# Platonic localisation: one ring to bind them

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## Abstract

In this paper, we present an asymptotic model describing localised flexural vibrations along a structured ring containing point masses or spring–mass resonators in an elastic plate. The values for the required masses and stiffnesses of resonators are derived in a closed analytical form. It is shown that spring–mass resonators can be tuned to produce a “negative inertia” input, which is used to enhance localisation of waveforms around the structured ring. Theoretical findings are accompanied by numerical simulations, which show exponentially localised and leaky modes for different frequency regimes.

## Keywords

Flexural waves Localisation Asymptotics Homogenisation## 1 Introduction

The interest in wave propagation and localised waveforms in structured media has led to analysis of defects in crystalline solids, dynamic Green’s kernels in lattices, transmission resonances for grating stacks, and novel designs of structured waveguides. While motivated by practical applications in acoustics and optics, these problems bring new challenges in elasticity: in particular, structured flexural elastic plates incorporate effects attributed to the interaction between evanescent and propagating modes. In turn, this may lead to localised flexural vibration modes within finite structured clusters. Lattice dynamics and analysis of point defects were extensively studied in [1, 2]. Analysis of propagating modes initiated by point defects was published in [3], and stop band Green’s functions representing exponentially localised waveforms were studied in [4]. A scalar problem governed by the Helmholtz operator for a circular cluster of small rigid disc-shaped inclusions was analysed in [5]. Localisation and waveguiding of flexural waves by structured semi-infinite grating stacks were studied in [6, 7], and transmission resonances for flexural waves in an elastic plate containing gratings of rigid pins were analysed in [8].

The recent papers [9, 10] present an asymptotic analysis of whispering Bloch modes around a circular array of inclusions for acoustic or electromagnetic waves. The results have been obtained for the case of the Helmholtz operator and the Neumann boundary conditions set at the boundaries of the scatterers. The thesis [11] has addressed flexural wave localisation and waveguiding by clusters of point masses inserted in an elastic plate. Clusters of resonators including springs and masses, embedded into an elastic plate, were studied in [6] in the context of wave scattering and trapping by semi-infinite structured gratings.

The asymptotic model developed in the present paper reduces the formulation to a spectral problem for an integral operator. The solution is obtained in a closed analytical form. Special attention is given to localised waveforms in a structured ring associated with spring–mass resonators of negative inertia, a condition we show to be necessary for their occurrence in the finite cluster represented by the structured ring.

## 2 Green’s function and vibration of an inertial cluster

*h*and

*D*being the mass density, the thickness and the flexural rigidity of the plate, respectively. The explicit form for \(G(r;\beta )\) is given in terms of Bessel functions (see, for example, [13, 14])

*N*points, each with mass

*m*, shown in Fig. 2, we consider a time harmonic flexural displacement \(u(\varvec{x}) e^{-i \omega t}\), where \(u(\varvec{x})\) satisfies the following equation governed by the bi-harmonic operator

*R*is the radius of the ring. The solution \(u(\varvec{x})\) satisfies the following relation

*N*becomes large, we propose a homogenisation model and find its solution in a simple analytical form. In particular, it is shown that for the case of a “negative inertia” of the structured ring, real roots \(\beta \) are identified for the homogenisation model. It is also shown that the effect of negative inertia is achieved by using spring–mass resonators instead of point masses.

## 3 Homogenisation approximation

*R*centred at the origin, we use the notations

*k*is an integer. The system of Eq. (6) can be rewritten in the form:

### 3.1 Spectral problem for an integral operator

*M*is constant and taking the limit as \(N\rightarrow \infty \), we deduce the homogenised spectral problem

*u*in the form \(u(R \mathrm {e}^{\mathrm {i}\theta }) = U \mathrm {e}^{\mathrm {i} k \theta }\), where

*k*is an integer and \(0 \le \theta < 2 \pi \). Then the equation takes the form

### 3.2 Normalisation

### 3.3 Evaluation of the integral in (16)

*n*is an integer, \({J}_{n}\) and \({Y}_{n}\) are Bessel’s functions, while \({I}_{n}\) and \({K}_{n}\) are modified Bessel’s functions.

Derivation of the above formula is based on the evaluation of integrals related to addition theorems for cylinder functions discussed in Chapter XI of Watson [15] and also linked to the classical work by Gegenbauer [16].

## 4 Frequencies of the localised modes

- First, the normalised frequency parameter is determined by solving the equation corresponding to the imaginary term in (17):$$\begin{aligned} \mathrm {J}_{n}\left( \frac{{\tilde{\beta }}}{2}\right) = 0 \, . \end{aligned}$$(22)
- Then, given \({\tilde{\beta }}\), the ring radius
*R*and the total mass*M*are derived from the equation$$\begin{aligned} \frac{{\tilde{\beta }}^{2}}{4}\mathrm {I}_{n}\left( \frac{{\tilde{\beta }}}{2}\right) \mathrm {K}_{n}\left( \frac{{\tilde{\beta }}}{2}\right) + 4 \pi \rho h \frac{R^{2}}{M} = 0 \, . \end{aligned}$$(23)

### 4.1 Asymptotic simplification of (23) for large \(\tilde{\beta }\)

### 4.2 Numerical illustrations

For positive values of *m*, equation (23) does not have real solutions, but complex roots can be readily identified. Hence, leaky flexural waveforms may be observed for the case of a ring of point masses *m* embedded into the elastic plate. Such waves were analysed in detail in [11].

