## 4 Mathematical Description for Linear Bearings
### Mathematical Model of Linear Contact Ball Bearing  
Pressure-induced stresses are important in the joints between elastic bodies in the design of bearings, gears or other components in contact. The field of science describing this phenomenon is known as contact mechanics. [PERSON] [25] described the basic mathematical relationships of surface stresses and deformations created during pressure. [PERSON]'s theory of pressure states that the stress on the contact surface usually significantly exceeds the yield point or even the strength limit, which can be explained by the occurrence of a state of triaxial compression [25]. [PERSON]'s equations have been supported by experimental studies carried out for gearboxes or ball and roller bearings [29]. We assume that the connection of the balls in the bearing with the raceways is so rigid that there is no slippage effect, so a ball bearing in a static state can be considered. The nonlinear properties of contact are based on the [PERSON] theory of pressures of two bodies in contact and loaded with a total force \(\text{F}_{\text{r}}\)[25], where \(\text{F}_{\text{r}}\) is the external load. The nonlinearity  
Figure 10: Elliptical contact model. Figure 9: Contact between two elastic bodies [25]. in contact between the linear bearing balls and raceways is caused by elastic deformation, friction or a situation in which there is a significant load causing yielding. From Equation (14), it follows that the preload depends on the contact angle. The contact angle of the balls with the raceways of the linear bearing is defined by the shape of the outer and inner raceway surfaces modeled by the Gothic arch, shown in Figure 11 (in practice, a V-shaped model is also used). Deep groove ball bearings have a high degree of contact, which gives them a relatively high load-carrying capacity. The tangential factor (t\({}_{I}\)) describes where two curves or surfaces meet, or where they share a tangent. In linear bearings, the contact coefficient describes the ratio of the raceway radius to the ball radius. Bearing housing diameter (C\({}_{\text{D}}\)), for a linear bearing, specifies the distance from the shaft axis to the center of the ball. The adopted model assumes that the linear bearing is mounted in the axis, as a result of which the bearing installation diameter is zero and the contact between the ball and the raceway is shown in Figure 12.  
For the adopted ball diameter and the adopted shape of the raceway, it is possible to select the general geometric properties of the connection, such as contact angle (a),  
Figure 11: Description of the angle of contact between the ball and the raceway for the adopted model of the shape of the raceway in the form of a Gothic arch. Figure 12: Linear bearing ball contact with raceway. tangential factor (t\({}_{\rm f}\)) and bearing cage diameter (C\({}_{\rm D}\)). From the above parameters, it is possible to determine the radii of contact curvature for a ball bearing with four-point connection (Equations (16)-(18)). \[{\rm R}_{1}=\frac{\rm D}{2} \tag{16}\]  
\[{\rm R}_{2}={\rm R}_{1}{\cdot}2{\cdot}{\rm t}_{\rm f} \tag{17}\]  
\[{\rm R}_{2}^{\prime}=\frac{{\rm C}_{\rm D}}{2}\left(\frac{+}{-}\right){\rm R}_ {2}{\cos\alpha} \tag{18}\]  
where  
a--angle of contact between the bearing ball and the raceway,  
D--linear bearing ball diameter,  
R\({}_{1}\)--minimum radius of curvature of the balls in a linear bearing,  
R\({}_{2}\)--minimum radius of contact between the inner and outer raceways,  
R\({}_{2}^{\prime}\)--maximum radius of contact between the inner and outer raceways,  
C\({}_{\rm D}\)--bearing cage diameter,  
t\({}_{\rm f}\)--tangential factor. After determining the values of the radii of the contact curvatures R\({}_{1}\) and R\({}_{2}\), the equivalent coefficient of the radii of contact curvature was determined (Equation (19)) [25].