One-way interfacial waves in a flexural plate with chiral double resonators

In this paper, we demonstrate a new approach to control flexural elastic waves in a structured chiral plate. The main focus is on creating one-way interfacial wave propagation at a given frequency by employing double resonators in a doubly periodic flexural system. The resonators consist of two beams attached to gyroscopic spinners, which act to couple flexural and rotational deformations, hence inducing chirality in the system. We show that this elastic structure supports one-way flexural waves, localized at an interface separating two sub-domains with gyroscopes spinning in opposite directions, but with otherwise identical properties. We demonstrate that a special feature of double resonators is in the directional control of wave propagation by varying the value of the gyricity, while keeping the frequency of the external time-harmonic excitation fixed. Conversely, for the same value of gyricity, the direction of wave propagation can be reversed by tuning the frequency of the external excitation. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.


Introduction
Bloch-Floquet waves in periodic systems are generally dispersive and may show interesting behaviour, such as dynamic anisotropy and localisation in different frequency regimes. In addition, waves in lattices having defects of a semi-infinite extent (such as cracks, for example) bring additional analytical and numerical challenges.
An elegant analytical approach to study cracks in lattices was developed by L.I. Slepyan [1], where the model is reduced to the analysis of a functional equation (or a system of functional equations) of the Wiener-Hopf type, whose kernel incorporates information about dispersion properties of the Bloch-Floquet waves in the periodic system. Quasi-static and dynamic problems for cracks in different types of media, both linear and non-linear, were investigated in depth by Slepyan in [2,1,3] (see also [4,5,6]). In particular, the specified direction of crack propagation brings a natural directional preference into the problem, and waves emanated from the crack tip may have different physical features ahead of the crack and behind the crack front.
Directional preference and dynamic anisotropy for waves in lattice systems were studied in [7,8]. Chiral systems bring new features complementing dynamic anisotropy with the special property of one-way wave propagation. In the present paper, we analyse directional preference phenomena for the case of a doubly-periodic system of specially designed resonators, made of elastic beams and gyroscopic spinners, connected to a Kirchhoff elastic plate. The chiral nature of gyroscopic resonators enables us to break the symmetry and to form "one-way waveforms" along interfaces. This unusual dynamic response of the structured elastic system is connected to the novel dispersion properties of Bloch-Floquet waves supported by the corresponding doubly periodic chiral systems, as discussed in the main text of the paper.
The concept of gyrobeams, developed in [9,10,11,12], highlights interesting features of elongated flexural elastic systems, which include active chirality and a special type of coupling between transverse displacements associated with flexural deformations. We note that active chirality is qualitatively different from geometrically chiral passive systems, also discussed in the literature (see, for instance, [13,14,15,16,17,18]).
Recent papers [19,20] have demonstrated that gyrobeams can be understood in the context of homogenisation approximations for elongated elastic solids containing multiple spinners connected along the axis of the solid. Although the classical models of gyroscopic conservative systems (see, for example, [21]) are commonly used in engineering practice, additional novel spectral properties have been studied for elastic systems including gyroscopic spinners, in the context of the theory of elastic waves in chiral metamaterials [22,23,24,25].
The present paper addresses new features in the dynamic response of gyroscopic systems with resonators that incorporate more than one spinner. The spectral problem for a beam with several spinners can be solved analytically in the linear approximation framework [26]. This provides a rich range of options to control vibrations of such systems by changing the gyricities of the spinners. The term gyricity was introduced in [19,20] to characterise the gyroscopic action of individual spinners. If an elastic flexural plate is attached to a doubly-periodic array of such resonators, an interface can be created by changing the sign of the gyricity in a sub-domain of the plate. Appropriate choice of the forcing frequency will lead to the generation of a one-way waveform, as shown in Fig. 1. Furthermore, the direction of wave propagation can be reversed by the appropriate tuning of the gyricity of the spinners. Changing the direction of wave propagation is not possible in a plate containing single-spinner resonators, investigated in [27]. The new work presented here delivers results which are substantially different from elastic periodic systems without chiral resonators (see, for example, [28,29]). In particular, the latter systems do not break time-reversal symmetry, which is the fundamental feature to generate one-way wave propagation.
