Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNanoptera in weakly nonlinear woodpile and diatomic granular chains
158
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study ``nanoptera, which are non-localized solitary waves with exponentially small but non-decaying oscillations, in two singularly-perturbed Hertzian chains with precompression. These two systems are woodpile chains (which we model as systems of Hertzian particles and springs) and diatomic Hertzian chains with alternating masses. We demonstrate that nanoptera arise from Stokes phenomena and appear as special curves, called Stokes curves, are crossed in the complex plane. We use techniques from exponential asymptotics to obtain approximations of the oscillation amplitudes. Our analysis demonstrates that traveling waves in a singularly perturbed woodpile chain have a single Stokes curve, across which oscillations appear. Comparing these asymptotic predictions with numerical simulations reveals that this accurately describes the non-decaying oscillatory behavior in a woodpile chain. We perform a similar analysis of a diatomic Hertzian chain, that the nanpteron solution has two distinct exponentially small oscillatory contributions. We demonstrate that there exists a set of mass ratios for which these two contributions cancel to produce localized solitary waves. This result builds on prior experimental and numerical observations that there exist mass ratios that support localized solitary waves in diatomic Hertzian chains without precompression. Comparing asymptotic and numerical results in a diatomic Hertzian chain with precompression reveals that our exponential asymptotic approach accurately predicts the oscillation amplitude for a wide range of system parameters, but it fails to identify several values of the mass ratio that correspond to localized solitary-wave solutions.
rate research
Read More
We use exponential asymptotics to study travelling waves in woodpile systems modelled as singularly perturbed granular chains with zero precompression and small mass ratio. These systems are strongly nonlinear, and there is no analytic expression for their leading-order solution. We instead obtain an approximated leading-order solution using a hybrid numerical-analytic method. We show that travelling waves in these nonlinear woodpile systems are typically nanoptera, or travelling waves with exponentially small but non-decaying oscillatory tails which appear as a Stokes curve is crossed. We demonstrate that travelling wave solutions in the zero precompression regime contain two Stokes curves, and hence two sets of tailing oscillations in the solution. We calculate the behaviour of these oscillations explicitly, and show that there exist system configurations which cause the oscillations to cancel entirely, producing solitary wave behaviour. We then study the behaviour of travelling waves in woodpile chains as precompression is increased, and show that there exists a value of the precompression above which the two Stokes curves coalesce into a single curve, meaning that cancellation of the tailing oscillations no longer occurs. This is consistent with previous studies, which showed that cancellation does not occur in chains with strong precompression.
The study of granular crystals, metamaterials that consist of closely packed arrays of particles that interact elastically, is a vibrant area of research that combines ideas from disciplines such as materials science, nonlinear dynamics, and condensed-matter physics. Granular crystals, a type of nonlinear metamaterial, exploit geometrical nonlinearities in their constitutive microstructure to produce properties (such as tunability and energy localization) that are not conventional to engineering materials and linear devices. In this topical review, we focus on recent experimental, computational, and theoretical results on nonlinear coherent structures in granular crystals. Such structures --- which include traveling solitary waves, dispersive shock waves, and discrete breathers --- have fascinating dynamics, including a diversity of both transient features and robust, long-lived patterns that emerge from broad classes of initial data. In our review, we primarily discuss phenomena in one-dimensional crystals, as most research has focused on such scenarios, but we also present some extensions to two-dimensional settings. Throughout the review, we highlight open problems and discuss a variety of potential engineering applications that arise from the rich dynamic response of granular crystals.
This article explores the excitation of different vibrational states in a spatially extended dynamical system through theory and experiment. As a prototypical example, we consider a one-dimensional packing of spherical particles (a so-called granular chain) that is subject to harmonic boundary excitation. The combination of the multi-modal nature of the system and the strong coupling between the particles due to the nonlinear Hertzian contact force leads to broad regions in frequency where different vibrational states are possible. In certain parametric regions, we demonstrate that the Nonlinear Schrodinger (NLS) equation predicts the corresponding modes fairly well. We propose that nonlinear multi-modal systems can be useful in vibration energy harvest- ing and discuss a prototypical framework for its realization. The electromechanical model we derive predicts accurately the conversion from mechanical to electrical energy observed in the experiments.
We numerically investigate and experimentally demonstrate an in-situ topological band transition in a highly tunable mechanical system made of cylindrical granular particles. This system allows us to tune its inter-particle stiffness in a controllable way, simply by changing the contact angles between the cylinders. The spatial variation of particles stiffness results in an in-situ transition of the systems topology. This manifests as the emergence of a boundary mode in the finite system, which we observe experimentally via laser Doppler vibrometry. When two topologically different systems are placed adjacently, we analytically predict and computationally and experimentally demonstrate the existence of a finite-frequency topologically protected mode at their interface.
We consider longitudinal nonlinear atomic vibrations in uniformly strained carbon chains with the cumulene structure ($=C=C=)_{n}$. With the aid of ab initio simulations, based on the density functional theory, we have revealed the phenomenon of the $pi$-mode softening in a certain range of its amplitude for the strain above the critical value $eta_{c}approx 11,{%}$. Condensation of this soft mode induces the structural transformation of the carbon chain with doubling of its unit cell. This is the Peierls phase transition in the strained cumulene, which was previously revealed in [Nano Lett. 14, 4224 (2014)]. The Peierls transition leads to appearance of the energy gap in the electron spectrum of the strained carbyne, and this material transforms from the conducting state to semiconducting or insulating states. The authors of the above paper emphasize that such phenomenon can be used for construction of various nanodevices. The $pi$-mode softening occurs because the old equilibrium positions (EQPs), around which carbon atoms vibrate at small strains, lose their stability and these atoms begin to vibrate in the new potential wells located near old EQPs. We study the stability of the new EQPs, as well as stability of vibrations in their vicinity. In previous paper [Physica D 203, 121(2005)], we proved that only three symmetry-determined Rosenberg nonlinear normal modes can exist in monoatomic chains with arbitrary interparticle interactions. They are the above-discussed $pi$-mode and two other modes, which we call $sigma$-mode and $tau$-mode. These modes correspond to the multiplication of the unit cell of the vibrational state by two, three or four times compared to that of the equilibrium state. We study properties of these modes in the chain model with arbitrary pair potential of interparticle interactions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
