Platonic crystal with low-frequency locally-resonant spiral structures: wave trapping, transmission amplification, shielding and edge waves

Abstract

We propose a new type of platonic crystal. The proposed microstructured plate includes snail resonators with low-frequency resonant vibrations. The particular dynamic effect of the resonators are highlighted by a comparative analysis of dispersion properties of homogeneous and perforated plates. Analytical and numerical estimates of classes of standing waves are given and the analysis on a macrocell shows the possibility to obtain localization, wave trapping and edge waves. Applications include transmission amplification within two plates separated by a small ligament. Finally we proposed a design procedure to suppress low frequency flexural vibration in an elongated plate implementing a by-pass system rerouting waves within the mechanical system. Dirac cones in platonic crystals equations, associated respectively to the presence of propagating and evanescent waves. Such waves can be coupled via the boundary or interface contact conditions. In most configurations the flexural waves are led by their Helmholtz component [5] and the homogenized equation can be of parabolic type at special frequencies [9,10]. However, short range wave scattering and Bragg resonance can be strongly influenced by the evanescent waves.Periodic structures play a major role in this field [18], since they create band gaps. These are frequency ranges where waves cannot propagate through the periodic system leading to possible application as acoustic and mechanical wave filters, vibration isolators, seismic shields. Partial band gap can lead to anisotropic wave response that can be used to obtain focusing and localization [19][20][21] as well as polarization properties [22,23].Two physical mechanisms can open band gaps: Bragg scattering and local resonance [24,25]. Bragg scattering is associated to the generation of band gaps at wavelengths of the same order of the unit cell around frequencies governed by the Bragg condition a = n(λ/2), (n = 1, 2, 3, · · · ), where a is the lattice constant of the periodic system and λ the wavelength [26]. Local resonances are associated to internal resonances due to the microstructures, they can be obtained from array of resonators as suggested in the seminal work [27]. Local resonances open tiny band gaps that can be at low frequencies [28][29][30] and inertial amplification mechanism that can widen stop band intervals have been proposed in [31,32]. The platonic system of snail resonatorsWe consider flexural vibrations in Kirchhoff plates. In the time-harmonic regime the transverse displacement W(x) satisfy the fourth-order biharmonic equation

Flexural vibration systems with gyroscopic spinners

“…Discrete systems have been studied to predict crack propagation speed and explain crack tip instabilities [37][38][39][40][41][42]. Enhanced transmission, trapping and wave control with different types of resonators were investigated in [43][44][45][46]. In the present paper, we show how gyroscopic spinners can be implemented in a periodic system supporting flexural waves to stabilize a structure and to tune the filtering properties of the system.…”

Directional Control of Rayleigh Wave Propagation in an Elastic Lattice by Gyroscopic Effects

“…Similar localized phenomena may play a substantial role in non-uniform crack propagation as evidenced in (Piccolroaz et al, 2020), where lattice dissimilarity has been shown to promote or diminish localized deformations around the faces of the crack. Trapped modes associated with Rayleigh-Bloch waves in systems incorporating periodic gratings or periodic arrays of resonators were analyzed by Porter and Evans (1999), Porter and Evans (2005), Linton and McIver (2002) and Antonakakis et al (2014) for scalar problems, by Colquitt et al (2015) for vector systems and by Haslinger et al (2017) and Morvaridi et al (2018) for plates.…”

2D Dynamic Directional Amplification (DDA) in Phononic Metamaterials

“…Li and Li [ 30 ] extended Frandsen et al’s model and included inertial amplification to infinite elastic beams. Other notable implementations include platonic crystals with spiral resonators [ 31 , 32 ], which could emerge as good alternatives due to their low-cost production merits.…”