Periodically Forced Nonlinear Oscillatory Acoustic Vacuum (original) (raw)
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Forced Nonlinear Oscillatory Acoustic Vacuum
2018
In this work, we study the in-plane oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system.
Oscillatory Instabilities of Standing Waves in One-Dimensional Nonlinear Lattices
Physical Review Letters, 2000
In one-dimensional anharmonic lattices, we construct nonlinear standing waves (SWs) reducing to harmonic SWs at small amplitude. For SWs with spatial periodicity incommensurate with the lattice period, a transition by breaking of analyticity versus wave amplitude is observed. As a consequence of the discreteness, oscillatory linear instabilities, persisting for arbitrarily small amplitude in infinite lattices, appear for all wave numbers Q fi 0, p. Incommensurate analytic SWs with jQj . p͞2 may however appear as "quasistable," as their instability growth rate is of higher order. 42.65.Sf, 45.05. + x, 63.20.Ry A well known and rather spectacular phenomenon occurring in many nonlinear media (e.g., fluids or optical waveguides) is the modulational (Benjamin-Feir) instability (MI), by which a traveling plane wave breaks up into a train of solitary waves (see, e.g., ). It is also well known that wave propagation in many continuous nonlinear media is well described by nonlinear Schrödingertype equations, where the solitary wave trains are described by spatially periodic and stable standing wave (SW) solutions, the so-called cnoidal envelope waves .
Localized Oscillations in Nonlinear Lattices: Existence and Stability
2005
Numerical methods using homoclinic orbits are applied to study the existence and stability of spatially localized and time-periodic oscillations of 1-dimensional (1D) nonlinear lattices, with linear interaction between nearest neighbors and a quartic on-site potential 4 2 4 1 2 1) (u Ku u V ± = where the (+) sign corresponds to "hard spring" and (-) to "soft spring" models. These localized oscillations-when they are stable under small perturbations-are very important for physical systems, since they seriously affect the energy transport properties of the lattice. We use Floquet theory to analyze their linear (local) stability, along certain curves in parameter space (α, ω), where α is the coupling constant and ω the frequency of the breather. We then apply the Smaller Alignment Index method (SALI) to investigate more globally their stability properties in phase space. Comparing our results for the ± cases of V(u), we find that the regions of existence and stability for simple breathers of the "hard spring" lattice are considerably larger than those of the "soft spring" system. The variation of the size of the regular region around a stable breather is investigated as the number of particles is increased.
Dynamics of metastable breathers in nonlinear chains in acoustic vacuum
Physical Review E, 2009
The study of the dynamics of 1D chains with both harmonic and nonlinear interactions, as in the Fermi-Pasta-Ulam (FPU) and related problems, has played a central role in efforts to identify the broad consequences of nonlinearity in these systems. Nevertheless, little is known about the dynamical behavior of purely nonlinear chains where there is a complete absence of the harmonic term, and hence sound propagation is not admissible, i.e., under conditions of "acoustic vacuum." Here we study the dynamics of highly localized excitations, or breathers, which have been known to be initiated by the quasi-static stretching of the bonds between the adjacent particles. We show via detailed particle dynamics based studies that many low energy pulses also form in the vicinity of the perturbation and the breathers that form are "fragile" in the sense that they can be easily delocalized by scattering events in the system. We show that the localized excitations eventually disperse allowing the system to attain an equilibrium-like state that is realizable in acoustic vacuum. We conclude with a discussion of how the dynamics is affected by the presence of acoustic oscillations.
Effective equation for a system of mechanical oscillators in an acoustic field
We consider a one dimensional evolution problem modelling the dynamics of an acoustic field coupled with a set of mechanical oscillators. We analyse solutions of the system of ordinary and partial differential equations with time-dependent boundary conditions describing the evolution in the limit of a continuous distribution of oscillators. MSC 2010: 76M50, 35D30, 35M33.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
We aim to study how the interplay between the effects of nonlinearity and heterogeneity can influence on the distribution and localization of energy in discrete lattice-type structures. As the classical example, vibrations of a cubically nonlinear elastic lattice are considered. In contrast with many other authors, who dealt with infinite and periodic lattices, we examine a finite-size model. Supposing the length of the lattice to be much larger than the distance between the particles, continuous macroscopic equations suitable to describe both low- and high-frequency motions are derived. Acoustic and optical vibrations are studied asymptotically by the method of multiple time scales. For numerical simulations, the Runge–Kutta fourth-order method is employed. Internal resonances and energy exchange between the vibrating modes are predicted and analysed. It is shown that the decrease in the number of particles restricts energy transfers to higher-order modes and prevents the equiparti...
