## 2 Finite Element Model of Cylindrical Roller Bearing
### Finite Element Model of Rolling Bearing  
Choosing N205 EM cylindrical roller bearings as the research object, this paper utilizes the finite element software ANSYS 2022 R2/LS-DYNA to simulate and calculate the dynamic characteristics of bearings with different failure sizes of the outer ring. The geometrical parameters of rolling bearings are presented in Table 1, and the geometrical schematic is shown in Scheme 1.  
To accurately represent the bearing's operational state, a plane strain element is employed for its effectiveness in capturing two-dimensional deformation characteristics. A plane strain element is chosen to simulate the two-dimensional shell element as the solid structure of the rolling bearing. The bearing materials are cast steel and austenitic stainless steel, with the specific materials for the inner and outer rings, rolling elements, and c The finite element models with element sizes of 0.75 mm, 0.5 mm, 0.25 mm, and 0.15 mm were thoroughly tested in this study to minimize the effect of the selected mesh size on the analysis results. The test results indicated that when the element sizes were 0.75 mm and 0.5 mm, the contact effect between the rolling elements and the raceway was suboptimal. However, with element sizes of 0.25 mm and 0.15 mm, ideal continuous contact between the rolling elements and the raceway was achieved. Therefore, a 0.25 mm quadrilateral mesh was used to discretize the components of the bearing, resulting in a final model with 47,475 nodes and 45,057 elements. The discretized rolling bearing finite element model is presented in Figure 1. The local spalling fault was simplified as a rectangular pit to investigate the dynamic response characteristics and crack propagation trends of the N205 EM bearing under different fault sizes on the outer ring. The fault is located on the inner raceway of the outer ring, with the fault length denoted as L and the depth as H. Specific fault size parameters are detailed in Table 3.  
\begin{table}
\begin{tabular}{c c c} \hline
**Geometric Parameters** & **Number** \\ \hline Roller diameter, D/mm & 6.75 \\ Bearing joint circle diameter, dm/mm & 38.5 \\ Bearing inner roller diameter, Di/mm & 31.75 \\ Bearing outer ring roller channel diameter, Do/mm & 45.25 \\ Outer circle thickness, T/mm & 15 \\ Contact angle, \(\alpha/^{\circ}\) & 0 \\ Number of rolling bodies, Z & 13 \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of N205 EM bearing.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Name** & **Part** & **Materials** & **Density/(kg\(\cdot\)m\({}^{-3}\))** & **Elastic** & **Poisson’s Ratio** \\ \hline \multirow{3}{*}{Cast steel} & Inter/outer ring & GCr15 & 7830 & 206 & 0.3 \\  & Rolling element & GCr15 & 7830 & 206 & 0.3 \\  & Cage & Brass & 8500 & 105 & 0.32 \\ \hline \multirow{3}{*}{Austenitic stainless steel} & Inter/outer ring & AISI304 & 7930 & 193 & 0.29 \\  & Rolling element & AISI304 & 7930 & 193 & 0.29 \\ \cline{1-1}  & Cage & Brass & 8500 & 105 & 0.32 \\ \hline \end{tabular}
\end{table}
Table 2: Material parameters of bearing.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Fault** & **Trouble-Free** & **Type 1** & **Type 2** & **Type 3** & **Type 4** & **Type 5** \\ \hline L/mm & 0 & 0.25 & 0.50 & 0.75 & 1.00 & 1.25 \\ H/mm & 0 & 0.3 & 0.3 & 0.3 & 0.3 \\ \hline \end{tabular}
\end{table}
Table 3: Bearing outer ring fault size.  
Then, loads, rotational speeds, and boundary conditions consistent with actual operating conditions were applied to ensure that the simulation model accurately reflected the real service state. Meanwhile, a fixed constraint was applied to the outer surface of the bearing's outer ring. Additionally, a radial load of F = 1500 N was applied to the lower half of the inner surface of the inner ring, acting in the negative Y direction. Furthermore, a rotational speed of \(n_{i}=1300\) r/min was applied to the inner surface of the inner ring, with the rotation direction being counterclockwise and a sampling frequency of 48 kHz.