## 5 Experimental Validation and Analysis
### Time-Domain Analysis
Furthermore, the distribution of these reconstructed rotational speeds exhibits a closer agreement with the theoretical rotational speed. Table 3 presents a quantitative evaluation of the synchronous sampling method's capability to mitigate runout interference in the time domain. Parameters such as RMSE (root mean square error), MAE (mean absolute error), MAX (maximum error), and R (correlation coefficient) were calculated by comparing the Raw and Optimized datasets against the Theoretical baseline. An analysis of the data in Table 3 indicates the following. 1. Analysis of the original rotational speed signal from a single sensor shows that under the 300 RPM condition, the rotational speed fluctuation was the mildest, corresponding to the least severe runout phenomenon observed during the experiment. The correlation coefficient (R-value) with the theoretical rotational speed reached 0.9875. Conversely, under the 500 RPM condition, the rotational speed fluctuation was the most severe, as was the experimental runout, with the R-value decreasing to 0.7557. This observation validates conclusion (b) drawn from the rotational speed curves. Figure 16.1 Error analysis of driving shaft at 500 RPM with a connection angle of 15\({}^{\circ}\). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**200 RPM**} & \multicolumn{3}{c}{**30 RPM**} & \multicolumn{3}{c}{**400 RPM**} & \multicolumn{3}{c}{**500 RPM**} \\ \hline  & **Optimized** & **Raw** & **Impervement** & **Optimized** & **Raw** & **Impervement** & **Raw** & **Impervement** & **Size** & **Optimized** & **Raw** & **Impervement** \\ \hline RMSE & 1.00 & 1.605 & 31.9\% & 1.899 & 1.299 & 16.2\% & 1.611 & 2.209 & 30.8\% & 2.260 & 3.292 & 31.3\% \\ MAE &It is also noted that the severity of rotational speed fluctuation does not exhibit a simple linear correlation with the input speed of the driving shaft. A preliminary inference is that this fluctuation characteristic is closely related to the proximity of the input excitation frequency to the natural modal frequencies of the test rig (i.e., resonance conditions). 2. Under different runout conditions, the rotational speed reconstructed by the synchronous sampling method consistently demonstrated a high degree of restoration. The most significant improvement in the R-value was observed at 500 RPM, with an increase of 0.1085 (from 0.7557 to 0.8642), elevating the correlation level. At 200 RPM and 400 RPM, the R-values improved by 0.0455 (from 0.9104 to 0.9559) and 0.0341 (from 0.9324 to 0.9665), respectively. Even under the 300 RPM condition, where the initial R-value was already high at 0.9875, a further improvement to 0.9930 was achieved. 3. A comparison of RMSE, MAE, and MAX parameters reveals varying degrees of improvement across all tested rotational speeds. This indicates that the proposed method demonstrates enhanced data fidelity across multiple statistical metrics. The aforementioned analysis validates that the synchronous sampling method can effectively enhance the accuracy of the reconstructed rotational speed signal and its correlation with the true rotational speed under different operational speeds and fluctuation conditions. This demonstrates its good universality and robustness, exhibiting a strong capability to counteract the effects of runout in the time domain.