# An Advanced DEM-FEM Method for Herringbone Gear in Shot Peening
## 3 Advanced DEM-FEM Modeling for Shot Peening
### Discrete Element Model
#### 3.1.3 Extraction of Impact Information  
Since the herringbone gear tooth surface is a curved surface, the angles between the individual unit positions on the tooth surface and the shot stream are inconsistent, and the impact velocity of the shot and the tooth surface in the DEM discrete element software (2021) is expressed in the global coordinate system. After extracting the impact information, it will be used as an input condition for the finite element software (2019), and the target plate in the finite element is a small part of the tooth surface. Therefore, it is necessary to convert the expressions of the impact velocity and impact position in the global coordinate system into those in the local coordinate system of the tooth surface (2019). As shown in Figure 10, during the shot peening process, the gear rotates around the \(Z_{0}\) axis with an angular velocity of \(\omega\). A new local coordinate system is established on the tooth surface grid units, with the X-axis of the local coordinate system set in the tooth width direction.  
In the global coordinate system (S\({}_{0}\)), the coordinates of the impact point are C\({}_{0}\), and the impact velocity is \(v_{0}\). The corresponding expressions are given by Equations (3) and (4).  
\[C_{0}=[x_{0},y_{0},z_{0}] \tag{3}\]  
\[v_{0}=[v_{x0},v_{y0},v_{z0}] \tag{4}\]  
In the gear coordinate system (S\({}_{g}\)), the coordinates of the impact point are C\({}_{g}\), and the impact velocity is represented as \(v_{g}\).  
\[C_{g}^{T}=M_{0g}\cdot C_{0}^{T} \tag{5}\]  
\[v_{g}^{T}=M_{0g}\cdot v_{0}^{T} \tag{6}\]  
Here, \(M_{0g}\) is the transformation matrix between \(S_{0}\) and \(S_{g}\) after the shot peening time \(t\), and at the initial moment, the gear coordinate system \(S_{g}\) coincides with \(S_{0}\).  
\[M_{0g}=\begin{bmatrix}\cos\alpha&0&-\sin\alpha\\ 0&1&0\\ \cos\alpha&0&\sin\alpha\end{bmatrix} \tag{7}\]  
where \(\alpha\) is the rotation angle of the gear at time \(t\).  
\[\alpha=\omega t \tag{8}\]  
As shown in Figure 10, three adjacent nodes (1, 2, 3) on the tooth surface are used to establish the local coordinate system S. The initial coordinates of the three nodes are represented in \(S_{0}\) as.  
Figure 10: Diagram of global coordinate system (S\({}_{0}\)), gear coordinate system (S\({}_{g}\)), and local coordinate system (S).  
\[\left\{\begin{array}{l}C_{1}=[x_{1},y_{1},z_{1}]\\ C_{2}=[x_{2},y_{2},z_{2}]\\ C_{3}=[x_{3},y_{3},z_{3}]\end{array}\right. \tag{9}\]  
After the passage of time, the node coordinates of the three adjacent nodes (1, 2, 3) are represented in \(S_{0}\) as  
\[\left\{\begin{array}{l}C_{1,0}{}^{T}=M_{t}\cdot C_{1}{}^{T}\\ C_{2,0}{}^{T}=M_{t}\cdot C_{2}{}^{T}\\ C_{3,0}{}^{T}=M_{t}\cdot C_{3}{}^{T}\end{array}\right. \tag{10}\]  
where the transformation matrix \(M_{t}\) is  
\[M_{t}=\begin{bmatrix}\cos\alpha&0&\sin\alpha\\ 0&1&0\\ \cos\alpha&0&-\sin\alpha\end{bmatrix} \tag{11}\]  
Therefore, the basis vector (\(X_{1}\), \(Y_{1}\), \(Z_{1}\)) in the local coordinate system \(S\) can be represented in the \(S_{0}\) coordinate system as  
\[\left\{\begin{array}{l}X_{1}=\dfrac{C_{2,0}-C_{1,0}}{|C_{2,0}-C_{1,0}|}\\ Y_{1}=\dfrac{(C_{2,0}-C_{1,0})\times(C_{3,0}-C_{1,0})\times(C_{2,0}-C_{1,0})}{ |(C_{2,0}-C_{1,0})\times(C_{3,0}-C_{1,0})\times(C_{2,0}-C_{1,0})|}\\ Z_{1}=\dfrac{(C_{2,0}-C_{1,0})\times(C_{3,0}-C_{1,0})}{|(C_{2,0}-C_{1,0})\times( C_{3,0}-C_{1,0})|}\end{array}\right. \tag{12}\]  
The impact velocity in the local coordinate system \(S\) can be expressed as  
\[v^{T}=\begin{bmatrix}X_{1}\\ Y_{1}\\ Z_{1}\end{bmatrix}\cdot v_{0}{}^{T} \tag{13}\]  
In the discrete element impact process, the impact frequency of a certain element on the tooth surface is recorded as \(N_{d}\), and the calculation formula of the unit area impact frequency \(N\) of the element is as follows  
\[N=\dfrac{N_{d}}{S} \tag{14}\]  
where \(S\) is the area of the grid element in EDEM, then the shot peening number is \(k\), the calculation formula of the impact number \(N_{f}\) in the finite element model is as follows  
\[N_{f}=N\times k \tag{15}\]