## 3 Simulation Model  
To analyse the joint kinematics using multi-body simulation, it is first necessary to define suitable coordinate systems. Therefore, a Cartesian inertial system is introduced, whose global z-axis corresponds to the rotational axis of the outer race. In addition, body-fixed coordinate systems are defined on the inner race \({}_{\text{IR}}\mathbf{e}\), the outer race \({}_{\text{OR}}\mathbf{e}\) and the cage \({}_{\text{CA}}\mathbf{e}\), whose origins lie in the joint system plane (see Figure 1). Starting from the body-fixed coordinate systems, several coordinate systems are introduced for each track and for each cage window. While the alignment of the coordinate systems in the cage window corresponds to the cage fixed coordinate system, the track coordinate systems \({}_{\text{tr}}\mathbf{e}\) follow the track orientation with the two track angles \(\mathbf{\alpha}_{incl}\) and \(\mathbf{\beta}_{incl}\). The transformation into the track coordinate system is performed sequentially using the matrix  
\[\mathbf{Q}_{\text{trI}}=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos(\mathbf{\alpha}_{incl})&-\sin(\mathbf{\alpha}_{incl})\\ 0&\sin(\mathbf{\alpha}_{incl})&\cos(\mathbf{\alpha}_{incl})\end{array} \right]\cdot \left[ \begin{array}{ccc}\cos(\mathbf{\beta}_{incl})&0&\sin(\mathbf{\beta}_{incl}) \\ 0&1&0\\ -\sin(\mathbf{\beta}_{incl})&0&\cos(\mathbf{\beta}_{incl})\end{array} \right]\quad. \tag{8}\]  
In Figure 2, the position of the described track coordinate systems is illustrated using the inner race as an example. Based on the defined coordinate systems, various coordinate transformations can be carried out to enable analytical contact determination for the entire joint. This applies to the ball track as well as to the ball cage contact, which is explained in detail in the following. Up to six potential contact points are possible per ball, two per track and two more in the cage window. In the entire joint, therefore, a large number of potential contact points must be evaluated for each time step of the numerical integration, which can lead to high computing times under arbitrary circumstances. The considered track geometry allows implementing an analytical contact determination, which is based on the observation of simple geometric shapes. Thus, a complex three-dimensional contact analysis can be obviated, ensuring a fast and stable contact routine. This also guarantees that an efficient contact routine is maintained for different variations of the track parameters,e.g., the ball size or the PCD. This is necessary because a low computing time with sufficient accuracy is of utmost importance for parameter studies and design optimisation in an early step of the product development process. The analytical contact determination is derived from the joint system plane for both ball track contact and ball cage contact for the undeflected state of the joint. Based on the considered track angles of each track, the balls move out of the joint system plane as a result of the deflection or axial displacement of the inner race. Hence, both the PCD and the distance between the ball centres along the PCD change. This leads to additional transformation steps. For ball track contact, the current ball position is evaluated with respect to the track coordinate system \({}_{\mathrm{tr}}e\), so that a corresponding contact plane \({}_{\mathrm{p}}e\) (red, Figure 2) can be defined orthogonal to the track axis as a function of the ball position. After the transformation of all necessary coordinates into the contact plane, the two-dimensional contact determination is performed by calculating the distance between the centres of two circles in order to identify the maximum plane penetration \(d\) between \({}_{\mathrm{P}}\mathbf{C}_{1}\) and \({}_{\mathrm{P}}\mathbf{C}_{2}\) of the contact bodies along the normal vector \({}_{\mathrm{P}}\mathbf{e}_{1}\) (see Figure 3a).