## 1 Introduction  
Fossil fuels are still at the first place among the most in-demand energy sources, and this despite the technological development in alternative energies. Therefore, drilling oil and gas wells remains the only way to reach hydrocarbon deposits.  
Various factors to which the drillstring is subjected favor the appearance of unwanted vibrations and oscillations, these vibrations are considered to be a major cause of inefficiency in the drilling process, leading to increased costs and expenses for the completion of the well [1]. These phenomena stress the elements of the drillstring, causing fatigue failure, drill bit wear, premature failure of pipes, and a decrease in the rate of penetration [2].  
Depending on the behavior of the drillstring under the effect of vibrations, three types of vibrations can be identified: (1) axial, (2) lateral, and (3) torsional. The latter is the most dangerous and the most destructive one [3]. Torsional vibrations generate a phenomenon called stick-slip, in which the angular speed of the bit periodically switches between two cycles: (1) a near-total sticking and (2) a sudden slip with a rotary speed two times greater or more than the surface speed [4].  
To understand the dynamic behavior of the system under stick-slip vibrations, several mathematical models have been proposed in the literature. The simplest and most widespread models are the lumped-parameter models; to imitate the behavior of drillstring, [PERSON] and [PERSON] [5] introduced an analogy between the behavior of the drillstring and that of a torsional pendulum, the bottom hole assembly (BHA) was regarded as a rigid body and the pipes as a torsional spring. [PERSON] and [PERSON] [6] presented a mathematical model for studying torsional oscillations of drillstring, wherein the obtained results showed that the top drive torque increases with rotary speed. Furthermore, [PERSON] et al. [7] studied the effect of friction uncertainties on the stick-slip response.  
Lumped-parameter model with multiple degrees of freedom (DOFs) has been an interesting research topic for many researchers; [PERSON] and [PERSON] [8] investigated the effect of weight on bit (WOB), rotary velocity, and drillstring stiffness on the stick-slip severity. [PERSON] et al. [9] conducted a study on the influence of slip motion on the drilling system by developing a more generic lumped-parameter model that takes into account the increase in drillstring length. [PERSON] and [PERSON] [10] presented two nonlinear models, the first with one DOF and the second with three DOFs, hence, different friction models were treated and analyzed. [PERSON] [11] evaluated a model with three DOFs, the relationship between the system equilibrium and the existing sliding motion was carefully studied, and thus, a controller strategy was designed in order to keep the rotational speed at a desired value. Lately, [PERSON] et al. [12] used a distributed model in which the wave equation was considered to describe the drill pipe model; the same friction model investigated by [11] was then evaluated.  
Risks caused by vibrations and other factors during drilling operation are considered to be challenging topics for researchers. Evaluating these risks and finding an optimal manner to carry out efficient preventive maintenance actions is problematic that has been subjected to many research works. Recently, [PERSON] et al. [13] investigated the failures of two critical components of the drilling system; they used the Bayesian approach to find optimal maintenance intervals.  
The aim of this paper is to analyze and investigate the relation between the occurrence of the stick-slip phenomenon and the deformations recorded in drill bits in wells located in Algerian oil field. For this purpose, a three-degree-of-freedom model was used to simulate the behavior of the drill string for different parameters of the rotary drilling system.