Yuli Starosvetsky

Strongly Nonlinear Beat Phenomena and Energy Exchanges in Weakly Coupled Granular Chains on Elastic Foundations

We study the dynamics of weakly interacting, strongly nonlinear one-dimensional granular chains mounted on elastic foundations. These chains are composed of a number of identical linearly elastic beads interacting with each other through Hertzian contact. No dissipative effects, such as plasticity or dry friction effects, are taken into account in our analysis. Assuming zero precompression between beads, the dynamics of the system under consideration is strongly (essentially) nonlinear, having no linear component. The complete absence of linear structural acoustics in these chains led to their characterization as “sonic vacua.” The two sources of strong nonlinearity in the considered granular chains are (i) the nonlinearizable Hertzian law interaction between adjacent beads in compression, and (ii) the possible separations between beads leading to bead collisions in the absence of compressive forces. In the current study we demonstrate that the weakly coupled granular chains possess complex dynamics leadi...

Solitary waves in diatomic chains.

We consider the mechanism of formation of isolated localized wave structures in the diatomic Fermi-Pasta-Ulam (FPU) model. Using a singular multiscale asymptotic analysis in the limit of high mass mismatch between the alternating elements, we obtain the typical slow-fast time scale separation and formulate the Fredholm orthogonality condition approximating a sequence of mass ratios supporting the formation of solitary waves in the general type of diatomic FPU models. This condition is made explicit in the case of a diatomic Toda lattice. Results of numerical integration of the full diatomic Toda lattice equations confirm the formation of these genuinely localized wave structures at special values of the mass ratio that are close to the analytical predictions when the ratio is sufficiently small.

Evolution of the primary pulse in one-dimensional granular crystals subject to on-site perturbations: analytical study.

The propagation of the primary pulse through an uncompressed granular chain subject to external on-site perturbation is studied. An analytical procedure predicting the evolution of the primary pulse is devised for the general form of the on-site perturbation applied on the chain. The validity of the analytical model is confirmed with several specific granular setups, such as chains mounted on the nonlinear elastic foundation, chains perturbed by the dissipative forces, as well as randomly perturbed chains. An additional interesting finding made in the present paper corresponds to the chains subject to a special type of perturbation, including the terms leading to dissipation and those acting as an energy source. In the paper, it is shown that an application of such a perturbation may lead to the formation of stable stationary primary pulses propagating with constant amplitudes and acting as attractors for the initially unperturbed Nesterenko solitary waves. Interestingly enough, the developed analytical procedure provides an extremely close estimation for the amplitudes of these stationary primary pulses as well as predicts zones of their stability. In conclusion, we would like to stress that the developed analytical model has demonstrated spectacular agreement with the results of the direct numerical simulations, and this is for various configurations considered in the current paper.

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

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.

Solitary waves in a general class of granular dimer chains

We report on a countable infinity of traveling solitary waves in a class of highly heterogeneous ordered one-dimensional granular media, in particular, granular dimers composed of an infinite number of periodic sets of “heavy” elastic spherical beads in contact with N “light” ones; these media are denoted as 1:N granular dimers. Perfectly elastic Hertzian interaction between beads is assumed and no dissipative forces are taken into account in our study; moreover, zero pre-compression is assumed, rendering the dynamics strongly nonlinear through complete elimination of linear acoustics from the problem. After developing a general asymptotic methodology for the 1:N granular dimer, we focus on the case N=2 and prove numerically and asymptotically the existence of a countable infinity of traveling solitary waves in the 1:2 dimer chain. These solitary waves, which can be regarded as anti-resonances in these strongly nonlinear media, are found to be qualitatively different than those previously studied in homog...

Wave propagation in a strongly nonlinear locally resonant granular crystal

Abstract 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.

Breather Solutions of the Discrete p-Schrödinger Equation

We consider the discrete p-Schrodinger (DpS) equation, which approximates small amplitude oscillations in chains of oscillators with fully-nonlinear nearest-neighbors interactions of order \(\alpha = p - 1> 1\). Using a mapping approach, we prove the existence of breather solutions of the DpS equation with even- or odd-parity reflectional symmetries. We derive in addition analytical approximations for the breather profiles and the corresponding intersecting stable and unstable manifolds, valid on a whole range of nonlinearity orders α. In the limit of weak nonlinearity (α → 1+), we introduce a continuum limit connecting the stationary DpS and logarithmic nonlinear Schrodinger equations. In this limit, breathers correspond asymptotically to Gaussian homoclinic solutions. We numerically analyze the stability properties of breather solutions depending on their even- or odd-parity symmetry. A perturbation of an unstable breather generally results in a translational motion (traveling breather) when α is close to unity, whereas pinning becomes predominant for larger values of α.

Emergence of non-stationary regimes in one- and two-dimensional models with internal rotators

In the present paper, we give a selective review of some very recent works concerning the non-stationary regimes emerging in various one- and two-dimensional models incorporating internal rotators. In one-dimensional models, these regimes are characterized by the intense energy transfer from the outer element, subjected to initial or harmonic excitation, to the internal rotator. As for the two-dimensional models (incorporating internal rotators), we will mainly focus on the two special dynamical states, namely a state of the near-complete energy transfer from longitudinal to lateral vibrations of the outer element as well as the state of a permanent, unidirectional energy locking with mild, spatial energy exchanges. In this review, we will discuss the recent theoretical and experimental advancements in the study of essentially nonlinear mechanisms governing the formation and bifurcations of the regimes of intense energy transfer. The present review is composed of two parts. The first part will be mainly devoted to the emergence of resonant energy transfer states in one-dimensional models incorporating internal rotators, while the second part will be mainly concerned with the manifestation of various energy transfer states in two-dimensional ones. This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

Targeted Energy Transfer

This chapter presents the analytical and numerical study of energy transport in a system of n linear impulsively loaded oscillators (a primary linear system), in which the nth oscillator is coupled with an essentially nonlinear attachment—the nonlinear energy sink (NES).

Traveling and solitary waves in monodisperse and dimer granular chains

We provide a review of propagating traveling waves and solitary pulses in uncompressed one-dimensional (1d) 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.