*Collaborators: G. Carta, M. Garau, I.S. Jones, A.B. Movchan and N.V. Movchan*

Chirality is the property of an object whereby it cannot be superimposed on to its mirror image through translation or rotation. This property can be found both in nature and various branches of science. It is currently being utilised to create novel metamaterials capable of controlling wave propagation. This research is concerned with the analysis, design and application of both continuous and discrete novel chiral systems. Research topics addresses in this direction that are discussed below include:

**Chiral elastic structures and the DASER****Gyrobeams and resonator systems and***Chiral elastic waveguides*

*Chiral elastic structures and the DASER*

*Chiral elastic structures and the DASER*

Here we use fundamental mechanical elements to design new discrete waveguides possessing chirality (for example see Fig. 1(a)). The chiral nature of the system is induced by attaching collections of gyroscopic spinners to structure as shown in Fig. 1(b).

**Fig 1: **(a) A triangular chiral elastic lattice composed of periodically placed masses (at the junctions) connected by elastic links. The junctions are attached to gyroscopic spinners shown in (b).

The first model of this kind appeared in [1] where a triangular lattice attached to a system of spinners was studied. This system is capable of producing fascinating and counter-intuitive wave propagation phenomena, such as wave polarisation and dynamic anisotropy [2]. They have also led to designs of materials capable of being used as effective cloaking or shielding devices.

Recently, in [3], it has been shown that by appropriately arranging the spinners attached to the lattice one can observe exponentially localised Gaussian beams as in Fig. 2. The path taken by these waveforms can be controlled and the frequencies at which they occur may be tuned through adjusting the array of spinners which is important for various applications. This phenomenon has been termed DASER (Dynamic Amplification via a Structured Elastic Reticulum).

**Fig 2. The DASER: ***The response of a triangular elastic lattice attached to a non-uniform array of spinners. The chiral lattices shown are partitioned into several regions (whose boundaries are indicated by the black dashed lines). In each region, the spinners are arranged so that the waves excited by the harmonic displacement (whose position is indicated by the black arrow) follow a predefined path in the form of a Gaussian beam. In *(a)*, we show an example where the chiral lattice can be designed to direct the Gaussian beam from its point of initiation at A to any other point in the lattice, where in this case the target point is B. In *(b)*, we demonstrate the design of a lattice where the Gaussian beams are forced to follow a closed path. Several more examples of the DASER can be found in *[3].

**References**

[1] M. Brun, I.S. Jones and A.B. Movchan, (2012): *Vortex-type elastic structured media and dynamic shielding*, Proc. R. Soc. A 468 3027-3046.

[2] G. Carta, M. Brun, A.B. Movchan, N.V. Movchan and I.S. Jones (2014), *Dispersion properties of vortex-type monatomic lattices*, Int. J. Solids Struct. 51 2213–2225.

[3] G. Carta, I.S. Jones, N.V. Movchan, A.B. Movchan and M.J. Nieves, (2017): *“Deflecting elastic prism” and unidirectional localisation for waves in chiral elastic systems*, Sci. Rep. 7, 26.

*Gyrobeams and resonator systems*

*Gyrobeams and resonator systems*

The *gyrobeam* is a theoretical element, which is interpreted as a beam with a *continuous distribution of stored angular momentum*. This mechanical element was originally introduced in [1] for the purposes of controlling the shape and attitude of spacecraft.

Here, we analyse the dynamic properties of these elements (for instance in Fig 1(a)) and their novel wave guiding properties. The stored angular momentum possessed by a gyrobeam is represented by what is called the gyricity parameter, which couples the transverse motions of the beam to produce rotational motion. In certain configurations, if the gyricity of the gyrobeams is high, its resonant frequencies cluster in the low frequency regime (see Fig 1(b).). This makes the gyrobeam an ideal tool for designing *resonator support systems* for civil engineering structures such as bridges and buildings. Such resonator systems act as energy sinks which redirect hazardous vibrations away from the main civil engineering structure, which has potential applications in earthquake protection [2].

