## 2 Drill String Dynamics
Flexural vibrations and the propagation of flexural waves are considered in the studies by [PERSON] and others [20; 21; 22; 23]. Due to the high flexibility of the drill string and its low rotation speed, it is generally accepted that the Bernoulli-Euler beam model accurately describes the dynamics of bending vibrations. An example of modeling the dynamics of drill string bending vibrations with  
Figure 2: The drill string scheme. the Timoshenko beam model can be found in the study by [PERSON] et al. [24]. The paper by [PERSON] et al. [25] is focused on the numerical aspects of modeling the drill string dynamics, and a description of modeling using the finite element approach is given. The bending vibrations of drill strings are divided into two main types: in-plane vibrations and full-scale spatial vibrations. Typically, in-plane vibrations are described in polar coordinates. Since the lower part of the column is in compression, and the upper part is in tension, the lower part is subject to basic vibrations. The upper pipes begin to vibrate at higher frequencies than the standard column rotation speeds. Flexural vibrations are the most dangerous because the flexural waves propagate along the drill string only at certain frequencies, so most of the resulting flexural vibrations cannot be captured at the surface where the sensors are commonly placed. Most of the literature is devoted to the modeling of torsional vibrations. Usually, the inverted pendulum model is used. The downhole mechanism is considered as a completely rigid body, and the drill string (drill pipes and extensions) is considered as a torsion spring with stiffness depending on the drill string parameters. A detailed description of such a model, as well as its modifications, can be found in the studies by [PERSON] and [PERSON] [26] and [PERSON] and [PERSON] [27]. In the same articles, one can also find a model of viscous friction between the downhole mechanism and the walls of the well, as well as the hydrodynamic damping by the drilling fluid. A discussion of the torsional dynamics stability can be found in the articles by [PERSON] et al. [28], [PERSON] et al. [29], and [PERSON] et al. [30]. An interesting effect of the interaction of the bit with the rock is considered in the study by [PERSON] et al. [31]. The phenomenon of \"jerky vibrations\" is considered in the studies by [PERSON] et al. [32] and [PERSON] et al. [33]. As a rule, the drill string vibrates in one of three modes: bending-axial, torsion-axial, or bending-torsional. When it comes to combined movement, nonlinear models need to be considered. This applies both to the equations themselves and to the quantities included in these equations. Nonlinearity, if any, should remain in the boundary conditions. The reason for combining directly axial and torsional vibrations is the interaction of the drill with the rock at the lower end of the drill string, which is expressed in nonlinear contact interaction and nonlinear friction. It is assumed that these factors also give rise to the phenomenon of \"jerky vibrations\". There are several papers describing the axial-torsional dynamics of the drill string. It is worth noting the studies by [PERSON] et al. [34], [PERSON] [35], and [PERSON] [27]. In the study by [PERSON] et al. [36], one can find a model with three degrees of freedom--one for the axial motion and two for the torsional one. [PERSON] et al. [37] studies the dynamics of the torsion-axial vibrations of a polycrystalline diamond drill. The model of the bending-torsional vibration dynamics is discussed in the studies by [PERSON] and [PERSON] [38; 39]. A model of dynamics and control over unwanted vibrations is presented by [PERSON] [40]. [PERSON] et al. [41] study the problem of \"vortex whirling\"--the transition of the phenomenon of \"jerky vibrations\" into \"vortex whirling\" using the bifurcation theory. The study considers the influence of the flowing fluid as well as contact interaction. The articles by [PERSON] and [PERSON] [42] and [PERSON] et al.