Gyro-elastic beams for the vibration reduction of long flexural systems

The paper presents a model of a chiral multi-structure incorporating gyro-elastic beams. Floquet–Bloch waves in periodic chiral systems are investigated in detail, with the emphasis on localization and the formation of standing waves. It is found that gyricity leads to low-frequency standing modes and generation of stop-bands. A design of an earthquake protection system is offered here, as an interesting application of vibration isolation. Theoretical results are accompanied by numerical simulations in the time-harmonic regime.


Introduction
Gyroscopic systems are always fascinating in their dynamic response and they are used in many applications in physics, engineering and aeronautics. In this paper, we show how to employ so-called gyrobeams (see, for example, [1]) in order to create low-frequency resonators, which can influence the behaviour of Floquet-Bloch waves in multi-structures subjected to dynamic excitations, such as earthquakes.
Many engineering structures (for example, bridges, pipelines and industrial warehouses) are designed as long chains of repetitive units, connected to each other by means of different types of joints. The dynamic study of such structures can be simplified by employing Floquet-Bloch theory, which reduces the problem to the analysis

Dynamics of a gyro-elastic beam
We consider an elastic Euler-Bernoulli beam of square cross section, which is clamped at one end and is free at the other end, as sketched in figure 1. The beam is characterized by a continuous distribution of gyricity, which is independent of the motion of the body and of the mass and stiffness distribution in the beam [1]; in addition, it is assumed for simplicity that gyricity is uniform along the beam length and is independent of time. Owing to this special feature, this structural element will be henceforth referred to as gyro-elastic beam or gyrobeam.

(a) Equations of motion
In the transient regime, the governing equations describing the flexural motion of the gyrobeam without external loads and pre-stress are given by Yamanaka et al. [34] EJ ∂ 4 u(z, t) ∂z 4 + ρA where z is the direction of the beam axis, u and v are the displacement components in the x-and y-directions, respectively, t is time, E is the Young's modulus, ρ is the density, A and J are the cross-sectional area and second moment of area (identical in the two directions), respectively, and h is the gyricity constant. We note that the second moments of area in the identical, i.e. J x = J y = J, because we have considered a beam with a square cross section. For a rectangular cross section J x = J y , and the equations (2.1) change accordingly.
The quantity h has the physical dimension of N s. The practical implementation of h depends on how a gyrobeam is modelled in engineering applications. In a different work, which is intended to provide a practical design for a gyrobeam, we have developed a concept based on the analysis of a multi-scale system of elastic links connecting gyroscopic spinners. It is assumed that the moments of inertia of each gyroscopic spinner with respect to the transverse axes are negligibly small in comparison with the moment of inertia I with respect to the spinner axis. The gyricity constant h is proved to be proportional to I and to the rate of spin Ω of individual gyroscopes, and it is inversely proportional to the distance d between neighbouring gyroscopes.
In the time-harmonic regime, the displacement components are expressed as u(z, t) = U(z) e iωt and v(z, t) = V(z) e iωt , where ω is the radian frequency. In this case, equations (2.1) take the form The general solutions of (2.2) are given by and when h = 0 and consequently, in this case, the solutions U(z) and V(z) take the form and are arbitrary constants. The latter expressions are the displacement components of a classical Euler-Bernoulli beam under time-harmonic conditions. The boundary conditions for a clamped-free gyrobeam are given by where L is the length of the gyrobeam, while φ j , M j and T j (j = x, y) are the rotations, bending moments and modified shear forces, respectively, with respect to the axes x and y. We note that the bending moments have the same expressions as for a classical Euler-Bernoulli beam, while the shear forces contain a term that depends on the gyricity constant h [34].

