## 2 Drill String Dynamics  
The main part of the drill string consists of long, thin-walled pipes, with a total length of up to several kilometers. The lower part includes collars or reamers, stabilizers, and various equipment for monitoring the drilling process. The length of the lower (heavy) part is about several hundred meters. As a rule, while drilling, the upper drill string part, consisting of lighter pipes, is kept under tension, and the lower, weighted part is under compression [1]. The rotational speed of the drill string is in the range of 50-200 rad/min. One of the important problems of the oil industry is finding out the causes of vibrations in the drill string and the development of ways to suppress them. Intensive uncontrolled vibrations can cause accidents. Such factors as damping by the drilling fluid, contact of the drill string with the borehole walls, and \"accidental\" interaction of the drill with the rock play a significant role in the dynamics of the drill string [9]. The most common application of drill string models is the prediction of the tool tuning effects (vibration sources) or the evaluation of key drilling parameters, such as rotation speed, torque, and axial load on the bit [10]. The main processes that need to be considered in the model of the drill string are shown in Figure 2, the drilling fluid flow is shown with blue arrows and the string possible movement with black ones. Drill string vibrations are usually divided into three main components: axial, lateral, and torsional. Dangerous cases of each of these vibrations are, respectively, \"bouncing\", \"vortex\" movement, and jerky vibrations. Their main causes are the interaction of the bit and the rock, eccentricity, variable points of contact between the drill string and the borehole walls, and vibrations generated by the tools. All three types of vibrations are related. Axial vibrations can cause lateral ones, while severe lateral vibrations in the bottom of the drill string can cause axial and torsional vibrations that can be measured in the top of drill string [11]. In vertical wells, the axial vibrations are much more noticeable and dangerous than torsional ones. In the past 70 years, the drill string vibration modeling has been actively developed, and the accuracy of predicting borehole vibrations and determining the effect of input parameters on drill string vibrations has been improved. However, despite the active development of modern computing systems, the modeling of a drill string (whose size ratios are comparable with the size ratio of a human hair) remains a complex physical and mathematical problem. Extensive theoretical material on modeling the dynamics of a drill string in a well is presented by [PERSON] et al. [12] and [PERSON] and [PERSON] [13]. A detailed review of drill string dynamics was made by [PERSON] et al. [5]. Equations and effects regarding the axial vibration of the drill string can be found in the studies by [PERSON] and [PERSON] [14] and [PERSON] et al. [15]. Drill string stability is discussed by [PERSON] et al. [16]. The positive effect of axial vibration on the drilling process is studied by [PERSON] [17]. In studies on axial vibrations, it is indicated that the length of the bottom part of the drill string plays a key role in the occurrence of vibrations, since this part is the heaviest, which means that by changing its length it is possible to adjust the position of the neutral point of the string (where tension gives way to compression). It is also important to understand that resonance must not be allowed to occur in the axial direction. If a harmonic force acts at the lower end of the drill string, and the drill beat frequency is close to the natural frequency of the string, this can lead to an uncontrolled increase in vibrations. Studies on the bending vibrations of drill strings and their bending dynamics can be found in papers by [PERSON] [18] and [PERSON] et al. [19].