# Analytical Electromechanical Modeling of Nanoscale Flexoelectric Energy Harvesting
## 4 Numerical Results and Discussion
### Power Density FRFs  
For energy harvesting, the power density is an important measure of performance. In the case of the flexoelectric beam, the power density FRFs could be expressed as follows  
\[\frac{P_{\mathrm{d}}(t)}{-\omega^{2}W_{0}e^{\mathrm{i}\omega t}}=\left[\frac{ \sum_{r=1}^{3}\frac{-\mathrm{j}\omega K_{r}\lambda_{r}}{\left(\frac{1}{\pi^{2} }-\omega^{2}\right)+2\mathrm{j}\omega\varepsilon_{r}\omega_{r}}}{\left(\frac{ 1}{\pi}+\frac{\chi_{3}BL}{h}\mathrm{j}\omega\right)+\sum_{r=1}^{3}\frac{ \mathrm{j}\omega K_{r}^{2}}{\left(\frac{\omega^{2}}{\sigma^{2}}-\omega^{2} \right)+2\mathrm{j}\omega\varepsilon_{r}\omega_{r}}}\right]^{2}\frac{1}{R}\frac {1}{Vol}\, \tag{37}\]  
Figure 5: Tip displacement FRFs of the 3 \(\upmu\)m thickness beam for the first three modes. Figure 6: Tip displacement FRFs of the 0.3 \(\upmu\)m thickness beam with the tip mass \(M_{\mathrm{t}}\) for the first mode. where \(Vol=B\times L\times h\) is the volume of the beam. Figure 7 shows the power density FRFs of the flexoelectric beams for the first mode. The frequency of peak power density is almost unchanged in Figure 7a for the 3 \(\upmu\)m thickness beam. The optimum power density was achieved with a finite load resistance of \(\sim\) 100 M\(\Omega\). For the 0.3 \(\upmu\)m thickness beam, Figure 7b shows the variation of the power density FRFs with the various load resistances for the external excitations. Due to enhanced electromechanical coupling in the small scale, the resonance frequency of the power density shifts from \(f_{1}^{\text{sc}}\) to \(f_{1}^{\text{oc}}\) with increasing load resistance, which is consistent with the result of Figure 3b. The corresponding peak power density increases to a certain value (at frequency \(\sim\)61,084 Hz and the load resistance \(\sim\)200 M\(\Omega\)) and then decreases with increasing load resistance. Comparing the results of Figure 7a,b, it should be noted that the maximum power density for the 0.3 \(\upmu\)m thickness beam was around 2 times that of the 3 \(\upmu\)m thickness beam. This suggests that the flexoelectric energy harvesting was enhanced in the smaller scale. Compared to piezoelectric energy harvesting, the peak power density of the flexoelectric energy harvesting was dependent on the structure size. Hence, we could find the optimal structure parameters to obtain the maximum power density. Figure 7: Power density FRFs of the beam for the first order mode for (**a**) \(h=3\)\(\upmu\)m thickness; (**b**) \(h=0.3\)\(\upmu\)m thickness beams. The behaviors of power density FRFs with various load resistances for the external excitations at the short and open circuit resonance frequencies are shown in Figure 8a. In log-log scale, the power density increases linearly with increasing load resistance for low values of load resistance, and then decreases for high values of load resistance. For low load resistances, the power density at the short circuit resonance frequency was larger than that at the open circuit resonance frequency. For high values of load resistance, there was the opposite phenomenon. There was an intersect point at the load resistance of about 100 M\(\Omega\). The solid symbols in Figure 8a indicate the peak power density with various resonance frequencies corresponding to the load resistances as shown in Figure 7b. For low values of load resistance, the peak values were almost equal to those at the short circuit resonance frequency. For high values of load resistance, the peak power density values were almost equal to those at the open circuit resonance frequency. However, in the transition region of about 100 M\(\Omega\), the optimal power density at the resonance frequency was larger than those at both the short and open circuit resonance frequencies. Figure 8b plots the variation of the optimum external frequencies with the load resistances.