Waves in one-dimensional quasicrystalline structures: dynamical trace mapping, scaling and self-similarity of the spectrum

Gei, Massimiliano

doi:10.1016/j.jmps.2018.06.007

Cited by 25 publications

(69 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We can deduce from expression (4.3) that the trajectories on the torus are periodic if the ratio β is a rational number. This condition is exactly the same as that introduced in [23] and necessary to realize Fibonacci structures with a periodic spectrum, which are called in that article canonical structures. Therefore, canonical configurations correspond to closed flow trajectories on the torus.…”

Section: Analysis Of the Flow Lines On The Reduced Torusmentioning

confidence: 95%

See 1 more Smart Citation

On the universality of the frequency spectrum and band-gap optimization of quasicrystalline-generated structured rods

Tetik

Gal

Gei

2019

Phil. Trans. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

The dynamical properties of periodic two-component phononic rods, whose elementary cells are generated adopting the Fibonacci substitution rules, are studied through the recently introduced method of the toroidal manifold. The method allows all band gaps and pass bands featuring the frequency spectrum to be represented in a compact form with a frequency-dependent flow line on the surface describing their ordered sequence. The flow lines on the torus can be either closed or open: in the former case, (i) the frequency spectrum is periodic and the elementary cell corresponds to a canonical configuration, (ii) the band gap density depends on the lengths of the two phases; in the latter, the flow lines cover ergodically the torus and the band gap density is independent of those lengths. It is then shown how the proposed compact description of the spectrum can be exploited (i) to find the widest band gap for a given configuration and (ii) to optimize the layout of the elementary cell in order to maximize the low-frequency band gap. The scaling property of the frequency spectrum, that is a distinctive feature of quasicrystalline-generated phononic media, is also confirmed by inspecting band-gap/pass-band regions on the torus for the elementary cells of different Fibonacci orders. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.

show abstract

Section: Analysis Of the Flow Lines On The Reduced Torusmentioning

confidence: 95%

“…As a consequence, the areas of subdomains D i 2 depend only on ratio A S /A L , while the direction of flow is defined by l S /l L . Moreover, according to the classification provided in [23], the analysed rods belong to the second family of canonical configurations.…”

Section: Analysis Of the Flow Lines On The Reduced Torusmentioning

confidence: 99%

On the universality of the frequency spectrum and band-gap optimization of quasicrystalline-generated structured rods

Tetik

Gal

Gei

2019

Phil. Trans. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

show abstract

“…This study contributes to the recent investigations of the dynamic response of QP continuous elastic media [27][28][29]. For example, the self-similar and invariant nature of stop and pass bands in beams and rods embedded with QP arrays of supports were analyzed in [27,28], while localization in plates with QP arrays of inclusions was demonstrated in [29]. While prior studies are notable in observing stop bands and localized modes, a formal treatment of their topological properties and connection to the existence of localized modes is currently missing.…”

mentioning

confidence: 79%

“…This pattern-generating procedure identifies families of structures ranging from periodic to QP, that are obtained through the smooth variation of the parameters defining the projection. This study contributes to the recent investigations of the dynamic response of QP continuous elastic media [27][28][29]. For example, the self-similar and invariant nature of stop and pass bands in beams and rods embedded with QP arrays of supports were analyzed in [27,28], while localization in plates with QP arrays of inclusions was demonstrated in [29].…”

