*3.1. Dynamics Simulation Results*

The results of the dynamics simulations contain time series that include dynamics of the cage as well as the rolling elements. Figure 5 shows an example of the dynamic behavior of a cage for different operating conditions of the bearing. In the qualitative assessment of cage dynamics, a fundamental differentiation is made between "unstable", "stable", and "circling" cage motions [21,34]. These types of movements could also be observed for the cages investigated. Figure 5a–c illustrates an example of an unstable cage motion (loading conditions *μ*<sup>c</sup> = 0.21, *F*<sup>x</sup> = −8058 N, *F*<sup>y</sup> = 1077 N, *n*<sup>i</sup> = 8263 rpm *T*<sup>z</sup> = 48 Nm), that is characterized by high dynamics as well as severe and high-frequency cage deformations. The cage was pressed against the outer ring and strongly deformed. This led to the diameter of the circular center of gravity trajectory being significantly larger than in the other two calculations. In addition, high contact forces caused frictional losses, which significantly impair the energy efficiency of the rolling bearing. In the case of stable cage motion (loading conditions *μ*<sup>c</sup> = 0.26, *F*<sup>x</sup> = −15,126 N, *F*<sup>y</sup> = 1636 N, *n*<sup>i</sup> = 4407 rpm *T*<sup>z</sup> = 14 Nm), no significant deformations occurred and the dynamics of the cage were generally low, see Figure 5d–f. The contact forces between the cage and the rolling element and outer ring were also significantly reduced compared to an unstable motion, and therefore the frictional losses were also lower. The circling cage motion (loading conditions *μ*<sup>c</sup> = 0.12, *F*<sup>x</sup> = −3030 N, *F*<sup>y</sup> = 377 N, *n*<sup>i</sup> = 6844 rpm *T*<sup>z</sup> = 12 Nm) is characterized by a circular motion of the cage center of mass that exhibits small variations in the rotational speed. The rotational speed of the cage center of mass corresponds to the speed of the rolling element set. The cage is pressed in a radial direction due to the centrifugal force acting, so that the number of contacts to the guidance rib and the contact force acting in the contact increase.

**Figure 5.** Dynamic behavior of a cage for different operating conditions: an unstable (**a**–**c**), stable (**d**–**f**), and circling (**g**–**i**) cage motion. The three-dimensional deformation of the cages, the center of gravity trajectory, and the amplitude spectrum of the node displacement are illustrated.

In addition to the load on the bearing, the geometry of the cage can also influence the dynamic response of the bearing. Figure 6 shows the cage dynamics for a load situation (*μ*<sup>c</sup> = 0.16, *F*<sup>x</sup> = −9655 N, *F*<sup>y</sup> = 3718 N, *n*<sup>i</sup> = 6844 rpm *T*<sup>z</sup> = 16 Nm) and three different cage geometries. The first cage variant performed a highly dynamic cage motion with severe deformations and a high rotational speed of the cage center of mass, see Figure 6a–c. A modification of the cage geometry (cross-section and shape of the cage pocket) for the other two variants and the same operating conditions led to circling cage motions in both cases. The amplitudes of the deformations were significantly smaller compared to the first cage variant and the larger amplitudes were shifted to the low frequency range, see Figure 6d–i.

**Figure 6.** Dynamic behavior of three different cage variants for the same loading conditions. The three-dimensional deformation of the cages (**a**,**d**,**g**), the center of gravity trajectory (**b**,**e**,**h**), and the amplitude spectrum of the node displacement (**c**,**f**,**i**) are illustrated.

An overview of the simulations performed and the resulting cage motion types is shown in Figure 7. Certain cage geometries (ID 02 or 05) had a high proportion of unstable cage motions, while other cage variants exhibited a much lower tendency to unstable cage motions (ID 14 or 10). In addition, differences in the proportion of circumferential and stable cage movements were also evident for the different cage variants. The dynamic behavior of the cage variants illustrates the potential of the geometry parameters to positively influence the dynamics of the cage. A clear influence could also be identified in the loading conditions, as was found, for example, by Schwarz et al. [14]. However, as the operating conditions often cannot be influenced, these serve only as a reference for comparing the dynamic behavior of the cage geometries.

**Figure 7.** Overview of the results of the dynamics simulation. (**a**) Number of motion types "unstable", "stable", and "circling" for each cage variant. (**b**) Motion type as a function of cage geometry and bearing load. (**c**) Number of motion types for each bearing load in the experimental design.

The simulation results were further processed so that the influence of cage geometry and bearing load was represented by a database consisting of input and target variables and could be used for machine learning.