One-way unidirectional waves immune to backscattering were firstly observed in photonic crystals [30,31,32,33,34], motivated by the analogy with the edge states in topological insulators linked to the quantum Hall effect. In elastic and mechanical systems, one-way edge and interfacial waves have been generated in plates [35,36,37,38], elastic lattices with spinners [39,40,41], granular media [42] and systems with coupled pendula [43]. We note that in most of these works, formation of oneway waveforms has been connected with "breaking the Dirac cones" of the dispersion surfaces via perturbation of the geometry and physical properties of the periodic system. In the present case of a doubly-periodic structure with double-spinner resonators, the mechanism of creating waveforms in the "time-reversal" mode does not require the presence of broken Dirac cones. The details of the analysis are given in Section 4.
Localisation phenomena in periodic systems have been extensively investigated in the literature. In particular, distributed rotational inertia has been exploited to create localised waveforms in arrays of Rayleigh beams [44,45,46]. Highly localised waves have been realised in grids of axially and flexurally deformable beams by introducing the effect of pre-stress [47]. Trapping, enhanced transmission and control of flexural waves in thin elastic plates incorporating periodic arrays of different types of resonators have been studied in [48,29,49,50,51]. In the present work, the chiral properties of the resonators attached to the plate allow the breakage of time-reversal symmetry of the system and, consequently, lead to the formation of one-way interfacial waves if the domain is split in two regions with different signs of gyricity.
The structure of the paper is as follows. Section 2 includes the formal description of the model and the special example of a spectral problem for an elastic beam connected to two gyroscopic spinners. The eigenvalues are obtained in the closed analytical form, and their dependence on the gyricity of the spinners is discussed in detail. In Section 3, we analyse Bloch-Floquet waves in a Kirchhoff plate connected to a doubly-periodic system of double-spinner resonators. Interfacial waves possessing one-way preferential directionality are constructed and analysed in Section 4. Concluding remarks and discussion are included in Section 5.

A plate incorporating a chiral flexural double resonator
We study flexural vibrations in a Kirchhoff plate connected to a chiral flexural resonator. As shown in Fig. 2, the middle plane of the plate lies in the -plane and the axis of the resonator is parallel to the -axis. The resonator consists of two Euler-Bernoulli beams, connected to two gyroscopic spinners at the points B and C. Accordingly, such a resonator will be referred to as a "double resonator" in the rest of the paper, to distinguish it from the "single resonator" investigated in [27], represented by a single beam and a gyroscopic spinner at the tip of the beam. As in [20,26], continuity of displacements and flexural rotations is prescribed at the junctions between the beams and the spinners (points B and C in Fig. 2); in addition, the spinning motions of the gyroscopic spinners are not transmitted to the beams, using the type of connection described in [20]. We also assume that at the junction A the beam axis always remains orthogonal to the plate.
The equation of motion of the Kirchhoff plate in the time-harmonic regime is given where Δ 2 is the biharmonic operator, p = p ( , ) is the amplitude of the plate's transverse displacement in the -direction, p = p ℎ 2 / 1/4 is the frequency-like parameter for the plate, is the angular frequency, p is the mass density of the plate, ℎ is the thickness and = p ℎ 3 /[12(1 − 2 p )] is the flexural stiffness of the plate, p and p being the Young's modulus and Poisson's ratio of the plate, respectively. The governing equations of the Euler-Bernoulli beams in the time-harmonic regime are expressed by with = 1, 2. In (2) the derivatives are taken with respect to and are the amplitudes of the transverse and longitudinal displacements of the beams.
