# Analytical Electromechanical Modeling of Nanoscale Flexoelectric Energy Harvesting
## 4 Numerical Results and Discussion
### Voltage FRFs  
The voltage FRFs in Equation (33) can be re-written as  
\[\frac{V(t)}{-\omega^{2}W_{0}\mathrm{e}^{\mathrm{i}\omega t}}=-\frac{\sum \limits_{r=1}^{3}\frac{\mathrm{j}\omega K_{r}\lambda_{r}}{\left(\omega_{r}^{2 }-\omega^{2}\right)+2\mathrm{j}\omega\xi_{r}\omega_{r}}}{\left(\frac{1}{R}+ \frac{\kappa_{3}BL}{h}\mathrm{j}\omega\right)+\sum\limits_{r=1}^{3}\frac{ \mathrm{j}\omega K_{r}^{2}}{\left(\omega_{r}^{2}-\omega^{2}\right)+2\mathrm{j} \omega\xi_{r}\omega_{r}}}, \tag{35}\]where \(\lambda_{r}=-m\int_{0}^{L}\Phi_{r}(x_{1})\mathrm{d}x-M_{\mathrm{t}}\Phi_{r}(L)\). In subsequent simulations, the first three modes were calculated and discussed. Figure 2 shows the multi-mode voltage FFRs of the flexoelectric beam with a thickness of 3 \(\upmu\)m for arbitrary frequency excitations. By increasing the loading resistance from a small value (\(R=1\) M\(\Omega\)) to open circuit (\(R=1\) G\(\Omega\), as an approximation for the open circuit condition) conditions, the voltage output increases monotonically at all frequencies. From the response around the first mode in Figure 2b, we found the resonance frequency (\(\sim\)5770 Hz) was almost identical for all loading resistances. This indicates that the resonance frequencies were insensitive to the variation of the loading resistances in the flexoelectric beam. It should be noted that, based on the present linear model, the electromechanical responses would perform well only in the narrow frequency zone close to the natural frequency. In practice, methods for performance enhancement have been developed by the use of nonlinearity [45; 46; 47; 48] and multimodal techniques [49; 50; 51; 52]. Figure 2: Voltage frequency response functions (FRFs) of the flexoelectric beam with 3 \(\upmu\)m thickness over the frequency range of (**a**) the first three modes; (**b**) around the first mode. When the flexoelectric beam was shrunk proportionally to 0.3 \(\upmu\)m in thickness, the voltage FRFs are plotted in Figure 3. It is clear to observe that the resonance frequency increases about 10 times compared to Figure 2 for the beam with thickness 3 \(\upmu\)m, as expected by Equation (25). The enlarged view of the voltage FRFs near the first mode is shown in Figure 3b. It is interesting to see that the resonance frequency corresponding to the peak voltage FRFs shifts from the short circuit resonance frequency (\(f_{1}^{\text{sc}}=57,703\) Hz) to the open circuit resonance frequency (\(f_{1}^{\text{sc}}=61,809\) Hz) as the loading resistance increases. The short circuit resonance frequencies can be obtained by solving Equations (23) and (25), and the open circuit resonance frequencies can be predicted using the previous result [53]. This result indicates that the resonance frequency of a small-scale flexoelectric energy harvesting device is dependent on the external loading resistance. Indeed, a resonance frequency shift was observed previously in piezoelectric and flexoelectric energy harvesting devices [37; 41; 42; 54]. Figure 4 gives the variation of the voltage FRFs for the external excitations at the short circuit resonance frequency and the open circuit resonance frequency, respectively. In log-log scale, both voltage  
Figure 3: Voltage FRFs of the flexoelectric beam with 0.3 \(\upmu\)m thickness over the frequency range of (**a**) the first three modes; (**b**) around the first mode. outputs increase with increasing loading resistance, and trend to constant values at large loading resistance. The two curves intersect at a point where the loading resistance is about 97 M\(\Omega\). If the loading resistance was greater than 97 M\(\Omega\), the voltage output at the open circuit resonance frequency was larger because the system was closer to the open circuit condition. On the other hand, for lower loading resistance, the voltage output at the short circuit resonance frequency was larger.