# 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
Consequently, the second non-symmetrical buckling mode is not completely prevented, which is detrimental for the bistability. On the other hand, bistability depends also on the ratio \(Q=h/t\): while the thickness varies randomly along the length of the bistable mechanism, the average thickness has a lower value than the expected nominal value, with an ensuing increase of \(Q\) and thus of bistability. Hence, the rotational motion of the bistable mechanism caused by the imperfections and the ensuing reduction of bistability are partially compensated by the increase of \(Q\). From the static nonlinear analysis under displacement control, the force-displacement and strain energy (\(\mathcal{E}\))-displacement curves are obtained, as shown in Figure 6. The static analysis is performed via arc-length control accounting for geometrical nonlinearities. The analysis consists of applying a displacement along the \(y\)-axis at the mid-span of the bistable mechanism, in correspondence with the central block, with a variation in the range \(0<d<d_{\text{end}}\), where \(d_{\text{end}}\approx 2h\). The reaction force is computed and the nonlinear curve is obtained. The bistable behavior of the mechanism is evidenced from the force-displacement curve: the latter, starting from the undeformed configuration and initially linear, becomes nonlinear as the stiffness of the bistable mechanism decreases. The STB force threshold \(F_{\text{top}}\) is hence reached, the stiffness becomes negative and the reaction force decays until becoming  
Figure 4: Microscope image of a portion of the manufactured curved beam. Figure 5: FEA of the device from the initial to the reversed stable configurations. (**a**) front view of the deformed shape during the transition; (**b**) contour plot of the Mises equivalent stress. null in correspondence with the unstable configuration for \(d=d_{\text{mid}}=\)4.23 mm. As the force becomes negative, equilibrium is allowed only with a force of reversed sign: the point of minimum \(F_{\text{bot}}\) is hence reached and corresponds to the the value of force which triggeres STB from the reversed to the initial configuration. The final point of the curve \(d_{\text{end}}\) indicates the reversed stable configuration. The \(\mathcal{E}\)-displacement curve is obtained from the same static nonlinear analysis as the sum of the strain energy of each finite element of the model. The curve is characterized by two points of local minimum, corresponding to the stable configurations, and a point of local maximum, corresponding to the transitional unstable configuration. As expected, the reversed configuration is less stable than the initial one, since the energy well corresponding to the reversed state (namely, the difference \(\Delta\mathcal{E}_{\text{reversed}}=\mathcal{E}_{\text{mid}}-\mathcal{E}_{ \text{end}}=1414.4\,\text{\mu J}-1216.6\,\text{\mu J}=197.8\,\text{\mu J}\)) is significantly more shallow than the one relative to the undeformed configuration (\(\Delta\mathcal{E}_{\text{undeformed}}=\mathcal{E}_{\text{mid}}-\mathcal{E}_{ d=0}=1414.4\,\text{\mu J}\)). This aspect can be explained in terms of forces: a lower magnitude of force is necessary to bring the mechanism from its reversed configuration to the initial one than vice versa. As a matter of fact, the values of forces obtained from the static analysis are \(F_{\text{top}}=630\,\text{mN}\) and \(F_{\text{bot}}=-220\,\text{mN}\), respectively. It is worth noting that the theoretical predictions, given by Equations (2) and (3), are different with respect to the numerical outcomes. As a matter of fact, the formulas are referred to the perfect case, with equal thickness for all the beams. Taking into account the overall average thickness, i.e., \(358.82\,\text{\mu m}\), and an equivalent [PERSON]'s modulus equal to 14 GPa for FR4, one finds \(F_{\text{top}}^{\text{an}}=748\,\text{mN}\) and \(F_{\text{bot}}^{\text{an}}=-374\,\text{mN}\).