveling wave solutions in coupled map lattices. The basic tool is the Gelfand transformation which enables us to determine the essential spectrum completely.

scholarly journals Evaluation of Spatial Variation of Nonlinear Energy Transfer by Use of Turbulence Diagnostic Simulator

2013 ◽  
Vol 8 (0) ◽  
pp. 2403070-2403070 ◽  
Author(s):  
Naohiro KASUYA ◽  
Satoru SUGITA ◽  
Makoto SASAKI ◽  
Shigeru INAGAKI ◽  
Masatoshi YAGI ◽  
...  
Keyword(s):  
Energy Transfer ◽  
Spatial Variation ◽  
Nonlinear Energy Transfer ◽  
Nonlinear Energy
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