# Influence of Nonstationary Processes in Drill Rigs on the Durability of Structural Elements
## 2 Mathematical Modeling of Dynamic Processes
\(\alpha=A_{r}\rho_{r}r_{p}\)3--constant factor (\(\rho_{r}\)--average wireline density); \(k\)--pulley multiplicity; \(\eta\)--absolute strain rate of the wireline, and  
Footnote 3: The \(\rho_{r}\)-dependence of the strain rate \(\rho_{r}\) is not a function of the strain rate \(\rho_{r}\). \[\eta=r_{t}\omega-(k+2)v_{1}+kv_{2}. \tag{4}\]  
Symbols \(v_{1}\), \(v_{2}\) denote the speed of movement for elements with masses \(m_{1}\) and \(m_{2}\). When the angular speeds of the driving and driven parts of the drive mechanism are equalized, the torque \(M_{f}\) is sufficient for a clutch between friction device elements, which means that, if the following conditions are fulfilled:  
\[\omega_{1}=\omega_{2};\ abs\bigg{(}\frac{J_{1}}{u}\frac{d\omega_{1}}{dt}-M_{E} \bigg{)}\leq M_{f}, \tag{5}\]  
then segments with inertia moments \(J_{1}\) and \(J_{2}\) proceed with joint movement as described by the following equations:  
\[J_{\Sigma}\frac{d\omega}{dt}+\frac{\alpha}{2}\omega^{2}+r_{p}c_{r}\delta \bigg{(}1+\frac{\delta}{2l_{r}}\bigg{)}+r_{p}v_{r}\xi=uM_{E};\ \frac{d\gamma}{dt}=\omega, \tag{6}\]  
where \(J_{\Sigma}\)--total moment of inertia of rotating drive parts; \(\gamma\) and \(\omega\)--coordinate and speed of rotation for a segment with inertia moment \(J_{\Sigma}\). Varying loads on elements in the lift system may lead to inequality in condition (5) being violated, which results in repeat slippages in the friction device. The equations of motion for upper and lower blocks of pulleys are written as  
\[ \begin{array}{c}m_{1}\frac{dv_{1}}{dt}-(k+2)c_{r}\delta+c_{d}y_{1}-(k+2)v_{ r}\xi+v_{d}v_{1}=m_{1}g;\\ m_{2}\frac{dv_{2}}{dt}+kv_{r}\delta+kv_{r}\xi-N_{1}(0,t)=m_{2}g;\\ \frac{dv_{1}}{dt}=v_{1};\ \frac{dv_{2}}{dt}=v_{2},\end{array}  \tag{7}\]  
where \(g\)--acceleration of gravity; \(N_{1}\)(0,\(t\))--longitudinal force in the initial cross-section in the first segment of the string. The equation of longitudinal motion for string segments is written as  
\[a_{i}^{2}\frac{\partial^{2}u_{i}}{\partial x_{i}^{2}}-2b_{i}\frac{\partial u _{i}}{\partial t}-\frac{\partial^{2}u_{i}}{\partial t^{2}}=-g_{i}\ (i=1,2,\ldots,n), \tag{8}\]  
where \(a_{i}\)--propagation speed of elastic deformations wave; \(b_{i}\)--factor of linear resistance to string movement in the well; \(g_{i}\)--function that considers distributed load. Quantities of \(a_{i}^{2}\), \(2b_{i}\), \(g_{i}\) are defined by the following relations:  
\[ \begin{array}{c}a_{i}^{2}=\frac{E_{i}}{\rho_{i}};2b_{i}=\frac{\kappa_{i}}{A _{i}\rho_{i}};\\ g_{i}=g\Big{(}1-\frac{\rho}{\rho_{i}}\sin^{2}\alpha_{0}\Big{)}\cos\alpha_{0}- \frac{a_{i}^{2}f}{\rho_{0}}\text{abs}\Big{(}\frac{\partial u_{i}}{\partial x_ {i}}\Big{)}\text{sign}\frac{\partial u_{i}}{\partial t},\end{array}  \tag{9}\]  
therewith \(E_{i}\), \(\rho_{i}\)--elasticity modulus of the first kind and drill pipe material density; \(\rho\)--density of flushing fluid; \(\kappa_{i}\)--factor equal to the force acting on unit length in the segment during its unit speed movement; \(\alpha_{0}\)--average vertical tilt angle of the string axis;\(f\)--dry friction factor of the string against walls in the well; \(\rho_{0}\)--average curvature radius of the well. Boundary conditions are given for the integration of Equation (9). For the upper end of the string (if \(x_{1}=0\)), second and fourth relations (8) are to be fulfilled, therewith:  
\[y_{2}=u_{1}(0,t);\ v_{2}=\frac{\partial u_{1}(0,t)}{\partial t}. \tag{10}\]On the borders of adjacent segments and in the lower end of drill pipe string boundary conditions are  
\[E_{i}A_{i}\frac{\partial u_{i}}{\partial x_{i}}-E_{i+1}A_{i+1}\frac{\partial u_{i +1}}{\partial x_{i+1}}=F_{i},\ u_{i}=u_{i+1}, \tag{11}\]  
\[(i=1,\ 2,\ \ldots,\ n-1),\ \text{if}\ x_{i}=l_{i},\ x_{i+1}=0;\]  
\[E_{n}A_{n}\frac{\partial u_{n}}{\partial x_{n}}+m_{3}\frac{\partial^{2}u_{n}}{ \partial t^{2}}=F_{n}+m_{3}g,\ \text{if}\ x_{n}=l_{n}, \tag{12}\]  
where \(F_{i}\)--concentrated forces of interaction between string and flushing fluid.