Multi‐parametric analysis of strongly inhomogeneous periodic waveguideswith internal cutoff frequencies

In this paper, we consider periodic waveguides in the shape of a inhomogeneous string or beam partially supported by a uniform elastic Winkler foundation. A multi‐parametric analysis is developed to take into account the presence of internal cutoff frequencies and strong contrast of the problem parameters. This leads to asymptotic conditions supporting non‐typical quasi‐static uniform or, possibly, linear microscale displacement variations over the high‐frequency domain. Macroscale governing equations are derived within the framework of the Floquet–Bloch theory as well as using a high‐frequency‐type homogenization procedure adjusted to a string with variable parameters. It is found that, for the string problem, the associated macroscale equation is the same as that applying to a string resting on a Winkler foundation. Remarkably, for the beam problem, the macroscale behavior is governed by the same equation as for a beam supported by a two‐parameter Pasternak foundation. Copyright © 2016 John Wiley & Sons, Ltd.


Introduction
Periodic structures with internal cutoff frequencies are of interest for numerous applications: As an example, we mention elastically supported periodic strings and beams [1][2][3], composite materials [4], phononic crystals [5][6][7], and vibration absorbers in fluid carrying pipes [8]. It is well renowned that a string supported by a Winkler foundation exhibits a cutoff frequency [9, §1.5.2] ! 0 D sˇ , whereˇis the Winkler foundation modulus and is the string linear mass density. As it is shown in [10], a two-phase piecewise periodic string supported by a uniform Winkler foundation cannot be treated by the conventional 'low-frequency' homogenization method [11,12]. For the latter, the sought for macroscale homogenized equation is of the same form as the original equation governing the behavior of the periodic system. Besides, a quasi-static uniform variation of the displacement field is retrieved at the microscale (see also [13,14]). On the other hand, the periodically supported string problem can be efficiently treated through a high-frequency asymptotic homogenization procedure, as established in [10,[15][16][17]. Within this approach and contrarily to the classical setup, the homogenized macroscale equation takes, as a rule, a different form than the original equation. Moreover, sinusoidal variations at the microscale are found, which correspond to the eigenforms of the unit cell. Dynamic homogenization has been the subject of a number of remarkable contributions among which we mention ( [18][19][20][21][22][23][24] and also [25][26][27]), dealing with the important case of periodic waveguides with contrast properties.
In this paper, we show that quasi-static uniform (or, possibly, linear) microscale variation over the high-frequency range is still possible for strongly inhomogeneous periodic structures with internal cutoffs. As an example, a two-phase periodic waveguide in the shape of a string or a beam supported by an elastic Winkler foundation is studied. The foundation is assumed to be periodically discontinuous. Such feature is crucial for the subsequent analysis, and to the best of our knowledge, it appears in the literature only in the shape of point supports [28].
A dimensional analysis brings up the relevant dimensionless quantities (three for the string and four for the beam), expressed in terms of relative lengths, stiffness, and densities. We develop a multi-parametric asymptotic approach assuming that two of the aforementioned parameters are small in each of the cases. The long-wave expansions of the obtained dispersion relations near the lowest cutoff frequencies are derived and the ODEs governing the macroscale behavior deduced.
Remarkably, although for the string problem, the macroscale equation takes the same form as that for a string on a Winkler foundation, the beam macroscale behavior is governed by the equation for a beam supported by a two-parameter Pasternak foundation (as opposed to a Winkler foundation, as it might be expected).
The associated quasi-static displacement fields are shown to be almost uniform at the microscale, as for a rigid body motion. Numerical testing of the asymptotic formulae for the lowest cutoff frequency and the displacement field exhibits excellent agreement.
The setup in which the string parameters vary along the unsupported region is also addressed using a two-scale approach. The derived macroscale equation reduces to the asymptotic expression previously obtained for constant parameters.

Periodically supported string
Let us consider a periodic waveguide constituted by a string in piecewise uniform tension periodically supported on a homogeneous Winkler elastic foundation ( Figure 1). The governing equation for the transverse displacement w.x, t/ is, for the supported regions S n D fx 2 . L 1 C nL, nL/g, n 2 Z, each with length L 1 , and, for the free string regions U n D fx 2 .nL, L 2 C nL/g, each with length L 2 , where i and T i are the constant linear mass density and tension of the string over the relevant regions i D 1, 2, respectively,ˇis the Winkler elastic modulus (whose dimension is force over length squared), and n 2 Z (see [9] for more details). These equations have periodic coefficients with period L D L 1 C L 2 , and they can be treated by means of the Floquet theory (e.g., [29,30]). Accordingly, we may restrict attention to the single cell S 0 [ U 0 D . L 1 , L 2 /. Let us rewrite the equations (1,2) in dimensionless form and having introduced the dimensionless positive ratios together with x i D x=L i , the dimensionless axial co-ordinates, and D t= p 1 =ˇ, the dimensionless time. We look for the harmonic motion of the system, that is,

