## 6 MATLAB Simulation  
In order to further study the influence of the substrate material on the electric output parameters of 1# piezoelectric brick, a series of programs were developed through MATLAB software based on the theoretical model in Section 4. RMS of voltages and average powers from LPCB made of different materials, including aluminum alloy, spring steel and Babbitt metal, are computed and depicted. In the simulation of three models, the force arm length and the footstep excitation amplitude all are set to increase from 19.5 mm to 33.5 mm, and from 250 N to 500 N, respectively. Other parameters are taken from Tables 1 and 6.  
Before the numerical research, the peak-to-peak voltages of 1#1 PZT4 patch from simulation results and experimental results are extracted and compared, which can be seen from Figure 18. The results show that the simulation values are slightly bigger than the experimental values. This is because the theoretical model in the simulation it is assumed that the force arm segment is a cantilever and its fixed end has no displacement. In actuality, its fixed end has a weakly displacement, especially the angular displacement which leads to a bigger bending moment compared to the experiment value. However, the errors between the simulation values and the experimental values are no more than 6%, which indicates the theoretical model and the simulation results are reliable and accuracy.  
According to the simulation results, the RMS of voltages and average powers corresponding to different substrate material are demonstrated in Figures 19-21. It is obvious that for three substrate materials, the RMS of voltages and average powers from the LPCB of 1# piezoelectric brick increase with the increases of the force arm length and the excitation force amplitude. For example, when the amplitude of the footstep excitation is kept as 500 N, the average output powers from 1#1 PZT4 patch of the LPCB with aluminum alloy substrate, spring steel substrate, and Babbitt metal substrate increase from 2.15 mW  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Substrate** & **Young’s** & **Mass Density** & **Maximum RMS** & **Maximum** \\
**Material** & **Modulus (GPa)** & **(kg/m\({}^{3}\))** & **of Voltage (V)** & **Average Power** \\ \hline Spring steel & 210 & 7820 & 48 & 11.39 \\ Aluminum alloy & 72 & 2742 & 53 & 13.24 \\ Babbitt metal & 58.8 & 7170 & 58 & 16.01 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Output electric parameters of 1#1 PZT4 patches on unimorph PCS with different substrate materials.  
Figure 18: Peak-to-peak voltages of 1#1 PZT4 patch of unimorph PCS from simulation and experiment versus the amplitude of footstep excitation.  
to 13.24 mW, 1.69 mW to 11.39 mW and 2.35 mW to 16.01 mW with the increase of the force arm length from 19.5 mm to 33.5 mm, respectively. This is because the increases of the force arm length and the excitation force amplitude all lead to the increase of the surface strain exerted on the PCS (see Equations (6)-(11)), which further causes the increase of the average power from 1#1 PZT4 patch (see Equations (13)-(17)). Specifically, the maximum RMS of the voltages and the average powers from LPCB with aluminum alloy, -spring steel, and Babbitt metal are extracted and listed in Table 6. Obviously, the RMS of voltage and the average power increase with the decrease of the Young's modulus of the substrate material. For example, compared with the aluminum alloy substrate and spring steel substrate, the maximum RMS of the voltages and the average powers from LPCB with Babbitt metal can reach up to 14.55 mW. This is because the Babbitt metal substrate has a smaller Young's modulus than that of the alloy substrate and spring steel substrate, which can be explained by Equations (4)-(11) and (