## 2 Materials and Methods  
### Concept  
A simplified equation for predicting the 1 st resonant frequency mode of a cantilever using point proof mass-based beam theory is given by the following equation [5; 20; 36]:  
\[f=\left\{\frac{1}{2\pi}\sqrt{\frac{E}{4m}}\right\}\sqrt{\frac{w\text{r}^{3}}{L^ {3}}} \tag{1}\]  
where \(E\), \(m\), \(w\), \(t\), and \(L\) are the elastic modulus, proof mass, width, thickness, and length of the beam, respectively. To tune the frequency one of these parameters would need to be altered. Altering the physical dimensions (\(L\), \(t\), and \(w\)) has been attempted using etching or laser dicing, but these methods are not practical for batch fabrication. Altering the elastic modulus can be done by adding additional layers to change the stiffness, but then this affects the neutral axis, which could alter piezoelectric performance. Therefore, altering the proof mass is the easiest method to change the resonant frequency during manufacturing. Equation (1) assumes the proof mass is a single point with a center of gravity at the centroid of the proof mass. The concept of altering the proof mass to reduce the resonant frequency can also be applied to other structures such as plates, fixed-fixed suspensions, and membranes. However, changing the location of the center of gravity of the proof mass will also affect the resonant frequency, as illustrated by Equation (2) which combines Equation (1) with the Rayleigh principle [19; 20; 37]:  
\[\text{f}=\frac{1}{2\pi}\sqrt{\frac{\text{Ewt}^{3}}{12\text{ mL}^{3}}\frac{6\text{r}^{2}+6\text{r}+2}{8\text{r}^{4}+14\text{r}^{3}+10.5 \text{r}^{2}+4\text{r}+\frac{2}{3}}} \tag{2}\]  
where \(\text{r}=\text{l}_{\text{c}}/\text{L}\), and \(\text{l}_{\text{c}}\) is the length location of the center of gravity. Thus, shifting the center of gravity towards the free end will reduce the frequency, and shifting center of gravity closer to the tethered end will increase the frequency. The center of gravity along the width also influences the resonant frequency and higher order resonant frequencies that have a twisting motion. However, the goal of this study was to investigate the 1 st order out-of-plane mode, as this is the most common mode used to harvest high-power-density energy harvesters. These equations illustrate that altering the magnitude and location of the mass will alter the resonant frequency.  
The tuning method investigated in this study involves creating a proof mass with an array of cavities which can be filled with a solid mass (microparticles) to alter the center of gravity, which should alter the resonant frequency. By modifying the type of particles (density), the volume, and the location of the filled cavities, the cantilever should be capable of being tuned. The resolution of tuning should be dependent on the volume of the cavity, whereas the range of tuning should be dependent on the change in the overall mass. Thus, higher density and volume should increase the potential range of tuning. Figure 1a,b illustrate the schematic of the proof mass with cavities. An example of the nomenclature used in this paper to identify the cavities is H24, where the first digit represents the row, and the second digit represents the column number.