#### 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*.

#### 3. Boundedness and Periodicity of the Solution

Equation (5) is known as Hill’s equation, after Hill who studied it in 1886 in the context of lunar dynamics [33]. For the sake of clarity, we shall write for the periodicity interval . It is easy to find a fundamental system of solutions , such thatThis is a set of two particular linearly independent solutions of the ODE (6); see [9] for further details. It is well known that, for any second-order ODE,where is the Wronskian of the fundamental solutions [34]. In light of the problem’s symmetry, is an* even* and an* odd* function. Let us introduce the* discriminant*beingHill’s equation is said to be* unstable* if all nontrivial solutions are unbounded in and* conditionally stable* if there is a nontrivial solution bounded in and* stable* if all solutions are bounded in .

By Theorem of [9], the solution is stable if and unstable when and special consideration is required for the case . Indeed, when , the characteristic multipliers and are complex conjugated and have unit modulus, whereupon the characteristic exponents are opposite; that is, . Figure 2 plots the bounding curves (dashed) and (solid) as a function of and for . Stable regions are shaded in gray. It is seen that Band Gaps (BGs) are bounded by dashed and solid curves in succession. When it is , respectively.

It is also well known [9, Section ] that the bounding curve corresponds to solutions in the formwhere and are periodic functions with period , provided that the following condition which grants the availability of two eigenvectors for the repeated eigenvalue holdsUnder the same condition, the bounding curve corresponds to solutions in the formwhere and are* semiperiodic* function with period ; that is,In Figure 3, the solution curves for the first and the second of the conditions (16) are plotted in dotted line style and they partly overlie the conditionally stable curves . However, it is observed that such conditions are satisfied only at some very special points. Indeed, we havewhence crossing is possible either whenor when . The case relates to a fully supported string for which and are both proportional toso that they vanish at the isolated points , . The first point, , corresponds to the pivotal frequency for the 0th BG. Conversely, the case corresponds to a free string, which is a nondispersive situation. Indeed, BGs collapse into single points which, according to (20), are located at . Other combinations exist which satisfy and (16), such as . In all such points two periodic (or semiperiodic) solutions exist and conditional stability is reverted to stability. However, in the general case, (16) never hold together and thus conditional stability remains. Indeed, the second of the conditions (15) is replaced bywhich is obviously unstable. By the same token, the second of the conditions (17) becomesThus, the* coexistence problem* for period is generally answered in the negative; that is, a single periodic (or semiperiodic) solution exists which is accompanied by a nonperiodic one. Besides, from a mechanical standpoint, it is observed that, in the general case, a periodic solution is not acceptable on physical grounds, as it conveys a jump discontinuity at the boundary, unlessLikewise, a semiperiodic solution is not acceptable unlessWe now prove that such conditions can always be met. To this aim we now take advantage of the problem’s symmetry and recall that a nontrivial even solution exists if and only ifrespectively, are periodic and semiperiodic, whereas an odd solution stands if and only ifagain for periodic and semiperiodic [9, Theorem ]. Now, when we have the nontrivial periodic solution and let us defineBy the periodicity of , clearly is proportional to the even part of and it is periodic. Besides,which are, respectively, (24) and the first of the conditions (26). With the choice for the denominator in (28), it is also by which we conclude that extended in periodic fashion (it is also easy to show that no slope jump discontinuity is admitted by Hill’s equation. Then, one can prove that and , where the sign is given in the periodic and semiperiodic situation, resp.). Likewise, when , we have the nontrivial semiperiodic solution and let us definewhich, in light of being semiperiodic, is again proportional to the* even* part of . It is the sum of two semiperiodic functions; is semiperiodic; besideswhich are, respectively, (25) and the second of the conditions (27). With the choice for the denominator in (30), it is and we conclude that extended in semiperiodic fashion. Through the analogous definitions of and , one gets the odd function extended in periodic and semiperiodic fashion, respectively.

The role of is illustrated comparing Figure 2 with Figure 4. It is seen that the frequency axis roughly scales with . Besides, reducing brings wider BGs which tilt and tend to cluster together. It is noticed that no symmetry exists about .

#### 4. Dispersion Relation

Imposing the BCs ((9), (10)) gives the dispersion relation:This relation can also be written as (see also [35])and it is plotted in Figure 5 for and . Equation (32) conforms to the form of () in [19].

It is easy to prove that quickly asymptotes from below for , whence we can give a simple expression for the dispersion relation:Such approximation is superposed onto the dispersion curves in Figure 8 whereupon it is seen that it is very effective already at close to , although it is unable to capture the BGs.

The relevant bounding values for each band gap (BG) are obtained considering the periodic and the semiperiodic eigenvalue problems, for which two sets of BCs need to be considered, respectively:(1)the periodicity conditions and ;(2)the semiperiodicity conditions and .The relevant eigenfrequencies are denoted by and , respectively, for the periodic and the semiperiodic eigenproblems. The eigenmode for the first periodic eigenfrequency is shown in Figure 6 at , . Likewise, the eigenmode for the first semiperiodic eigenfrequency is shown in Figure 7. Such eigenfrequency is the lower boundary of the system’s first BG.

The dispersion curve intersection with the axis is given by while the intersection with the axis is given by , where . BGs’ size is obtained by and .

#### 5. Conclusions

In this paper, the Floquet theory of periodic coefficient ODEs is applied to describe the behavior of a mechanical waveguide. A homogeneous elastic string periodically supported by an elastic Winkler foundation is considered and it is found that the governing equation is given by a second-order Hill equation. Floquet-Bloch periodic boundary conditions are enforced. The stability regions together with the dispersion relation are found in terms of the Floquet theory through the discriminant. An asymptotic approximation to the lowest cutoff frequency is given. The fundamental eigensolutions of the periodic and of the semiperiodic problem are also determined. It is remarked that the present methodology can be extended to tackle functionally graded beams [36–39] or plates [40, 41]. Applications in the realm of civil engineering are also possible [28–32, 42]. A nice extension of the present theory could be related to temperature [43, 44] or viscoelastic effects in the fiber [45–48].

#### Competing Interests

The author declares that there are no competing interests.