## 1 Introduction  
Driven by the demands of decarbonization and sustainable development strategies, it is foreseeable that unconventional energy sources are progressively occupying a central position in the modern energy structure [1]. One of the core aspects of promoting their application is the efficient and high-quality development of deep target reservoirs. To meet this demand, drilling technology has evolved from traditional vertical drilling to advanced horizontal well drilling and precise trajectory control techniques. Specifically, small-diameterhorizontal well technology, owing to its significant advantages in enhancing drilling efficiency, reducing operational costs, achieving precise trajectory control, and expanding the effective contact area with the reservoir, has gained increasingly widespread application in the exploitation of unconventional oil and gas resources such as shale gas [2]. Indeed, the modeling of drill string dynamics in directional wells, as explored by [PERSON] et al. using [PERSON]'s method for real-time simulation, and the analysis of nonlinear rotordynamics in curved wells by [PERSON] et al., underscore the industry's focus on these advanced drilling scenarios [3,4]. However, the prevalent phenomena of drill pipe vibration and friction during horizontal well drilling not only severely constrain drilling efficiency but also pose potential threats to operational safety. Furthermore, their impact tends to exacerbate with increasing well depth [5], representing a major contributor to tool failure, component wear, and reduced rate of penetration (ROP) [6]. This is a central concern in many studies, including the investigation of drill string-casing collision by [PERSON] et al. and the analysis of vibrations in composite drill strings by [PERSON] et al. [7,8]. Therefore, in-depth analysis, effective control, and optimization of the vibration behavior within the drill string in horizontal wells are of crucial theoretical and practical significance for ensuring efficient, safe, and economical drilling operations, as well as the efficient recovery of oil and gas resources. Regarding drill string vibration, numerous scholars have dedicated efforts to uncovering its physical mechanisms and exploring effective vibration mitigation strategies [9]. These models range from simplified lumped-parameter approaches to complex continuous models, with [PERSON] et al. critically comparing their efficacy and highlighting the limitations of lumped models in capturing distributed phenomena like wave reflection [10]. In terms of theoretical modeling and analysis, researchers have constructed various mathematical models to characterize the complex vibration behavior of drill strings: for instance, investigating force distribution under strong vibration and optimizing drill pipe combinations by establishing a \"drill string-wellbore\" mechanical model [11]; deriving a lateral vibration model considering the collision effect between the drill string and the wellbore wall to analyze contact vibration characteristics and natural frequencies [12], a theme also explored by [PERSON] et al. who modeled drill string-casing collision with a focus on rubbing damage, and by [PERSON] et al. who employed a penalty method for contact interactions in realistic geometries and effects of drill bit manufacturing errors and material choices on tool stress and operational safety [7,13,14]; and proposing a comprehensive vibration model incorporating axial load, drilling clearance, and contact forces to accurately analyze vibration characteristics for guiding bottom hole assembly (BHA) design [15]. The influence of BHA design and its interaction with the wellbore are critical, as emphasized by [PERSON] et al. in their study of composite drill strings considering bit-rock interaction and drill string-wellbore contact, and by [PERSON] et al. who modeled BHA interactions using a component-based approach with assumed modes [8,16].