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31.
Y. Zhang D. Pozharskiy D. M. McFarland P. G. Kevrekidis I. G. Kevrekidis A. F. Vakakis 《Experimental Mechanics》2017,57(4):505-520
We experimentally study a one-dimensional uncompressed granular chain composed of a finite number of identical spherical elastic beads with Hertzian interactions. The chain is harmonically excited by an amplitude- and frequency-dependent boundary drive at its left end and has a fixed boundary at its right end. Such ordered granular media represent an interesting new class of nonlinear acoustic metamaterials, since they exhibit essentially nonlinear acoustics and have been designated as “sonic vacua” due to the fact that their corresponding speed of sound (as defined in classical acoustics) is zero. This paves the way for essentially nonlinear and energy-dependent acoustics with no counterparts in linear theory. We experimentally detect time-periodic, strongly nonlinear resonances whereby the particles (beads) of the granular chain respond at integer multiples of the excitation period, and which correspond to local peaks of the maximum transmitted force at the chain’s right, fixed end. In between these resonances we detect a local minimum of the maximum transmitted forces corresponding to an anti-resonance in the stationary-state dynamics. The experimental results of this work confirm previous theoretical predictions, and verify the existence of strongly nonlinear resonance responses in a system with a complete absence of any linear spectrum; as such, the experimentally detected nonlinear resonance spectrum is passively tunable with energy and sensitive to dissipative effects such as internal structural damping in the beads, and friction or plasticity effects. We compare the experimental results with direct numerical simulations of the granular network and deduce satisfactory agreement. 相似文献
32.
P.G. Kevrekidis 《Physics letters. A》2009,373(40):3688-3693
We revisit the theme of non-nearest-neighbor interactions in nonlinear dynamical lattices, in the prototypical setting of the discrete nonlinear Schrödinger equation. Our approach offers a systematic way of analyzing the existence and stability of solutions of the system near the so-called anti-continuum limit of zero coupling. This affords us a number of analytical insights such as the fact that, for instance, next-nearest-neighbor interactions allow for solutions with nontrivial phase structure in infinite one-dimensional lattices; in the case of purely nearest-neighbor interactions, such phase structure is disallowed. On the other hand, such non-nearest-neighbor interactions can critically affect the stability of unstable structures, such as topological charge S=2 discrete vortices. These analytical predictions are corroborated by numerical bifurcation and stability computations. 相似文献
33.
We study the azimuthal modulational instability of vortices with different topological charges, in the focusing two-dimensional nonlinear Schrödinger (NLS) equation. The method of studying the stability relies on freezing the radial direction in the Lagrangian functional of the NLS in order to form a quasi-one-dimensional azimuthal equation of motion, and then applying a stability analysis in Fourier space of the azimuthal modes. We formulate predictions of growth rates of individual modes and find that vortices are unstable below a critical azimuthal wave number. Steady-state vortex solutions are found by first using a variational approach to obtain an asymptotic analytical ansatz, and then using it as an initial condition to a numerical optimization routine. The stability analysis predictions are corroborated by direct numerical simulations of the NLS. We briefly show how to extend the method to encompass nonlocal nonlinearities that tend to stabilize such solutions. 相似文献
34.
We consider the question of existence of ??bell-shaped?? (i.e., nonincreasing for x>0 and nondecreasing for x<0) traveling waves for the strain variable of the generalized Hertzian model describing, in the special case of a p=3/2 exponent, the dynamics of a granular chain. The proof of existence of such waves is based on the English and Pego (Proc. Am. Math. Soc. 133:1763, 2005) formulation of the problem. More specifically, we construct an appropriate energy functional, for which we show that the constrained minimization problem over bell-shaped entries has a solution. We also provide an alternative proof of the Friesecke?CWattis result (Commun. Math. Phys. 161:391, 1994) by using the same approach (but where the minimization is not constrained over bell-shaped curves). We briefly discuss and illustrate numerically the implications on the doubly exponential decay properties of the waves, as well as touch upon the modifications of these properties in the presence of a finite precompression force in the model. 相似文献
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Mason A. Porter Martin Centurion Ye Pu P. G. Kevrekidis D. J. Frantezeskakis Demetri Psaltis 《PAMM》2007,7(1):2030029-2030030
We study nonlinearity management in optics by investigating the propagation of localized pulses and plane waves in a layered, cubically nonlinear (Kerr) medium that consists of alternating layers of glass and air. We show that such nonlinearity management delays the blow-up/collapse of pulses and leads to a band structure of modulationally unstable regions for plane waves. We find excellent agreement between experiments, numerical simulations, and theory. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
37.
Tomohiro Yanao Wang S. Koon Jerrold E. Marsden Ioannis G. Kevrekidis 《PAMM》2007,7(1):1080503-1080504
Reduction of dimensionality is crucial for the deeper understanding of the mechanism for large-amplitude conformational transitions of complex molecules. By taking up a six-atomcluster as an illustrative example, we present a general methodology to understand conformational transitions of molecules in terms of the low-dimensional dynamics of molecular gyration radii. The dynamics of gyration radii is generally governed by the interplay between the ordinary potential force and a dynamical force called the internal centrifugal force. We show that the internal centrifugal force can be more important than the original potential barrier and gives rise to a new dynamical barrier that truly dominates the conformational transitions of the system. This kind of dynamical effect should be crucially important in a wide class of molecular reaction dynamics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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