Elastodynamical properties of Sturmian structured media

TL;DR

This work introduces Sturmian sequences as a principled method to design 1D elastic waveguides with quasiperiodic parameter distributions. By mapping Sturmian words, generated from a generator $\alpha\in[0,1]$, onto binary parameter patterns and employing transfer-matrix analysis, the authors derive dispersion relations and construct the Sturmian bulk spectrum, highlighting fractal self-similarity across scales. Closed-form 2×2 transfer-matrix results and a Chebyshev-based recursion underpin the observed spectral self-similarity, complemented by 3 numerical examples: a discrete spring-mass chain, a rod with varying sectional stiffness, and a Timoshenko beam, each confirming the theoretical predictions. The findings offer a framework for engineering mechanical structures with tailored band structures and fractal spectral features, with potential applications in vibroacoustic filtering and metamaterial design.

Abstract

In this paper, wave propagation in structured media with quasiperiodic patterns is investigated. We propose a methodology based on Sturmian sequences for the generation of structured mechanical systems from a given parameter. The approach is presented in a general form so that it can be applied to waveguides of different nature, as long as they can be modeled with the transfer matrix method. The bulk spectrum is obtained and its fractal nature analyzed. For validation of the theoretical results, three numerical examples are presented. The obtained bulk spectra show different shapes for the studied examples, but they share features which can be explained from the proposed theoretical setting.

Figures (12)

• Figure 1: Three different examples of dynamical systems based on Sturmian blocks for number $\alpha = 2/7 = [0;3,2]$. Above: a discrete spring-mass system, $\Theta \equiv k$ (spring coefficients). Middle: a continuous rod (axial waves), $\Theta \equiv \rho A$ (mass per unit of length). Bottom: a continuous beam (flexural waves), $\Theta \equiv EI$ (sectional bending stiffness)
• Figure 2: Two rectangles $S_1 (u \times \theta_p)$ and $S_2(v \times \theta_q)$ forming a tiling by periodic translation of basis $\mathbf{e}=(v,\theta_p), \ \mathbf{f}=(-u,\theta_q)$. The examples shown has been built for $\alpha = v/u = 2/7 = [0;3,2]$.
• Figure 3: Lattice plane tiling using rectangles $\mathcal{S}_1$ and $\mathcal{S}_2$. Sturmian blocks associated to $\alpha = 2/7 = [0;3,2]$ and the corresponding Sturmian-like spring-mass chain. Right: sequence of bases $\{\mathbf{g}_k\}_{k=1}^n$ and length of the supercell in terms of the quasiperiodic parameter $\sum_{j=1}^n\Theta(j) = \nu_n \theta_q + \delta_n \theta_p$
• Figure 4: A one-dimensional Sturmian structured system associated to the number $\alpha$. The binary parameter $\Theta(j) \in \{\theta_p,\theta_q\}$ changes its value according to the Sturmian pattern given by the block $\mathcal{B}(\alpha)$
• Figure 5: (Example 1) Discrete spring-mass system with Sturmian quasiperiodic distribution of rigidities $K_j$.