## 2 Mathematical Modeling of the Drill String System
### Nonlinear Bit-Rock Interaction Model  
The frictional force is recognized as the cause of the self-excited stick-slip phenomenon. To include the influence of torque in the drill string system, some mathematical torque equations are combined to simulate the dynamics of the drill string under various conditions. One of the major contributions associated with the existing models is the description  
Figure 3: Schematic diagram of a bit–rock interaction during full operation and the full torque profile. (**a**) Full bit-rock torque contact. (**b**) Full bit-rock torque plot in drilling rig work.  
of some of the parameters involved, such as the presence of kinetic friction mechanisms, which leads to discontinuous differential equations and results in consistency with the field data. Therefore, the parameters that influence the friction mechanism, such as the angular displacement and angular velocity, must be considered in a proper stick-slip model (Figure 3). For this analysis, it is assumed that the bit is in continuous contact with the bottom and is laterally restrained. As a result, a viscous damping torque is considered both at the upper drive system (\(T_{en}\)) and the bottom hole of the drive bit (\(T_{bt}\)) caused by the interaction of the bit and rock level [17]:  
\[T_{bit}=\left(\eta_{cb}(\omega_{Rb})+(\eta_{sb}-\eta_{cb})\times e^{-\lambda| \omega_{Rb}|}\right)\times(W_{0}+W_{1})R_{Rb}-\xi_{b}\times\omega_{Rb} \tag{12}\]  
where \(\lambda\) = 0.9 is the decaying factor and \(\omega_{Rb}\) is the angular velocity of the drill-bit (i.e., \(\omega_{Rb}\) = \(\Omega\)) \(W_{1}=k_{f}\chi_{0}(1-\sin(2\pi f_{b}t))\) stands for the amplitude value of the WOB and varies with the type of bit used, \(\chi_{0}\) is the depth of the cut in one turn, and \(f_{b}\) stands for the frequency, which is determined by the depth of the cut \(\chi_{0}\) and the rate of penetration. In the above expression, \(T_{M}\) operates alone when there is no bit contact with the ground. When frictional contact occurs, the system switches to the mixed total torque mode. The electrical motor's surface torque \(T_{M}\) is represented by this symbol [18]:  
\[T_{M}=-K_{0}\theta+c_{r}\omega_{Rb} \tag{13}\]  
The total nonlinear torque on the drill string bit (\(T_{Tb}\)) is modeled as the sum of the total torque. Unlike the engine torque, the friction contact is modeled as a dynamic lateral and axial discontinuous dry friction contact (\(T_{Rb}\)) expressed using Equation (9) by:  
\[T_{Rb}=F_{Z\_rub}R_{Rb}=-H(\Delta+\delta_{ax}-\delta)\frac{(\Delta+\delta_{ ax}-\delta)K_{s}}{\Delta}\mu Z\times R_{Rb} \tag{14}\]  
The torque on the bit (TOB), generated in the (\(\theta\), \(X\), \(Y\), \(Z\)) direction of the drill associated with the cutting torque \(T_{c}\), can be expressed in the following equations:  
\[T_{Tb}=\left\{\begin{array}{l}T_{M}\\ T_{M}+T_{bit}+T_{Rb}\end{array}\right.\begin{array}{l}no\ contact\\ if\ full\ contact\end{array} \tag{15}\]  
The data relating to an actual drill string design given in [17] are used in the simulations for the following: \(W\)o (base case) = 30 kN, \(k_{f}\) = 23 \(\times\) 10\({}^{6}\) N/m, \(f_{b}\) = 0.005 m, \(R_{Rb}\) = 0.005 m, \(\eta_{\rm sb}\) = 0.8, and \(\eta_{\rm cb}\) = 0.45.  
The developed torque profile presented in Figure 3b shows that the fluctuations in the torque could have a detrimental effect on the drill bit and downhole equipment. The evolution of the torque during the stick-slip oscillation demonstrated that there are periodic fluctuations in the torque profile at startup around the mean value of 3400 Nm. The amplitude of this fluctuation is relatively low at the start but remains significant throughout the drilling process. Suddenly, the torque means the value drops to the minimum value, indicating that the drill bit is about to stick. At stick RPM, the bit stops momentarily causing the peak torque and TOB to build up almost linearly to a very high value, resulting in an extreme drop in torque, and the system becomes relatively stable.