# Development of Automatic Electric Drive Drilling System for Core Drilling
## 2 Modeling and Theory  
A drilling rig is composed of a variety of machines and equipment, with a variety of functions regarding the joint working unit. It is mainly composed of a rotation system, circulation system, and hoisting system. There are many undetermined factors in the drilling process, such as the concealability and complexity of the formation and the operation characteristics of the drilling rig, which can cause harmful vibration impact on the rotation system of the drilling rig. ROP is the core of drilling, and the commonly used empirical model from [PERSON] and [PERSON] [17] is  
\[ \begin{array}{l}\text{ROP}=\exp[a_{1}+a_{2}(3048-\text{TVD})+a_{3}\text{TVD}^{ 0.69}(g_{p}-1.078)\\ \qquad\qquad+a_{4}\text{TVD}(g_{p}-\text{ECD})+a_{5}(\frac{\text{RPM}}{60})+a_{6} (-h)\\ \qquad\qquad+a_{7}\frac{\varrho q}{350\mu a_{n}}+a_{8}\ln[\frac{\frac{\text{WOB} }{60}-\frac{\text{WOB}}{60}}{0.7-(\frac{\text{WOB}}{60})}t]]\end{array}  \tag{1}\]  
where \(a_{1}\)-\(a_{7}\) are the empirical coefficients of formation, well depth, well pressure, drill bits and fluid; TVD is well depth; \(g_{p}\) is pore pressure gradient of the formation; ECD is equivalent circulating mud density at the hole bottom; \(\rho\) is mud density; \(\mu\) is the apparent viscosity at 10,000 s\({}^{-1}\); \(d_{n}\) is bit nozzle diameter; \(d_{b}\) is bit diameter. Although all the terms in the formula are empirical values, it is obvious that WOB and RPM are two important quantities. The current study of a drill string motion model can be summarized as a spring mass damping system, which simplifies the whole drill string motion process into a single degree of freedom or multidegree of freedom system model, and the interaction model between drill bit and rock is introduced as the lower boundary condition. Based on these factors, the basic dynamic equation of the system is established. By solving the basic equations of the system dynamics, the variables describing the dynamic characteristics of the system are obtained [18; 19]. Generally, the drill string system of a drilling rig can be described as a torsional pendulum [20; 21]. The main components are two mechanical devices with attenuated inertia, connected by an elastic drill pipe [22]. Table 1 shows the parameters used in the model of the rotation system. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number** & **Parameter** & **Definition** & **Units** \\ \hline
1 & \(\varphi\) & Angular displacements between \(\varphi_{r}\) and \(\varphi_{b}\) & rad \\
2 & \(\varphi_{r}\) & Angular displacement of power head & rad \\
3 & \(\varphi_{b}\) & Angular displacement of drill bit & rad \\
4 & \(J_{b}\) & Moment of inertia of drill bit & kg \(\cdot\) m\({}^{2}\) \\
5 & \(\bar{\varphi}_{b}\) & Angular velocity of drill bit & rad/s \\
6 & \(\bar{\varphi}_{b}\) & Angular acceleration of drill bit & rad/s\({}^{2}\) \\
7 & C\({}_{1}\) & Attenuation coefficient of drill string & N \(\cdot\) m \(\cdot\) s/rad \\
8 & \(T_{tob}\) & Nonlinear friction torque of bit torque & N \(\cdot\) m \\
9 & K & Torsional stiffness coefficient of drill string & N \(\cdot\) m/rad \\
10 & \(J_{r}\) & Equivalent moment of inertia of power head & kg \(\cdot\) m\({}^{2}\) \\
11 & \(J_{m}\) & Moment of inertia of motor & kg \(\cdot\) m\({}^{2}\) \\
12 & \(J_{rot}\) & Moment of inertia of power head & kg \(\cdot\) m\({}^{2}\) \\
13 & C\({}_{2}\) & Attenuation coefficient of power head & N \(\cdot\) m \(\cdot\) s/rad \\
14 & u & Unit step input & N \(\cdot\) m \\
15 & \(\Omega_{ref}\) & Given speed of bit & rad/s \\
16 & \(T_{m}\) & Motor driving moment & N \(\cdot\) m \\
17 & \(T_{rbd(dyn)}\) & Moment of dynamic friction & N \(\cdot\) m \\
18 & \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\) & Empirical coefficient of drilling & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters and variables characterizing the rotation system.