*2.3. Simulation Plan for the Cage Geometry and Bearing Load*

Twenty cage variants were generated for the angular contact ball bearing, and their dynamic behavior was calculated for 100 different operating conditions using multi-body simulations as described in Section 2.2. In total, 2000 dynamics simulations were performed. Figure 3 illustrates the geometry parameters used to generate different cage designs. The chosen parametrization of the cage geometry enables the shape to be represented as generically as possible by 7 parameters. This allows cages with different properties to be created in the given design space and their dynamic behavior to be investigated. Using the parameter *d*g, the clearance between the cage and the outer ring and thus the guidance clearance can be influenced, see Figure 3a. The cross-section of the cage is defined by the height *h*<sup>c</sup> and width of the cage *b*c, see Figure 3a. Both parameters affect important properties such as mass, moment of inertia, and stiffness of the cage. The shape of the cage pocket was varied using the parameters *c*0, *c*1, *c*2, and *c*3, which represent the pocket clearance along the circumference, see Figure 3b.

By choosing the pocket shape parameters, the pocket clearance of the cage on the one hand and the contact point between cage and the rolling element on the other hand can be influenced. The pocket clearance has a significant effect on the cage dynamics, as the number of contacts to the rolling elements increases with decreasing pocket clearance and can cause highly dynamic cage movements [21]. The contact point between the rolling element and the cage defines the direction of the normal and frictional force vector in the contact and finally the direction of the cage acceleration.

**Figure 3.** (**a**) Cross-section of the angular contact ball bearing cage. (**b**) Three-dimensional view of cage pocket and a rolling element. The blue area represents the geometry of the cage pocket and shows an exemplary shape defined by four parameters *c*0, *c*1, *c*2, and *c*3.

Using the geometry parameters, a total of 20 different cage variations were created using Latin hypercube sampling. The boundaries for the sampling shown in Table 3 were chosen in such a way that there are no dependencies between the cage design parameters. For the smallest guidance diameter *d*g and largest cage height *h*c, the clearance cage/inner ring is greater than the clearance cage/outer ring, and the same guidance type is provided.

Besides the modifications of the cage geometry, the load on the rolling bearing was also modified using an additional Latin hypercube sampling. The forces acting on the inner ring were varied using the load ratio *R* and the equivalent dynamic bearing load *P*. Based on the two parameters in Equations (1) and (2), the forces *F*x and *F*y to be defined in the simulation can be calculated. In addition to the forces, the inner ring was also loaded by the torque *T*<sup>z</sup> acting around the z-axis, see Figure 2. The frictional force in the rolling element/cage contact was varied via the coefficient of friction *μ*c.

$$P = X \cdot F\_{\mathbf{x}} + Y \cdot F\_{\mathbf{y}} \tag{1}$$

$$R = \frac{F\_{\chi}}{F\_{\chi}} \tag{2}$$

The speed of the inner ring *n*<sup>i</sup> was also taken into account in the sampling. The kinematic speed of the rolling elements *n*r and the cage *n*c at the beginning of the simulation were determined for the initial time step by the Equations (3) and (4) depending on the defined inner ring rotational speed [33].

$$m\_{\rm tr} = -\frac{n\_{\rm i}}{2} \cdot \left(\frac{d\_{\rm P}}{d\_{\rm re}} - \frac{d\_{\rm re} \cdot \cos^2(a)}{d\_{\rm P}}\right) \tag{3}$$

$$m\_{\mathbb{C}} = \frac{n\_{\mathbb{I}}}{2} \cdot \left(1 + \frac{d\_{\text{re}} \cdot \cos(a)}{d\_{\mathbb{P}}}\right) \tag{4}$$

A simulation plan consisting of a total of 100 operating conditions (inner ring rotational speed, force and torque on the inner ring, and friction coefficient in the rolling element/cage contact) was created using the boundary values in Table 3 and Latin hypercube sampling. Simulation models were generated for each of the 20 cage variants according to the same operating conditions defined by the created simulation plan, so that a total of 2000 dynamics simulations were performed.

**Table 3.** Minimum and maximum values of the parameters for the Latin hypercube sampling.

