## 1 Introduction  
Drill string machines are typically used to drill up to several hundred meters into the ground as well as into the sea. The vibrations of the drill string due to recurrent contact with an external element disturb the optimal control of the pressure at the drill head and considerably compromise the quality of the drilling, in particular in the upper tens of meters. The drill string system, typically used in a non-Newtonian fluid (mud) where the viscosity can change with applied shear, is also used in a Newtonian fluid such as inviscid water (sea). Different fluid models, namely, the Navier-Stokes equations, have been used to describe the interaction of Newtonian fluids with various structures. The industry is constantly researching new techniques to predict and estimate the remaining useful life of machinery operations based on newly developed monitoring technologies. Most of the techniques used to extract distinguishing features from rotating machinery models consist of classifying different types of faults based on the available system information. An important study conducted by [PERSON] (2010) assessed the relationship between head loss and the rotation speed of the well according to the geometry of the well and the flow regime [1]. According to research in the literature, premature wear of rotors such as drill strings is a predominant cause of mechanical failure as well as the geological environment of marine structures, which can significantly alter the mechanical properties of the fluid interacting with the rotor mass of the rotor, making the fluid density near the working area extremely uneven. Drill strings are designed for tight, hard-to-reach access. In addition, strong vibrations and axially induced bit excitations cause early deterioration of the drill string. The structural durability and degradation of a drilling system operating in a fluid medium are influenced by the characteristics of the fluid. The unwanted drill string vibrations waste some of the energy that should be sent to the drill bit [2]. Monitoring the condition of drill rods is difficult due to their size and accessibility; therefore, failure prediction is one of the most complicated and misunderstood difficulties a drilling operator can face. Although the drill string system normally operates in a non-Newtonian fluid medium, there is little research on the behavior of the drilling system operating in a Newtonian fluid [3]. Modeling and monitoring a drill string interacting with an inviscid fluid are two of the most challenging topics in mechanical science that have piqued the interest of professionals in the field. Mud damping, drill-wellbore contact, stochastic bit-rock interaction forces, and various external sources all contribute to the complexity of the dynamic responses of the drill string [4]. Unfortunately, due to their use in a variety of locations and accessibility issues, drilling parameters (such as feed force, drill bit diameter, rotational speed, etc.) significantly affect the torque of the drill, and the underlying mechanisms have not yet been studied in depth. However, further research is needed to determine how torque responds to changes in these mechanical characteristics. Many field applications relying on vertical rotors have been deployed in a variety of industries, including oil and gas exploration and petroleum drilling engineering. Any exciting force can cause a vertical drilling system to vibrate and react. The drill rods are always in a state of longitudinal compression and inclination at parametric resonance under axial excitation. As a result, the impact of these drill strings on the borehole wall is substantially increased. Impact force fluctuates rapidly during a collision. Thus, a lateral stress wave will arise and propagate in the drill string. In practice, there is a high risk of friction between the rotor stabilizer and the static section of the machine for machines such as a rotating drill string [5].