Certain aspects of the acoustics of a strongly nonlinear discrete lattice

Mojahed, Alireza; Vakakis, Alexander F.

doi:10.1007/s11071-019-05080-9

Cited by 12 publications

(8 citation statements)

References 24 publications

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“…One observes that the breather travels through the lattice without substantial decay. Similar results for the breather propagation in conservative chain were reported in recent paper [40] for the case of linear oscillators with cubic coupling.…”

Section: Description Of the Model And Numeric Resultssupporting

confidence: 89%

“…The dependence between energy of the undamped motion and the action is determined by well-known formula [40]:…”

Section: Simplified Model Of the Breather Propagationmentioning

confidence: 99%

“…It means, in turn, that also the internal displacement can be treated in the quasilinear approximation. To this end, we use the well-known actionangle transformation for linear oscillator [40]. These transformations allow one to find the relationship between the displacement and the averaged action:…”

Section: Simplified Model Of the Breather Propagationmentioning

confidence: 99%

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Breather arrest in a chain of damped oscillators with Hertzian contact

Strozzi

Gendelman

2021

Wave Motion

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Breather propagation in a damped oscillatory chain with Hertzian nearest-neighbour coupling is investigated. The breather propagation exhibits an unusual two-stage pattern. The first stage is characterized by power-law decay of the breather amplitude. This stage extends over finite number of the chain sites. Drastic drop of the breather amplitude towards the end of this finite fragment is referred to as breather arrest. At the second stage, the breather exhibits very small amplitudes with hyper-exponential decay. Numeric results are rationalized by considering a simplified model of two damped linear oscillators coupled by Hertzian contact forces. Initial excitation of one of these oscillators results in a finite number of beating cycles in the system. This simplified model reliably predicts main features of the breather arrest. More general coupling potentials and effect of pre-compression on the breather propagation are also discussed.

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Section: Description Of the Model And Numeric Resultssupporting

confidence: 89%

“…The dependence between energy of the undamped motion and the action is determined by well-known formula [40]:…”

Section: Simplified Model Of the Breather Propagationmentioning

confidence: 99%

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Breather arrest in a chain of damped oscillators with Hertzian contact

Strozzi

Gendelman

2021

Wave Motion

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“…Most previous work on nonlinear topological systems have relied on asymptotic methods to study the effect of the nonlinearity [5,15,30,31], however these are only applicable when the nonlinearity is weak so that the solutions can be regarded as small perturbations of the linearized ones. Methods such as complexification averaging [26,32] have been developed for certain cases of strongly nonlinear or essentially nonlinear systems, however the application of this particular method is restricted to approximating the slow flow evolution component of dynamics. In our system, we have a strongly nonlinear lattice for which asymptotic techniques generally do not apply.…”

Section: B Numerical Continuationmentioning

confidence: 99%

“…However, this framework does not address the nonlinearity in the bulk-bands. Indeed, it is known that the bulk bands of nonlinear lattices evolve with energy [25][26][27], and this evolution should be considered for the topological system as well.…”

Section: Introductionmentioning

confidence: 99%

Topological Protection in a Strongly Nonlinear Interface Lattice

Tempelman,

Matlack,

Vakakis

2021

Preprint

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The application of topological phases in classical mechanical systems is unlocking previously unattainable topologically insulated waves in acoustic, phonic, and elastic media. Mechanical topological insulators are well understood for linear and weakly nonlinear systems, however traditional analysis methods break down for strongly nonlinear systems since linear methods can not be applied in that case. We study one such system in the form of a one-dimensional mechanical analog of the Shu-Schrieffer-Heeger interface model with strong nonlinearity of the cubic form. Two nonlinear half-lattices make a topological interface system with a nonlinear coupling added to the stiff spring, while linear grounding springs are added on all oscillators. The frequency-energy dependence of the nonlinear bulk modes and topologically insulated mode is explored using Numerical continuation of the system's nonlinear normal modes (NNMs), i.e., of standing waves. Moreover, the linear stability of the NNMs are investigated using Floquet Multipliers (FMs) and Krein signature analysis. We find that the nonlinear topological lattice supports a family of topologically insulated NNMs that are parameterized by the total energy of the system and are stable within a range of frequencies. Using numerical simulations, we empirically recover the geometric Zak phase to determine at which energies the nontrivial phase conditions cease to exists. We compare the predictions from FM analysis and the numerical phase analysis by numerically investigating the interface system for harmonic excitation applied to the interface site. It is shown that empirical calculations of the geometric Zak Phase lead to reliable measures for predicting the existence of the topological mode in the nonlinear system, and that while the FMs of the NNMs are dependable predictors of the NNMs stability, they are inferior to the empirical phase calculations for predicting at which energies the lattice supports waves localized at the interface. Furthermore, the results are shown to be robust to parametric perturbations of both chiral and achiral forms. Thus, we provide a new and improved method for analyzing and predicting the existence of topologically insulated modes in a strongly nonlinear lattice.

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Nonlinear wave propagation in locally dissipative metamaterials via Hamiltonian perturbation approach

et al. 2022

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The cellular microstructure of periodic architected materials can be enriched by local intracellular mechanisms providing innovative distributed functionalities. Specifically, high-performing mechanical metamaterials can be realized by coupling the lowdissipative cellular microstructure with a periodic distribution of tunable damped oscillators, or resonators, vibrating at relatively high amplitudes. The benefit is the actual possibility of combining the design of wavestopping bands with enhanced energy dissipation properties. This paper investigates the nonlinear dispersion properties of an archetypal mechanical metamaterial, represented by a one-dimensional lattice model characterized by a diatomic periodic cell. The intracellular interatomic interactions feature geometric and constitutive nonlinearities, which determine cubic coupling between the lattice and the resonators. The nondissipative part of the coupling can be designed to exhibit a softening or a hardening behavior, by inde-

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Certain aspects of the acoustics of a strongly nonlinear discrete lattice

Cited by 12 publications

References 24 publications

Breather arrest in a chain of damped oscillators with Hertzian contact

Breather arrest in a chain of damped oscillators with Hertzian contact

Topological Protection in a Strongly Nonlinear Interface Lattice

Nonlinear wave propagation in locally dissipative metamaterials via Hamiltonian perturbation approach

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