## 2 Theory  
Figure 1 illustrates the basic design of a constant velocity ball joint of this type. The inner race and the outer race are each connected to a rotating shaft and have several tracks, so that a positive transmission of the torque takes place in the joint between the name-giving balls and the tracks. Further, for numerical analysis using multi-body simulation, it is necessary to define fixed coordinate systems for all joint components in addition to the inertial system \({}_{1}\epsilon\), which are also shown in Figure 1. For the undeflected state of the joint, all ball centres in the system plane are located on the pitch circle diameter (PCD) of the joint. As the joint rotates, the balls are positioned over the ball cage in the bisecting plane to ensure uniform rotary transmission [12]. For this purpose, it has recesses called cage windows. In addition, the joint is filled with grease to achieve a better tribological behaviour and to reduce wear. Depending on the intended use of the joint, the designs of the track geometries differ. The analysis of the CVJ is carried out by applying the multi-body simulation under the assumption of rigid bodies. The algorithm was implemented in the MBS software EMD, which was developed at the Institute of Mechanics of the Otto von Guericke University Magdeburg. A detailed description of the functionality can be found in the dissertation of [PERSON] [13]. First, the differential equations of motion for spatial dynamics are established  
\[\mathbf{M}\cdot\mathbf{a}+\mathbf{h}_{\omega}=\mathbf{h}_{a}\quad, \tag{1}\]  
where \(\mathbf{M}\) is the mass matrix, \(\mathbf{a}\) is the acceleration vector, \(\mathbf{h}_{a}\) is the vector of external forces and moments and \(\mathbf{h}_{\omega}\) is the vector of centrifugal, gyroscopic and Coriolis forces. A derivation of the individual components can be found in [14,15]. Considering only free bodies, the analysis is performed using an ODE solver for the numerical solution of the differential equation. For this reason, the ODE of second order must be transformed into a differential equation system of first order. For this purpose, the state space vector and its time derivative  
\[{}_{1}\mathbf{z} =\left[ \begin{array}{cccc}\mathbf{\mathrm{i}}\mathbf{r}_{A}&\mathbf{p}&\bm {\mathrm{i}}\dot{\mathbf{r}}_{A}&\mathbf{\mathrm{i}}\mathbf{\omega}\end{array} \right]^{T}\quad, \tag{2}\] \[{}_{1}\mathbf{\dot{z}} =\left[ \begin{array}{cccc}\mathbf{\mathrm{i}}\dot{\mathbf{r}}_{A}&\dot {\mathbf{p}}&\mathbf{\mathrm{i}}\dot{\mathbf{r}}_{A}&\mathbf{\mathrm{i}}\dot{\mathbf{\omega}} \end{array} \right]^{T}\quad, \tag{3}\]  
are defined, which contain the positions \(\mathbf{\mathrm{i}}\mathbf{r}_{A}\), the translational velocities \(\mathbf{\mathrm{i}}\dot{\mathbf{r}}_{A}\) and the accelerations \(\mathbf{\mathrm{i}}\dot{\mathbf{r}}_{A}\) relative to the inertial system \(\mathbf{\mathrm{i}}\mathbf{\mathrm{e}}\) as well as the orientation parameters \(\mathbf{p}\) and the angular velocities \(\mathbf{\mathrm{i}}\mathbf{\omega}\) and their time derivatives. The translational parameters are specified in Cartesian coordinates  
\[\mathbf{r}_{A}=\left[ \begin{array}{cccc}x&y&z\end{array} \right]^{T}\quad, \tag{4}\]  
while different forms are implemented for the orientation parameters \(\mathbf{p}\). Cardan angles are used to analyse the results, while quaternions are used to solve the system of equations in order to avoid the gimbal lock [16]  
\[\mathbf{p}=\left[ \begin{array}{cccc}\alpha&\beta&\gamma\end{array} \right]^{T} \quad\text{and}\quad\mathbf{p}=\left[ \begin{array}{cccc}p_{0}&p_{1}&p_{2}&p_{3} \end{array} \right]^{T}\quad. \tag{5}\]  
Figure 1: Components of a CVJ. After the transformation into the state space and the conversion according to the unknown time derivative, the equation  
\[\dot{\mathbf{z}}=\mathbf{M}^{-1}\left(\mathbf{h}_{a}(\mathbf{z})-\mathbf{h}_{\omega}(\mathbf{z})\right)\quad, \tag{6}\]  
results, which can be committed to the solver.