## 1 Introduction  
Torsional vibrations are very concerning for drilling operations, especially in deep wells where drill string length, speed of rotation, weight, flow rate, and torque conditions can result in undesirable effects such as stick-slip phenomena, tool wear and decreased drilling efficiency [1, 2, 3, 4]. Understanding and mitigating torsional vibrations is critical for improving the reliability and performance of drilling operations, particularly in challenging environments [5, 6]. While extensive work has been done in torsional vibration studies, mainly in large-scale operations [7, 8], and drill string behavior in a downscaled experimental setup [2], a significant gap exists in direct comparative studies that evaluate the scalability of downscale models to real-world drilling conditions. Various methods have been adopted in studying torsional drilling vibrations, such as analytical, numerical, and experimental approaches. [PERSON] and [PERSON] [9] put forward a lumped parameter model that included axial as well as torsional and lateral vibrations. They showed that feedback control can be quite effective in suppressing stick-slip vibrations. [PERSON] et al. [10] compared 3-mode (axial, torsional, and lateral) and 2-mode coupled models; the results pointed toward the absence of torsional coupling, which produced greater lateral impacts. Such vibrational stability models that consider bit-rock interaction have been studied in great detail for both finite-dimensional [11; 12; 13; 14; 15; 16] and distributed systems [17; 18]. Other studies [19; 20; 21; 22; 23] have carried out comparative analysis for controller design aiming to suppress torsional vibrations, which proved that adjustments in the parameter of weight on bit (WOB), top-drive torque, and rotational speed (RPM) can be equally used as mitigation strategies. Controller design for drilling torsional vibrations has been addressed for finite-dimensional models [24; 25; 26; 27; 28]. Attenuation of axial-torsional drilling vibrations was studied in [3; 29] for lumped-parameter models, in [30] for distributed parameter models, and in [31] for a partial differential equation-ordinary differential equation (PDE-ODE) coupled model. Experimental studies on small-scale drilling setups have provided valuable insights into drilling vibrations. [PERSON] et al. [32] and [PERSON] et al. [33; 34] developed small vertical rig to investigate vibration phenomena, while [PERSON] et al. [35] designed an experimental rig replicating undesired drill string behaviors such as stick-slip, whirling, and buckling. [PERSON] [36] studied friction-induced vibrations using a DC motor, rotating discs, and a low-stiffness string. Recent efforts by [PERSON] and [PERSON] [37] initiated an experimental setup that enabled the investigation of torsional behavior in different wellbore sections with the downscaling ratio set as 1:40. [PERSON] and [PERSON] [38] and [PERSON] et al. [34] considered in more detail torsional wave propagation by means of their experimental rigs, focusing on the effects of various operational parameters, such as WOB and RPM. Additionally, [PERSON] et al. [39] examined geometric nonlinear interactions and mechanical friction effects in drill string dynamics under compressive loads. A complete look at drill string vibrations shows how important non-linear dynamics are in knowing what a system does. To gain a better understanding, researchers have used non-linear analysis tools. [PERSON] [40] applied bifurcation theory to look at a coupled torsional-lateral model of the drill string, and to compare results against experimental field data. Similarly, [PERSON] et al. [41] studied axial-torsional coupling dynamics by nonlinear analysis. Their parametric study using bifurcation theory showed that it could be a useful tool for assessing how the system behaves under different operating conditions. In a related study, [PERSON] et al. [42] conducted a parametric assessment of drill string motion to enhance understanding of dynamic responses.