## 3 Results and Discussion
### Tuning Range  
Frequency as a function of power output was used to illustrate the tuning range that was achievable for the given cantilever with various fillers. The cantilever with an empty proof mass was used to represent the maximum frequency, while the proof mass with all the cavities filled (100%) was used to illustrate the minimum frequency that could be achieved with the various powders. The results are shown in Figure 3a for the V25W cantilever, which had an initial resonant frequency of 125 Hz without a proof mass. Adding an empty proof mass/cap decreased the resonant frequency to 49.1 Hz due to the additional weight. Filling the proof mass with the various powders decreased the resonant frequency. The resonant frequencies for the 100% filled cavities were 29.5, 25.4, and 21.7 Hz for Silica, Molybdenum, and Tungsten, respectively. The Silica-filled proof mass had the least weight per cavity and resulted in the lowest potential tuning range of 19.6 Hz (40% tuning reduction), while the Tungsten-filled proof mass demonstrated a tuning range of 27.4 Hz (55.8% tuning reduction). The output power also increased as mass increased due to increased stress/strain, which resulted in increased power. The FEM-simulated resonant frequency for Tungsten-filled solid proof mass resulted in a frequency of 22.55 Hz (3.92% error from the experimental data). The error is likely due to the potential error in filling each cavity with the same mass, as the fill factor and density of the powders had some variation as shown in Table 1.  
The tuning potential of the V25W cantilever was relatively low as the initial frequency with proof mass was low (49.1 Hz), which limited the potential tuning range as adding mass can only decrease the resonant frequency. However, high-frequency cantilevers should demonstrate increased tuning range. To demonstrate this, a higher resonant frequency (440 Hz) cantilever (Mide, S233-H5 FR-1107 XB) was investigated with the same empty proof, and then the cavities were filled with 100% Tungsten. The results are shown in  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Filler Material (Avg. Size)** & **Average Density (g cm\({}^{-3}\))** & **Average Fill Factor (\%)** \\ \hline Tungsten (75 \(\upmu\)m) & 4.33 \(\pm\) 0.152 & 61.8\(\%\)\(\pm\) 2.8\% \\ Molybdenum (40 \(\upmu\)m) & 2.19 \(\pm\) 0.085 & 73.2\(\%\)\(\pm\) 1.5\% \\ Silica (45 \(\upmu\)m) & 1.48 \(\pm\) 0.088 & 69.4\(\%\)\(\pm\) 1.6\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Filler Density Measurements.  
Figure 3b, where the cantilever with empty proof mass had a frequency of 144.6 Hz. When the cavities were filled with Tungsten powder, the resonant frequency decreased to 30.2 Hz, thus demonstrating a 114.4 Hz tuning range potential (79.1% tuning reduction). The tuning range could be increased further by decreasing the mass of the initial proof mass, increasing the initial frequency of the cantilever, or increasing the amount/density of the filler. The power output values between the two different cantilevers were different due to the dimensional changes which affect the frequency and voltage output.  
To further illustrate methods to increase tuning range, the cavity dimensions were increased from 3 \(\times\) 3 mm to 4.8 by 4.8 mm (5 \(\times\) 7 array) and the gap spacing between the cavities was reduced to 0.5 mm instead of 1 mm. This resulted in a decrease in the initial proof mass, but allowed for increased mass from the filler. For the V25W cantilever, this resulted in an empty proof mass frequency of 53.7 Hz (increase of 4.6 Hz or 9.37%) and a 100% Tungsten-filled resonant frequency of 19.2 Hz (decrease of 2.5 Hz or 11.5%), illustrating a tuning range of 34.5 Hz (64.3% reduction) compared to the 7 \(\times\) 9 array, which had a tuning range of 27.4 Hz (55.8% reduction). Therefore, optimizing the cavity array dimensions could be used to further increase the tuning range in the future.