## 2 Generalized Method for Calculation of Metal-Polymer Bearings
The unknown angle of initial contact \(2\alpha_{0}\) is determined from the equilibrium of the forces applied to shaft 2, i.e., the load \(F\) and the reaction to it in the form of contact pressure \(p_{a}\):  
\[F=R_{2}\int_{-\alpha_{0}}^{\alpha_{0}}p_{a}cos\alpha d\alpha=4\pi R_{2}E_{ \lambda}\varepsilon sin^{2}(\alpha_{0}/4). \tag{2}\]  
Hence, taking into account the expression for \(E_{\lambda}\), the formula for the semi-angle of the angle \(\alpha_{0}\) will be  
\[\alpha_{0}=2 arcsin\sqrt{F/\pi e_{\lambda}\varepsilon}. \tag{3}\]  
The dependence of (1) on \(p_{max}\) and of (3) on \(\alpha_{0}\) can be used to determine these parameters in ball joints. For this purpose, instead of the radius \(R_{2}\) of the shaft, the conventional radius \(R^{*}=0.5\sqrt{R_{1}R_{2}}\) of the joint should be introduced, where \(R_{1}=R_{2}+\varepsilon\) is the radius of the ball-shaped skull and \(R_{2}\) is the radius of the ball-shaped head. The solution of this problem was presented for a ball-shaped hip endoprosthesis in [29; 30]. Based on the mathematical model of material wear under sliding friction [18; 19] and the method for calculating contact pressures in a bearing [18; 19; 20; 21; 22], the durability of a sliding bearing at the maximum linear wear \(h_{1}=h_{1 max}\) of the bushing is calculated using the following formula [19; 28]:  
\[t=\frac{-B_{1}\tau_{10}^{m_{1}}}{vc_{h}\tau_{b}\Sigma_{1}(1-m_{k})K_{t}^{(1)} }\Big{\{}\big{[}\tau-\tau_{10}\big{]}^{1-m_{1}}-\big{[}(\tau-\tau_{10})+c_{h}h _{1 max}\Sigma_{1}\tau_{h}\big{]}^{1-m_{1}}\Big{\}}, \tag{4}\]  
where \(B_{k},m_{k},\ \tau_{k0}\) are characteristics of the wear resistance of the materials in a hybrid tribopair under the accepted external research conditions, which are determined by the results of model triboexperimental studies [23; 28] (Tables 1-3); \(v=\omega R_{2}\) is the sliding speed; \(c_{h}\) is the wear rate coefficient; \(\Sigma_{1}=\left(-K_{t}^{(1)}+h_{1}^{\prime}\right)\); \(K_{t}^{(1)}=1\) and \(K_{t}^{(2)}=\alpha_{0}/\pi\) are the coefficients of mutual overlap of the sleeve and shaft, respectively; \(h_{1}^{\prime}=h_{2}/h_{1}\) is the relative linear wear of the shaft to the wear of the sleeve. \[h_{1}^{\prime}=\frac{B_{1}\tau_{10}^{m_{1}}(\tau-\tau_{20})^{m_{2}}}{B_{2} \tau_{20}^{m_{2}}(\tau-\tau_{10})^{m_{1}}}K_{t}^{(2)} \tag{5}\]  
The relationship between the specific friction force \(\tau\) on a tribocontact and the contact pressure \(p_{max}\) is taken according to the Amonton-Coulomb law:  
\[\tau=fp_{max}, \tag{6}\]  
where \(f\) is the sliding friction coefficient. Due to the wear of the bearing bushing and shaft, the pressure \(p_{max}\) will decrease. In an MP bearing, the polymer bushing wears out 50-100 times faster than the steel shaft, and therefore the shaft's wear can be neglected. The current pressure \(p_{thmax}\) is determined as follows:  
\[p_{thmax}=p_{max}-|p_{h}|, \tag{7}\]  
where \(p_{h}\) is the change in the initial pressures when the bushing experiences wear in a mechanical bearing. The value of \(p_{h}\) is calculated according to the following dependence [19]:  
\[p_{h}\approx E_{\lambda}c_{h}\varepsilon_{h}\tan\Bigl{(}\frac{\alpha_{0h}}{2} \Bigr{)}, \tag{8}\]  
where \(c_{h}>0\) is the wear rate; \(\varepsilon_{h}=h_{1 max}\Sigma_{1}\); and \(\alpha_{0h}>\alpha_{0}\) is an unknown contact angle when the bushing wears. Therefore, during the wear process, the current contact pressure is determined by the formula  
\[p_{thmax}\approx E_{\lambda}\Bigl{[}\varepsilon\tan\Bigl{(}\frac{\alpha_{0}}{ 2}\Bigr{)}-c_{h}h_{1}\Sigma_{1}\tan\Bigl{(}\frac{\alpha_{0h}}{2}\Bigr{)} \Bigr{]}. \tag{9}\]  
When the polymer sleeve experiences wear, the current specific friction force is \(\tau_{thmax}<\tau\). Accordingly, \(\tau_{thmax}=fp_{thmax}\) and \(\tau_{h}=f|p_{h}|\).