## 4 Mathematical Description for Linear Bearings
### Mathematical Model of Linear Contact Ball Bearing
\[{\rm K}_{\rm D}=\frac{1.5}{\frac{1}{{\rm R}_{1}}+\frac{1}{{\rm R}_{2}}+\frac{ 1}{{\rm R}_{1}^{\prime}}+\frac{1}{{\rm R}_{2}^{\prime}}} \tag{19}\]  
\[{\cos\theta}=\frac{{\rm K}_{\rm D}}{1.5}\sqrt{\left(\frac{1}{{\rm R}_{1}}- \frac{1}{{\rm R}_{1}^{\prime}}\right)^{2}+\left(\frac{1}{{\rm R}_{2}}-\frac{1 }{{\rm R}_{2}^{\prime}}\right)^{2}+2\left(\frac{1}{{\rm R}_{1}}-\frac{1}{{\rm R }_{1}^{\prime}}\right)\left(\frac{1}{{\rm R}_{2}}-\frac{1}{{\rm R}_{2}^{ \prime}}\right)\cos 2\varphi} \tag{20}\]  
where  
K\({}_{\rm D}\)--equivalent radius factor of contact curvatures,  
\(\varphi\)--the angle between the planes tangent to the bodies in contact at the point of contact (Figure 12),  
cos\(\theta\)--angular dependence on the radii of contact curvatures and the angle of contact occurrence. Taking into account the defined material properties of balls and raceways, using the expression (21) [22], we determine the equivalent material coefficient in contact C\({}_{\rm E}\). Figure 9:0  
where  
C\({}_{\rm E}\)--equivalent material factor in contact,  
\(\vartheta_{1}\)--Poisson ratio for the ball,  
\(\vartheta_{2}\)--[PERSON]'s ratio for treadmills,  
E\({}_{1}\)--[PERSON]'s modulus for the ball,  
E\({}_{1}\)--[PERSON]'s modulus for treadmills. Taking into account the values set K\({}_{\rm D}\) and C\({}_{\rm E}\) and the values of the angular parameter \(\lambda\)[26], from Equations (22)-(25) [25] we can determine the contact parameters, such as the maximum and minimum semi-axis of elliptical contact, maximum stress in contact, and strain in contact caused by a given load, understood as relative motion in the direction of the load axis of two points of the bodies in contact [25]. Figure 9:1  
Figure 9:2Figure 9:3  
Figure 9:4  
where  
\(\sigma_{\rm{cmax}}\)--maximum Hertz stress in contact,  
y--deformation in contact,  
a--maximum semi-axis of elliptical contact of the ball with the raceway,  
b--minimum half-shaft of elliptical contact between the ball and the raceway,  
\(\kappa\), \(\lambda\) and \(\mu\)--transcendental auxiliary functions (Figure 13), whose values (contact coefficients) are selected on the basis of the determined value cos\(\theta\) described by Equation (21). The values of the function were determined experimentally [29]. Assuming that the contact force increases linearly, it is possible to estimate the contact strain for a linear ball bearing with four-point connection as the sum of the deformations in individual contacts of the balls with opposite raceways based on Equations (22)-(25). Taking into account the total contact strain for the linear bearing and the forces acting in contact, it is possible to determine the exponential contact load factor (n), which is described by Equation (26). The load factor of one ball (C\({}_{1}\)) describes Equation (27). On the basis of Equations (28) and (29), the following are determined, respectively: the exponential stiffness factor (k) and the stiffness coefficient (C\({}_{2}\)). Figure 9:5  
Figure 9:6  
Figure 9:7  
Figure 9:8  
where  
n--exponential load factor,  
k--exponential stiffness factor,  
C\({}_{1}\)--load factor of one ball,  
C\({}_{2}\)--coefficient of stiffness of one ball. Figure 9:9 Dependence of angular parameters on cos\(\theta\) in the selection of contact properties [29]. In the finite element method, contact occurs when the nodes of field 1 penetrate the surface formed from the nodes of field 2. The depth of penetration of the nodes of body 1 caused by force determines the stiffness of the contact (Equation (30)). Equation (30) is the equation for the spring stiffness in a linear ball bearing with four-point connection (Figure 14). Based on Equations (26)-(29) and Equation (30) (spring stiffness), the nonlinear contact stiffness for the corresponding force is calculated as \(\mathrm{F_{n}}\) (Equations (31) and (32)).