Nonlinear dispersion relation in anharmonic periodic mass-spring and mass-in-mass systems

The study of wave propagation in chains of anharmonic periodic systems is of fundamental importance to understand the response of dynamical absorbers of vibrations and acoustic metamaterials working in nonlinear regime. Here, we derive an analytical nonlinear dispersion relation for periodic chains of anharmonic mass-spring and mass-in-mass systems resulting from considering the hypothesis of weak anharmonic energy and a periodic distribution function as ansatz of a general solution of the nonlinear equations of motion. Numerical simulations show that this expression is valid for anharmonic potential energy up to 50% of the harmonic one. This work provides a simple tool to design and study nonlinear dynamics for a class of seismic metamaterials.Comment: 18 pages, 5 figure

Seismic isolation of buildings using composite foundations based on metamaterials

Metamaterials can be engineered to interact with waves in entirely new ways, finding application on the nanoscale in various fields such as optics and acoustics. In addition, acoustic metamaterials can be used in large-scale experiments for filtering and manipulating seismic waves (seismic metamaterials). Here, we propose seismic isolation based on a device that combines some properties of seismic metamaterials (e.g., periodic mass-in-mass systems) with that of a standard foundation positioned right below the building for isolation purposes. The concepts on which this solution is based are the local resonance and a dual-stiffness structure that preserves large (small) rigidity for compression (shear) effects. In other words, this paper introduces a different approach to seismic isolation by using certain principles of seismic metamaterials. The experimental demonstrator tested on the laboratory scale exhibits a spectral bandgap that begins at 4.5 Hz. Within the bandgap, it filters more than 50% of the sei..

Scalable synchronization of spin-Hall oscillators in out-of-plane field

A strategy for a scalable synchronization of an array of spin-Hall oscillators (SHOs) is illustrated. In detail, we present micromagnetic simulations of two and five SHOs realized by means of couples of triangular golden contacts on the top of a Pt/CoFeB/Ta trilayer. Results highlight that the synchronization occurs for the whole current region that gives rise to the excitation of self-oscillations. This is linked to the role of the magnetodipolar coupling, which is the phenomenon driving the synchronization when the distance between oscillators is not too large. Synchronization turns out to be also robust against geometrical differences of the contacts, simulated by considering variable distances between the tips ranging from 100nm to 200nm. Besides, it entails an enlargement of the radiation pattern that can be useful for the generation of spin-waves in magnonics applications. Simulations performed to study the effect of the interfacial Dzyaloshinskii-Moriya interaction show nonreciprocity in spatial propagation of the synchronized spin-wave. The simplicity of the geometry and the robustness of the achieved synchronization make this design of array of SHOs scalable for a larger number of synchronized oscillators

Wave amplitude decay driven by anharmonic potential in nonlinear mass-in-mass systems

Periodic or quasi-periodic arrangements of artificial structures can be used to design a class of materials, i.e., metamaterials, with intriguing properties. Recently, it has been proposed to use periodic systems with internal resonances for the attenuation of acoustic/seismic waves. However, large input displacements due to seismic waves can drive the working point of these systems in a nonlinear regime. Here, we have studied the nonlinear dynamics of periodic chain of mass-in-mass systems, which can be used to model composite foundations, where the external spring is characterized by an anharmonic potential. The main finding of this work is the identification of two attenuation mechanisms, one is characterized by an exponential amplitude decay and is observed in the strongly anharmonic regime, whereas the other has a linear decay pattern and characterizes the weak anharmonic dynamics. This result has a direct impact in the design of low frequency seismic metamaterials

Design of non-linear seismic metamaterials - Invited talk by R. Zivieri

A model used to understand seismic metamaterials from a theoretical point of view is based on the concept of the periodic sub-wavelength resonant mass-in-mass system, see Fig. 1. We have already proposed a continuous implementation of those type of seismic metamaterials based on the use of isochronous mechanical oscillators. However, the bandgap of the this device is centered at the resonance frequency of the atomic mass-in-mass element. A key challenge is to achieve a broad extension of the bandgap and a bandgap starting at a frequency as low as possible To reach this result, it has been proposed to exploit the non-reciprocity feature of the chiral materials, introducing in the system hybrid modes. Here, we focus on the possible engineering of the non-linearity of the external spring of a mass-in-mass system, we evaluate three cases including an hysteretic ke value. Starting by the Lagrangian equation of the energy, we obtain the dynamical equations taking into account the relationship of ke relate to the displacement of the me. The dynamical equations are then solved numerically. Our results point out that the start frequency of the bandgap can be decreased by 25% by considering ke=A atan(Bue) with proper values of A and B

Modeling and design of non-linear seismic metamaterials - Invited talk by R. Zivieri

A model used to understand seismic metamaterials from a theoretical point of view is based on the concept of the periodic sub-wavelength resonant mass-in-mass system, see Fig. 1. Figure 1 : Sketch of the system studied. We have already proposed a continuous implementation of those type of seismic metamaterials based on the use of isochronous mechanical oscillators. However, the bandgap of this device has its centre at the resonance frequency of the atomic mass-in-mass element. A key challenge is to achieve a broad extension of the bandgap and a bandgap starting at a frequency as low as possible. Here, we focus on the possible engineering of the non-linearity of the external spring of a mass-in-mass system. In order to do that, first we define an anharmonic force exerted on the mass me resulting from a potential energy developed up to the fourth-order. Second, starting from the Lagrangian equation we obtain the dispersion relation in the presence of the anharmonic contributions. The acoustical branch of the dispersion relation is strongly downshifted with respect to that obtained in the linear case