# Influence of Nonstationary Processes in Drill Rigs on the Durability of Structural Elements
## 2 Mathematical Modeling of Dynamic Processes  
The mathematical model is designed to describe dynamic processes that emerge in the mechanical system of a drill rig during the lifting of a pipe string. The calculation scheme of the lifting system is represented in Figure 1, where \(J_{1}\), \(J_{2}\)--moments of inertia for an engine rotor with mechanical transmissions and drawworks drum with attached drive parts considered about the axis of the lifting shaft; \(m_{1}\)--the mast reduced mass evaluated with respect to the crown block mass; \(m_{2}\)--the mass of block with a hook and suspended equipment; \(m_{3}\)--the mass of the lower part of a drill string weighted with a tool; \(c_{d}\) and \(c_{r}\)--stiffness coefficients of mast and wireline; \(\
u_{d}\), \(\
u_{r}\)--factors that characterize energy dissipation in correspondent segments, therewith \(\
u_{r}\) is physically the force of linear resistance to wireline deformation, which corresponds to unit relative strain rate. The drill string is considered as a homogeneous straight rod with a step change in the cross-section. Lengths and cross-sectional areas of string segments, within which the elastic and dissipative characteristics of the rod are constant, are denoted by \(l_{i}\), \(A_{i}\) (\(i=1,\,2,\,\ldots,\,n\)). Motion coordinates of discrete inertial elements in the calculation model are denoted by \(\gamma_{1}\), \(\gamma_{2}\), \(y_{1}\), \(y_{2}\), \(y_{3}\). Translational movements of string sections are denoted by functions \(u_{i}\) dependent on time and longitudinal coordinates \(x_{i}\) with origins in upper extreme cross-sections in each string segment. During driving system operation, the body with the moment of inertia \(J_{1}\) is subjected to the torque \(M_{E}=uM_{e}\), where \(M_{e}\)--electromagnetic torque of the motor, \(u\)--ratio of mechanical gears connecting the engine rotor to the drawworks drum; \(M_{e}\)--is the moment of interacting forces of the friction coupling halves. In the presence of slippage in the coupling, \(M_{c}=M_{f}\)-sign(\(\omega_{1}-\omega_{2}\)), wherein \(M_{f}\) is the moment of friction forces in the coupling; \(G_{1}=m_{1}g\), \(G_{2}=m_{2}g\), and \(G_{3}=m_{3}g\) are the weight forces of the crown block, the block with a hook and suspended equipment, and the weighted lower part of the drill string with the tool (\(g\) is the acceleration of free fall); \(q_{1}\), \(q_{2}\), \(\ldots\), \(q_{n}\) are the weight forces of drill pipes distributed along the length of the string. The equation of motion for discrete inertial elements of a mechanical system is written by the scheme of [PERSON]'s equations of the second kind. In the initial start-up period for the drive system, the equation of rotational motion for elements with moments of inertia \(J_{1}\) and \(J_{2}\) is written as  
\[J_{1}\frac{d\omega_{1}}{dt}=u[M_{e}-M_{f}\text{sign}(\omega_{1}-\omega_{2})];\]  
\[J_{2}\frac{d\omega_{2}}{dt}+\frac{\alpha}{2}\omega_{2}^{2}+r_{p}c_{r}\delta \bigg{(}1+\frac{\delta}{2l_{r}}\bigg{)}+r_{p}v_{r}\varepsilon=uM_{f}\text{ sign}(\omega_{1}-\omega_{2});\]  
\[\frac{d\gamma_{1}}{dt}=\omega_{1};\,\frac{d\gamma_{2}}{dt}=\omega_{2}, \tag{1}\]  
where drawworks drum moments of inertia \(J_{2}\), length \(l_{r}\), and stiffness \(c_{r}\) of wireline vary by the following laws:  
\[J_{2}=J_{20}+\alpha\gamma_{2};\,l_{r}=l_{r0}-r_{p}\gamma_{2},\,\,c_{r}=E_{r}A _{r}/l_{r}, \tag{2}\]  
and absolute strain \(\delta\) and relative strain rate \(\varepsilon\) of the wireline are defined as  
\[\delta=r_{p}\gamma_{2}-(k+2)y_{1}+ky_{2};\xi=(\eta\,\,l_{r}+r_{p}\,\,\delta\, \omega_{2})\,\,l_{r}^{2}, \tag{3}\]  
therewith, \(J_{20}\) and \(r_{p}\)--initial moment of inertia and radius of the drawworks drum; \(E_{r}\), \(A_{r}\), \(I_{r0}\)--elasticity modulus, cross-sectional area and initial length of the wireline;  
Figure 1: Calculation schemes for the lift system of the drill rig.