## 4 Synchronous Sampling Method Based on the Differential Measurement Principle  
Building upon the stable rotational speed calculation algorithm described previously, it is crucial to eliminate the interference in speed measurement caused by toothed disc runout, as analyzed in Section 2. This paper develops an algorithm based on the differential measurement principle [25; 26; 27], leveraging symmetry and the common-mode rejection characteristics of dual-channel signal acquisition to eliminate systematic errors [6; 7; 9]. Based on the model described in Section 2, an ideal speed measurement scenario is envisioned: the speed probe would no longer rely on time intervals and toothed disc indexing but could directly accumulate and record the angular displacement information of the rotating toothed disc. Rotational speed could then be expressed as the rate of change of angular displacement. Thus, at time t, the recorded angular displacement of the toothed disc on the positive \(x\)-axis can be expressed as  
\[\phi_{+}(t)=\theta(t)+\frac{d_{y}(t)}{R} \tag{21}\]  
Differentiating this yields the measured value of instantaneous rotational speed \(\omega_{+}\), derived by  
\[\omega_{+}(t)=\omega_{true}(t)+\frac{1}{R}\frac{dd_{y}(t)}{dt} \tag{22}\]  
Under this hypothetical measurement condition, the measured instantaneous rotational speed consists of the true rotational speed \(\omega_{\text{true}}\) and an error term proportional to the rate of change of runout in the Y-direction. Now, consider simultaneous measurements on both the positive and negative x-axes. At the same instant t, based on geometric relationships, the angular displacement recorded on the negative \(x\)-axis is  
\[\phi(t)=\theta(t)-\frac{d_{y}(t)}{R} \tag{23}\]  
The instantaneous rotational speed \(\omega_{-}\) measured on the negative \(x\)-axis is  
\[\omega_{-}(t)=\omega_{true}(t)-\frac{1}{R}\frac{dd_{y}(t)}{dt} \tag{24}\]  
Taking the average then yields  
\[\omega_{avg}(t)=\frac{\omega_{+}(t)+\omega_{-}(t)}{2}=\omega_{true} \tag{25}\]  
From the analysis above, it is evident that when the toothed disc moves in the positive y-direction,  
\[\frac{dd_{y}(t)}{dt}>0. \tag{26}\]  
The rotational speed perceived at the positive \(x\)-axis measurement point decreases \(\omega_{+}(t)<\omega_{true}\), while that perceived at the negative \(x\)-axis measurement point increases \(\omega_{-}(t)>\omega_{true}\), and vice versa. Evidently, irrespective of runout variations, the average of the instantaneous rotational speeds measured on both sides always equals the true rotational speed. Under actual sampling conditions, interpolation methods can be employed to approximate the aforementioned ideal speed measurement model by fitting an ideal rotational speed profile based on available speed information. By installing sensors on both sides of the toothed disc to synchronously acquire tooth-passing times and separately calculate instantaneous rotational speeds, the true rotational speed can be computed. However, due to runout, a phenomenon occurs, as illustrated in Figure 6; the recorded times for speeds acquired over the same measurement interval are not aligned between the two sensors, exhibiting a runout-induced time difference. To enhance calculation accuracy, the two sets of rotational speed data must be time-aligned through mutually referenced interpolation. The midpoint of a pair of corresponding speed measurement times is taken as a new time point, and rotational speeds are interpolated with reference to these new common time points. Let the rotational speed-time series for the positive and negative x-axes be, respectively,  
\[\omega_{+}\{t_{+,j}\},\;i=1,2,\ldots,N \tag{27}\]  
\[\omega_{-}\{t_{-j}\},\;\mathrm{j=1,2,\ldots,N} \tag{28}\]  
The midpoint time is taken as  
\[T_{\mathrm{z}}=\frac{t_{+,x}+t_{-,x}}{2},\;\mathrm{z=1,2,\ldots,N} \tag{29}\]  
A new common time series \(\{T_{z}\}\) is thus obtained.