(b) Eigenfrequency analysis
The eigenfrequencies and eigenmodes of the clamped-free gyrobeam are obtained by solving the spectral problem (2.2), (2.6). The results are shown in figure 1 for different values of the gyricity constant. The quantitiesω andĥ in the diagram represent the normalized eigenfrequency and the normalized gyricity, respectively, which are defined aŝ (2.8) Figure 1 shows the eigenfrequencies for a beam with square cross section. Forĥ = 0, the eigenfrequencies have multiplicity 2; forĥ = 0, each pair of eigenfrequencies splits into two, one smaller and one larger than the double eigenfrequency forĥ = 0. This phenomenon was already observed in [1] for a gyrobeam with a rectangular cross section and a non-uniform gyricity distribution. This will be exploited in § §4 and 5 to design an efficient system of resonators to reduce the vibrations of an elastic multi-structure. As the number of eigenfrequencies which accumulate in a prescribed low-frequency range becomes larger as the gyricity constant is increased, we can take a large value of the gyricity constant in order to have many eigenfrequencies in that low-frequency range. When the structure is subjected to a dynamic force whose Fourier spectrum has one or more frequencies falling inside the considered lowfrequency range, the gyrobeams act as resonators because their eigenfrequencies are close to the frequencies of the external force. Consequently, they start vibrating and thus divert the energy coming from the external excitation away from the main structure which, as a result, undergoes smaller vibrations.
The first three types of eigenmodes of the clamped-free gyrobeam (denoted as modes I, II and III), calculated forĥ = 1, are illustrated in the bottom part of figure 1. The two deformed shapes associated with each mode have the same number of nodes, which are the points where the displacement is zero. However, they differ in the number and positions of the points of inflection, which are defined as the points along the beam axis where the curvature changes sign and the bending moment is zero. The locations of the inflection points are identified with small crosses in the insets of figure 1. This observation was not discussed in [1].
In order to better visualize the three-dimensional deformed shapes presented in the bottom part of figure 1, we have applied to the top of the gyrobeam a harmonic force having a frequency equal to one of the eigenfrequencies of the beam for the caseĥ = 1. We have then computed the responses of the gyrobeam in the steady-state regime for the six eigenfrequencies indicated in figure 1a-f. The motion of the beam for each frequency is shown in electronic supplementary material, Video S1. There, the arrow represents the direction of the harmonic force. The trajectory of each point of the gyrobeam is a circle, since the amplitudes of u and v are identical and their phase difference is ±π/2. We point out that the aim of the video is only to show how the gyrobeam deforms at different frequencies.
It is interesting to observe that the curves √ω versus ĥ in figure 1 intersect at a number of points where the gyrobeam possesses a double eigenfrequency. The eigenmodes associated with each double eigenfrequency are different, being characterized by a different number of nodes.

Analysis of a frame with a gyro-elastic column
We study the structural frame sketched in figure 2, consisting of a continuous Euler-Bernoulli beam (A-C) and a column with distributed gyricity (B-D). Both ends of the horizontal beam and the bottom end of the column are clamped; the connection between the beam and the column is assumed to be rigid, which implies continuity of displacements, rotations, moments and forces at the junction. For this structure, we also take into account the longitudinal displacements of both structural elements. We assume that the two beams have length L = 2 l = 2 m and a square cross section of side length 0.05 m, and that they are made of steel, having Young's modulus E = 210 GPa, Poisson's ratio ν = 0.3 and density ρ = 7850 kg m −3 . The eigenfrequencies ω of the frame are plotted in figure 2 against the gyricity constant h. It is apparent that gyricity affects the dynamic behaviour of the system, making it more flexible or rigid depending on the vibration mode. The most interesting feature is that the eigenfrequencies cluster at low frequencies for large values of h, as observed for a single gyrobeam.
The eigenmodes associated with the lowest four eigenfrequencies for the cases h = 0 N s and h = 3000 N s are illustrated in the bottom part of figure 2 (modes a-d). The eigenmodes were obtained by using the finite-element software Comsol Multiphysics , while the eigenfrequencies were calculated analytically and verified with the finite-element model. It is interesting to observe how the gyricity constant modifies the deformed shapes of the frame at the different frequencies.
The motion of the gyrobeam under an external excitation is displayed in the video included in electronic supplementary material, Video S2. There, the gyricity constant is h = 3000 N s and the frame is subjected to a harmonic force applied to the junction, acting along the x-direction and having a radian frequency ω = 314.8 rad s −1 (which corresponds to the second eigenfrequency of the frame for that value of h). The video shows that the horizontal beam moves in the y-direction though the force acts in the x-direction (the u displacement of the junction point is very small because of the high axial stiffness of the horizontal beam). Moreover, it can be noticed from the top view of the frame that the gyro-elastic column rotates around the z-axis, hence its points move both in the x-and y-directions.
Finally, we observe that some lines in figure 2 are horizontal, which means that some eigenfrequencies are not affected by a variation in the gyricity constant. This occurs for the eigenmodes in which the gyro-elastic column is not deformed (see, for example, modes f in the bottom part of figure 2).

