# A Systematic Approach for Energy-Efficient Design of Rolling Bearing Cages
## 2 Materials and Methods
### Heat Generation Model
The force interactions of the cage with the parts of an ideal geometric shape (Figure 1) were considered in the following zones: A--pressing of the driving rolling element on the jumper by force \(F_{c}(\varphi)\); B--lack of contact between the rolling elements and the cage; C--pressing the brake (driven) rolling element on the jumper by force \(F^{\prime}{}_{c}(\varphi)\); D--braking of the movement of the cage by rolling elements that are not loaded with radial forces and slide along the raceways under the action of resistance from centrifugal forces \(F_{p}\). The rolling element of mass \(m_{r}\), driven by the friction force from the side of the moving inner ring \(F_{\text{fin}}\), perceives the resistance of the friction force from the side of the stationary outer ring \(F_{\text{fout}}\), the forces of interaction with the cage jumper \(F_{c}(\varphi)\), \(f_{c}\cdot F_{c}(\varphi)\), and moments from the components of the external load \(F_{b}(\varphi)\cdot(D_{w}-h_{b})\), \(k\cdot[F_{\text{fin}}(\varphi)+F_{\text{rout}}(\varphi)]\cdot\cos\alpha_{i}\). The system of equations of motion of such an element at the exit from area A of zone \(\psi_{F_{c}}\) (see Figure 2) can be written as follows:  
\[m_{r}\cdot\ddot{x}=F_{\text{fin}}(\varphi)-F_{\text{fout}}(\varphi)+m_{r}\cdot g \cdot\sin\varphi-F_{c}(\varphi)+[F_{\text{rin}}(\varphi)+F_{\text{rout}}( \varphi)]\cdot\sin\alpha_{i}, \tag{6}\]  
\[m_{r}\cdot\ddot{y}=m_{r}\cdot g\cdot\cos\varphi+[F_{\text{rout}}(\varphi)-F_{ \text{rin}}(\varphi)]\cdot\cos\alpha_{i}+f_{c}\cdot F_{c}(\varphi), \tag{7}\]  
\[I_{r}\cdot\varphi=\Big{[}F_{\text{fin}}(\varphi)+F_{\text{fout}}(\varphi) \Big{]}\cdot\frac{D_{w}}{2}-k\cdot[F_{\text{rin}}(\varphi)+F_{\text{rout}}( \varphi)]\cdot\cos\alpha_{i}-F_{b}(\varphi)\cdot(D_{w}-h_{b})-f_{c}\cdot F_{c}( \varphi)\cdot\frac{D_{w}}{2}, \tag{8}\]  
\[\frac{3}{16}m_{r}\cdot D_{w}^{2}\Big{(}\omega_{p1}^{2}-\omega_{p0}^{2}\Big{)} =m_{r}g\cdot(\cos\varphi\ ^{\prime}{}_{0}-\cos\varphi\ ^{\prime}{}_{1})\frac{d_{0}}{2}+\Big{[}F_{\text{ fin}}(\varphi)+F_{\text{fout}}(\varphi)\Big{]}S-\frac{k\cdot S}{D_{w}}[F_{\text{rout}}( \varphi)+F_{\text{rin}}(\varphi)]\cdot\]  
\[M_{c}=M_{br}+M_{j}, \tag{2}\]0 The scheme of force interactions in zone A: 1, 2—outer ring and a rib; 3, 4—inner ring and a rib; 5—rolling body; 6—cage. \[M_{c}=M_{br}+M_{j}, \tag{2}\]1  
\[M_{c}=M_{br}+M_{j}, \tag{2}\]2  
Expressions (6)-(8) are the differential equations of rolling elements motion, expression (9) is the equation of the change in kinetic energy of the rolling element during the movement from the middle of area B to the beginning of area A (see Figure 1), and expression (10) corresponds to the condition of no slippage of the rolling element. Taking into account that the cage perceives the greatest loads at the beginning of the sliding of the rolling element and having adopted the following designations: \(\varphi^{*}\)--the angle of the beginning of sliding of the rolling element in the zone \(\psi_{r}\); and \(\varphi\ ^{\prime}\ _{0}\), \(\varphi\ ^{\prime}\ _{1}\)--the angular positions, \(\omega_{p1}\), \(\omega_{p2}\)--the angular velocities of the rolling element at the beginning and at the end of the movement. By solving the equation of the change in the kinetic energy of the cage during the cycle of its load during the 2S movement, the pressing forces of the driving and driven rolling elements on the jumpers were obtained:  
\[