## 2 Discussion about the Importance of Volume Consideration  
In order to investigate the importance of volume consideration, the outcome of four piezoelectric energy harvesters (PEHs) of different volumes were investigated (see Figure 1a). The PEHs A, B, C, and D are in the form of the bimorph piezoelectric cantilever displayed in Figure 1b, where \(l\) is the free length, \(h\) is the total thickness, and \(b\) is the width. The piezoelectric layer has a thickness of \(h_{p}\) and a total length of \(l_{c}\). The density of the piezoelectric layers and the modulus of elasticity are \(\rho_{p}\) and \(E_{p}\), respectively. The non-piezoelectric layer has a thickness of \(h_{s}\), a density of \(\rho_{s}\), and a modulus of elasticity of \(E_{s}\). The ratio between the thickness of the middle layer and the thickness of the piezoelectric layer is \(h_{sp}\), where \(h_{sp}=h_{s}/h_{p}\). A tip mass of \(m_{t}\) is attached at the end of the beam. The mass ratio between the tip mass and the mass of the entire beam is \(\beta\), where \(\beta=m_{t}/(\rho_{s}bl_{h}+2\rho_{p}llbl_{p})\). The transversal displacement at the tip at any time \(t\) is given by \(x(t)\) for an input displacement of \(u(t)\). The material properties displayed in Table 3 are assumed for all the PEHs, where \(s_{11}^{E}\) is the elastic compliance under constant electric field (\(E_{p}\) equals \(1/s_{11}^{E}\) at the short circuit condition (resonance)), \begin{table}
\begin{tabular}{c c c} \hline
**Reference** & **Formula** & **Variables** \\ \hline \multirow{3}{*}{Roundy [17] Effectiveness \(e\)} & \(\epsilon=\frac{k_{2}^{2}}{2}\frac{2^{6}\rho_{p}}{\frac{1}{2}}\)\(\lambda_{max}\) & \(\lambda\): transmission coefficient at applied load \\  & \(k_{2}^{2}|_{2\times T}=\frac{k_{2}^{2}}{4-2k_{2}^{2}}\) & \(\lambda_{max}\): maximum transmission coefficient at optimal load \\  & \(\lambda_{max}=\frac{k_{2}^{2}}{4-2k_{2}^{2}}\) & \(d\): piezoelectric strain coefficient \\  & \(\epsilon\): dielectric constant \\ \hline [PERSON] et al. [19] Normalized power density (NPD) & NPD \(=\frac{E_{p}}{\tau^{\prime}}\) & \(\rho_{p}\): effective mass density (mass per entire volume) \\ \hline [PERSON] et al. [20] Power normalized mass acceleration (PNMA) & PNMA \(=\frac{P_{max}}{M\tau}\) & \(\rho_{p}\): baseline material density \\  & \(\tau\): ambient acceleration \\ \hline [PERSON] et al. [21] Figure of merit (FoM) & FoM \(=\frac{P_{max}}{\tau^{\prime}}\frac{1}{\tau^{\prime}}\frac{1}{\tau^{\prime}}\)\(\rho_{ent}\): power output \\  & \(M\): proof mass \\ \hline [PERSON] et al. [22] Volume figure of merit (FOM\({}_{\text{V}}\)) & FoM\({}_{\text{V}}=\frac{P_{max}}{\frac{P_{max}}{2\pi\mu\mu\mu\
u\
u\
u\
u\
u\
u\
u\
u}}\) & \(\chi_{\text{E}}\): volume including the displacement \\  & \(\chi_{\text{O}}\): mechanical quality factor & \(\rho_{\text{rad}}\): density of gold \\  & FoM\({}_{\text{BW}}\) = FoM\({}_{\text{V}}\) \(\frac{\chi_{\text{O}}\mu\mu\
u\
u\
u\
u\
u\
u}\) & \(\omega\): excitation frequency \\  & \(\delta\omega_{\text{rad}}\): is the 1 dB power bandwidth \\  & \(\Delta\gamma\): bandwidth of the generator \\  & \(\delta\): center frequency of the bandwidth \\ \hline \end{tabular}
\end{table} 
Table 2: Figures of merit to evaluate between the vibration energy harvesters. Figure 1: (**a**) The piezoelectric energy harvesters (PEHs) under investigation (**b**) Cantilevered bimorph piezoelectric energy harvester. \(\varepsilon_{33}^{T}\) is the permittivity of the piezoelectric material at a constant stress, and \(d_{31}\) is the piezoelectric charge constant. The mechanical damping ratio is assumed to be 0.01. The PEHs A, B, C, and D have different geometrical parameters (dimensions and tip masses). However, it is taken into account that the geometrical parameters are tuned in order for all the PEHs have their first eigenfrequency at 154.8 Hz.