## 3 Numerical and Experiment Results of Metamaterials
### Curved Beam with Metamaterials  
For the third example, the curved beams with the cylindrical and the conical metamaterials are considered in Figure 13. Metamaterials with periodic rod structures can be applied in a variety of engineering applications. For example, periodic rod-shaped metamaterials can stop vibrations in a specific frequency range, so they can be installed as supporting structures that prevent the transmission of vibrations and noise. However, in some cases, straight metamaterials are often not suitable for installation in tight spaces. To overcome this, it is sometimes necessary to install a curved metamaterial. Note that the cylindrical metamaterials may have some issues in installation at junctions. Thus, the conical metamaterials can be a solution as it allows some geometrical gaps among the metamaterials than the cylindrical metamaterials. To test this characteristic, Figure 13 shows the simulations of the bend bars with the cylindrical and the conical metamaterials. To our best knowledge, some relevant researches that consider wave transmission with metamaterials only investigate waves in straight bar as well as the wave transmission at curved structures and the bandgap phenomenon with the cylindrical and the curved conical metamaterials. Figure 14 shows the simulation results with the bar bended by certain angles with the metamaterials and compares the wave transmissions. The frequency response results for the curved bar with the cylindrical metamaterials are shown in Figure 14a. The frequency responses at \(0^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), and straight bar show that the bandgaps appear at around 8700 Hz with these simulation conditions. The frequency responses with the conical metamaterials are drawn in Figure 14b. As it allows some geometrical gaps among the unit cells, it is possible to install the metamaterials with a large bended angle, i.e., \(50^{\circ}\). The band gaps occur at around 17,000 Hz in the straight bar with the conical metamaterials, and the band gaps appear at around 13,000 Hz the curved bar. Figure 15a shows the displacements at certain frequencies. One of the benefits of the metamaterials in the curved structure lies in the fact that it shows the bandgaps in x and y directions. In other words, as the bending loads in x and y directions become the axial loads at the curved section of the structure, the bending loads can be stopped with the present structure. For an example, Figure 14c shows the bandgaps of the curved cylindrical metamaterials (\(\theta=30^{\circ}\)) with respect to the bending loads in x and y directions. As illustrated, due to the curved structure, it is possible to exploit the bandgaps for the bending loads. Figure 16 shows some mechanical deformations at some frequencies. Figure 13: Model of (**a**) the curved cylindrical metamaterials and (**b**) the curved conical metamaterials. Figure 17 shows examples of the curved bars with the cylindrical, while the conical metamaterials and their frequency responses are shown in Figure 18. With 0.03 m for the radius (\(r_{p}\)), the bandgaps appear between 8000 Hz to 15,000 Hz, and at around 16,000 Hz. By increasing the radius, the bandgap becomes lower. The same tendency can be observed in the case of the conical metamaterials in Figure 18b. Some wave propagations are shown in Figures 19 and 20. Figure 15: (**a**) Displacement at certain frequency of the cylindrical metamaterials, (**b**) the frequency responses (\(\theta=45^{\circ}\)). Figure 16: (**a**) Displacement at certain frequency of the conical metamaterials and (**b**) the frequency responses (\(\theta=30^{\circ}\)). Figure 17: Model of (**a**) the round cylindrical metamaterial and (**b**) the round conical metamaterial. Figure 19: (**a**) Displacement at a specific frequency of the round cylindrical metamaterials and (**b**) the frequency responses (\(r_{p}=0.03\) m, \(h_{p}=0.02\) m). Figure 18: Frequency response results of (**a**) the round cylindrical metamaterials and (**b**) the round conical metamaterials.