# Influence of Nonstationary Processes in Drill Rigs on the Durability of Structural Elements
## 3 Mathematical Model of Electromagnetic Processes in an Asynchronous Motor
In the moment of time \(t=0\), these functions are given on one layer only. Therefore, let the finite-difference scheme (20) be transformed, taking into account the initial conditions, into such a form that allows the first step in the integration of Equation (8) to be performed:  
\[ \begin{split} u_{i}(x_{i},\ \Delta t)=d_{2i}f_{1i}(x_{i}-\Delta l)+d_{3i}f_ {1i}(x_{i})+d_{4i}f_{1i}(x_{i}+\Delta l)+d_{5i}f_{2i}(x_{i})+d_{0i}g_{1i}\\ (i=1,2,\ \ldots,n),\end{split}  \tag{24}\]  
where  
\[d_{0i}=\frac{(\Delta t)^{2}}{2};\ \ d_{2i}=\frac{(\Delta t)^{2}\left(a_{i}^{2 }+g_{2i}\Delta l\right)}{2(\Delta l)^{2}};\ \ d_{3i}=\frac{a_{i}(\Delta t)^{2}+(\Delta l)^{2}}{ (\Delta l)^{2}};\]  
\[d_{4i}=\frac{(\Delta t)^{2}\left(a_{i}^{2}-g_{2i}\Delta l\right)}{2(\Delta l) ^{2}};\ \ d_{5i}=-\Delta t(b_{i}\Delta t+1).\]  
Taking into account boundary conditions and Equations (21)-(23), formulas for determining unknown functions at the first step of integration in extreme sections of the stepped rod are written as  
\[u_{1}(0,\ \Delta t)=p_{20}f_{11}(0)+p_{30}f_{11}(\Delta l)+p_{40}; \tag{25}\]  
\[u_{n}(l_{n},\ \Delta t)=p_{2n}f_{1n}(l_{n})+p_{3n}f_{1n}(l_{n}-\Delta l)+p_{4n}; \tag{26}\]  
\[u_{i}(l_{i},\ \Delta t)=u_{i+1}(0,\ \Delta t)=p_{2i}f_{1i}(l_{i}-\Delta l)+p_{3i}f_ {1i}(l_{i})+p_{4i}f_{1,\ i+1}(h)+p_{5i}\ (i=1,\ 2,\ \ldots,\ n-1), \tag{27}\]  
where  
\[p_{20}=r_{0}s_{20},\ \ p_{30}=r_{0}s_{30},\ \ p_{40}=r_{0}[s_{40}-2s_{00} \Delta tf_{21}(0)];\]  
\[p_{2n}=r_{n}s_{2n},\ \ p_{3n}=r_{n}s_{3n},\ \ p_{4n}=r_{n}[s_{4n}-2s_{0n} \Delta tf_{2n}(l_{n})];\]  
\[p_{2i}=r_{i}s_{2i},\ \ p_{3i}=r_{i}s_{3i},\ \ p_{4i}=r_{i}s_{4i},\ \ p_{5i}=r_{i}[s_{5i}-2s_{0 0}\Delta tf_{2i}(l_{i})],\]here:  
\[r_{1}=(b_{1}\Delta t+1)\Delta I/\Big{[}2E_{1}A_{1}(\Delta t)^{2}\Delta I+4(m_{0}+m _{2})\big{(}a_{1}^{2}+g_{21}\Delta I\big{)}(\Delta t)^{2}\Big{]};\]  
\[r_{2}=(b_{n}\Delta t+1)\Delta I/\Big{[}2E_{n}A_{n}(\Delta t)^{2}\Delta I+4m_{3} \big{(}a_{n}^{2}-g_{2n}\Delta I\big{)}(\Delta t)^{2}\Big{]};\]  
\[d_{4i}=\frac{(\Delta t)^{2}\left(a_{i}^{2}-g_{2i}\Delta l\right)}{2(\Delta l) ^{2}};\ \ d_{5i}=-\Delta t(b_{i}\Delta t+1).\]0  
The obtained Formulas (20)-(27) provide a comprehensive analysis of stepped drill string unsteady vibrations by solving a system of partial differential equations of hyperbolic type (8) on given initial and boundary conditions. Therefore, the drill rig start-up operation modes are described by the set of Equations (1), (5), (9)-(11), (14) and (15), which are integrated taking into account relations (2)-(4), (6)-(8), (12), (13) and (16). To solve the problem, initial conditions are determined based on deformed state analysis for the elastic system being in equilibrium. For moments of time when slippage in the friction device stops or reappears, the initial conditions for integrating Equations (5) and (9) are found as correspondent kinematic characteristics of the segments at the end of the previous stage of the movement. The represented mathematical model is based on the joint integration of the equations of motion for an elastic mechanical system, which has members with concentrated and distributed parameters, and the equations of the electromagnetic state for an asynchronous motor. Detailed representation of elastic and inertial properties for drill rig elements in the calculation scheme and a refined account of dynamic properties of motors provide a high degree of adequate correspondence for mathematical modeling results in the real physical process. This opens up possibilities for in-depth study of extreme operating modes of technological equipment, performing refined calculations of drive system elements and bearing structures for fatigue strength and durability.