# Derivation and Validation of Bandgap Equation Using Serepentine Resonator
## 2 Theory
### Serpentine Resonator  
Figure 2 shows the serpentine resonator that consists of a combination of a rigid mass and serpentine beams, which correspond to the mass and spring of the system, respectively. The serpentine resonator can be divided into three parts. Parts 1 and 3 are in contact with the mass and plate, respectively. Part 2 connects parts 1 and 3 and it consists of \(N\) repeated segments; additionally, the serpentine resonator in Figure 2 is repeated twice. The symbols representing the length of each beam are shown in Figure 2 and Table 1. To simplify the calculation, we have assumed that \(l_{ini}=l_{fin}=l_{p}\). Figure 3 shows the schematic of each part and their segment beams. The serpentine beams of each part are composed of horizontal and vertical beams. Part 1 consists of three horizontal beams and one and a half vertical beams; part 2 consists of two horizontal beams and two vertical beams, and part 3 consists of one horizontal beam, and half of a vertical beam. The section enclosed in a box with dotted lines in Figure 3 is the free-body diagram with forces and moments of the horizontal and vertical beams. The spring constant of the serpentine beam is derived using the principle of virtual work. The red arrow represents the virtual force exerted on the beam. The formula for total energy in the bending beam is derived by integrating the strain energy density for each beam as follows:  
\[U=\sum_{i=1}^{m}\left[\int_{0}^{l_{p}}\left(\frac{M^{2}}{2 EI_{zp}}\mathrm{d}x \right)\right]_{i}+\sum_{j=1}^{n}\left[\int_{0}^{l_{o}}\left(\frac{M^{2}}{2 EI_{ zo}}\mathrm{d}y\right)\right]_{j}, \tag{15}\]  
where \(M\) is the moment of the beam, \(E\) is the Young's modulus, and \(I_{zp}\) and \(I_{zo}\) are the moments of inertia of the horizontal and vertical beams, respectively. The first and second terms represent the energies of the horizontal and vertical beams, respectively. The moments of inertia of the horizontal and vertical beams are given as:  
\[I_{zp}=\frac{1}{12}tb^{3}, \tag{16a}\] \[I_{zo}=\frac{1}{12}tl_{p}^{3}, \tag{16b}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Definition** \\ \hline \(L_{x}\) & Length of the plate in x direction \\ \(L_{z}\) & Length of the plate in z direction \\ \(T\) & Thickness of the plate \\ \(L_{mm}\) & Length of the rigid mass in x direction \\ \(L_{ym}\) & Length of the rigid mass in y direction \\ \(t\) & Thickness of the serpentine beams \\ \(l_{imi}\) & Length of the initial beam \\ \(l_{fin}\) & Length of the final beam \\ \(\dot{l}_{o}\) & Length of the beam orthogonal to the y-axis (vertical) \\ \(l_{p}\) & Length of the beam parallel to the y-axis (horizontal) \\ \(\dot{b}\) & Width of the serpentine beams \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols representing the length of the serpentine resonator. Figure 2: Concept of serpentine resonator. Using the principle of virtual work, the displacement of the serpentine beams in the x-direction can be derived as:  
\[\delta_{x}=\frac{\partial U}{\partial F_{x}}=\sum_{i=1}^{m}\biggl{[}\int_{0}^{l_{p }}\biggl{(}\frac{M}{EI_{zp}}\frac{\partial M}{\partial F_{x}}\mathrm{d}x\biggr{)} \biggr{]}_{i}+\sum_{j=1}^{n}\biggl{[}\int_{0}^{l_{o}}\biggl{(}\frac{M}{EI_{zo}} \frac{\partial M}{\partial F_{x}}\mathrm{d}y\biggr{)}\biggr{]}_{j} \tag{17}\]  
Using [PERSON]'s law to derive the spring constant, the displacement of the total serpentine beams can be expressed as:  
\[k_{x}=\frac{F_{x}}{\delta_{x}}, \tag{18}\]  
\[\delta_{x,tot} = \delta_{1,1}+\delta_{1,2}+N(\delta_{2,1}+\delta_{2,2}+\delta_{2, 3}+\delta_{2,4})+(\delta_{3,1}+\delta_{3,2}+\delta_{3,3}+\delta_{3,4}+\delta_ {3,5}), \tag{19}\] \[= \frac{(N+1)F_{x}^{2}}{6 EI_{zo}}+\frac{(N+1)F_{x}^{2}l_{p}^{2}}{2 EI_{zp}},\]  
where \(N\) is the number of times each part is repeated and \(F_{x}\) is the virtual force.