## 2 Background
### Ball Bearings  
The presented work uses data from ball bearings with different damages, types, and states of wear. All of them are created in the laboratory. To create different types and states of wear that can occur in normal operation, material is removed manually with a grinder to create pits. Bearings may corrode depending on the environment during operation. This is reproduced by degreasing the bearings with subsequent repeated dipping in saltwater. The result is shown in Figure 1. If a ball bearing is overloaded, this can lead to rolling bodymarks. This case is reproduced with a static overload of the bearing. The used bearing type is 6203 from SKF. This is a single-row ball bearing without sealing that enables access to the running surface and the balls. The whole list of the tested damages is shown in Table 1.  
The evaluation of bearings related to the VDI regulation 3832 [16] can be done by considering the envelope curve spectrum. Each defect has characteristic frequencies visible in the frequency spectrum. A ball bearing with an advanced damage on the outer ring has characteristic lines at rollover frequency (\(f_{A}\)) for the outer ring and its multiples (see Figure 2). The rollover frequency can be calculated with the formula below with the rotational frequency of the rotor, \(f_{n}\); the number of rolling elements, Z; diameter of the rolling element, \(D_{W}\); contact angle, \(\alpha\); and pitch circle diameter, \(D_{PW}\)[16]:  
\[f_{A}=\frac{1}{2}f_{n}\cdot Z\left[1-\frac{D_{W}\cdot cos\alpha}{D_{PW}}\right] \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Class** & **Damage** \\ \hline
0 & Undamaged \\ \hline
1 & Degreased \\ \hline
2 & Light wear outer ring \\ \hline
3 & Medium wear outer ring \\ \hline
4 & Heavy wear outer ring \\ \hline
5 & Heavy wear inner ring \\ \hline
6 & Medium wear inner ring \\ \hline
7 & Light wear inner ring \\ \hline
8 & Light Corrosion \\ \hline
9 & Medium corrosion \\ \hline
10 & Heavy corrosion \\ \hline
11 & Rolling body marks \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Classes of damage. Figure 1: Ball bearing with heavy corrosion. The characteristic line of a damaged inner ring is completely different. The Sidebands are distributed around the rollover frequency (\(f_{I}\)) of the inner ring and its multiples with the distance of \(f_{n}\) (the rotor rotational frequency), see Figure 3. The rollover frequency can be calculated with the formula below with the rotational frequency of the rotor, \(f_{n}\); the number of rolling elements, Z; diameter of the rolling element, \(D_{W}\); contact angle, \(\alpha\); and pitch circle diameter, \(D_{PW}\)[16]:  
\[f_{I}=\frac{1}{2}f_{n}\cdot Z\bigg{[}1+\frac{D_{W}\cdot cos\alpha}{D_{PW}}\bigg{]} \tag{2}\]  
Figure 3: Envelope curve spectrum of a ball bearing with advanced inner ring damage. Figure 2: Envelope curve spectrum of a ball bearing with advanced outer ring damage. Ball bearings with a rolling element damage have sidebands around the rollover frequency of the rolling element (\(f_{W}\)) and its multiples with the distance of \(f_{K}\) (the rotational frequency of the cage), see Figure 4. The rollover frequency of the rolling element can be calculated with the formula below with the rotational frequency of the rotor, \(f_{n}\); diameter of the rolling element, \(D_{W}\); contact angle, \(\alpha\); and pitch circle diameter, \(D_{PW}\)[16]:  
\[f_{W}=\frac{f_{n}\cdot D_{PW}}{D_{W}}\Bigg{[}1-\left(\frac{D_{W}\cdot cos\alpha }{D_{PW}}\right)^{2}\Bigg{]} \tag{3}\]  
The rotational frequency of the cage can be calculated with the formula below with the rotational frequency of the rotor, \(f_{n}\); diameter of the rolling element, \(D_{W}\); contact angle, \(\alpha\); and pitch circle diameter, \(D_{PW}\)[16]:  
\[f_{K}=\frac{1}{2}f_{n}\bigg{[}1-\frac{D_{W}\cdot cos\alpha}{D_{PW}}\bigg{]} \tag{4}\]  
By comparing Figures 2-4, it becomes obvious that the envelope curve is completely different for each type of damage.