# Dynamic Modeling and Output Characteristics Analysis of the Hub-Drive Reduction System
## 4 Structural Parameters and Characteristic Frequency Analysis  
In the wheel hub transmission reduction system, a two-stage planetary gear reducer serves as the core speed-reduction component, determining the operational stability and reliability of the entire system. During the design phase, theoretical parameters were used for matching calculations. However, due to the large number of components, complex connections, and stringent manufacturing and assembly precision requirements, the dynamic performance of the system could fluctuate during actual operation. Simple theoretical models cannot accurately describe these behaviors. Therefore, before conducting numerical simulations, the structural parameters and characteristic frequencies of the two-stage planetary gear reducer had to be analyzed to support the subsequent research on the system's dynamic performance. In a two-stage planetary gear reducer, achieving a stable transmission ratio is critical for both reducing speed while increasing torque and ensuring smooth operation. Figure 4 illustrates the transmission principle of the two-stage planetary gear reducer. Here, R1, P1, S1, and C1 represent the primary ring gear, primary planetary gear, primary sun gear, and primary planet carrier, respectively. Similarly, R2, P2, S2, and C2 represent the secondary ring gear, secondary planetary gear, secondary sun gear, and secondary planet carrier, respectively. From the transmission principal diagram, it is evident that the primary and secondary ring gears are fixed and connected to the ground. The primary sun gear meshes with the three primary planetary gears, transmitting power to the primary planet carrier. This carrier, in turn, engages with the secondary sun gear through internal and external ring gear meshing, transmitting power to the secondary planetary gears. Finally, the secondary planet carrier outputs the power. The transmission ratio can be calculated using Equations (1) and (2):  
\[i_{10}=\frac{n1}{n0} \tag{1}\]  
\[i_{0}=1+\frac{n_{w}}{n0} \tag{2}\]  
where \(i_{10}\) represents the transmission ratio between the sun gear and the planetary gear, \(i_{0}\) is the transmission ratio between the sun gear and the planet carrier, \(n1\) is the number of teeth on the planetary gear, \(n_{w}\) is the number of teeth on the ring gear, and \(n0\) is the number of teeth on the sun gear. The specific parameters of the two-stage planetary gear reducer in the high-torque wheel hub drive system under study are shown in Table 2. The gear accuracy grade refers to the AGMA 2015 A standard [33]. Additionally, to reduce interference during gear meshing, the gears of the reducer underwent a certain degree of profile modification. Based on the transmission ratio calculation formulas and the motor's input angular velocity, the transmission ratios and corresponding angular velocities of each planetary gear stage are listed in Table 3.  
In the dynamic analysis of the two-stage planetary gear system, attention was given not only to the stability of the transmission ratio and the angular velocity output but also to the meshing frequencies between the gears. The meshing frequency was crucial for both understanding the system's dynamic characteristics and for later stages of fault diagnosis, stiffness matching, decoupling calculations, and reliability analysis. Both stages of the planetary gear reducer take the sun gear as the input and the planet carrier as the output, with the ring gear fixed. The calculation of the meshing frequency follows the formula:  
\[f=z\frac{n-n_{p}}{60} \tag{3}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Gear Component** & **Number of Teeth** & **Module (mm)** & **Pressure Angle (\({}^{\circ}\))** \\ \hline Primary Sun Gear & 19 & 7.65 & 25 \\ Primary Planetary Gear & 53 & 7.65 & 25 \\ Primary Ring Gear & 128 & 7.65 & 25 \\ Secondary Sun Gear & 17 & 12.89 & 25 \\ Secondary Planetary Gear & 30 & 12.89 & 25 \\ Secondary Ring Gear & 79 & 12.89 & 25 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of the two-stage planetary gear reducer.