# A Systematic Approach for Energy-Efficient Design of Rolling Bearing Cages
## 2 Materials and Methods  
### Heat Generation Model  
Energy losses due to the design of the cage in the rolling bearing are more convenient for determining the effect of heat generation on the friction surfaces by calculating the temperature gradient [26]:  
\[\Delta T=T_{b}-T_{e}=P_{fr}/K_{T}, \tag{1}\]  
where \(T_{b}\), \(T_{e}\)--respectively, the temperature of the bearing and the environment, \(K_{T}\)--cooling factor, \(P_{fr}\)--power loss due to friction, \(P_{fr}=\frac{M_{c}\cdot n}{950}\), where \(n\)--cage rotation frequency. The frictional moment due to the design of the cage \(M_{c}\), without taking into account losses in lubrication and sealing, can be defined as the sum of the frictional moment that is caused by the contact of the cage with the sides of the bearing base ring \(M_{br}\) and the frictional moment that is caused by the contact of the rolling elements with the jumpers of the cage \(M_{j}\)[27; 28]:  
\[M_{c}=M_{br}+M_{j}, \tag{2}\]  
where  
\[M_{br}=1.38\cdot 10^{-4}\cdot W_{c}\cdot f_{b}\cdot n^{2}\cdot d_{out}\cdot \varepsilon\left(\frac{d_{o}-D_{br}\cdot\cos\alpha}{d_{o}}\right)^{2}, \tag{3}\]  
\[M_{j}=\frac{d_{o}}{4}\left(1-\frac{D_{br}^{2}}{d_{o}^{2}}\cos^{2}\alpha\right) \cdot\sin\left[\alpha+\text{arctg}\left(\frac{D_{br}\cdot\sin\alpha}{2\cdot r _{g}}\right)\right]W_{c}\cdot f_{c}. \tag{4}\]where \(W_{c}\)--cage weight, \(f_{b}\)--cage with the ring friction coefficient, \(f_{c}\)--cage with the rolling element friction coefficient, \(\varepsilon\)--cage eccentricity, \(d_{out}\)--outer ring diameter, \(d_{o}\)--rolling element center diameter, \(D_{w}\)--rolling element diameter, \(r_{g}\)--bearing ring groove radius, \(\alpha\)--bearing contact angle. The obtained expressions take into account the cage weight, the main geometric parameters of the bearing parts, the rotation frequency of the cage, and the friction coefficients of the parts. At the same time, the forces of interaction between the cage and the parts of the bearing are not considered. Such expressions can be used for calculations of light-load bearings. Taking into account the forces of interaction between the driving and driven rolling elements with the jumpers of the cage in the radial load zone of the bearing \(F_{c}(\varphi)\), \(F^{\prime}_{c}(\varphi)\), the frictional moment of the bearing that is caused by the design of the cage, without taking into account losses in lubrication and sealing, can be written:  
\[M_{c}=0.5d_{bs}\cdot f_{b}\big{[}F_{c}(\varphi)+F^{\prime}_{c}(\varphi)]\cos \varphi-0.5D_{w^{\prime}}f_{c}\big{[}F_{c}(\varphi)+F^{\prime}_{c}(\varphi) \big{]}, \tag{5}\]  
To determine the corresponding pressure forces on the driving and driven rolling elements in the radial load zone, a flat model of the interaction of the rolling elements with the cage is proposed (Figures 1 and 2). Figure 1: The scheme of the interaction of rolling elements with the cage. When determining the forces, we relied on the bearing kinematics model developed in [20]. The flat model is built in the zone of the bearing's radial load, where the transfer of motion from the rolling elements to the cage occurs intermittently because of the gradual reduction of the gaps in the windows of the cage. The transformation of the brake (driven) rolling element, which relies on the movement of the cage, into the driver, which pushes the cage, is carried out in the radial load zone of the bearing. The intermittent nature of the cage's movement that is caused by the relative difference in the speeds of the cage and the rolling elements leads to shock loading of the cage. This determines the internal dynamics of the rolling bearing.