## 3 Results and Discussions
### Stick Slip Behavior at 100 m (Upscale) and 3.3 m (Downscale) Depth  
This analysis helps to understand the propagation of torsional forces through the drill string and their effects on the rotational behavior at different scales. Figures 5-7 present  
Figure 4: Upscale and downscale bit RPM at torque values of 4500 kgfm and 0.09 kgfm. Figure 3: Upscale and downscale bit RPM at torque values of 1500 kgfm and 0.04 kgfm. comparative results for both the upscale and downscale drill string models under different torque conditions, showcasing the emergence and behavior of stick-slip oscillations. A torque of 500 kgfm was applied to the upscale string, and 0.01 kgfm to the downscale string (Figure 5). From the onset, it can be seen that both strings closely mimicked the top RPM, indicating that the rotational energy was effectively transmitted through the drill string without significant disruption. However, as time progressed, the upscale string started to exhibit stick-slip behavior, where periods of sticking (when the rotation momentarily stops) alternate with slipping (sudden rotational jumps). Interestingly, the stick-slip effect was more severe in the downscale string, which reflects its higher sensitivity to torque variations due to its scaled-down geometry and reduced mass. Both strings entered the stick-slip phase simultaneously, suggesting that the onset of torsional oscillations in both models was influenced by a common factor--likely the transmission of top-end rotational fluctuations. However, after approximately 5 s, both the upscale and downscale strings managed to overcome the stick-slip effect, gradually stabilizing and synchronizing with the top RPM. This indicates that the systems could reach a steady rotational state after overcoming initial torsional disturbances. The applied torque was increased to 1500 kgfm for the upscale string and 0.04 kgfm for the downscale string (Figure 6). As observed in the first case, both strings initially mirrored the top RPM's rotation without noticeable torsional instability. However, after a brief period, both strings entered a stick-slip phase, but the severity of the oscillations was much greater than in the first scenario. This suggests that the magnitude of the applied torque has a direct influence on the intensity and persistence of stick-slip behavior. The downscale string continued to exhibit more severe stick-slip oscillations compared to the upscale string. This outcome aligns with the expectation that smaller-scale systems, due to their reduced inertia and stiffness, are more prone to sudden rotational disturbances. Moreover, increased torque levels amplify the system's vulnerability to torsional vibrations, further validating the importance of optimized torque management during drilling operations to minimize mechanical wear and instability. Figure 5: Upscale and downscale string RPM at torque values of 500 kgfm and 0.01 kgfm at 100 m (upscale) and 3.3 m (downscale). Torque was increased to 4500 kgfm for the upscale string and 0.09 kgfm for the downscale string, as shown in Figure 7. As in the previous tests, both strings maintained synchronization with the top RPM for a longer period before experiencing stick-slip oscillations. This implies that at elevated levels of torque, there could be a transient delay in the development of torsional vibration in the drill string until stick slip sets in. Even though both strings underwent stick slip, the downscale string had a more drastic stick-slip phenomenon owing to its smaller dimensions and reduced structural rigidity. This shows how scaling can affect the dynamic response of drill strings, with smaller systems being more vulnerable to torsional instabilities. Nonetheless, it is important to note that both strings stabilized and resumed rotation in time with the top RPM, showing that the system achieved the steady-state condition after it overcame transient instabilities. Figure 6: Upscale and downscale string RPM at torque values of 1500 kgfm and 0.04 kgfm at 100 m (upscale) and 3.3 m (downscale).