whence Equations (3) and (4) become the linear ODEs with constant coefficients
having let the shorthand notation u 1 for u.x 1 / and u 2 for u.x 2 /. Besides, it is that let Equation (6a) clearly shows that the supported string region possesses the internal cutoff frequency D 1. The conditions expressing continuity of displacement and tension at the supported/unsupported interface x 1 D x 2 D 0 are where s D The Floquet-Bloch conditions read The general solution of Equation (6a) is where A 1 and B 1 are real constants provided > 1. The general solution of Equation (6b) is The dispersion relation reads cos q cos and it gives rise to the usual pass/block bands depicted in Figure 2. The focus of the paper is on quasi-static uniform (or, possibly, linear) displacement variations at leading order long-wave approximation (q 1). As it can be seen from (6) and (10,11), these are obtained whenever s is small and is close to the internal cutoff frequency, that is, In this case, from the dispersion relation (12), it follows that Here and later, it is assumed that Ä 1 is of order unity. Thus, Equations (13) and (14) together show that the fundamental assumption for quasi-static behavior at leading order for q 1 is that s and ı s be both small, which entails while it only demands s to be not too large, namely, s 2 s . The asymptotic behavior of the lowest cutoff frequency D in small ı s and s follows from the transcendental Equation 12 taken at q D 0, and it is given by It is remarked that the cutoff frequency is close to the internal cutoff frequency of the string in supported region, that is, 1. The related eigenforms are which show that u 1 .x 1 / and u 2 .x 2 / undertake a rigid body motion at leading order. It should be remarked that the aforementioned eigenform formulae are valid provided that s is not a large parameter. Figure 3 compares the eigenforms evaluated through a numerical procedure with their asymptotic expressions (16), for the parameter set s D 0.4, s D 0.6, and Ä 1 D 1. For this choice of parameters, ı s D 0.144 is small, and the lowest cutoff frequency numerically occurs at 2 D 0.932716, as opposed to Equation (15), which gives 2 D 0.856. Nonetheless, very good agreement is met for the eigenforms and an almost rigid body behavior found.
The leading order long-wave approximation (q 1) of the dispersion relation (12) can be written as and its accuracy is shown in Figure 4 for the usual parameter set. The corresponding macro-model ODE is a continuously supported string equation where

Two-scale procedure for a periodically supported string
Let us now consider the case when the tension along the unsupported region is given by a periodic function with period L, that is, It is observed that this condition implies either horizontal motion or a restraining device to prevent it and, for simplicity, we assume the latter. Conversely, it is further assumed that the string tension T 1 is still constant along the supported region and, therefore, a cutoff frequency can still be clearly defined. Then, governing equation for the transverse displacement w.x, t/ in the supported interval is again Equation (1), while the governing equation for the free string becomes The set of dimensionless governing Equations (6) is replaced by the following pair of ODEs, the second of which is an equation with variable coefficients, where it is understood that Ä D Ä.x 2 / and s D s .x 2 /; in addition, d ln Ä dx2 is the logarithmic derivative of Ä.x 2 /. The same set of equations governs the behavior of a string whose linear mass density is constant along the supported region and periodically variable along the unsupported one, that is, 1 D const and 2 D 2 .x 2 / with 2 .x 2 / D 2 .x 2 C L/. The conditions expressing continuity at x 1 D x 2 D 0 are where s D Let us, for the sake of definiteness, assume the following asymptotic relation between the small parameters of the previous section: where it is recalled that 0 D 0 .x 2 /. Then, Equation (21b) becomes In this section, we adopt a two-scale approach [31] setting being X D s x 2 the slow variable, i D x i , i D 1, 2. Then, with the usual understanding for the integer power of a linear operator, The conditions expressing continuity at D 0 are while periodicity yields Let us take the regular expansions and, likewise, for the frequency (e.g., [10]) Here and in the succeeding discussion, we understand D 1 ( D 2 ) when the (un)supported string is dealt with. Then, we obtain the usual succession of linear problems in the expansion terms. Indeed, at order zero in , we obtain where prime is short for the total derivative d=d 2 . The first equation admits the linear polynomial solution while the second equation is linear and first order in @ q 0 , whence its solution is having let Plugging the solutions (31,32) into the expansions (29), we obtain from the conditions (27,28) at leading order whereupon the zero-order solution is just a rigid body motion At first order in , we obtain where a 1 , b 1 , c 1 , and d 1 are (yet) undetermined functions of X and Solvability of the first-order problem [31] yields 1 D 0, and after tedious calculations, we find At second order in , we obtain whose general solution is and a 2 , b 2 , c 2 , d 2 are yet unknown functions of the slow variable. In the case of constant coefficients (i.e., for constant string tension, Ä D const, and constant linear mass density, 0 D const), Solvability of the second-order problem yields Finally, at third order in , it is found that where, for constant coefficients, In this case, we obtain from the solvability of the third-order problem which is identical to Equation (19). Indeed, setting in Equation (33) (18) is retrieved, which is associated with Equation (19). In the general case of a string with variable parameters, we also arrive at a second-order macroscale governing equation with messy expressions for its coefficients. For example, in the case of constant tension, Ä D const, and variable linear mass density, 0 D 0 .x 2 /, it is found that where

Periodically supported beam
Let us consider bending of a piecewise homogeneous beam periodically supported by a Winkler elastic foundation. Similarly to the string case earlier (Figure 1), the governing equations for the transverse displacement w.x, t/ are given by .EI/ 2 @ 4 xxxx w C 2 @ 2 tt w D 0, where i and .EI/ i are, respectively, the linear mass density and the flexural rigidity of the beam, which are constant in the relevant regions i D 1, 2, whileˇis the Winkler foundation modulus. These equations may be rewritten in dimensionless form for a single cell as follows: having introduced the dimensionless ratios together with the dimensionless axial co-ordinates x i D x=L i . Here, as before, D t= p 1 =ˇis the dimensionless time. We look for the harmonic behavior of w, that is, w.  where the problem's parameters of interest are The dispersion relation is presented in the Appendix in the form of a 8 8 determinant. A plot is given in Figure 5 for the parameter set 1 D 1,˛D 0.9, b D 0.4, and b D 0.6.
As in Section 2, the lowest cutoff frequency ( D 1, q D 0) can be expanded from the dispersion relation for ı b D b 4 b and b small, namely, provided that 1 and˛are of order unity. We emphasize that the analogous formula for the string (15) does not involve the geometric parameter˛explicitly. The associated eigenforms are