Modelling and Simulation in Engineering

Volume 2016 (2016), Article ID 2651953, 7 pages

http://dx.doi.org/10.1155/2016/2651953

## Floquet Theory for Discontinuously Supported Waveguides

Dipartimento di Scienze e Metodi dell’Ingegneria (DISMI), Universitá degli Studi di Modena e Reggio Emilia, Via G. Amendola 2, 42122 Reggio Emilia, Italy

Received 24 December 2015; Accepted 5 May 2016

Academic Editor: Julius Kaplunov

Copyright © 2016 A. Sorzia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We apply Floquet theory of periodic coefficient second-order ODEs to an elastic waveguide. The waveguide is modeled as a uniform elastic string periodically supported by a discontinuous Winkler elastic foundation and, as a result, a Hill equation is found. The fundamental solutions, the stability regions, and the dispersion curves are determined and then plotted. An asymptotic approximation to the dispersion curve is also given. It is further shown that the end points of the band gap structure correspond to periodic and semiperiodic solutions of the Hill equation.

#### 1. Introduction

Periodic structures often appear in several mechanical systems, ranging from strings and beams [1, 2] to phononic crystals [3–5] to name just a few. Such systems exhibit a typical pass/block band structure when wave propagation is considered. Indeed, periodic structures are especially relevant when employed as waveguides [6, 7] or energy scavenging devices [8]. The analysis of the transmission property of waveguides is best carried out through Floquet theory of periodic coefficient ODEs, although this fact is seldom neglected in favor of a more direct approach by means of the Floquet-Bloch boundary conditions. In this paper, an analysis of the mechanical problem of a uniform string periodically supported on a Winkler foundation is presented form the standpoint of the stability theory of Hill’s equation [9]. Besides, a high-frequency asymptotic homogenization procedure is presented, following [10, 11]. The discontinuous character of the support may be due to crack propagation [12–15] or debonding in composite materials [16, 17]. It could also be due to the tensionless character of the substrate [18]. This study follows upon a very vast body of literature on elastic periodic structures [2, 19–24]. The situation of wave propagation through a thin coating layer [25–27] could also be considered. Applications in the realm of civil engineering are also possible [28–32]. The paper is structured as follows: Section 2 sets up the mechanical model and the governing equations. Section 3 discusses stability of the solution of Hill’s equation. The dispersion relation and its asymptotic approximation are presented in Section 4. Finally, conclusions are drawn in Section 5.

#### 2. The Mechanical Model

Let us consider a homogeneous elastic string in uniform tension , periodically supported on a Winkler elastic foundation (Figure 1). The governing equation for the transverse displacement in the absence of loading is periodic with period :where is the mass linear density of the string, assumed constant, is the Winkler subgrade modulus (with physical dimension of stress), and is Heaviside step function; that is,This equation may be rewritten in dimensionless formhaving introduced the dimensionless positive ratios:together with , the dimensionless axial coordinate, and , the dimensionless time. Here, prime denotes differentiation with respect to and dot differentiation with respect to and and are assumed. We shall look for the harmonic behavior of ; that is, , whence (3) becomes the constant coefficient ODE for :We shift the unit period to range in the interval in order to consider an even/odd problem; namely,The general harmonic solution of (3) in the supported region is given bywhile the solution in the free region iswhere , and , are integration constants to be determined through the boundary conditions (BCs). The BCs for the system require continuity at the supported/unsupported transition:where prime stands for differentiation. Furthermore, consideration of the Floquet-Bloch waves lends the periodicity conditionswhere is the* characteristic multiplier*. For a second-order ODE, there are two nonnecessarily distinct characteristic multipliers, which are denoted by and . Besides, let with ; then is the* characteristic exponent*.