## 5 Experimental Validation and Analysis
### Frequency-Domain Analysis  
To investigate the effectiveness of the synchronous sampling method in mitigating runout interference in the frequency domain, a spectral analysis of the rotational speed signal was performed. To more clearly illustrate the interference of the runout phenomenon across different frequency bands and the algorithm's restoration efficacy in these bands, the frequency-domain analysis of the rotational speed was segmented into low, medium, and high-frequency ranges. Consistent with the time-domain analysis, the theoretical rotational speed served as the baseline for comparison. The comparative results are depicted in Figures 25-28, and the quantitative results are presented in Table 4.  
Figure 26: Speed frequency domain comparison of driving shaft at 300 RPM with a connection angle of 15\({}^{\circ}\).  
Figure 25: Speed frequency domain comparison of driving shaft at 200 RPM with a connection angle of 15\({}^{\circ}\).  
Figure 28: Speed frequency domain comparison of driving shaft at 500 RPM with a connection angle of 15\({}^{\circ}\).  
Figure 27: Speed frequency domain comparison of driving shaft at 400 RPM with a connection angle of 15\({}^{\circ}\).  
Observation of the frequency-domain analysis plots (Figures 25-28) reveals the following:  
1. The runout phenomenon induced during the experiments primarily interferes with the rotational speed signal in the medium and high-frequency bands, with relatively less interference observed in the low-frequency band.
2. In the medium and high-frequency bands, the synchronous sampling method demonstrates commendable noise reduction capabilities, effectively mitigating the substantial noise introduced by the runout phenomenon in the medium-frequency range.
3. In the high-frequency band, where noise frequencies overlap with the inherent rotational speed frequencies, the method achieves effective separation and reduction of noise. It successfully diminishes the impact of noise without compromising the original frequency components of the rotational speed.  
Table 4 presents a comparative analysis of the deviation of the Raw and Optimized rotational speed datasets from the Theoretical rotational speed in the frequency domain. Key metrics, including the R-value and RMSE for each frequency band relative to the baseline and the RMSE improvement rate of the reconstructed rotational speed compared to the original rotational speed, were calculated. These metrics serve to quantitatively assess the capability of the synchronous sampling method to counteract runout interference in the frequency domain. An analysis of the data in Table 4 indicates the following:  
1. Analysis of the R-values between the Raw rotational speed signals and the Theoretical rotational speed across different experimental groups generally shows the highest correlation in the low-frequency band, the lowest correlation in the medium-frequency band, and a relatively high correlation in the high-frequency band. This observation substantiates conclusion (a) drawn from the frequency-domain comparison plots.
2. In the low-frequency band, despite being less affected by runout, the synchronous sampling method still achieves minor optimization. This is evident from the improvements in R-values and the RMSE improvement rates for all experimental groups.
3. In the high-frequency band, runout introduces noise with frequency characteristics similar to the inherent frequencies of the rotational speed. The synchronous sampling method effectively reduces this noise while preserving the original frequency components. This is reflected by an increase in R-values, with notable improvements observed at 200 RPM and 500 RPM. The substantial RMSE improvements across all groups further corroborate this finding.  
The aforementioned analysis demonstrates that the synchronous sampling method exhibits a significant capability to mitigate the interference of runout on torsional vibration measurements in the frequency domain under various experimental conditions.