## 3 Mathematical Model  
The developed model in this study is based on'mass-spring-damper' approach; it represents a mechanism in the form of a proxy system containing masses attached to springs (dampers can be added too). The obtained proxy system may contain several DOFs depending on the number of used masses and springs. This approach allows us to obtain a reduced and discrete model of the studied system due to the division of the system into two parts; therefore, it facilitates the resolution of non-linear equations of motion that will be obtained at the end of this section. The main purpose of a mathematical model is to express and represent, in the form of equations, the various external forces acting on the system dynamic.  
In this research work, we will focus on studying the different vibrations that affect an operating drillstring. Many researchers have demonstrated that torsional vibrations are the most dangerous type of vibrations; its sever appearance can lead to the stick-slip phenomenon. To properly assimilate this phenomenon, several methods can be used, including the mass-spring-damper model. The drillstring, in this model, is considered as a torsional pendulum shown in Figure 3. The drill pipes represent the torsional spring with the torsional stiffness k, the BHA is represented by a mass M, the moment of inertia \(I\) represents the inertia of the BHA and that of the drill pipes. The viscous friction between the drillstring and the drilling fluid behaves as a damper with a damping coefficient C.  
Figure 3: Three elements proxy model of rotary drilling system.  
Figure 2: Types of drill bit: (**a**) bit anatomy, (**b**) matrix body of PDC bit, (**c**) natural diamond tricone bit.  
Assuming that the system rotates with a constant velocity \(\Omega_{0}\) applied by the top drive, and that \(\Phi\) represents the angular position of the bit, the equation of motion according to the principle of dynamics (the sum of the moments of the external forces is equal to the derivative of the angular momentum) is given in (1).  
\[I\ddot{\Phi}+C\dot{\Phi}+k(\Phi-\Omega_{0}t)=T_{b}, \tag{1}\]  
\(T_{b}\) is the torque generated by the frictional forces along the drillstring; it is a function of the rotational velocity of the bit [16]. Equation (1), called classical equation, represents a lumped system of one degree of freedom. This model contains three parameters: The moment of inertia \(I\), the damping coefficient C, and the torsional stiffness \(k\) for two elements: the spring that represents the drill pipes and the disk of mass m that represents the BHA. This model is the basic model that can be considered for rotary drilling systems; however, several hypotheses can be added, for example, considering that each of the drill pipes represents a disk of mass m and inertia \(I\) instead of considering them as a single spring.  
\(J_{r}\) is the inertia of the top drive; \(J_{p}\) is the inertia of the drill pipes, \(J_{b}\) is the inertia of the BHA (including drill bit), \(k_{t}\) is the torsional stiffness of the spring between the top drive and drill pipes, \(k_{tb}\) is the torsional stiffness of the spring between drill pipes and the BHA, \(c_{t}\) is the viscous damping along drill pipes, \(c_{tb}\) is the viscous damping along the BHA, \(Tab\) is the viscous damping torque at the bit, and \(T_{fb}\) is the friction torque at the bit [16].