Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

Abstract

In-plane wave propagation in a periodic rectangular grid beam structure, which includes rotational inertia (so-called 'Rayleigh beams'), is analyzed both with a Floquet-Bloch exact formulation for free oscillations and with a numerical treatment (developed with PML absorbing boundary conditions) for forced vibrations (including Fourier representation and energy flux evaluations), induced by a concentrated force or moment. A complex interplay is observed between axial and flexural vibrations (not found in the common idealization of out-of-plane motion), giving rise to several forms of vibration localization: 'X-', 'cross-' and 'star-' shaped, and channel propagation. These localizations are triggered by several factors, including rotational inertia and slenderness of the beams and the type of forcing source (concentrated force or moment). Although the considered grid of beams introduces an orthotropy in the mechanical response, a surprising 'isotropization' of the vibration is observed at special frequencies. Moreover, rotational inertia is shown to 'sharpen' degeneracies related to Dirac cones (which become more pronounced when the aspect ratio of the grid is increased), while the slenderness can be tuned to achieve a perfectly flat band in the dispersion diagram. The obtained results can be exploited in the realization of metamaterials designed to control wave propagation. and the solution techniques are well-known, many interesting features still remain to be explored. This exploration is provided in the present article, where an exact Floquet-Bloch analysis is performed and complemented with a numerical treatment of the forced vibrations induced by the application of a concentrated force or moment, including presentation of the Fourier transform and energy flow (treated in [26] for free vibrations). It is shown that (i.) aspect ratio of the grid, (ii.) slenderness and (iii.) rotational inertia of the beams decide the emergence of several forms of highly-localized waveforms, namely, 'channel propagation', 'X-', 'cross-', 'star-' shaped vibration modes. Moreover, these mechanical properties of the grid can be designed to obtain flat bands and degeneracies related to Dirac cones in the dispersion diagram and directional anisotropy or, surprisingly, dynamic 'isotropization', for which waves propagate in a square lattice with the polar symmetry characterizing propagation in an isotropic medium.The presented results open the way to the design of vibrating devices with engineered properties, to achieve control of elastic wave propagation.2 In-plane Floquet-Bloch waves in a rectangular grid of beams An infinite lattice of Rayleigh beams is considered, periodically arranged in a rectangular geometry as shown in Fig. 1a, together with the unit cell, Fig. 1b.

Experimental full wavefield reconstruction and band diagram analysis in a single-phase phononic plate with internal resonators

“…The investigation of elastic wave propagation phenomena in artificially structured composite materials is an active research topic in the scientific community. Shortly after the introduction of photonic crystals and electromagnetic metamaterials, their elastic counterpart, i.e., phononic crystals (PCs) and elastic metamaterials [1][2][3], have attracted increasing attention due to the possibility of reproducing in elasticity an abundant set of unusual physical properties [4], such as stop-band filtering [5,6], negative refraction [7-9], acoustic lensing [10], ordinary [11,12] and topologically protected [13][14][15][16][17] wave localization / splitting, and fluid elasticity [18]. Among these, the ability to attenuate elastic waves over entire frequency ranges, often referred to as phononic band gaps (BGs), is among the most attractive and studied properties.…”

Analytical treatment of the transient motion of inertial beams attached to coupling inertial resonators

“…The growth of interest in flexural systems with uncommon properties ("flexural metamaterials") has led to the design of innovative structures consisting of beams and rods. Dispersion degeneracies and localisation phenomena in lattices of Rayleigh beams have been studied in [39][40][41]. Applications in soft robotics and structural folding of beams undergoing large deformations have been considered in [42][43][44].…”

Flexural vibration systems with gyroscopic spinners

“…In particular, new designs of lattice structures, exhibiting unusual dynamic properties, have been proposed. Dispersion degeneracies and localization in a grid of beams with rotational inertia, referred to as Rayleigh beams, were analysed in [33][34][35]. Pre-stress in two-dimensional arrays of axially and flexurally deformable beams was exploited in [36] to create negative refraction, total reflection and wave channelling.…”