## 2 Rotary Drilling System  
Drilling systems are designed to make a link between the surface of the earth and the deposit that may contain gas and oil reserves. This connection is carried out in the form of a hole from the surface to a certain depth. There are several types of drilling systems, but the most efficient and widespread is the rotary drilling system (Figure 1). Its operating principle is based on the fact of rotating a system while applying weight on its bottom from the top, so that drilling can proceed. It is mandatory to have a draw winch which aims at suspending the drillstring through a crown block and a traveling block with a hook at its end [14]. The set of draw works, crown block, traveling block, and hook, can apply the necessary weight on the drill bit (WOB).  
The rotational motion is provided by two mechanisms: (1) a top drive operating with an integrated electric motor; (2) a rotary table driven by an electric motor through a mechanical transmission mechanism. Both mechanisms are connected to the top of the drillstring. The rotary table consists of a drive shaft that contains at one end a pinion driven either by a drilling winch or by an independent motor; the shaft is supported by rolling bearings. At the other end of the drive shaft, there are level gear meshes with a ring gear on the rotating part of the rotary table. Most of the rotation tables are controlled by chains from the winch. The shaft drive is very expensive and requires more maintenance because it has bearings and gears at the right angles, but it is reliable and can often be installed under the drill floor.  
During drilling operations, the rotary table and the drive shaft may be subjected to various dysfunctions; as the bottom vibrations reoccur, the bearings at the rotary table may deform, which would then lead to vibrations at the floor. Considering the dynamic loads that the rotary table supports, and the torsional vibrations that propagate to the surface, the service life of the ball bearings could decrease [15].  
The drill pipes compose the majority of the drillstring; they have a nominal length of 9 m and a wall thickness of 9 mm; their role is essentially limited to the transmission of the rotational movement from the top drive to the bit [14].  
The weight on the bit is provided mainly by the last section of the drillstring, which is the bottom hole assembly (BHA). The BHA is a thick-walled tube with a higher weight that is located below the bit, which can be in several forms and designs according to the conditions of use. The two most commonly used types of drill bits are: the PDC (Polycrystalline diamond compact) bits and Tricone bits (Figure 2). The WOB and the applied torque values are well set so that they allow an efficient drilling of the rock. The different debris obtained after drilling the rock are transported by a drilling mud fluid, which is injected into the well by mud pumps located on the surface. The injection is done through the drillstring and the ascent is guaranteed through the annulus between the drillstring and the wellbore. The injected fluid mud is intended to lubricate the bit and ensure pressure equilibrium in order to avoid any undesired fluid eruption from the penetrated hydrocarbon reservoirs.  
To improve drilling efficiency, other equipment is often added to the drillstring, for example stabilizers that reduce the lateral displacements, heavyweight drill pipes that are used to make the transition from drill collars to drill pipes, and drilling jars that are used to deliver an impact load to  
Figure 1: A general schema of typical rotary drilling system.  
the bit when it is stuck [14]. In order to study the dynamics of the system, it is better to develop its mathematical model as explained in Section 3.