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 T, periodically supported on a Winkler elastic foundation (Figure 1). The governing equation for the transverse displacement w(x,t) in the absence of loading is periodic with period L:(1)-T∂xxw+ρA∂ttw+1-Hβw=0,x∈-L1,L-L1,where ρA is the mass linear density of the string, assumed constant, β is the Winkler subgrade modulus (with physical dimension of stress), and H is Heaviside step function; that is,(2)Hx=1,x>0,0,x<0.This equation may be rewritten in dimensionless form(3)-κ2w′′+w¨+1-Hw=0,x1∈-α,1-α,having introduced the dimensionless positive ratios:(4)κ=TβL2,α=L1L≤1,together with x1=x/L∈[0,1], the dimensionless axial coordinate, and τ=t/ρA/β, the dimensionless time. Here, prime denotes differentiation with respect to x1 and dot differentiation with respect to τ and w(x1,τ) and H(x1) are assumed. We shall look for the harmonic behavior of w; that is, w(x1,τ)=ux1exp(iΩτ), whence (3) becomes the constant coefficient ODE for u(x1):(5)-κ2u′′-Ω2u+1-Hu=0,x1∈-α,1-α.We shift the unit period to range in the interval (-α/2,1-α/2) in order to consider an even/odd problem; namely,(6)-κ2u′′-Ω2u+1-Hx1-α2u=0,x1∈-α2,1-α2.The general harmonic solution of (3) in the supported region x1∈(-α/2,α/2) is given by(7)usx1=A1exp1-Ω2κx1+A2exp-1-Ω2κx1,while the solution in the free region x1∈(α/2,1-α/2) is(8)ufx1=B1sinΩκx1+B2cosΩκx1,where A1, A2 and B1, B2 are integration constants to be determined through the boundary conditions (BCs). The BCs for the system require continuity at the supported/unsupported transition:(9)us0=uf0,us′0=uf′0,where prime stands for x1 differentiation. Furthermore, consideration of the Floquet-Bloch waves lends the periodicity conditions(10)us-α2=uf1-α2ρ,us′-α2=uf′1-α2ρ,where ρ≠0 is the characteristic multiplier. For a second-order ODE, there are two nonnecessarily distinct characteristic multipliers, which are denoted by ρ1 and ρ2. Besides, let ρk=exp(mk) with k∈{1,2}; then mk=iq is the characteristic exponent.

A homogeneous string periodically supported by an elastic foundation.

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 (0,a) for the periodicity interval (-α/2,1-α/2). It is easy to find a fundamental system of solutions ϕ1(x1), ϕ2(x1) such that(11)ϕ10=1,ϕ1′0=0,ϕ20=0,ϕ2′0=1.This 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,(12)Wϕ1x1,ϕ2x1=detϕ1x1ϕ2x1ϕ1′x1ϕ2′x1=ϕ1x1ϕ2x1′-ϕ2x1ϕ1′x1≡1,where W is the Wronskian of the fundamental solutions [34]. In light of the problem’s symmetry, ϕ1 is an even and ϕ2 an odd function. Let us introduce the discriminant(13)D=ϕ1a+ϕ2′a=ρ1+ρ2=2fΩsin1-αΩκsinαΩ2-1κ+2cos1-αΩκcosαΩ2-1κ,being(14)fΩ=1-2Ω22ΩΩ2-1=-1-1/2Ω21-1/Ω2.Hill’s equation is said to be unstable if all nontrivial solutions are unbounded in R and conditionally stable if there is a nontrivial solution bounded in R and stable if all solutions are bounded in R.

By Theorem 1.3.1 of [9], the solution is stable if D<2 and unstable when D>2 and special consideration is required for the case D=2. Indeed, when D<2, the characteristic multipliers ρ1 and ρ2 are complex conjugated and have unit modulus, whereupon the characteristic exponents are opposite; that is, q1=-q2. Figure 2 plots the bounding curves D=2 (dashed) and D=-2 (solid) as a function of Ω and α for κ=0.5. Stable regions are shaded in gray. It is seen that Band Gaps (BGs) are bounded by dashed and solid curves in succession. When D=±2 it is ρ1=ρ2=±1, respectively.

Stability regions (shaded in gray) for Hill’s equation (5) at κ=0.5. Solid and dashed curves represent the level lines D=±2, respectively.

