## 2 Generalized Method for Calculation of Metal-Polymer Bearings  
A plain bearing (Figure 1) consisting of a polymer sleeve 1, a steel shaft 2, and a housing 3 was considered. Shaft 2 is loaded with a static load \(N\) and rotates with an angular velocity \(\omega=const\). The diagram of the bearing (Figure 1b) shows the force reduced to the unit width \(l\) of bushing 1: \(F=N/l\). When the bearing is loaded, an unknown contact pressure \(p_{\alpha}\) occurs, which is distributed along an unknown arc of contact \(2R_{2}a_{0}\). The maximum contact pressure \(p_{max}\) occurs between the shaft and the sleeve along the action line of the force \(F\). A radial clearance \(\epsilon=R_{1}-R_{2}>0\) must be guaranteed between the polymer sleeve 1 with inner radius \(R_{1}\) and shaft radius \(R_{2}\). The polymer sleeve has a certain thickness \(S\), which is selected structurally. It is known that the mechanical properties of polymeric materials for sleeves of sliding bearings have significantly lower properties than structural steel for shafts: an 8-10 times lower tensile strength, 60-260 times lower Young's modulus, and 50-150 times lower wear resistance. Consequently, at the limit friction of such bearings, their load-bearing capacity, wear resistance, and durability will also be significantly lower than those of plain bearings with metal sleeves. Metal bearings, on the other hand, cannot operate under dry friction, in water, or in process liquids. Therefore, MP bearings are indispensable under these conditions, which leads to their widespread use within a reasonable limitation of bearing capacity based on the strength properties of the tribopolymers. However, except for the authors' method, there are no such methods available for use at the design stage. The generalized analytical method for studying MP sliding bearings includes the following partial methods for calculation:  
* Contact pressures \(p_{\alpha}\) and the area of their distribution, that is, the initial contact angle \(2\alpha_{0}\);
* Maximum contact pressure \(p_{max}\);
* Taking into account bushing wear to reduce contact pressures \(p_{\alpha h}\) and increase the contact angle \(2\alpha_{0h}\);  
Figure 1: Metal-polymer plain bearing: (**a**) general view and (**b**) diagram. Taking into account the wear of the bushing to reduce the initial maximum contact pressure;
* Durability \(t\) of the MP bearing at the accepted permissible linear wear \(h_{1}\) of the polymer sleeve;
* Maximum linear wear \(h_{1 max}\) of the bushing at a given bearing life \(t\). These methods are described in detail in the authors' previous publications [18; 19; 20; 21; 22]. Below are the basic formulas for calculating the initial maximum contact pressure \(p_{max}\) in the bearing and its durability at a given wear \(h_{1}=h_{1 max}\) of the polymer bushing. Here, a modified method was used to solve the contact problem with bushing wear, which takes into account the effect of the bushing thickness \(S\) on the contact parameters. The maximum contact pressure \(p_{max}\) occurs along the force line \(F\) and is determined using the following formula:  
\[p_{max}\approx E_{\lambda}\varepsilon\tan\left(\frac{\alpha_{0}}{2}\right), \tag{1}\]  
where \(E_{\lambda}\approx(e_{\lambda}/R_{2})cos^{2}(\alpha_{0}/4)\),\(e_{\lambda}=eL\), \(e_{4}=4E_{1}E_{2}/Z\), \(L=\left(1+\lambda^{4}\right)\), \(\lambda=R_{1}/(R_{1}+S)\), \(0<\alpha_{0}<90^{\prime}\), \(Z=(1+\kappa_{1})(1+\
u_{1})E_{2}+(1+\kappa_{2})(1+\
u_{2})E_{1}\), and \(\kappa=3-4\
u\). \(E_{k}\) and \(\
u_{k}\) are the Young's modulus and [PERSON]'s ratio of the materials of shaft 2 and sleeve 1 of the bearing, and \(k\) is the numbering for the bearing elements (Figure 1).