# Derivation and Validation of Bandgap Equation Using Serepentine Resonator
## 3 Numerical Validation
### Bandgap  
Based on the design described in Section 3.1, the design parameters are set as \(N=2\), \(l_{p}\)/\(b\) = 1.5, and \(l_{o}\)/\(b\) = 10. This section describes the dispersion curve analysis that was conducted for the unit structure of the parameter in the design point of Figure 5 to compare the bandgap between the theoretical implementation and simulation. The dispersion curve is calculated by combining finite element (FE)-based unit cell modeling and Floquet-Bloch boundary conditions using COMSOL. For the analysis setting, a 25 mm steel plate was used as the host structure and a serpentine resonator was attached to the host structure, and this structure was repeated infinitely.  
The dispersion curve analyzed with the above setting is shown in Figure 6. The shaded part highlighted in black is the bandgap generated based on FEM (COMSOL Multiphysics) analysis and the section enclosed in the box in red is the bandgap generated based on the theoretical implementation. The bandgap of the FEM ranges from 158 to 228 Hz and that for theoretical implementation ranges from 151 to 217 Hz. Figure 6b shows the mode shapes of the serpentine resonator unit structure. A and B are the Bloch mode shapes at the lower and upper bounds of the bandgap, respectively. A is the in-phase mode and B is the out-of-phase mode; C is the bending mode in x-axis and D is the bending mode in y-axis.  
Figure 5: Relative error map for the resonance frequency of the serpentine resonator.  
The bandgaps for all the mode shapes are different from each other. The error of the bandgap result may be based on the following two reasons. The first is the simplification of the resonator model and the second is the simplification of the bandgap estimation theory. In the former, the analysis model is simplified to reduce the cost of the analysis, which induces the error. However, the error is negligible. In the latter, the conventional bandgap estimation theory begins with the unit structure modeling in which a mass-spring resonator is combined on a thin plate. This unit structure is deduced from the calculation of mass and spring matrices through the FEM analysis, and the dispersion relation is inferred based on the Floquet-Bloch theorem. Additionally, the dispersion curve is decided in the irreducible Brillouin-zone (IBZ) to estimate the bandgap.  
However, the bandgap estimation theory proposed in this study is simply a theory based on the condition that the wave does not propagate through the wave equation. Therefore, the theory is prone to inevitable error. This can be attributed to the process of estimating the spring constant. It is assumed that the joint between the beams is ideal, but the larger the joint of the beam or the larger the thickness, the more invalid is this assumption. Since the bandgap estimation theory is related to the spring constant of the serpentine resonator, this error cannot be ignored.  
Figure 7 shows the relative error of the resonance frequency and bandgap for \(N\) fixed at two and constant \(l_{p}\) when \(l_{o}/b\) is modified. The figure indicates that as the values of the parameters \(l_{o}\) and \(l_{p}\) increase, the relative bandgap error decreases, which is similar to the phenomenon when \(N\) changes. Therefore, for the experimental validation, we have considered \(l_{p}=1.5\) and \(l_{o}/b=10\), which have the lowest relative error.  
Figure 6: (a) Dispersion curve and (b) mode shapes of the analysis model.