## 3 Numerical and Experiment Results of Metamaterials
### Straight Beam with Cylindrical and Conical Metamaterials  
In this subsection, the combination of the cylindrical and the conical metamaterials is considered. Figures 10 and 11 show the frequency response functions and the dispersion relation of the combined metamaterials. By combining the two hetero-metamaterials, the frequency ranges of the band gaps are observed between the frequency ranges of the cylindrical structures and the conical structures. The studies imply that the combinations of different unit cells can be used for the wave attenuation of the longitudinal stress wave. Figure 4: (**a**) Cylindrical metamaterials, (**b**) the displacement at certain and (**c**) the frequency responses (\(r_{p}=0.03\) m, \(h_{p}=0.02\) m, \(s=0.02\) m, \(n=10\)). Figure 5: (**a**) Floquet periodic condition of unit cell and (**b**) the dispersion relation for the unit cell in cylindrical metamaterials (\(r_{p}=0.03\) m, \(h_{p}=0.02\) m, \(s=0.02\) m, \(n=10\)). Figure 6: Frequency response analysis results for each parameter of straight beam with repeated cylinders of the metamaterials for the longitudinal wave: (**a**) \(h_{p}=0.02\) m, \(n=10\), \(\mathrm{d}=0.02\) m (**b**) \(r_{p}=0.03\) m, \(n=10\), \(\mathrm{d}=0.02\) m (**c**) \(r_{p}=0.03\) m, \(h_{p}=0.02\) m \(n=10\) (**d**) \(r_{p}=0.03\) m, \(h_{p}=0.02\) m \(d=0.02\) m.  
Figure 7: (**a**) Model of conical metamaterials, (**b**) displacement at certain frequency and (**c**) the frequency responses (\(r_{p}=0.03\) m, \(h_{p}=0.02\) m, \(s=0.02\) m, \(n=10\)). Figure 8: (**a**) Floquet periodic condition of unit cell and (**b**) dispersion relation for a unit cell in conical metamaterials (\(r_{p}=0.03\) m, \(h_{p}=0.02\) m, \(s=0.02\) m, \(n=10\)). Figure 9: Frequency response analysis results for each parameter of straight beam with repeated conical of the metamaterials for the longitudinal wave: (**a**) \(h_{p}=0.02\) m, \(n=10\), \(\text{d}=0.02\) m (**b**) \(r_{p}=0.03\) m, \(n=10\), \(\text{d}=0.02\) m (**c**) \(r_{p}=0.03\) m, \(h_{p}=0.02\) m \(n=10\) (**d**) \(r_{p}=0.03\) m, \(h_{p}=0.02\) m \(d=0.02\) m.  
Figure 11: (**a**) Floquet periodic condition of unit cell and (**b**) dispersion relation for a unit cell in combining cylindrical and conical metamaterials (\(r_{p}=0.03\) m, \(h_{p}=0.02\) m, \(s=0.02\) m, \(n=5\)). Figure 10: (**a**) Model of combining cylindrical and conical metamaterials, (**b**) the displacement at certain frequency and (**c**) the frequency responses (\(r_{p}=0.03\) m, \(h_{p}=0.02\) m, \(s=0.02\) m, \(n=5\)). Figure 12 shows the frequency response of the cylindrical and the combined conical structures. The analysis about four parameters are performed. The results of the change in the radius of the two shapes are shown in the Figure 12a. As the radius increases, the band gap occurs in the lower frequency range. With 0.03 m for the radius, the band gap occurs at around 8700 Hz. The frequency range of the band gap is observed from 6000 Hz to 19,000 Hz with 0.04 m for the radius, and from 4500 Hz to 13,500 Hz with 0.05 m for the radius. With the 0.03 for the radius of two shapes, the band gap occurs at about 14,000 Hz or more. The frequency range of the band gap is observed at 8300-20,000 Hz with 0.04 m for the radius, and at 4800-14,000 Hz with 0.05 m for the radius. It can be seen that this frequency range corresponds to the intermediate level of the radius in the cylindrical and the conical structures. With 0.03 m for the radius of the cylinder, the band gap is 8700 Hz or more, and with 0.03 m for the radius of the cone, the band gap observed 17,300 Hz or more. When using combined structures, the band gap occurs at 14,000 Hz or more. Figure 12b shows the frequency response functions with the different heights of the combined metamaterials. With 0.03 m for the heights of the cylinder and the cone, the band gap is formed at 12,600 Hz or more. The frequency range of the band gap is observed at 11,300 Hz or more with 0.04 m for the height, and at 12,600 Hz or more with 0.05 m for the height of combined metamaterials.