224 research outputs found
Breathers in oscillator chains with Hertzian interactions
We prove nonexistence of breathers (spatially localized and time-periodic
oscillations) for a class of Fermi-Pasta-Ulam lattices representing an
uncompressed chain of beads interacting via Hertz's contact forces. We then
consider the setting in which an additional on-site potential is present,
motivated by the Newton's cradle under the effect of gravity. Using both direct
numerical computations and a simplified asymptotic model of the oscillator
chain, the so-called discrete p-Schr\"odinger (DpS) equation, we show the
existence of discrete breathers and study their spectral properties and
mobility. Due to the fully nonlinear character of Hertzian interactions,
breathers are found to be much more localized than in classical nonlinear
lattices and their motion occurs with less dispersion. In addition, we study
numerically the excitation of a traveling breather after an impact at one end
of a semi-infinite chain. This case is well described by the DpS equation when
local oscillations are faster than binary collisions, a situation occuring e.g.
in chains of stiff cantilevers decorated by spherical beads. When a hard
anharmonic part is added to the local potential, a new type of traveling
breather emerges, showing spontaneous direction-reversing in a spatially
homogeneous system. Finally, the interaction of a moving breather with a point
defect is also considered in the cradle system. Almost total breather
reflections are observed at sufficiently high defect sizes, suggesting
potential applications of such systems as shock wave reflectors
Wave Propagation in a Strongly Nonlinear Locally Resonant Granular Crystal
In this work, we study the wave propagation in a recently proposed acoustic
structure, the locally resonant granular crystal. This structure is composed of
a one-dimensional granular crystal of hollow spherical particles in contact,
containing linear resonators. The relevant model is presented and examined
through a combination of analytical approximations (based on ODE and nonlinear
map analysis) and of numerical results. The generic dynamics of the system
involves a degradation of the well-known traveling pulse of the standard
Hertzian chain of elastic beads. Nevertheless, the present system is richer, in
that as the primary pulse decays, secondary ones emerge and eventually
interfere with it creating modulated wavetrains. Remarkably, upon suitable
choices of parameters, this interference "distills" a weakly nonlocal solitary
wave (a "nanopteron"). This motivates the consideration of such nonlinear
structures through a separate Fourier space technique, whose results suggest
the existence of such entities not only with a single-side tail, but also with
periodic tails on both ends. These tails are found to oscillate with the
intrinsic oscillation frequency of the out-of-phase motion between the outer
hollow bead and its internal linear attachment
Micropterons, Nanopterons and Solitary Wave Solutions to the Diatomic Fermi-Pasta-Ulam-Tsingou Problem
We use a specialized boundary-value problem solver for mixed-type functional
differential equations to numerically examine the landscape of traveling wave
solutions to the diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) problem. By using a
continuation approach, we are able to uncover the relationship between the
branches of micropterons and nanopterons that have been rigorously constructed
recently in various limiting regimes. We show that the associated surfaces are
connected together in a nontrivial fashion and illustrate the key role that
solitary waves play in the branch points. Finally, we numerically show that the
diatomic solitary waves are stable under the full dynamics of the FPUT system
Tunable Vibrational Band Gaps in One-Dimensional Diatomic Granular Crystals with Three-Particle Unit Cells
We investigate the tunable vibration filtering properties of one-dimensional
diatomic granular crystals composed of arrays of stainless steel spheres and
cylinders interacting via Hertzian contact. The arrays consist of periodically
repeated three-particle unit cells (steel-cylinder-sphere) in which the length
of the cylinder is varied systematically. We apply static compression to
linearize the dynamic response of the crystals and characterize their linear
frequency spectrum. We find good agreement between theoretical dispersion
relation analysis (for infinite systems), state-space analysis (for finite
systems), and experiments. We report the observation of up to three distinct
pass bands and two finite band gaps and show their tunability for variations in
cylinder length and static compression
Nonlinear Waves in Disordered Diatomic Granular Chains
We investigate the propagation and scattering of highly nonlinear waves in
disordered granular chains composed of diatomic (two-mass) units of spheres
that interact via Hertzian contact. Using ideas from statistical mechanics, we
consider each diatomic unit to be a "spin", so that a granular chain can be
viewed as a spin chain composed of units that are each oriented in one of two
possible ways. Experiments and numerical simulations both reveal the existence
of two different mechanisms of wave propagation: In low-disorder chains, we
observe the propagation of a solitary pulse with exponentially decaying
amplitude. Beyond a critical level of disorder, the wave amplitude instead
decays as a power law, and the wave transmission becomes insensitive to the
level of disorder. We characterize the spatio-temporal structure of the wave in
both propagation regimes and propose a simple theoretical interpretation for
such a transition. Our investigation suggests that an elastic spin chain can be
used as a model system to investigate the role of heterogeneities in the
propagation of highly nonlinear waves.Comment: 10 pages, 8 figures (some with multiple parts), to appear in Physical
Review E; summary of changes: new title, one new figure, additional
discussion of several points (including both background and results
Nonlinear Low-to-High-Frequency Energy Cascades in Diatomic Granular Crystals
We study wave propagation in strongly nonlinear one-dimensional diatomic granular crystals under an impact load. Depending on the mass ratio of the “light” to “heavy” beads, this system exhibits rich wave dynamics from highly localized traveling waves to highly dispersive waves featuring strong attenuation. We demonstrate experimentally the nonlinear resonant and antiresonant interactions of particles, and we verify that the nonlinear resonance results in strong wave attenuation, leading to highly efficient nonlinear energy cascading without relying on material damping. In this process, mechanical energy is transferred from low to high frequencies, while propagating waves emerge in both ordered and chaotic waveforms via a distinctive spatial cascading. This energy transfer mechanism from lower to higher frequencies and wave numbers is of particular significance toward the design of novel nonlinear acoustic metamaterials with inherently passive energy redistribution properties
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