Floquet-Bloch waves in a periodic structure with gyrobeams
In this section, we consider a system made of a very large number of frames, attached to each other. One of the frames is shown in figure 3. This system can be studied as a periodic structure by imposing the quasi-periodicity (or Floquet-Bloch) conditions at the ends of a repetitive frame (or periodic cell). The quasi-periodicity conditions are expressed by An alternative is presented in figure 4: the main structure (namely, the horizontal beam and the column) is made of Euler-Bernoulli beams without gyricity, and two gyro-elastic columns are connected to it by means of a rigid beam (depicted by a grey line). Displacements, rotations, moments and forces are continuous at the junction between the main structure and the rigid beam. The gyricity constants of the two gyrobeams are equal in absolute value but have opposite sign, hence they rotate in opposite directions. The absolute value of the gyricity constants of the two gyrobeams is indicated by h * . The gyrobeams are connected to the rigid beam by hinges (drawn as empty circles), which have proved to be the best internal constraint for our purposes after a thorough study of the structure.
The dispersion curves for the cases when the entire structure has zero gyricity and when the gyrobeams have gyricity constant h * = ±50 000 N s are shown on the right of  seen that the dispersion diagram is characterized by very narrow pass-bands at low frequencies and flat dispersion curves at higher frequencies. Accordingly, we expect that transverse waves do not propagate in such a system, except for small frequency intervals. Vibrations associated with the upper dispersion curves displayed in figure 4 are mainly in the z-direction, as shown by the eigenmode in figure 5d. The frequencies of the flat dispersion curves can be estimated analytically. They coincide with the eigenfrequencies of a single gyrobeam with a clamped end and a pinned end, which were calculated by using the analytical approach described in §2. An example of an eigenmode of  figure 5, it is apparent that, at this frequency, the main structure does not vibrate, and all the energy of deformation is confined within the gyrobeams. Above each flat dispersion curve, there is a stop-band and, at higher frequencies, a small pass-band appears, where the entire structure vibrates (see figure 5, mode b). However, this pass-band is very narrow.
The dispersion curves determined for some intermediate values of h * between h * = 0 N s and h * = 50 000 N s are presented in appendix A.
The structural configuration sketched in figure 4 will be used in the next section to show how gyrobeams can be employed to reduce the vibrations of the main horizontal beam when the structure is subjected to an external excitation.

Transmission problem in a multi-structure with gyrobeams
In order to test the isolation device proposed in §4, we study the structure illustrated in the top part of figure 6, which consists of an assembly of many repetitive units, as that depicted in figure 4. At the left end of the structure we impose a harmonic displacement of amplitude d 0 and frequency f , acting in the y-direction. At the right end, we introduce PMLs (perfectly matched layers), which are used to minimize reflections of waves at the boundary and to model a semiinfinite system. The PML are designed to gradually dissipate the energy of the impinging waves [37]. The absorption effect is introduced by implementing a complex Young's modulus in the beams, given by In the formula above, η is the absorption factor or isotropic loss factor, and x PML is the x coordinate of the interface between the non-dissipating frames and the PML. The coefficients η 1 and η 2 are tuned to make the absorption coefficient increase slowly, since a high-contrast interface would generate reflected waves. The part of the structure with PML needs to be long enough to dissipate all the energy travelling through it. An alternative type of PML in a flexural system is discussed in [38]. We have built a finite-element model in Comsol Multiphysics to determine the displacement field in the structure produced by the imposed displacement. The computations have been performed in the frequency domain. As earlier, we have examined two scenarios: in the first one, the entire structure is made of classical Euler-Bernoulli beams, while in the second one gyricity is introduced in the lateral columns, which have gyricity constant ±h * (the reader is referred to the periodic unit of figure 4 for the position of the gyrobeams in each frame). In both cases, the beams connecting the gyrobeams to the main structure are rigid. Figure 6 shows the displacement amplitude of a point of the structure, indicated by P in the top part of the figure, for both scenarios. Additional computations have shown that very similar diagrams are found for the other points of the structure, provided that they are not too close to the external excitation. These results are included in electronic supplementary material, Fig. S6. The deformed shapes of the system in the two scenarios are presented in figure 7 for two different frequencies, at which waves propagate.
The displacement amplitudes plotted in figure 6 are different from zero at the frequencies within the pass-bands of the corresponding periodic structure, determined from figure 4. This shows that the forced response of the semi-infinite structure can be deduced from the behaviour of the infinite system, represented by the dispersion diagram. The very narrow pass-bands calculated for the system with gyrobeams at higher frequencies were not detected by the model in the frequency domain, though the frequency step adopted in the computations was very small ( f = 0.0001 Hz).
The most important outcome of figure 6 is that the frequency ranges where waves can propagate are reduced significantly if gyrobeams are employed. This demonstrates that gyrobeams are an efficient tool to mitigate the transverse vibrations of a structural system if the frequency of the applied force is not very low. An enhanced design, combining alternating frames with and without gyrobeams, is discussed below.
In the numerical computations, we have not considered an external excitation acting along the z-direction, because the ensuing waves would not trigger the gyricity effect in the columns. Indeed, the dispersion curves corresponding to this type of waves are not altered by changing the gyricity constant in the columns (compare the upper dispersion curves in figure 4 for h * = 0 N s and h * = 50 000 N s). On the other hand, an external excitation acting in the x-direction would generate compressional and extensional waves in the horizontal beams which, however, occur at higher frequencies due to the high longitudinal stiffness of the beams. In addition, rigid motions along the x-direction are usually prevented in practice by introducing appropriate constraints (e.g. hinges) at the abutments of the structure.
The structural responses of the system for other values of the gyricity constant are included in appendix B.

