## 2 Materials and Methods
### Two-Stage Planetary Gearbox  
The most important specifications of the used two-stage planetary gear should be presented. The gearbox has a maximum input speed rotation of 4170 rpm and a maximum input torque of 450 Nm. The specifications of each stage are presented in Table 1.  
The total gear ratio is \(i=37.2\). A planetary gear consists of three main components, namely the ring gear, sun gear and planet gears. In total, there are three types of planetary gearbox designs [9]. They differ in the component that is fixed and therefore not rotating. The presented design is based on a fixed ring gear. The sun gear is in the center, rotating around its own center. The planets are placed between the sun and the fixed ring gear and mesh with both components [10,11]. They rotate orbitally to the sun and, in addition, they rotate around their own center. The used planetary gearbox is a two-stage design. Hence, the planet gears of the first stage are connected to the sun of the second stage by using a planet carrier shaft. The construction design is shown in Figure 1.  
In addition to the gearing components, the two-stage planetary gearbox has two sets of taper roller bearings. Both are mounted between the ring gear of the second stage and the ring gear carrier as depicted in Figure 1. The most important specifications of the applied taper roller bearings are listed in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **First Stage** & **Second Stage** \\ \hline number of teeth sun gear & 15 & 12 \\ number of teeth planet gear & 31 & 23 \\ number of teeth ring gear & \(-78\) & \(-60\) \\ number of planets & 3 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of the investigated two-stage planetary gearbox. Figure 1: Construction draft of a two-stage planetary gear box. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Dimensions** & \multicolumn{2}{c}{**Performance Data**} \\ \hline Inner diameter & 150 mm & \begin{tabular}{c} Basic dynamic load \\ rating, radial \(C_{\mathrm{r}}\) \\ \end{tabular} & 455,000 N \\ \hline Diameter between two opposite rolling elements & 187.8 mm & \
\begin{tabular}{c} Basic static load \\ rating, radial \(C_{\mathrm{0r}}\) \\ \end{tabular} & 680,000 N \\ \hline Outer diameter & 225 mm & Limiting speed \(n_{\mathrm{G}}\) & 3550 1/min \\ \hline Width & 48 mm &
\begin{tabular}{c} Operating \\ temperature range \\ \end{tabular} & \(-30\)–\(200\,^{\circ}\) C \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specifications of the investigated taper roller bearings. A taper roller bearing consists of an outer ring, taper rollers and an inner ring. In the presented application, the outer ring is rotating and the inner ring is stationary. Since both bearing sets are rotating with dependence on the wheel hub, they are investigated in conjunction. For each bearing component, an individual frequency can be calculated. The first factor is a characteristic constant for each component, which shows design dependencies. The second factor is the rotation frequency of the wheel hub \(f_{\rm n}\), which depends on the input rotation speed:  
\[\text{Ball spin frequency }BSF=BSFF\cdot f_{\rm n}=5.06\cdot f_{\rm n}, \tag{1}\]  
\[\text{Ball pass-frequency outer race }BPFO=BPFFO\cdot f_{\rm n}=13.58\cdot f_{\rm n}, \tag{2}\]  
\[\text{Ball pass-frequency inner race }BPFI=BPFFI\cdot f_{\rm n}=16.42\cdot f_{\rm n}, \tag{3}\]  
\[\text{Fundamental train frequency }FTF=FTFF\cdot f_{\rm n}=1.09\cdot f_{\rm n}. \tag{4}\]  
Planetary gearboxes are gaining increasing attention. They are used in transportation systems, aerospace, automotive applications and wind turbines [9; 12]. The main advantages are a compact structure, a stable transmission, a large transmission ratio and high efficiency. Due to rough conditions, the bearings and gearings often show faults like wear, fatigue cracks or pittings. The resulting faulty gearbox can cause operation hazards, with a shutdown as the worst-case scenario and the associated high costs.