## 2 Analysis of the Runout Interference Mechanism During Sampling  
To eliminate the impact of random toothed disc runout on torsional vibration measurements, an analysis of the error generation mechanism is essential. The analysis begins with the measurement principle itself. Typically, the zero-crossing detection method is employed to obtain the fluctuating rotational speed, which embodies the torsional vibration characteristics. During the measurement process, the sensor detects the rotation of the toothed disc, and the output voltage waveform is as illustrated in Figure 1. A periodic waveformis generated for each indexing interval traversed, and the time \(\Delta t\) taken to traverse this interval is used to calculate the rotational speed. To precisely determine this period \(\Delta t\), interpolation of sample points in the vicinity of zero voltage is required to accurately identify two adjacent zero-crossing instants. Let the two sample points immediately above and below the zero-voltage level be denoted \(T_{n}\) and \(T_{n+1}\), with corresponding voltage values of \(V_{n}\) and \(V_{n+1}\), respectively. Then, a zero-crossing instant \(T_{i}\) can be determined as follows:  
\[T_{i}=T_{n}-V_{n}\frac{T_{n+1}-T_{n}}{V_{n+1}-V_{n}} \tag{1}\]  
Let the time difference between the zero-crossing instants of two consecutive rising edges be \(\Delta t\):  
\[\Delta t=T_{i+1}-T_{i} \tag{2}\]  
In this formula, \(T_{i+1}\) represents the zero-crossing instant of the subsequent cosine wave's rising edge. If the number of teeth on the toothed disc is \(N\), then the indexing angle \(\alpha\) is  
\[\alpha=\frac{2\pi}{N} \tag{3}\]  
The rotational speed for the i-th tooth on the toothed disc is calculated as  
\[\omega_{i}=\frac{\alpha}{\Delta t}=\frac{2\pi}{N\Delta t} \tag{4}\]  
The average of two consecutive zero-crossing instants is taken as the recording time for the i-th rotational speed measurement, i.e.,  
\[t_{i}=\frac{T_{i}+T_{i+1}}{2} \tag{5}\]  
In practice, during the sampling process, the toothed disc undergoes a compound motion resulting from the combined effects of rotation and runout, as illustrated in Figure 2. The frequency and amplitude of this runout are relatively random. Assuming its amplitude does not exceed the sensor's detection range, the runout component primarily responsible  
Figure 1: Typical voltage waveform from the magneto-electric sensor during tooth passage. for errors in speed measurement is that perpendicular to the sensor's horizontal detection line. If the calculation relies solely on the time difference \(\Delta t\) and the toothed disc indexing angle \(\alpha\), the resultant rotational speed will contain spurious information introduced by the runout. In the following, a model is developed to analyze the mechanism by which runout introduces errors during the speed measurement process. Assume the toothed disc rotates in the X-Y plane around the origin with an angular displacement \(\theta(T)\), accompanied by a small-amplitude random runout \(d(T)\):  
\[d(T)=\big{(}d_{x}(T),d_{y}(T)\big{)} \tag{6}\]  
Then, the time-varying coordinates of the k-th tooth in the plane can be expressed as  
\[\left\{ \begin{array}{l}x_{k}(T)=\text{Rcos}(\theta(T)+\alpha(k-1))+d_{x}(T) \\ y_{k}(T)=\text{Rsin}(\theta(T)+\alpha(k-1))+d_{y}(T)\end{array} \right. \tag{7}\]  
In this formula, R represents the pitch circle radius of the toothed disc. In this model, the instant the sensor records a tooth passing is considered equivalent to the tooth's coordinate point crossing the positive \(x\)-axis.