It is also well known [9, Section 1.2] that the bounding curve D=2 corresponds to solutions in the form(15)ψ1x1=p1x1,ψ2x1=p2x1,where p1(x1) and p2(x1) are periodic functions with period 1, provided that the following condition which grants the availability of two eigenvectors for the repeated eigenvalue ρ1,2=1 holds(16)ϕ2a=ϕ1′a=0.Under the same condition, the bounding curve D=-2 corresponds to solutions in the form(17)ψ1x1=P1x1,ψ2x1=P2x1,where P1(x1) and P2(x1) are semiperiodic function with period 1; that is,(18)Pix1+1=-Pix1,i=1,2.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 D=±2. However, it is observed that such conditions are satisfied only at some very special points. Indeed, we have(19)1-Ω2ϕ2a-κ2ϕ1′a=κΩsin1-αΩκcoshα1-Ω2κ,whence crossing is possible either when(20)Ω=nκπ1-α,n∈0,1,2,3,…or when α=1. The case α=1 relates to a fully supported string for which ϕ1′(a) and ϕ2(a) are both proportional to(21)Ω2-1sinhΩ2-1κ,so that they vanish at the isolated points Ω=1+n2κ2π2, n∈{0,1,2,…}. The first point, Ω=1, corresponds to the pivotal frequency for the 0th BG. Conversely, the case α=0 corresponds to a free string, which is a nondispersive situation. Indeed, BGs collapse into single points which, according to (20), are located at Ω=nκπ. Other (Ω,α,κ) combinations exist which satisfy D=2 and (16), such as (3.22313,0.512649,0.5). 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 by(22)ψ2x1=x1p1x1+p2x1,which is obviously unstable. By the same token, the second of the conditions (17) becomes(23)ψ2x1=x1P1x1+P2x1.Thus, the coexistence problem for period 1 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 p(x) is not acceptable on physical grounds, as it conveys a jump discontinuity at the boundary, unless(24)p0=pa.Likewise, a semiperiodic solution P(x) is not acceptable unless(25)P0=-Pa.We 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 if(26)ϕ1′a2=0or ϕ1a2=0,respectively, are periodic and semiperiodic, whereas an odd solution stands if and only if(27)ϕ2′a2=0or ϕ2a2=0,again for periodic and semiperiodic [9, Theorem 1.3.4]. Now, when D=2 we have the nontrivial periodic solution p1(x1) and let us define(28)p1+x1=p1x1+p1a-x1p10+p1a.By the periodicity of p1, clearly p1+ is proportional to the even part of p1 and it is periodic. Besides,(29)p1+0=p1+a,p1+′a2=0,which are, respectively, (24) and the first of the conditions (26). With the choice for the denominator in (28), it is also p1+(0)=1 by which we conclude that p1+≡ϕ1 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 ϕ1′(0)=ϕ1′(a)=0 and ϕ2′(0)=±ϕ2′(a), where the sign is given in the periodic and semiperiodic situation, resp.). Likewise, when D=-2, we have the nontrivial semiperiodic solution P1(x1) and let us define(30)P1-x1=P1x1-P1a-x1P10-P1a,which, in light of P1 being semiperiodic, is again proportional to the even part of P1. It is the sum of two semiperiodic functions; P1-(x) is semiperiodic; besides(31)P1-0=-P1-a,P1-a2=0,which are, respectively, (25) and the second of the conditions (27). With the choice for the denominator in (30), it is P1-(0)=1 and we conclude that P1-≡ϕ1 extended in semiperiodic fashion. Through the analogous definitions of p1- and P1+, one gets the odd function ϕ2 extended in periodic and semiperiodic fashion, respectively.

Periodic and semiperiodic solutions for Hill’s equation (5) at κ=0.5. The red and the blue dotted curves represent the solution of the first and the second of the conditions (16), respectively.

The role of κ is illustrated comparing Figure 2 with Figure 4. It is seen that the frequency axis roughly scales with κ. Besides, reducing κ brings wider BGs which tilt and tend to cluster together. It is noticed that no symmetry exists about α=0.5.

Stability regions (shaded in gray) for Hill’s equation (5) at κ=0.25.

4. Dispersion Relation

Imposing the BCs ((9), (10)) gives the dispersion relation:(32)12D-cosq=0.This relation can also be written as (see also [35])(33)121+fΩcos1-αΩ-αΩ2-1κ+121-fΩcos1-αΩ+αΩ2-1κ-cosq=0and it is plotted in Figure 5 for κ=0.5 and α=0.25. Equation (32) conforms to the form of (4) in [19].

Dispersion relation for a string on a periodic support (κ=0.5, α=0.25).

It is easy to prove that f(Ω) quickly asymptotes -1 from below for Ω>1, whence we can give a simple expression for the dispersion relation:(34)1-αΩ+αΩ2-1κ=q+kπ,k∈0,1,2,….Such approximation is superposed onto the dispersion curves in Figure 8 whereupon it is seen that it is very effective already at Ω close to 1, 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:

the periodicity conditions u(0)=u(a) and u′(0)=u′(a);

the semiperiodicity conditions u(0)=-u(a) and u′(0)=-u′(a).

The relevant eigenfrequencies are denoted by λ0,λ1,… and μ0,μ1,…, respectively, for the periodic and the semiperiodic eigenproblems. The eigenmode for the first periodic eigenfrequency λ0 is shown in Figure 6 at =0.5, α=0.25. Likewise, the eigenmode for the first semiperiodic eigenfrequency μ0 is shown in Figure 7. Such eigenfrequency is the lower boundary of the system’s first BG.

The first eigenmode, relative to the eigenfrequency λ0=0.488445 at κ=0.5, α=0.25, and q=0, is a periodic function.

The eigenmode at the lower end of the first BG, relative to the eigenfrequency μ0=1.57858 at κ=0.5, α=0.25, and q=π, is a semiperiodic function.

Dispersion relation (solid) superposed onto its first approximation (34) (dashed).

The dispersion curve intersection with the axis q=0 is given by λm while the intersection with the axis q=π is given by μm, where m∈N. BGs’ size is obtained by μ2m-μ2m+1 and λ2m+1-λ2m+2.

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.

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