mentioning

confidence: 84%

Topological bands and localized vibration modes in quasiperiodic beams

2019

View full text Add to dashboard Cite

We investigate a family of quasiperiodic continuous elastic beams, the topological properties of their vibrational spectra, and their relation to the existence of localized modes. We specifically consider beams featuring arrays of ground springs at locations determined by projecting from a circle onto an underlying periodic system. A family of periodic and quasiperiodic structures is obtained by smoothly varying a parameter defining such projection. Numerical simulations show the existence of vibration modes that first localize at a boundary, and then migrate into the bulk as the projection parameter is varied. Explicit expressions predicting the change in the density of states of the bulk define topological invariants that quantify the number of modes spanning a gap of a finite structure. We further demonstrate how modulating the phase of the ground springs distribution causes the topological states to undergo an edge-to-edge transition. The considered configurations and topological studies provide a framework for inducing localized modes in continuous elastic structural components through globally spanning, deterministic perturbations of periodic patterns defined by the considered projection operations. systems [19] by exploiting the notion that QP lattices are projections of higher dimensional manifolds onto lower dimensional lattices [20,21]. Notable examples include the investigations in [22,23], where photonic waveguides are used to achieve adiabatic pumping of waves between opposite ends of one-dimensional (1D) QP lattices. With further studies, a dynamically generated four-dimensional (4D) quantum Hall system was experimentally demonstrated using two-dimensional (2D) periodic lattices [24]. In the classic mechanics realm, localized modes at the boundary of a QP chain of magnetic spinners were shown in [25], while topological boundary and interface modes were experimentally demonstrated in acoustic waveguides with QP patterning of the walls [26]. Although open questions still remain regarding how eigenmodes may localize in any region of the domain, these studies provide insight into modes that are localized at edges or interfaces, and suggest new methodologies for wave localization and transport based on higher dimensional topological properties.Motivated by these contributions, we here investigate how localized modes arise in continuous elastic media with QP stiffness modulations. Such modulations are introduced through arrays of ground springs embedded in structural beams undergoing transverse vibrations (figure 1). The locations of the springs are determined by projecting from periodic array of circles. This pattern-generating procedure identifies families of structures ranging from periodic to QP, that are obtained through the smooth variation of the parameters defining the projection. This study contributes to the recent investigations of the dynamic response of QP continuous elastic media [27][28][29]. For example, the self-similar and invariant nature of stop and pass bands in beams and rods embedd...

show abstract

“…so that the quasi-periodic motion of the infinite periodic structural system is defined once the amplitudes defining the motion of the 0-th cell are given. In particular, the application of the Bloch-Floquet condition to the boundary conditions (33), (34), and (36) leads to the following linear system in the twelve amplitudes of the axial motion in the 0-th cell, U [0] JKj (J = A, B, K = L, C, R,, and j = 1, 2), to be solved for given values of the two transverse amplitudes V [0] AC1 and V [0] BC1 , both corresponding respectively to modes n A and n B ,…”

Section: Bloch-floquet Analysis and Resonancementioning

confidence: 99%

Nested Bloch waves in elastic structures with configurational forces

Corso

Tallarico

Movchan

et al. 2019

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

Small axial and flexural oscillations are analyzed for a periodic and infinite structure, constrained by sliding sleeves and composed of elastic beams. A nested Bloch-Floquet technique is introduced to treat the non-linear coupling between longitudinal and transverse displacements induced by the configurational forces generated at the sliding sleeve ends. The action of configurational forces is shown to play an important role from two perspectives. First, the band gap structure for purely longitudinal vibration is broken so that axial propagation may occur at frequencies that are forbidden in the absence of a transverse oscillation and, second, a flexural oscillation may induce axial resonance, a situation in which the longitudinal vibrations tend to become unbounded. The presented results disclose the possibility of exploiting configurational forces in the design of mechanical devices towards longitudinal actuation from flexural vibrations of small amplitude at given frequency.

show abstract

Waves in one-dimensional quasicrystalline structures: dynamical trace mapping, scaling and self-similarity of the spectrum

Cited by 25 publications

References 41 publications

On the universality of the frequency spectrum and band-gap optimization of quasicrystalline-generated structured rods

On the universality of the frequency spectrum and band-gap optimization of quasicrystalline-generated structured rods

Topological bands and localized vibration modes in quasiperiodic beams

Nested Bloch waves in elastic structures with configurational forces

Contact Info

Product

Resources

About