## 5 Experimental Validation and Analysis  
### Experimental Design  
To validate the effectiveness of the algorithms described above, this study utilized a universal joint torsional vibration test rig. An experiment was designed to simulate the sampling conditions characterized by the rotational runout of the toothed disc. A universal joint enables the transmission of angular displacement between the input and output shafts [28,29]. The basic working principle of the universal joint is illustrated in Figure 9.  
The angle formed between the input and output shafts is defined as the joint angle \(\delta\). Through theoretical analysis, the relationship between the input rotational speed of the driving shaft \(\omega_{in}\) and the output rotational speed of the driven shaft \(\omega_{out}\) can be obtained as follows [30]:  
\[\omega_{out}=\omega_{in}\frac{2\cos\delta}{2-\sin^{2}\delta+\sin^{2}\delta \cdot\cos 2\omega_{in}t} \tag{33}\]  
According to Equation (33), given the preset driving shaft rotational speed and the universal joint angle, the theoretical output rotational speed curve can be calculated. This curve serves as a baseline in subsequent experiments to evaluate the effectiveness of the proposed algorithm in rotational speed restoration. The schematic of the universal joint torsional vibration test rig is illustrated in Figure 10. Its main components, including the motor, universal joint driving shaft, universal joint driven shaft, and sensors, are mounted on a base platform via vertical mounting plates, ensuring the planarity of the transmission axis. The test rig is driven by an electric motor, whose rotational speed is controlled by a servo controller. The motor's output shaft is connected to the driving shaft of the universal joint via a flexible coupling. The driven shaft, following the universal joint, is connected to the speed-measuring toothed disc. Both the driving and driven shafts are supported by bearings mounted on their respective mounting plates. Each mounting plate is precisely positioned and vertically connected to the base platform using a pinhole fit to ensure alignment. A pair of eddy current sensors for torsional vibration measurement are installed on the final mounting plate. The mounting plate for the driven shaft and the sensor bracket mounting plate are affixed to a turntable. Figure 9: Working principle of the Cardan universal joint used to induce defined torsional vibrations. This turntable, situated on the base platform, can be rotated within a defined angular range and features an angular scale. By rotating the turntable to the desired scale marking, the joint angle of the Cardan universal joint can be quantitatively adjusted. As illustrated in Figure 11, to simulate the random runout phenomenon of the toothed disc, flexible silicone pads were placed beneath the driven shaft mounting plate. All other mounting plates and the shaft mounting plates were correspondingly shimmed to maintain alignment. Leveraging the elasticity of these silicone pads, the rotational motion of the shaft induced a simultaneous random micro-runout of the toothed disc in the vertical direction. The mounting precision of the pair of sensors used for synchronous sampling is ensured by a custom-designed sensor bracket, whose coaxial accuracy is controlled through integral machining. A 3D model of the sensor bracket is depicted in Figure 12. A data acquisition card records the voltage signals from the two sensor channels as digital information, which is then transmitted to a PC for subsequent processing using a dedicated program. Figure 11: Test bench modification. Figure 10: Universal shaft torsional vibration test bench. The hardware architecture of the torsional vibration measurement system is illustrated in Figure 13.  
The key equipment and its specifications used in the experimental setup are listed in Tables 1 and 2 and Figures 14 and 15.