We have constructed illustrative examples of leaky flexural waveforms, as shown in Fig. 4, which are close to the eigenmodes of a finite elastic plate containing a ring of point masses. The values of the physical and geometrical quantities used in the simulations are detailed in the caption of the figure.

## 5 Resonators with negative inertia

*c*and mass

*m*, as shown in Fig. 5. The dynamic response of gratings consisting of spring–mass resonators has been analysed in [6]. Compared to the case of point masses, for the case of spring–mass resonators it has been shown that in the equations of motion the inertia term \(m \omega ^{2}\) should be replaced by \(c {{m}} \omega ^{2} / (c - {{m}} \omega ^{2})\). Consequently, Eq. (8) becomes

*R*, \({\mathscr {C}}\) and \({\mathscr {M}}\) are chosen so that

*M*. Namely,

*M*and the value of the spectral parameter \({\tilde{\beta }}\), Fig. 7 shows \({\mathscr {C}}\) versus \({\mathscr {M}}\) according to the formula (33).

### 5.1 Exponentially localised waveforms

Exponentially localised waveform solutions in a ring of spring–mass resonators are shown in Fig. 6. In these computations, we have considered negative inertia, as discussed above. Strong localisation is observed in this case, as compared to Fig. 4.

The spring–mass resonators are tuned according to (33) to choose the appropriate mass \({{\mathscr {M}}}\) and the stiffness \({{\mathscr {C}}}\) to achieve the required negative inertia *M*. For the computations of Fig. 6, the range of admissible parameters \({{\mathscr {M}}}\) and \({{\mathscr {C}}}\) is shown in Fig. 7, where the formula (33) has been implemented numerically.

## 6 Infinite grating of masses

*m*embedded in an infinite elastic plate, as shown in Fig. 8. These are often referred to as Rayleigh–Bloch waves [13]. Although it is tempting to think of this grating as the limit case of a circular ring as \(R \rightarrow \infty \), we show that the trapped waves observed here cannot be obtained as the limit of the localised waveforms studied in Sect. 3.

*d*denotes the distance between the masses. We note that \(\left| \varvec{a}^{(k)}-\varvec{a}^{(p)}\right| = \left| k-p\right| d\). We seek the solution

*u*in the form \(u\left( \varvec{a}^{(k)}\right) = U \mathrm {e}^{\mathrm {i} \alpha k d}\), where \(\alpha \) is the Bloch parameter. Then, Eq. (34) becomes

*N*has no meaning for the infinite array, but is instead the limit as \(d \rightarrow 0\) and \(m/d = O(1)\).

An example of a trapped flexural waveform for the infinite grating in the elastic plate is shown in Fig. 9, where quasi-periodic boundary conditions have been imposed on the vertical sides of the square domain. In this example, the wavenumber is chosen to be \(\alpha = 2\) 1/m, which corresponds to the numerical value of the frequency \(f = 5.3\) Hz. We note that this value of the frequency is close to the analytically predicted value, that is \(f = 5.5\) Hz.

We point out that Eqs. (36) and (37) do not have real and positive solutions \(\alpha , \omega (\alpha )\) such that \(\omega '(\alpha )=0\), i.e. the homogenisation approximation does not capture localised standing waves supported by the infinite grating of point masses in the elastic plate. This statement also holds true if point masses are replaced by spring–mass resonators. We also note that localised standing waves can be observed when the wavelength is comparable with *d*, which is outside the framework of the homogenisation approximation. This is described in Sect. 6 of [13] and illustrated in Fig. 5a of the same paper.

## 7 Discussion and concluding remarks

We have presented a homogenisation model, which leads to closed-form solutions that correspond to exponentially localised waveforms in structured plates. The cases considered here include flexural plates containing structured rings as well as infinite gratings.

Compared to the leaky waves, studied in detail in [11], we have obtained exponentially localised ring-shaped waveforms by identifying the regime of negative inertia, which can be achieved by implementing specially tuned spring–mass resonators, as discussed in Sect. 5. It has also been derived that a non-unique choice of resonator parameters [as shown in formula (33)] can be used to achieve the same value of the effective negative inertia.

The work has interesting extensions to the design of energy absorbers in flexural plates and construction of flexural elastic waveguides.

## Notes

### Acknowledgements

A.B.M., G.C. and R.V.C. would like to thank the EPSRC (UK) for its support through Programme Grant No. EP/L024926/1. R.C.M. gratefully acknowledges support for travel to Liverpool from the same grant.

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