parameter for the beams, b and b are the Young's modulus and mass density of the beams, and b and b are the beams' cross-sectional area and second moment of inertia, respectively. Here, we assume that the two beams have the same geometrical and material properties. The lengths of the beams are fixed. The junction conditions defining the connection between the plate and the beam were derived in [27] and employ the notion of "logarithmic rotational spring". The derivation of these junction conditions is presented in the Appendix. After imposing continuity of displacements and rotations between the beam and the plate, the beam can be studied on its own, substituting the junction conditions at the connection between the beam and the plate with boundary conditions for the beam. For a circular plate of radius , clamped at its boundary and connected at its centre to a beam with circular cross-section of radius = , the boundary conditions at point A are given by where In (3) we have taken into account that in a Kirchhoff plate the in-plane displacement components are assumed to be zero. When = / 1, the conditions (4) and (5) respectively. Since the effect of the boundary becomes negligible when → 0, the conditions (7) and (8) can also be used when the beam is attached to any point of the plate and when the plate has a non-circular shape and other boundary conditions, provided that is understood as an "equivalent" radius. In particular, the junction conditions (7) and (8) can be employed for the elementary cell with quasi-periodicity boundary conditions, analysed in Section 3. Each gyroscopic spinner is characterised by mass , moment of inertia 1 about its axis of revolution and moments of inertia 0 about the two axes perpendicular to the axis of revolution and passing through the base of the spinner. We assume that the lengths of the gyroscopic spinners are negligible in comparison with the lengths of the beams. We indicate by Ω (1) and Ω (2) the gyricities of the two spinners, which are generally different. As discussed in [19,20], the gyricity Ω of a spinner is given by where and are the precession and spin rates, respectively. According to (9), the gyricity remains constant throughout the motion. At the junction point B, we prescribe continuity of displacements and flexural rotations: According to the formulation developed in [19,20], the effect of the gyroscopic spinners is replaced by the following effective junction conditions, representing the balance of bending moments, shear forces and axial forces: where At the point C, we impose the following effective boundary conditions: We remark that the matrix C ( ) ( = 1, 2) couples the equations of angular momentum balance when Ω ( ) ≠ 0.

An auxiliary spectral problem for the double resonator
It is important to note that the boundary conditions (4) and (5) include and log( ). This reflects on the singular perturbation associated with the junction of the one-dimensional beam and the two-dimensional flexural plate. A simplified approach without the singular perturbation would lead to the boundary conditions corresponding to a fixed hinge. This approach is still useful for predicting one-way wave propagation discussed in Section 4 and it will be considered in this section.
The plate provides translational and rotational stiffness to the double resonator, thus behaving as a flexural foundation. In the limit case when → 0 in (4) or in (7) and b is constant, the bending moments at the junction between the plate and the double resonator tend to zero. Accordingly, the conditions (4) and (7) take the form In this situation, the eigenfrequencies and eigenfunctions of the double resonator are determined by solving the differential equations (2) with the boundary conditions (3), (14) and (5) (or (8)) at A, the junction conditions (10) and (11) at B, and the boundary conditions (13) at C. We note that the amplitudes of the transverse displacements b and b and of the longitudinal displacement b are decoupled. In the following, we focus attention on the flexural vibrations of the double resonator, associated with b and b . In order to obtain a closed form solution for the eigenfrequencies, we assume that the mass of the beams is negligible in comparison with the mass of the gyroscopic spinners. Consequently, the transverse displacement amplitudes b and b are cubic functions in (see (2) when b = 0). When Ω (1) = Ω (2) = Ω, the double resonator has fourteen non-zero eigenfrequencies, seven of which are positive while the other seven are negative and with the same absolute values. The two remaining eigenfrequencies are zero. The relations between 7 the gyricity Ω and the eigenfrequency can be written explicitly as The functions Ω ( ) ( = 1, .., 4) are plotted in Fig. 3b. For the choice of parameters detailed in the caption of Fig. 3, the curves intersect at four points on the horizontal axis Ω = 0, which represent the double eigenfrequencies for the case when the spinners neither spin nor precess. The curves also intersect when Ω ≠ 0; one of these common points is shown in the inset of Fig. 3b.
