## 1 Introduction
Additionally, research has also focused on suppressing specific vibration modes and energy conversion, such as active control techniques based on the concept of negative damping to suppress torsional stick-slip vibration [17]. [PERSON] et al. specifically designed a sliding mode controller (SMC) using an Extended Kalman Filter (EKF) to suppress coupled axial and torsional vibrations in horizontal drill strings. The challenge of stick-slip and other torsional vibrations was also a key focus for [PERSON] et al. in their comparative study of downscale and upscale drill strings [18,19]. Recent studies have further refined these models, such as models differentiating between drill pipes and BHA and including nonlinear bit-rock friction to analyze the influence of parameters like rotational speed and weight on bit (WOB) on torsional vibration and ROP [20]. [PERSON] et al. also incorporated detailed bit-rock interaction models in their FEM analysis [8]. Concurrently, efficient and accurate modeling methods are also evolving, such as improved semi-analytical methods capable of simulating the dynamics of fluid-conveying pipelines under arbitrary geometric and boundary conditions [21]. The challenges of modeling drill strings in complex geometries like curved and directional wells are addressed by [PERSON] et al. using FEM with Craig-Bampton reduction and by [PERSON] et al. using [PERSON]'s method for real-time simulation [3; 4]. The fluid-structure interaction (FSI) effect between the drilling fluid and the drill string, along with its impact on the drill string's dynamic behavior, is a core element influencing drill string vibration and friction. Scholars often employ methods like finite element simulation and intelligent algorithms for analysis. Research areas cover the following: developing new methods for calculating the coupled modal response of flexible structures and fluids [22], and establishing governing equations for bending vibration that consider internal flow, gyroscopic effects, and external loads to fully describe the FSI field [23]. For specific systems, scholars have conducted dynamic modeling, simulation, and optimization, such as for offshore drilling riser systems and analyzing the natural frequencies of pipes containing two-phase flow [24; 25]. Flow-induced instability is also a major concern, with studies analyzing the instability risk of drill strings conveying mud [26], exploring the stability of pipe columns under the combined action of internal and external flow, and particularly highlighting the dominant role of annulus flow in stability [27; 28]. [PERSON] et al. modeled hydrodynamic forces in inviscid fluid using linearized Navier-Stokes equations and explored advanced modeling of thermal and hydrodynamic conditions for specialized downhole tool deployment [7; 29]. Furthermore, research has examined the influence of fluid parameters on drill string performance, revealing the complexity of friction variations under vibration and buckling conditions [30], demonstrating that adjusting drilling fluid flow rate and density can alter fluid damping and consequently affect lateral amplitude [31], and improving the accuracy of pressure loss prediction under complex conditions using neural networks [32]. The lubricating effect of drilling fluid is equally important; adding suitable lubricants can effectively reduce friction, improve force transmission, and decrease buckling risk [33; 34]. [PERSON] and [PERSON] considered mud buoyancy and viscous damping in their torque and drag model [35].