## 3 Simulation Model
Subsequently, the identified contact points \({}_{\mathrm{P}}\mathbf{C}_{1}\) on the track and \({}_{\mathrm{P}}\mathbf{C}_{2}\) on the ball surface are transformed into the inertial system and the maximum spatial penetration \(\delta\) is determined for further force calculation  
\[{}_{\mathrm{P}}\mathbf{C}_{1} ={}_{\mathrm{P}}\mathbf{M}_{tr}+{}_{\mathrm{P}}\mathbf{e}_{1}\cdot R _{tr}\quad, \tag{9}\] \[{}_{\mathrm{P}}\mathbf{C}_{2} ={}_{\mathrm{P}}\mathbf{M}_{ball}+{}_{\mathrm{P}}\mathbf{e}_{1}\cdot R _{ball}\quad,\] (10) \[{}_{\mathrm{P}}\mathbf{e}_{1} =\frac{{}_{\mathrm{P}}\mathbf{M}_{ball}-{}_{\mathrm{P}}\mathbf{M}_{tr}}{ ||_{\mathrm{P}}\mathbf{M}_{ball}-{}_{\mathrm{P}}\mathbf{M}_{tr}||}\quad,\] (11) \[{}_{\mathrm{d}} ={}_{\mathrm{1}}\mathbf{C}_{2}-{}_{\mathrm{1}}\mathbf{C}_{1}\quad,\] (12) \[\delta =||\mathbf{d}||\quad. \tag{13}\]  
Here, \({}_{\mathrm{P}}\mathbf{M}_{tr}\) is the centre of the track flank, \({}_{\mathrm{P}}\mathbf{M}_{ball}\) is the centre of the ball, \(R_{tr}\) is the track radius, \(R_{ball}\) is the ball radius, \(d\) is the vectorial penetration between the contact bodies and \(||\mathbf{d}||\) is the Euclidean norm of it. Figure 2: Position of an exemplary track coordinate system and the contact plane using the example of the inner race: (left) spatial presentation; and (right) top view. An analytical approach is also chosen for the cage ball contact, but this is implemented three-dimensionally (see Figure 3b). First, a contact plane is defined on the respective window edge by a normal vector \({{}_{\mathrm{CA}}}n_{P}\) and a receiving point \({{}_{\mathrm{CA}}}{{}_{\mathrm{CF}}}_{\mathrm{CF}}\) in the body fixed cage coordinate system \({{}_{\mathrm{CA}}}e\). Then, a potential contact point on the sphere surface  
\[{{}_{\mathrm{CA}}}{{}_{\mathrm{2}}}={{}_{\mathrm{CA}}}{{}_{\mathrm{{\bf r}} ball}}+{{}_{\mathrm{CA}}}n_{P}\cdot{{}_{\mathrm{R}ball}} \tag{14}\]  
is defined from the normal vector of the plane \({{}_{\mathrm{CA}}}{{}_{\mathrm{{\bf r}}}}_{P}\), the centre of the sphere \({{}_{\mathrm{CA}}}{{}_{\mathrm{{\bf r}}}}_{ball}\) and the radius of the sphere \({{}_{\mathrm{R}ball}}\). After a transformation of the potential contact point on the ball \({{}_{\mathrm{CA}}}{{}_{\mathrm{2}}}\) and the normal vector of the contact plane \({{}_{\mathrm{CA}}}{{}_{\mathrm{{\bf r}}}}_{P}\) into the inertial system \({{}_{\mathrm{1}}}e\) (resulting in \({{}_{\mathrm{1}}}{{}_{\mathrm{C}}}_{\mathrm{2}}\) and \({{}_{\mathrm{1}}}n_{P}\)), the distance between plane and potential contact point can be determined  
\[\delta=\frac{{{}_{\mathrm{1}}}n_{P,x}\cdot{{}_{\mathrm{C}}}_{2,x}+{{}_{\mathrm{ 1}}}n_{P,y}\cdot{{}_{\mathrm{1}}}{{}_{\mathrm{C}}}_{2,y}+{{}_{\mathrm{1}}}n_{ P,z}\cdot{{}_{\mathrm{1}}}{{}_{\mathrm{C}}}_{2,z}}{||{{}_{\mathrm{1}}}n_{P}||}\quad. \tag{15}\]  
Finally, if the contact is positive for \(\delta>0\), the contact point in the cage window can be calculated as the point for the force application  
\[{{}_{\mathrm{1}}}{{}_{\mathrm{C}}}_{1}={{}_{\mathrm{1}}}{{}_{\mathrm{C}}}_{2}- {{}_{\mathrm{1}}}n_{P}\cdot\delta\quad. \tag{16}\]  
In the present work, a one-dimensional approach is used to calculate the contact forces with high efficiency, since this approach provides sufficient accuracy while maintaining a low level of complexity. In this case, the contact force acts exclusively along a direction vector between two contact points. The elastic component of the contact normal force is therefore calculated with the previously determined penetration based on the Hertzian contact theory [19], so that the penetration between the bodies can be interpreted as a local elastic deformation (see Figure 4). Mathematically, this can be described as a power law according to [PERSON] [20].