## 2 Dynamic Model of Drilling System  
### Dynamic Equations of Drilling System  
In deep hole drilling, the vibration of the system mainly refers to the vibration of the drill bit and the drill pipe, which could also be called a drilling system (see Figure 1b). The rotational velocity \(\Omega_{rot}\) and feed velocity \(V_{0}\) decided by the rig are the most important control parameters in deep hole drilling. After establishing the dynamic model, the output characteristic of the drill bit at any time could be analyzed by the geometric structure of the drill, the rock characteristics, and the rock breaking mechanism of the drill bit, i.e., the actual drilling speed \(V\) and rotating speed \(\Omega\) of the drill bit. Also, the control parameters could be changed to achieve the desired output characteristics of the drilling system and avoid stick-slip vibration, which is the purpose of this paper. The stick-slip vibration frequency is similar to its torsional vibration frequency of the drilling system through test analysis and field observation. On the other hand, the axial drilling speed and torsional rotational speed can directly affect the drilling process of the drill bit. So a discrete model of the drilling system is characterized by two degrees of freedom, \(Z\) and \(\phi\), corresponding to the axial and angular position of the bit, respectively. This model has six mechanical elements from Table 1, namely point mass \(m\), moment of inertia \(I\), torsional stiffness \(K_{T}\) and axial stiffness \(K_{A}\), torsional damping \(\beta_{T}\) and axial damping \(\beta_{A}\) (see Figure 2). The mass \(m\) and the moment of inertia \(I\) are taken to represent the bit, while the spring \(K\) and the damper \(\beta\) are assumed to stand for the drill pipe. The drilling system is affected by two boundary conditions, the input of the drill rig (constant input drilling speed \(V_{0}\) and rotational speed \(\Omega_{rot}\)) and the bit-coal interaction (the bit weight \(W\) and torsion \(T\) of the drill bit). The perturbation dynamic equations of the drilling system can thus be written in the form:  
\[m\ddot{z}(t)+\beta_{A}\dot{z}(t)+k_{A}z(t) =-\Delta W(t) \tag{3}\] \[\ddot{\phi}(t)+\beta_{T}\dot{\phi}(t)+k_{T}\phi(t) =-\Delta T(t), \tag{4}\]  
where \(z\) and \(\phi\) are the disturbed axial displacement and angular displacement under stable drilling of the drilling system. \(\Delta W\) and \(\Delta T\) are the non-linear disturbance cutting force, which is determined by the nature of the coal, the structure of the drill and the depth of the cutting depth, i.e., the difference between the instantaneous cutting force and steady-state cutting force at any time of the drilling system. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Quantities** & **Variable** & **Value** & **Unit** \\ \hline Mass of Bit & \(m\) & \(2.0\times 10^{4}\) & kg \\ Rotary inertia of Bit & \(I\) & \(2.5\times 10^{2}\) & kg\(\cdot\)m\({}^{2}\) \\ Axial damping & \(\beta_{A}\) & \(2.4\times 10^{4}\) & N\(\cdot\)s/m \\ Torsion damping & \(\beta_{T}\) & \(72\) & N\(\cdot\)m\(\cdot\)s/rad \\ Axial stiffness & \(k_{A}\) & \(7\times 10^{5}\) & N\(\cdot\)m \\ Torsion stiffness & \(k_{T}\) & \(5.2\times 10^{2}\) & N\(\cdot\)m/rad \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters in the discrete model. Figure 2: Dynamic model of the drill string system. The PDC (Polycrystalline Diamond Compact) bit is widely used in deep hole drilling in coal seam with good impact toughness and the ability to handle minor accidents. According to the single blade cutting experiment of PDC bit [18], the action of such a bit consists of two independent processes, i.e., a cutting process taking place on the cutting face and a frictional contact process taking place on the wearflat.