**Fig 1.** (a) *A gyrobeam with a clamped base and a tip which is free to move. Vibration frequencies of the gyrobeam and their dependency on the gyricity are presented in* (b). *For a given gyricity one can determine the vibration frequencies where the gyrobeam undergoes a periodic motion. As an example, when the gyricity of the beam is equal to unity, the frequencies for such vibrations are indicated by the markers *a-f* (as intersections of the black dashed lines with the curves). In* (c), *we show the corresponding deformations of the gyrobeam for the case when its tip is subjected to an oscillating force with a frequency equal to the one of the frequencies *a-f* in* (b). *The motion of the gyrobeam in the cases shown in* (c) *are shown in the video below, where it can be observed the gyrobeam gyrates and undergoes a circular motion in each case. For further details see *[2]*.*

**References**

[1] G.M.T. D’Eleuterio and P.C. Hughes, (1984).* Dynamics of gyroelastic continua*. J. Appl. Mech. 51, 415-422

[2] G. Carta, I.S. Jones, N.V. Movchan, A.B. Movchan and M.J. Nieves (2017): *Gyro-elastic beams for the vibration reduction of long flexural systems*, Proc. R. Soc. A 473 20170136.

*Chiral elastic waveguides*

*Chiral elastic waveguides*

As mentioned above, the gyrobeam is a theoretical mechanical element and how to constructs this element is a very interesting question. Here, we design novel elastic waveguides with the potential of reproducing similar behaviour to the gyrobeam.

An example of such a design involves connecting a Euler-Bernoulli beam to gyroscopes. Fig 1. shows the a beam, with a clamped base, and its tip is connected to a special hinge where the beam tip is attached to a gyroscopic spinner. The displacement of the tip is fixed but the gyroscopic spinner induces a moment on the beam tip that couples the transverse displacements in the beam.

**Fig. 1. ***A beam with a clamped base at* A *and its tip at* B *is supported by a special type of hinge where the tip is also connected to a gyroscopic spinner. This special hinge is referred to as a gyro-hinge.*

If the nutation of the gyroscope is sufficiently small, an approximate model for the system can be used to determined the influence of the gyroscope on the beam. In fact, the response of the beam can be controlled through a parameter we call the *gyricity* of the system, which is the linear combination of the gyroscopes initial precession and nutation. Fig. 2(a) and (b) show the effect of gyricity on the vibration frequencies for the chiral system. The presence of the gyricity (through the gyroscope) causes the beam to produce a rotational motion indicated in Fig 2(c). An example showing the motion of the system for one of these vibration frequencies is given in the video below. A complete investigation of the behaviour of this system has appeared in [1].

**Fig. 2.*** In* (a) *the variation of the vibration frequencies of the elastic chiral system composed of a beam with a clamped base and with a tip connected to a gyro-hinge. The computations are carried out for the case when the gyroscopic spinner embedded in the system is cylindrical. The influence of the gyricity (which is the sum of the cylindrical spinner’s initial precession and nutation rates) is shown. The vibration frequencies of the system when the gyricity is zero are those of a beam with a clamped base and a simply supported tip (represented by the dashed lines). The presence of the gyicity causes these frequencies to split, as indicated in* (a) *by the formation of branches emanating from the vertical axis. As gyricity increases, the elastic chiral system stiffens and the branches approach asymptotes represented by dot-dashed lines that correspond to represent the frequencies of a beam with clamped ends. In* (c) *the vibration modes of the beam corresponding to the frequencies highlighted in* (a) *by the red and blue crosses are presented. The presence of the gyroscope causes the beam to not only deform but also rotate in a clockwise or anticlockwise direction indicated by the circular arrows at different vibration frequencies. As an example we show the motion of the beam for case *d, *shown in* (c),* in the video below.*

**Video:** On the left, the motion of the elastic chiral system is shown. The system is composed of a beam that is clamped at the base and its tip is connected to a gyro-hinge, having a cylindrical gyroscopic spinner. A plan view of the system is also provided on the right. The motion of the system corresponds to the vibration mode “d” discussed in Fig. 2. During the periodic motion of the system, all points perform an anticlockwise rotation.

**References**

[1] G. Carta., M.J. Nieves, I.S. Jones, N.V. Movchan and A.B. Movchan, (2018): *Elastic chiral waveguides with gyrohinges*, Quarterly Journal of Mechanics and Applied Mathematics, doi: 10.1093/qjmam/hby001.