# Snap-Through Buckling Mechanism for Frequency-up Conversion in Piezoelectric Energy Harvesting
## 4 Results  
### Finite Element Analyses and Numerical Results of the STB-EH Device in Open-Circuit Configuration  
The proposed device is modeled using the software ABAQUS for finite element analysis (FEA), in which it is possible to account for the electromechanical coupling and the nonlinearity of the bistable mechanism. A single deformable solid part is considered in the tridimensional modeling space and the various components are extruded from bidimensional shape. A linear elastic constitutive law is assumed for the different components, the values of [PERSON]'s modulus are \(E_{\text{FR4}}=14\,\text{GPa}\), \(E_{\text{P}}=60\,\text{GPa}\) and \(E_{\text{t}}=115\,\text{GPa}\) for FR4, titanium and PZT, respectively; [PERSON]'s ratio \(\
u_{\text{FR4}}=0.16\), \(\
u_{\text{P}}=0.33\) and \(\
u_{\text{t}}=0.30\); mass density \(\rho_{\text{FR4}}=1560\,\text{kg}/\text{m}^{3}\), \(\rho_{\text{P}}=7500\,\text{kg}/\text{m}^{3}\) and \(\rho_{\text{t}}=11\,340\,\text{kg}/\text{m}^{3}\). For the sake of simplicity, an equivalent isotropic model is adopted for FR4, in view of the fact that the curved beams are thin elements almost aligned with the material direction characterized by the abovementioned uniaxial stiffness. The restraint conditions applied to the model are of perfect clamps in correspondence with the four ends of the curved beams of the bistable mechanism. The model are meshed by means of quadratic hexaedral finite elements; in particular, those employed to model the piezoelectric layers belong to the piezoelectric family, whilst the remainder of the device is modeled with 3D stress elements. In order to model the bistable mechanism, it is necessary to preliminarily measure the thickness of the curved beams. Forty thickness measurements are made by means of an optical microscope along each of the four curved portions, as shown in Figure 4. The measurements show how significantly the thickness varies along the length of the single curved beam: for instance, for the lower right-end \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**a**} \\ \hline \(h\) [mm] & 3 \\ \(t\) [mm] & 0.55 \\ \(Q\) [] & 5.45 \\ \(b\) [mm] & 1.6 \\ \(L\) [mm] & 80 \\ clamp’s width \(b_{\text{cl}}\) [mm] & 2.5 \\ clamp’s length \(l_{\text{cl}}\) [mm] & 2.5 \\ \hline \multicolumn{1}{c}{**b**} \\ \hline \(b_{\text{b}}\) [mm] & 1.5 \\ \(L_{\text{b,tot}}\) [mm] & 15 \\ \(L_{\text{b,net}}\) [mm] & 12 \\ \(t_{\text{p}}\) [mm] & 0.28 \\ \(t_{\text{s}}\) [mm] & 0.065 \\ \(t_{\text{tot}}\) [mm] & 0.625 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nominal dimensions of the bistable mechanism (a) and of the bimorph piezoelectric beam (b). Figure 3: (a) bimorph cantilever beam with two piezoelectric layers electrically connected in series and (b) equivalent circuit of the piezoelectric converter. beam, the maximum measured value is 384.23 \(\upmu\)m whilst the minimum measured value is 345.21 \(\upmu\)m. The mean values and coefficient of variations (CoV) are (refer to Figure 2): top left beam 352.88 \(\upmu\)m (CoV 2.38%); bottom left beam 365.94 \(\upmu\)m (CoV 2.65%); top right beam 365.44 \(\upmu\)m (CoV 2.49%); bottom right beam 351.02 \(\upmu\)m (CoV 2.49%). In all the cases, the real thickness is lower than the nominal thickness of about 550 \(\upmu\)m. The asymmetry arising from the thickness variation caused by the tolerances of the fabrication process is modeled in the FEA simulation tool by assigning to the four curved element, for the sake of simplicity, the abovementioned mean value of thickness, uniform along each curved beam but varying from one beam to the other. The asymmetric nature of the bistable mechanism affects its expected bistable behavior, resulting in a rotation of the central clamp when a transition from the initial to the reversed state is triggered, as shown in Figure 5. Similar results have been obtained with the execution of different analyses, e.g., by introducing some local variation in the thickness of each beam, according to the measurements.