When Ω ≠ 0, the eigenfunctions of the double resonator show coupling between flexural and rotational motions. The direction of rotation depends on the curve where the eigenfrequency is computed. As an example, we study the dynamic behaviour of the resonator in the neighbourhood of the intersection displayed in the inset of Material. It can be seen that at the eigenfrequencies P and S the resonator rotates in the counter-clockwise direction in the -plane, while at the eigenfrequencies Q and R it rotates in the clockwise direction. Fig. 3b shows that the same eigenfrequency can be obtained with two different values of gyricity, and the corresponding modes of vibration are characterised by opposite directions of rotation. This result will be exploited in Section 4 to change the direction of one-way wave propagation, while keeping the frequency of the time-harmonic excitation constant. Fig. 3c illustrates how gyricity varies with angular frequency when the beams possess inertia. In particular, it is assumed that their density is b = 2700 kg/m 3 . Fig. 3c was obtained with a finite element model built in Comsol Multiphysics (version 5.3). The main effect of adding distributed inertia to the beams is to move the curves to lower values of . However, this effect is not significant in the range of frequencies displayed in Fig. 3c, for the choice of parameters considered here. Accordingly, the simplified

Doubly-periodic array of double resonators attached to a plate
In this section, we study propagation of Bloch-Floquet flexural waves in an infinite plate connected to a doubly-periodic array of double resonators, described in Section 2. The double resonators are arranged in a square pattern, where the side length of the elementary cell is denoted by (see Fig. 4). The position of each resonator is defined by the vector r = ( , ), where , ∈ Z.
In the elementary cell x = ( , ) ∈ (− /2, /2) × (− /2, /2), the governing equation for the plate's displacement amplitude is given by where ( , ), ( , ) and ( , ) represent the axial force and bending moments transmitted by the beam, and (x) is the Dirac delta function. Bloch-Floquet quasiperiodicity conditions relate the displacements and rotations at the junctions between the plate and the double resonators: where k = ( , ) T is the wave vector. Non-trivial solutions of (16) and (17) lead to the dispersion relation for the periodic system, which shows how the angular frequency depends on the wave vector k.
The analytical approach formulated above has been described in detail in [27]. It has also been demonstrated in [27] that the boundary conditions (4) and (5) produce equivalent results to the numerical computations performed in Comsol Multiphysics, for the case of massless plate and beams and of a single spinner per beam.
Here, the dispersion diagrams are determined numerically with a bespoke finite element model developed in Comsol Multiphysics. The governing equation of the plate is given by (1) and the rod is modelled as an Euler-Bernoulli beam. In the finite element model the singular perturbation based on the small parameter is not used, and the connection between the two-dimensional plate and the one-dimensional beam incorporates continuity of displacements and rotations. The gyroscopic spinners are modelled implementing the effective junction and boundary conditions (10)- (13). At the boundaries of the elementary cell, we apply Bloch-Floquet conditions: We also note that the number of dispersion surfaces is infinite since the system possesses distributed inertia.
