K. Vorotnikov | Technion - Israel Institute of Technology (original) (raw)

Papers by K. Vorotnikov

Research paper thumbnail of OUP accepted manuscript

The IMA Journal of Applied Mathematics, 2019

A new method is introduced for the simulation of multiple impacts in granular media using the Kuw... more A new method is introduced for the simulation of multiple impacts in granular media using the Kuwabara-Kono (KK) contact model, a nonsmooth (not Lipschitz continuous) extension of Hertz contact that accounts for viscoelastic damping. We use the technique of modified equations to construct time-discretizations of the nondissipative Hertz law matching numerical dissipation with KK dissipation at different consistency orders. This allows us to simulate dissipative impacts with good accuracy without including the nonsmooth KK viscoelastic component in the contact force. This tailored numerical dissipation is developed in a general framework, for Newtonian dynamical systems subject to dissipative forces proportional to the time-derivative of conservative forces. Numerical tests are performed for the simulation of impacts in Newton’s cradle and on alignments of alternating large and small balls. Resulting wave phenomena (oscillator synchronization, propagation of dissipative solitary wave...

Research paper thumbnail of Regimes of two-dimensional energy channeling in the inertially coupled unit-cell model subjected to an asymmetric potential

Journal of Sound and Vibration, 2019

We analyze the stationary and non-stationary states emerging in a two-dimensional (2D) model cons... more We analyze the stationary and non-stationary states emerging in a two-dimensional (2D) model consisting of an outer element that incorporates an internal rotator and is subjected to a 2D anharmonic and non-symmetric local potential. We focus on the effect of the asymmetry of the potential on the mechanism of formation and bifurcations of various stationary and non-stationary regimes in the system. The non-stationary regimes are characterized by an intense bi-directional energy transfer between the axial and lateral vibrations of the main element as well as the regimes of unidirectional energy locking. We show that the transitions between the different states of the system are controlled by the motion of the internal rotator. Our analysis is based on the complexification-averaging procedure and the singular multiscale asymptotic method. Investigation of the system's dynamics in the vicinity of the 1:1:1 resonance enables us to reproduce the bifurcation structure of the stationary regimes and describe the formation mechanisms of highly non-stationary states of intense energy transfer. Our analysis shows that the most important effect of the asymmetric potential is the breakdown of a unique complete bi-directional energy channeling mechanism into two separate mechanisms that correspond to incomplete horizontal-to-vertical and vertical-tohorizontal energy transfer, respectively, and are enabled at two different critical values of the main bifurcation parameter. Results of the analysis are in good agreement with the numerical simulations of the full model.

Research paper thumbnail of Nonlinear energy channeling in the two-dimensional, locally resonant, unit-cell model. I. High energy pulsations and routes to energy localization

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015

ABSTRACT

Research paper thumbnail of Nonlinear mechanisms of two-dimensional wave-wave transformations in the initially coupled acoustic structure

Journal of Applied Physics

Research paper thumbnail of Wave propagation in a strongly nonlinear locally resonant granular crystal

Physica D: Nonlinear Phenomena

In this work, we study the wave propagation in a recently proposed acoustic structure, the locall... more 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.

Research paper thumbnail of Bifurcation structure of the special class of nonstationary regimes emerging in the 2D inertially coupled, unit-cell model: Analytical study

Journal of Sound and Vibration

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

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

In the present paper, we give a selective review of some very recent works concerning the non-sta... more 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 firs...

Research paper thumbnail of Nonlinear energy channeling in the two-dimensional, locally resonant, unit-cell model. I. High energy pulsations and routes to energy localization

Chaos (Woodbury, N.Y.)

ABSTRACT

Research paper thumbnail of OUP accepted manuscript

The IMA Journal of Applied Mathematics, 2019

A new method is introduced for the simulation of multiple impacts in granular media using the Kuw... more A new method is introduced for the simulation of multiple impacts in granular media using the Kuwabara-Kono (KK) contact model, a nonsmooth (not Lipschitz continuous) extension of Hertz contact that accounts for viscoelastic damping. We use the technique of modified equations to construct time-discretizations of the nondissipative Hertz law matching numerical dissipation with KK dissipation at different consistency orders. This allows us to simulate dissipative impacts with good accuracy without including the nonsmooth KK viscoelastic component in the contact force. This tailored numerical dissipation is developed in a general framework, for Newtonian dynamical systems subject to dissipative forces proportional to the time-derivative of conservative forces. Numerical tests are performed for the simulation of impacts in Newton’s cradle and on alignments of alternating large and small balls. Resulting wave phenomena (oscillator synchronization, propagation of dissipative solitary wave...

Research paper thumbnail of Regimes of two-dimensional energy channeling in the inertially coupled unit-cell model subjected to an asymmetric potential

Journal of Sound and Vibration, 2019

We analyze the stationary and non-stationary states emerging in a two-dimensional (2D) model cons... more We analyze the stationary and non-stationary states emerging in a two-dimensional (2D) model consisting of an outer element that incorporates an internal rotator and is subjected to a 2D anharmonic and non-symmetric local potential. We focus on the effect of the asymmetry of the potential on the mechanism of formation and bifurcations of various stationary and non-stationary regimes in the system. The non-stationary regimes are characterized by an intense bi-directional energy transfer between the axial and lateral vibrations of the main element as well as the regimes of unidirectional energy locking. We show that the transitions between the different states of the system are controlled by the motion of the internal rotator. Our analysis is based on the complexification-averaging procedure and the singular multiscale asymptotic method. Investigation of the system's dynamics in the vicinity of the 1:1:1 resonance enables us to reproduce the bifurcation structure of the stationary regimes and describe the formation mechanisms of highly non-stationary states of intense energy transfer. Our analysis shows that the most important effect of the asymmetric potential is the breakdown of a unique complete bi-directional energy channeling mechanism into two separate mechanisms that correspond to incomplete horizontal-to-vertical and vertical-tohorizontal energy transfer, respectively, and are enabled at two different critical values of the main bifurcation parameter. Results of the analysis are in good agreement with the numerical simulations of the full model.

Research paper thumbnail of Nonlinear energy channeling in the two-dimensional, locally resonant, unit-cell model. I. High energy pulsations and routes to energy localization

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015

ABSTRACT

Research paper thumbnail of Nonlinear mechanisms of two-dimensional wave-wave transformations in the initially coupled acoustic structure

Journal of Applied Physics

Research paper thumbnail of Wave propagation in a strongly nonlinear locally resonant granular crystal

Physica D: Nonlinear Phenomena

In this work, we study the wave propagation in a recently proposed acoustic structure, the locall... more 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.

Research paper thumbnail of Bifurcation structure of the special class of nonstationary regimes emerging in the 2D inertially coupled, unit-cell model: Analytical study

Journal of Sound and Vibration

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

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

In the present paper, we give a selective review of some very recent works concerning the non-sta... more 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 firs...

Research paper thumbnail of Nonlinear energy channeling in the two-dimensional, locally resonant, unit-cell model. I. High energy pulsations and routes to energy localization

Chaos (Woodbury, N.Y.)

ABSTRACT