Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook

The aim of this article is to investigate the wave propagation in one-dimensional chains with attached non-linear local oscillators by using analytical and numerical models. The focus is on the influence of non-linearities on the filtering properties of the chain in the low frequency range. Periodic systems with alternating properties exhibit interesting dynamic characteristics that enable them to act as filters. Waves can propagate along them within specific bands of frequencies called pass bands, and attenuate within bands of frequencies called stop bands or band gaps. Stop bands in structures with periodic or random inclusions are located mainly in the high frequency range, as the wavelength has to be comparable with the distance between the alternating parts. Band gaps may also exist in structures with locally attached oscillators. In the linear case the gap is located around the resonant frequency of the oscillators, and thus a stop band can be created in the lower frequency range. In the case with non-linear oscillators the results show that the position of the band gap can be shifted, and the shift depends on the amplitude and the degree of non-linear behaviour.

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Low-frequency band gaps in chains with attached non-linear oscillators

Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis

The interaction of waves in nonlinear media is of practical interest in the design of acoustic devices such as waveguides and filters. This investigation of the monoatomic mass–spring chain with a cubic nonlinearity demonstrates that the interaction of two waves results in different amplitude and frequency dependent dispersion branches for each wave, as opposed to a single amplitude-dependent branch when only a single wave is present. A theoretical development utilizing multiple time scales results in a set of evolution equations which are validated by numerical simulation. For the specific case where the wavenumber and frequency ratios are both close to 1:3 as in the long wavelength limit, the evolution equations suggest that small amplitude and frequency modulations may be present. Predictable dispersion behavior for weakly nonlinear materials provides additional latitude in tunable metamaterial design. The general results developed herein may be extended to three or more wave–wave interaction problems.

Multiple scales analysis of wave–wave interactions in a cubically nonlinear monoatomic chain

Wave propagation through a structured medium has attracted the attention of researchers for centuries due to its relevance to problems in condensed matter physics, chemistry, optics, phononics, composite, acoustics and mechanics. Wave containing certain band of frequencies can either propagate, known as transmission, or attenuated, known as attenuation band. This band structure for a continuum and its equivalent lumped spring mass model are not identical, although continuum medium is often modelled as a chain of discrete periodic structures because the continuous and discrete is depends on the scale. These band characteristics are dependent on the properties of the units, thus the effects of different parameters, such as damping, stiffness and mass ratios, nonlinearity, on the bandwidth are compared with each other in this review. To cloak, modulate, guide, filter out or attenuate unwanted frequencies from the propagating waves, metamaterials are widely investigated as a special form of the periodic structures from the past 2 decades. The main aim of this review is to compare the bandwidth for one-dimensional periodic structures. Waves through two and three-dimensional periodic medium are not considered in the review because the key band characteristics of periodic system can be perceived in one dimensional. The methods for computing the wave transmission are evaluated in the non-dimensional domain and the band characteristics of different one-dimensional periodic structures are critically assessed in this review. This review will help to the future researchers to choose a proper periodic medium for getting a specific band phenomenon.

Waves in Structured Mediums or Metamaterials: A Review

Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales

An inverted pendulum with asymmetric elastic restraints (e.g. a one-sided spring), when subjected to harmonic vertical base excitation, on linearizing trigonometric terms, is governed by an asymmetric Mathieu equation. This system is parametrically forced and strongly nonlinear (linearization for small motions is not possible). However, solutions are scaleable: if x ( t ) is a solution, then so is αx ( t ) for any real α >0. We numerically study the stability regions in the parameter plane of this system for a fixed degree of asymmetry in the elastic restraints. A Lyapunov-like exponent is defined and numerically evaluated to find these regions of stable and unstable behaviour. These numerics indicate that there are infinitely many possibilities of instabilities in this system that are missing in the usual or symmetric Mathieu equation. We find numerically that there are periodic solutions at the boundaries of stable regions in the parameter plane, analogous to the symmetric Mathieu equation. We compute and plot several of these solution branches, which provide a relatively simpler means of computing the stability transition curves of this system. We prove theoretically that such periodic solutions must exist on all stability boundaries. Our theoretical results apply to the asymmetric Hill9s equation, of which the pendulum system is a special case. We demonstrate this with numerical studies of a more general asymmetric Mathieu equation.

Asymmetric mathieu equations

Multiple scales analysis of early and delayed boundary ejection in Paul traps

Unimportance of geometric nonlinearity in analysis of flanged joints with metal-to-metal contact

This introduction to nonlinear systems is written for students of fluid mechanics, so connections are made throughout the text to familiar fluid flow systems. The aim is to present how nonlinear systems are qualitatively different from linear and to outline some simple procedures by which an understanding of nonlinear systems may be attempted. Considerable attention is paid to linear systems in the vicinity of fixed points, and it is discussed why this is relevant for nonlinear systems. A detailed explanation of chaos is not given, but a flavor of chaotic systems is presented. The focus is on physical understanding and not on mathematical rigor.

Nonlinear Dynamical Systems, Their Stability, and ChaosLecture notes from the FLOW-NORDITA Summer School on Advanced Instability Methods for Complex Flows, Stockholm, Sweden, 2013

Amol Marathe

Papers

Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales

Asymmetric mathieu equations

Multiple scales analysis of early and delayed boundary ejection in Paul traps

Unimportance of geometric nonlinearity in analysis of flanged joints with metal-to-metal contact

Nonlinear Dynamical Systems, Their Stability, and ChaosLecture notes from the FLOW-NORDITA Summer School on Advanced Instability Methods for Complex Flows, Stockholm, Sweden, 2013