*2.2. Calculation of Bearing Cage Dynamics*

The multi-body simulation software Caba3D [29] developed by SCHAEFFLER Technologies AG & Co. KG was used to determine the rolling bearing dynamics. This tool allows the calculation of the dynamics of all rolling bearing components for a previously defined time step and simulation time using a Runge–Kutta-method for performing the numerical time step integration. The results of the multi-body simulation include the kinematics (position, velocity, and acceleration in all degrees of freedom) of the rolling bearing components as well as contact results (pressures, relative velocities, etc.) and node displacements of the elastically modeled cage [23].

The discretization of the contacts rolling element/raceway as well as cage/ring was achieved by means of slices. Contact results such as pressure or forces were calculated for each of the slices and thus resolved locally [22]. For the contact calculation between rolling element and cage pocket, the 'node-to-surface model' was used. This approach determines the contact results using the surface nodes of the finite element (FE) model of the cage and the slices of the rolling element. This allowed the elastic deformations of the cage and their effects on the contact conditions to be determined during the calculation [22]. An elastohydrodynamic model with consideration of mixed friction and the surface roughness was used for the calculation of the friction between rolling elements and raceways. The lubricant film thickness was calculated according to Dowson–Higginson [30]. Coulomb's friction law was used to calculate the frictional force in the contact between the cage and the other bearing elements.

The calculation of the node displacements of a FE model of the rolling bearing cage with several thousand degrees of freedom would be too computationally intensive in the context of a multi-body simulation. Therefore, a model order reduction according to Craig and Bampton [31] was performed to consider the node displacements of the FE model during the dynamics simulation. This allows the number of degrees of freedom to be significantly reduced without a meaningful degradation in accuracy [29]. For the reduction in the FE model, eigenfrequencies up to 20 kHz and a maximum of 100 eigenmodes were considered. The deviation of the eigenfrequencies from the original model and thus the quality of the reduced FE model was verified using various quality criteria (e.g., modal assurance criteria and normalized relative eigenfrequency difference [29,32]).

The modeling considered the angular contact ball bearing without adjacent machine elements consisting of two bearing raceways, the rolling elements, and the outer ring guided cage, see Figure 2.

**Figure 2.** (**a**) Cross-section of the angular contact ball bearing as typically used in machine tools. (**b**) Exploded view of the three-dimensional dynamics simulation model consisting of two raceways, 19 rolling elements, and the outer ring-guided window cage.

The degrees of freedom of the outer ring were disabled, while the other rolling bearing components could move along all six degrees of freedom. The angular contact ball bearing was loaded axially (*F*x) and radially (*F*y) by a force on the inner ring. Further parameters important for the calculation can be found in Table 2.


**Table 2.** Calculation information of the simulation model.

The data-driven approach to employ machine learning methods for cage dynamics prediction requires a high-quality set of data. The source of the data is the multi-body simulation software Caba3D, for which a high correlation with the real cage motion has already been found several times [10,14,21] and is therefore considered as a reliable source for the generation of datasets. Schwarz et al. used a test rig specially developed for testing cages of rolling bearings and high-speed cameras for optical measurement of cage dynamics. As in the calculations, cage instability could be observed in the experiment. For the shape, amplitude, and frequency of the cage deformation, high agreement was found with the measurement results [21].
