## 4 Time-Series Analysis
### Assessment of the Potential of wMPS for Estimating the Oscillation Amplitude  
The estimation of the accuracy of the wMPS for the determination of the oscillation amplitude was based on a comparison of the oscillation amplitude derived from the wMPS time series with the real amplitude (designed value). The real amplitude (designed value) is the radius of each driving wheel, marked by red lines in Figure 7 in the representative examples for the wMPS. The wMPS time-series amplitude was computed from the mean values of the maximum and minimum peaks using a simple peak-picking technique [19]. Absolute deviations and mean-square errors of amplitudes of different segments are listed in Table 1, which shows that, when the deviation value ranged from 0.05 \(\pm\) 0.06 mm to 4.13 \(\pm\) 6.01 mm, the absolute deviations increased with increases in amplitude and frequency. For the cases shown in Figure 7a-c, the wMPS described a nearly ideal sinusoidal oscillation; however, in other cases, i.e., Figure 7d-j, several cycles were misrecorded and clearly deviated from sinusoidal curves. Based on the analysis of these signals, the quality of the measurements mainly depends on two parameters [15]. Figure 7: _Cont._Firstly, the wMPS can collect enough signals (Figure 7a-c) to recover the displacement curve in the low-frequency stage; however, with increasing frequency, certain cycles were poorly described or even lost, because there were few displacement signals recorded in one period (Figure 7d-j). The second parameter is the maximum velocity of the oscillation, \(\upsilon_{\mathrm{max}}=2\pi fR\). If \(\upsilon_{\mathrm{max}}<13\)  cm/s, the oscillation amplitude was described well (Figure 7a-c) corresponding to 3, 6, and 12.5 cm/s, respectively. On the contrary, for \(\upsilon_{\mathrm{max}}>18\)  cm/s, the oscillation amplitude was poorly described (Figure 7d-j). Figure 8 shows a more intuitive error distribution than that presented in Table 1. As shown in Figure 8, the absolute deviation of the displacement of different segments under the red line was similar and reliable (the effect of the sampling frequency on the absolute deviation can be ignored for these segments), because the absolute deviation was very close to the sensitivity of the instrument (approximately 0.2 mm). Above the red line, the uncertainties were relatively large. \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Frequency (Hz)**} & \multicolumn{5}{c}{**Amplitude (mm)**} \\ \cline{2-6}  & **3** & **5** & **10** & **20** & **30** \\ \hline
0.5 & 0.05 \(\pm\) 0.06 & 0.05 \(\pm\) 0.06 & 0.09 \(\pm\) 0.11 & 0.18 \(\pm\) 0.19 & 0.35 \(\pm\) 0.36 \\
1 & 0.08 \(\pm\) 0.09 & 0.08 \(\pm\) 0.09 & 0.12 \(\pm\) 0.17 & 0.21 \(\pm\) 0.25 & 0.41 \(\pm\) 0.44 \\
2 & 0.09 \(\pm\) 0.11 & 0.13 \(\pm\) 0.17 & 0.22 \(\pm\) 0.31 & 0.55 \(\pm\) 0.71 & 0.89 \(\pm\) 1.27 \\
3 & 0.21 \(\pm\) 0.23 & 0.22 \(\pm\) 0.33 & 0.39 \(\pm\) 0.60 & 1.04 \(\pm\) 1.43 & 1.65 \(\pm\) 2.19 \\
4 & 0.25 \(\pm\) 0.37 & 0.32 \(\pm\) 0.46 & 0.84 \(\pm\) 1.11 & 1.85 \(\pm\) 2.65 & 3.28 \(\pm\) 4.40 \\
5 & 0.31 \(\pm\) 0.44 & 0.54 \(\pm\) 0.78 & 1.26 \(\pm\) 1.83 & 2.61 \(\pm\) 3.38 & - \\
6 & 0.43 \(\pm\) 0.64 & 0.73 \(\pm\) 1.09 & 1.45 \(\pm\) 2.14 & 4.13 \(\pm\) 6.01 & - \\
7 & 0.52 \(\pm\) 0.69 & 0.96 \(\pm\) 1.23 & 1.94 \(\pm\) 2.44 & - & - \\
8 & 0.89 \(\pm\) 1.16 & 1.62 \(\pm\) 2.05 & 3.25 \(\pm\) 4.08 & - & - \\
9 & 1.06 \(\pm\) 1.35 & 1.94 \(\pm\) 2.39 & 3.92 \(\pm\) 4.80 & - & - \\ \hline \end{tabular}
\end{table} 
Table 1: Absolute deviation of amplitude of different segments. Figure 7: Oscillation displacements of different segments (amplitude: 10 mm; frequency: 0.5–9 Hz) obtained through the wMPS. Horizontal lines indicate the real oscillation amplitude.