When Ω = 0, namely when the spinners neither spin nor precess, no stop-bands are observed at low frequencies (see Fig. 5a). Conversely, when Ω ≠ 0 narrow stop-bands are created in the low-frequency regime, as shown in Figs. 5b and 5c. We point out that the value Ω = 268.8 rad/s, used in Fig. 5c, has been chosen so that the lower limit for Ω = 200 rad/s, at the frequencies indicated by "a" and "b" respectively in Fig. 5b. These frequencies correspond to the limits of the pass-bands. In Video 6 the rotation of the double resonator about the -axis is counter-clockwise, while in Video 7 it is clockwise. This observation is of fundamental importance for the generation of one-way waves, as described in Section 4. We also point out that the eigenfunctions in Videos 6 and 7 are similar, respectively, to those shown in Videos 1 and 2, which were determined for a double resonator hinged at the bottom end and made of massless beams. Videos 8 and 9 show the eigenfunctions of the periodic cell at the frequencies denoted by "c" and "d" in Fig. 5c, corresponding to Ω = 268.8 rad/s. These videos highlight that the rotation of the double resonator at the lower (higher) frequency is clockwise (counter-clockwise), in contrast to that observed for Ω = 200 rad/s. The eigenfunctions in Videos 8 and 9 have shapes similar to those in Videos 3 and 4, respectively. Fig. 6 shows the eigenfrequencies of the periodic system at point M (where k = ( / , / ) T ), calculated for different values of Ω. The direction of rotation of the double resonator remains constant as we move along a single curve. Similar features have been observed with reference to Fig. 3. The horizontal dashed line indicates the frequency defining the lower limit of the second stop-band for Ω = 200 rad/s, which coincides with that corresponding to Ω = 268.8 rad/s. However, these frequencies lie on two different lines, that is why the directions of rotation in Videos 6 and 8 are opposed though the frequency is identical.
The dispersion analysis performed in this section will be useful in predicting the frequencies of external harmonic excitations at which one-way wave propagation will occur, as discussed in the following section.

One-way interfacial waves
Gyricity introduces preferential directionality into the system. In the periodic structure studied in Section 3, the eigenfunctions calculated at the lower and upper limits of a stop-band generally show a different direction of rotation of the double resonators (see, for instance, the eigenfunctions calculated at the points "a" and "b" in Fig. 5b and illustrated in Videos 6 and 7 of the Supplementary Material). Accordingly, gyricity can be exploited to force waves to propagate in one direction, determined by the direction of rotation of the double resonators. When the direction of rotation is changed (refer, for example, to the vibration modes associated with points "c" and "d" in Fig. 5c and presented in Videos 8 and 9 of the Supplementary Material), we expect waves to travel in the opposite direction.
In this section, we consider an infinite structure made of square cells, characterised by the same geometrical and material parameters taken for the periodic cell analysed in Section 3. In the numerical simulations, the computational domain consist of a plate connected to a 50 × 50 array of chiral double resonators. A time-harmonic force of amplitude 1 N and parallel to the -plane is applied to one resonator, in particular to the upper gyroscopic spinner. We note that the direction of application of the force is not important. This force generates bending moments, which are transmitted to the plate. Due to the gyroscopic effect, the force produces two bending moments, in both the perpendicular and parallel directions with respect to the direction of the force. PML (Perfectly Matched Layers) are inserted at the boundaries of the structure. They are designed as plate elements with damping, whose parameters are tuned in order to minimise reflections of waves from the boundaries (as in [27]).
We start by studying the system shown in Fig. 7a. The domain is divided into two regions, characterised by equal and opposite values of gyricity, as indicated by the circular arrows. In this case, the interface between the two regions is horizontal and straight (see the dot-dashed line in the figure). The force is applied to the upper gyroscopic spinner of the double resonator in the cell highlighted in grey, just below the interface. PML are set at the boundaries of the computational domain.