(a) Effect of damping on the dynamic response
We have also computed the response of the semi-infinite system shown in the top part of figure 6 when damping is introduced in the lateral columns of each frame (i.e. the thick lines in the periodic unit of figure 4). Damping was introduced by defining Young's modulus of all lateral columns as We have considered a uniform absorption factor equal to 5%, namely η = 0.05, which is very common in civil engineering structures [13]. The structural response of the system for different values of the frequency is plotted in figure 8 for the cases where the lateral columns are classical Euler-Bernoulli beams and gyrobeams. To simplify the comparison with figure 6, we have also shown in figure 8 the displacement amplitudes determined without damping (dotted lines). It can be noticed from the figure that the main advantage of adding damping to the structure is obtained when h * = 0, because in this case the displacement amplitudes decrease significantly.

(b) Enhanced design of a multi-structure
We have shown that a periodic system made of an infinite array of frames as that of figure 4 creates a low-frequency band-gap when h * = 0 N s and a large band-gap at higher frequencies when h * = 50 000 N s. Accordingly, we can create a very efficient filter for transverse waves if we design the structure as an alternating series of frames with and without gyricity. An example of such a structure is depicted in the top part of figure 9. It consists of an array of eight frames  without gyricity and eight frames with gyricity, and it is modelled as a semi-infinite system by inserting PML. We again apply a harmonic displacement at the left end of the structure and we compute the resulting displacement field. In figure 9, we plot the displacement amplitudes determined at four different points, indicated by P 1 -P 4 , which are located after 16 frames, namely after one set of eight frames without gyricity and one set of eight frames with gyricity. The diagram reveals that, apart from a small peak at f = 0.7 Hz, the displacement amplitudes are negligible, implying that waves cannot travel in this structure. Further computations have shown that, if the number of frames in each set is reduced, the frequency intervals where waves can propagate are increased.
The deformed shape of the structure at f = 0.7 Hz is illustrated in the top part of figure 9. It is apparent that the displacement amplitudes are sizable only in the first set of frames with h * = 0 N s, where they are given as the superposition of the incident waves and the waves reflected at the interface between the two sets of eight frames. In the second set of frames, where h * = 50 000 N s, the displacement amplitudes are much smaller, because f = 0.7 Hz is within the stopband of the structure with zero gyricity.

Conclusion
A highly efficient wave filtering system, whose design includes gyro-elastic beam resonators, has been proposed in this paper. It has been demonstrated that gyrobeams provide a new tuning mechanism, different from the known high-contrast resonators or conventional tuned mass dampers.
The analysis of the dynamic response of a single gyrobeam has shown that a large number of eigenfrequencies tend to cluster within a low-frequency interval as the gyricity constant is increased. In a periodic system made of an infinite array of elastic gyroscopic frames, gyroelastic beams contribute to very interesting new dispersion properties of waves and, in particular, standing modes. We have proposed a novel design, in which two systems of gyrobeams rotating with the same speed and in opposite directions are attached to the main structure, such as a bridge. In this case, very narrow pass-bands corresponding to transverse modes have been observed from the dispersion analysis. Accordingly, when determining the response of the same system under an external excitation, we have found that the frequency intervals where waves propagate are reduced significantly.
As discussed in §5a, dissipative structures are very important for the reduction of vibrations initiated by seismic loads. We have also demonstrated in this paper that the role of gyrobeams is primarily to create low-frequency 'energy sinks', in which waves generated by external excitations are channelled. As a consequence, energy is diverted away from the main structure, which undergoes smaller displacements and smaller stresses. Additional dampers may then be attached to the gyrobeams to reduce their vibrations.
Gyrobeams offer a practical alternative to methods currently used to reduce the effects caused by seismic waves. This work opens a new perspective in chiral metamaterial design and in a wide range of applications in earthquake wave filtering.