# Development of Automatic Electric Drive Drilling System for Core Drilling
## 2 Modeling and Theory
In most of the literature, the downhole mechanism is considered as a completely rigid body, and the drill string (drill pipes and extensions) is considered as a torsion spring with stiffness depending on the drill string parameters [20]. As shown in Figure 1, the main components of the model are two damped inertias [23]. It is assumed that the drill pipe is of the same mass and the whole length is simply considered as a single linear spring system, with an elastic coefficient of \(K\) and no lateral motion. The mathematical model of drilling can be described by the motion equation of the drilling tool in downhole part, the motion equation of the driving system in overground part, and the friction equation of the bit. (1) Equation of motion of the downhole part  
According to the assumption that the drill string moves similar to a torsional pendulum, the drill pipe is equivalent to a torsional spring, the power head rotates at a constant speed, and the drill collar is stiff. The equation of motion of downhole part is  
\[J_{b}\ddot{\varphi}_{b}+C_{1}\dot{\varphi}_{b}-K\varphi=-T_{tob}(\dot{\varphi} _{b}) \tag{2}\]  
(2) Equation of motion of the drive system  
The mechanical system characteristics of the drive system consist of three parts: the power head, the gearbox, and the motor. The equation of motion of the system is:  
\[ \begin{array}{c}J_{r}\ddot{\varphi}_{r}+C_{2}\dot{\varphi}_{r}+K\varphi=T_{ m}\\ C_{2}\Omega_{ref}+u=T_{m}\end{array}  \tag{3}\]  
where \(J_{r}=J_{rot}+n^{2}J_{m}\). (3) Equation of friction moment  
Drilling is limited by the conditions of the downhole part. Because the static friction coefficient of the drill string is greater than the sliding friction coefficient, the torque of the drill pipe storage energy will convert the energy of inertia of the drill string. Then, the  
Figure 1: Simplified model of rotation system of drilling rig (model of drive system is in the green box, and that of the downhole part is in the red box). speed of the bit will be beyond the steady speed. In this condition, the rotational speed is in critical condition, the interaction of rock and bit will produce harmful vibrations. Torsional vibrations are considered as the most dangerous vibrations to which the drill string is vulnerable. The phenomenon responsible for the regeneration of these vibrations is known as'stick-slip'. Stick-slip occurs when the drill bit is frictionally engaged with the formation in a nonlinear manner [24]; it makes the whole drill pipe slide or stop moving. However, if the accumulated torsional energy reaches a certain level, the drill will be released suddenly and will rotate at a high speed. The resulting torsion wave will be passed to the running at the top of the rotation system. However, because the power head has strong inertia, it can hold the drill pipe in place, so that the torsion wave returns to the bit through the drill pipe. In this way, the bit will stall again, and the torque wave will repeat, which causes a very large and harmful vibration around the drill pipe. If out of control, the drill pipe and the drill bit will be damaged together. In the two-degree-of-freedom (2-DOF) model of the drilling rig rotation system, the friction torque is simply considered as the sum of the counter torque of the drill bit and the nonlinear friction torque on the drilling bottom device. The bit velocity function can be given by the nonlinear function of Equation (4):  
\[T_{tob}(\dot{\varphi}_{b})=T_{tob(dyn)}\frac{2}{\pi}(\alpha_{1}\dot{\varphi}_ {b}\exp(-\alpha_{2}\big{|}\dot{\varphi}_{b}\big{|})+\arctan(\alpha_{3}\dot{ \varphi}_{b})) \tag{4}\]  
Equation (4) is used for the nonlinear equations of motion of the rotation system of a drilling rig with 2-DOF. 1. Nonlinear equation of the drilling system The nonlinear state space representation of the rotary drilling system is \[ \begin{array}{l}\dot{x}(t)=f[x(t)]+g(x)u(t)\\ y(t)=h[x(t)]\end{array}