# 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 reduction of limit loads is a consequence of the sensitivity to imperfection of the considered structure [36]: the asymmetry of beams' thickness that characterizes the FE model entails a premature snap with respect to the theoretical framework. For model A, the total mass, sum of the mass of the bimorph beam, and the mass of the bistable mechanism is \(m_{\text{tot,A}}\approx 340\,\text{mg}\) and thus the value of input acceleration triggering STB, respectively, from the initial to the reversed state and vice versa are \(a_{\text{top,A}}=F_{\text{top}}/m_{\text{tot,A}}=189\,\text{g}\) and \(|a_{\text{bot,A}}|=|F_{\text{bot}}|/m_{\text{tot,A}}=66\,\text{g}\). Vibrational piezoelectric energy harvesters are generally employed to scavenge energy from ambient vibrations, characterized by accelerations acting at low frequencies and magnitude. It is possible to decrease the magnitude of the acceleration necessary for STB by simply increasing the mass of the device: as a matter of fact, for model B, the total mass is \(m_{\text{tot,B}}=m_{\text{tot,A}}+m_{\text{tip}}\approx 1100\,\text{mg}\). Given the increase of mass, in the simplifying hypothesis of invariance of snap loads for the models, a proportional reduction of the acceleration is expected. Furthermore, an increase of mass can be useful if the EH device has to scavenge energy under the effect of impacts or, in case the increase of mass is sufficiently high, if the device is tilted. In addition to the static analysis, a modal analysis is carried out, by means of a frequency--linear perturbation analysis around the initial configuration of the bistable mechanism. The Lanczos eigenvalue solver is adopted and the obtained modal frequencies are listed in Table 2 for model A and in Table 3 for model B. Clearly, in the second case, the modal frequencies are relevantly lower due to the presence of the lead tip mass on the free end of the bimorph beam. The examination of the first  
Figure 6: Force–displacement curve (**a**) and strain energy–displacement curve (**b**) obtained from the nonlinear static analysis. mode shape indicates a significant out-of-plane motion of the bistable mechanism, together with the oscillation of the cantilever beam. Subsequently, a dynamic implicit analysis accounting for geometrical nonlinearities and the electro-mechanical coupling is performed. In the FE model, the electromechanical coupling is accounted for by means of the piezoelectric coefficient \(d_{31}=242\times 10^{-12}\,\mathrm{C/N}\). In addition, the following electrical boundary conditions are imposed: the bottom surface of the lower piezoelectric layer (lower electrode) is set to zero voltage, whereas the voltage on the top surface of the upper layer (upper electrode) is let free to vary in time but is set constant over the whole surface. The damping is introduced via the Rayleigh model, tuned in order to have an arbitrarily chosen damping ratio \(\xi=0.005\) for the first eigenmode. The test is here described with the help of the force-displacement curve shown in Figure 6a. First, it is possible to verify that the structure does not show an abrupt transition from the initial to the reversed configuration. Then, the snap in the opposite sense is exploited. To that purpose, in the first phase, the bistable mechanism is brought to its reversed stable configuration (\(d\approx d_{\mathrm{end}}\)). From this configuration, in the second phase, an opposite displacement is applied at mid-span beyond the point of inversion of the force sign (\(d<d_{\mathrm{mid}}\)), so that the bistable mechanism, in the third stage, automatically and impulsively snaps back to its initial configuration. On the FE model, the first two phases are modeled by means of nonlinear static analyses with assigned displacement. As \(d<d_{\mathrm{mid}}\), the displacement restraint is released and a dynamic implicit analysis starts, with the mechanism switching back to its initial configuration, inducing oscillations of the bimorph beam. The dynamic step runs over 200 ms, with a time increment equal to 0.1 ms.