## 5 Experimental Validation and Analysis
### Time-Domain Analysis  
First, the rotational speed was calculated directly from the tooth signals obtained in the torsional vibration measurement experiment using the zero-crossing detection method. This revealed anomalous fluctuations in the rotational speed, as illustrated in Figure 3. The dual-line approximation method was then applied to calculate the rotational speed signal. The resulting rotational speed is shown in Figure 16. It can be observed that the application of the dual-line approximation method successfully overcame the spurious data points in the signal and eliminated the anomalous rotational speed values. An observational analysis of the rotational speed acquired from a single sensor revealed the following at various motor speeds:  
Figure 16: Rotational speed calculated using the dual-line approximation method at various motor speeds (driving shaft input) with a 15\({}^{\circ}\) joint angle: (**a**) 200 RPM; (**b**) 300 RPM; (**c**) 400 RPM; (**d**) 500 RPM. Under the same joint angle, different input speeds from the driving shaft resulted in distinct rotational speed fluctuation trends. These trends were broadly consistent with the theoretical rotational speeds calculated using Equation (33). 2. Runout-induced fluctuations in rotational speed were evident in all cases. More severe runout of the toothed disc during the experiment corresponded to more pronounced rotational speed fluctuations. Subsequently, the synchronous sampling method was employed to fuse the data from the dual sensors and restore the rotational speed. In the subsequent time-domain and frequency-domain analyses, figures, and tables, the following designations are used: 'Raw' refers to the original rotational speed acquired by a single sensor prior to compensation. 'Optimized' denotes the reconstructed rotational speed after applying the synchronous sampling method. 'Theoretical' represents the theoretically calculated rotational speed derived from Equation (33), serving as the baseline. The Optimized speed was then compared with the Theoretical speed and the Raw speed. This comparative analysis aimed to quantitatively evaluate the anti-interference capability of the synchronous sampling method against runout. All calculation results are shown in Figures 17-24. The comparison results are depicted in Figures 17, 19, 21, and 23. The error comparison and analysis of the two rotational speed datasets against the theoretical rotational speed are presented in Figures 18, 20, 22, and 24. The theoretical rotational speed and quantitative analysis results are summarized in Table 3.  
Figure 17: Speed comparison of driving shaft at 200 RPM with a connection angle of \(15^{\circ}\). Figure 19: Speed comparison of driving shaft at 300 RPM with a connection angle of 15\({}^{\circ}\). Figure 18: Error analysis of driving shaft at 200 RPM with a connection angle of 15\({}^{\circ}\). Figure 21: Speed comparison of driving shaft at 400 RPM with a connection angle of 15\({}^{\circ}\). Figure 20: Error analysis of driving shaft at 300 RPM with a connection angle of 15\({}^{\circ}\). Figure 23: Speed comparison of driving shaft at 500 RPM with a connection angle of 15\({}^{\circ}\). Figure 16.0 Error analysis of driving shaft at 400 RPM with a connection angle of 15\({}^{\circ}\). Observation and analysis of the rotational speed curves presented in Figures 17-24 reveal the following:  
1. A comparison of the rotational speeds (as depicted in Figures 17, 19, 21, and 23) demonstrates that, at all tested motor speeds, the reconstructed rotational speed obtained via the synchronous sampling method effectively mitigates runout-induced interference and aligns more closely with the theoretical rotational speed curve than the original rotational speed acquired from a single sensor. 2. The error analysis plots (Figures 17, 20, 22, and 24) indicate that, in comparison, the rotational speed errors associated with the synchronous sampling method are more tightly clustered around the zero-error level.