Using a finite element model built in Comsol Multiphysics, we have determined the response of the chiral structure to the time-harmonic force. In Fig. 7b we show the displacement amplitude p in the plate when the gyricity is Ω = 200 rad/s and when the angular frequency of the time-harmonic force is = 26.67 rad/s (top part) and = 30.85 rad/s (bottom part). The frequency = 26.67 rad/s falls within the second stop-band of the dispersion diagram for the corresponding periodic system (see Fig.  5b), above point "a". The frequency = 30.85 rad/s is instead located inside the third stop-band, above point "b". It is apparent that, in both cases, a one-way interfacial wave is produced by the excitation. In Fig. 1 the gyricity was the same (Ω = 200 rad/s) and the angular frequency of the external excitation was = 5.03 rad/s; this frequency lies in the first stop-band of the dispersion diagram of the corresponding periodic system. Figs. 1 and 7b demonstrate that the direction of wave propagation can be changed by varying the frequency of the external force, while keeping the gyricity the same. Fig. 7c presents the contour diagram of the plate's displacement amplitude p for the same frequencies and for a different value of gyricity, namely Ω = 268.8 rad/s. The dispersion diagram for the corresponding periodic system is shown in Fig. 5c. The frequency = 26.67 rad/s lies in the second stop-band above point "c", while = 30.85 rad/s falls within the third stop-band above point "d". We note that the frequencies = 26.67 rad/s and = 30.85 rad/s are located in the second and third stop-bands of both dispersion diagrams in Figs. 5b and 5c. This is the reason that one-way interfacial waves are generated for both values of Ω. However, the direction of propagation changes with the gyricity, because the direction of rotation of the double resonators is different for Ω = 200 rad/s and Ω = 268.8 rad/s, as discussed in Section 3 (see also Fig. 6). Figs. 5b and 5c demonstrate that the direction of one-way wave propagation can be reversed if the gyricity is changed while the frequency of the external force is kept constant. This may be very important for practical applications.
For a similar structure, an interface has been introduced containing corners, as shown in Fig. 8a. Figs. 8b and 8c present the fields of transverse displacement amplitude in the plate when the angular frequency of the external excitation is = 26.67 rad/s and the gyricity of the spinners is either Ω = 200 rad/s (part b) or Ω = 268.8 rad/s (part c). The directions of wave propagation for the two values of gyricity are identical to those shown in Figs. 7b and 7c, respectively. When = 30.85 rad/s, the direction of propagation is the opposite to that when = 26.67 rad/s; the results are not given here for brevity. The simulations of Figs. 8b and 8c demonstrate the robustness of interfacial waves. It is also demonstrated how efficiently interfacial waves may be guided around corners with minimal backscattering.

Conclusions
In this paper, we have shown that a plate with a doubly-periodic array of gyroscopic resonators offers the possibility of creating one-way interfacial waveforms. The dynamic response to an external force can be predicted from the dispersion analysis of the corresponding infinite periodic system, in particular from the determination of the stop-bands and the analysis of the eigenfunctions at the edges of the pass-bands.
The simulations presented in Section 4 demonstrate the versatility of the proposed flexural system, where one-way waves can be generated for any frequency of the external excitation by tuning the gyricity and where the direction of propagation can be chosen ad libitum, depending on the needs.
The mechanism of creating one-way waveforms does not require breaking Dirac cones. It has also been demonstrated that there is minimal backscattering from the corners along the interfaces.  Although the analysis has been presented for resonators containing two spinners of identical gyricities, the work allows for a significant extension to the case of multiple spinners of different gyricities. We assume that the rigid inclusion is subjected to a time-harmonic displacement of amplitude 0 in the -direction and a time-harmonic rotation of amplitude Ψ around the -axis. Hence, the conditions at the boundary of the inclusion are expressed by p = = 0 − Ψ cos( ) , For the sake of simplicity, we assume zero rotation around the -axis. Of course, the relationship between bending moment and rotation around the -axis is similar to that derived in the following for the -axis. The displacement in the plate, satisfying the governing equation (1), has the form p ( , ) = 0 0 ( p ) + 0 0 ( p ) + 0 0 ( p ) + 0 0 ( p ) + 1 1 ( p ) + 1 1 ( p ) + 1 1 ( p ) + 1 1 ( p ) cos( ) ,  respectively. Calling = / , we finally derive the conditions (4) and (5), with the signs chosen in accordance with the adopted convention for the bending moments and axial force in the beam.