Classification of Spatially Localized Oscillations in Periodically Forced Dissipative Systems
SIAM Journal on Applied Dynamical Systems, 2008
Formation of spatially localized oscillations in parametrically driven systems is studied, focusing on the dominant 2:1 resonance tongue. Both damped and self-excited oscillatory media are considered. Near the primary subharmonic instability such systems are described by the forced complex Ginzburg-Landau equation. The technique of spatial dynamics is used to identify three basic types of coherent states described by this equation-small amplitude oscillons, large amplitude reciprocal oscillons resembling holes in an oscillating background, and fronts connecting two spatially homogeneous states oscillating out of phase. In many cases all three solution types are found in overlapping parameter regimes, and multiple solutions of each type may be simultaneously stable. The origin of this behavior can be traced to the formation of a heteroclinic cycle in space between the finite amplitude spatially homogeneous phase-locked oscillation and the zero state. The results provide an almost complete classification of the properties of spatially localized states within the one-dimensional forced complex Ginzburg-Landau equation as a function of the coefficients.
Damping and pumping of localized intrinsic modes in nonlinear dynamical lattices
Physical review, 1994
It has been recently demonstrated that dynamical models of nonlinear lattices admit approximate solutions in the form of self-supported intrinsic modes (IM s). In this work, the intensity of the emission of radiation ("phonons") from the oue-dimensional IM is calculated in an analytical approximation for the case of a moderately strong anharmonicity. Contrary to the emission in nonintegrable continuum models, which may be summarized as fusion of several vibrons into a phonon, the emission in the lattice may be described in terms of fission of a vibron into several phonons: as the IM's internal frequency lies above the phonon. band of the lattice, the radiative decay of the IM in the discrete system can be only subharmonic. It is demonstrated that the corresponding lifetime of the IM may be very large. Then, the threshold (minimum) value of the amplitude of an external ac 6eld, necessary to support the IM in a lattice with dissipative losses, is found for the limiting cases of the weak and strong anharmonicity.
Steady Solitary and Periodic Waves in a Nonlinear Nonintegrable Lattice
Symmetry
In this paper, stationary solitary and periodic waves of a nonlinear nonintegrable lattice are numerically constructed using a two-stage approach. First, as a result of continualization, a nonintegrable generalized Boussinesq—Ostrovsky equation is obtained, for which the solitary-wave and periodic solutions are numerically found by the Petviashvili method. In the second stage, discrete analogs of the obtained solutions are used as initial conditions in the numerical simulation of the original lattice. It is shown that the initial perturbations constructed in this way propagate along the lattice without changing their shape.
Generation of acoustic solitary waves in a lattice of Helmholtz resonators
Wave Motion, 2015
This paper addresses the propagation of high amplitude acoustic pulses through a 1D lattice of Helmholtz resonators connected to a waveguide. Based on the model proposed by Sugimoto (J. Fluid. Mech., 244 (1992), 55-78), a new numerical method is developed to take into account both the nonlinear wave propagation and the different mechanisms of dissipation: the volume attenuation, the linear viscothermal losses at the walls, and the nonlinear absorption due to the acoustic jet formation in the resonator necks. Good agreement between numerical and experimental results is obtained, highlighting the crucial role of the nonlinear losses. Different kinds of solitary waves are observed experimentally with characteristics depending on the dispersion properties of the lattice. a wide range of areas, including the theory of solitons and the dynamics of 4 discrete networks. Works have been led in electromagnetism and optics [1], 5 and numerous physical phenomena have been highlighted, such as dynamical 6 multistability [2, 3, 4], chaotic phenomena [5, 6], discrete breathers [7, 8, 9] 7 and solitons or solitary waves [10, 11]; for a review, see [12]. Solitary waves 8 have been observed and studied firstly for surface wave in shallow water [13].