## 4 Calculating Time-Varying Mesh Stiffness
### Experiment Result Analysis  
To validate the model, an experimental setup for the level gear was constructed, and experiments were conducted. The level gear test rig, as shown in Figure 11, is powered by a 30 kW electric motor with a maximum speed of 2950 rad/min. The power is transmitted to the gearbox through a coupling, which is equipped with vibration, speed, and torque sensors. The sensor shown in the figure is a vibration sensor that extracts real-time vibration data. The extracted data is transmitted in real time to the central control system for storage. Some of the data is illustrated in Figure 13. The level angle of the level gear is 90\({}^{\circ}\), and power is transmitted to the output side through a reduction gearbox. An eddy current loader is installed on the output side, and the central control system can control the loaded torque by setting the output torque. Figure 12: Time-varying meshing stiffness under different fractal dimensions. Figure 13: Experimental process illustration. During the experiment, the gear speed was controlled in real time through the inverter, with an experimental speed set at 1200 rpm and applied torques of 77 N\(\cdot\)m and 105 N\(\cdot\)m. Data were recorded at intervals of 1 s after starting the experiment. From the experiments, it was observed that as the torque increased, the vibration of the gears intensified. This was reflected in the laboratory by the continuous increase in shaft vibration, the growing noise of the gearbox, and an accelerated rise in oil temperature. The obtained experimental vibration data has been systematically organized. Figure 14 shows that under a 77 N\(\cdot\)m load, the average experimental vibration velocity registers at 4.195 mm/s. Concurrently, the simulation yields an average velocity of 4.06 mm/s under identical conditions, resulting in a deviation of 3.22% when compared to the experimental results. As shown in Table 4, at a rotational speed of 1200 rpm and torque of 105 N\(\cdot\)m, the experimental average vibration is 4.384 mm/s. Under analogous conditions, the simulated vibration speed averages 4.116 mm/s, resulting in a deviation of 6.113% between the simulation and experiment. Comparative analysis between the two sets of experimental and simulated results indicates a minimal discrepancy, substantiating the accuracy of the model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Time/s**} & **Simulation Data under** & **Experimental Data** & **Simulation Data under** & **Experimental Data** \\  & **77 N\(\cdot\)m (mm/s)** & **under 77 N\(\cdot\)m (mm/s)** & **105 N\(\cdot\)m (mm/s)** & **under 105 N\(\cdot\)m (mm/s)**  
Compared to the smooth surface spiral bevel gear, the maximum vibrational velocity increases by 5.53% for the spiral bevel gear with a roughness of Ra0.4. Similarly, for the spiral bevel gear with a roughness of Ra0.4, the maximum vibrational velocity experiences a 12.04% increase compared to the smooth surface counterpart. Furthermore, the vibrational velocity of the spiral bevel gear with Ra0.8 roughness is found to increase by 16.44% compared to the maximum vibrational velocity under smooth conditions. \[D=\frac{1.528}{Ra^{0.042}} \tag{41}\]  
Using the proposed computational method, the maximum vibration accelerations for four different roughness levels are determined to be 13.517 m/s\({}^{2}\), 21.525 m/s\({}^{2}\), 23.953 m/s\({}^{2}\), and 27.503 m/s\({}^{2}\). The results reveal a consistent trend where the maximum vibration acceleration increases with greater surface roughness. The relationship between roughness and fractal dimension is reflected in Equation (41). Specifically, for the roughness of Ra0.2, the spiral bevel gear exhibits a 59.2% increase in maximum vibration acceleration compared to its smooth counterpart. Similarly, for the roughness of Ra 0.4, the maximum vibration acceleration increases by 77.21%, and for Ra 0.8, the vibration acceleration surpasses the smooth condition by 103.47%.