## 3 Results and Discussion  
### Dynamic Response Under Drilling Parameters  
As illustrated in Figure 12, three tracer particles were selected within the drill string fluid to analyze their trajectories over a 1 s timeframe. The particle motion paths under different drilling parameters are depicted in Figure 13. Driven by drill string rotation and centrifugal forces, the fluid exhibits helical flow patterns, with particles at different radial positions demonstrating motion independence--indicating negligible radial interference. Notably, particles closer to the drill string axis (Particle 1) experience reduced rotational and centrifugal effects, exhibiting prolonged rotational periods and elongated axial pitch. Specifically, the rotational periods in Figure 13a are 0.18 s, 0.24 s, and 0.25 s; in Figure 13b, 0.15 s, 0.20 s, and 0.21 s; and in Figure 13c, 0.10 s, 0.15 s, and 0.17 s. These results demonstrate a radial gradient in rotational influence, with periods lengthening from the wall toward the center. Elevating the rotational speed or axial flow rate effectively accelerates helical flow velocities. Figure 14 shows the effect of drill string internal fluid on the drill string under different rotational speeds. The average contact force between the drill string and wellbore wall increases with rotational speed, exhibiting a 9.65% rise at 800 rpm compared to static conditions (0 rpm). Under combined gravitational and rotational coupling, sustained contact predominantly localizes in the lower-right quadrant, while elevated centrifugal loading degrades stability and intensifies collision frequency at higher speeds. Additionally, the centrifugal force-induced radial pressure gradient within the fluid intensifies with increasing rotational speed, consequently amplifying the wall forces exerted on the drill string. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Condition** & **Parameter** & **Explicit Dynamics** & **Recursive Algorithm** & **Efficiency Improvement (vs. Explicit Dynamics)** \\ \hline \multirow{3}{*}{Condition 1} & Mesh Size (mm) & 1.8 & 1.8 & \\  & Element Count & 422,682 & 356,520 & 59.3\% \\  & Total Sim. Time (Days) & 4.3 & 1.75 & \\ \hline \multirow{3}{*}{Condition 2} & Mesh Size (mm) & 8 & 8 & \\  & Element Count & 132,400 & 124,800 & 56.3\% \\ \cline{1-1}  & Total Sim. Time (Days) & 0.96 & 0.42 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Cost comparison of two solution methods. Figure 12: Selection rules for tracer particles in drill string internal fluid. Figure 15 shows the effect of drill string internal fluid on the drill string under varying feed pressures. The increase in feed pressure exacerbates the deformation-induced squeezing of the drill string against the borehole wall, leading to a corresponding increase in average contact force. The average contact force exhibits an 8.4% elevation at 15 kN feed pressure relative to the zero-load condition. However, the elevated feed pressure does not directly affect the internal fluid dynamics, and there was no obvious increasing trend of the wall force of the fluid on the drill string. Figure 14: Effect of rotational speeds on drill string dynamics considering drill string internal fluid interaction. Figure 13: Particle motion paths under different drilling parameters: (**a**) 30 L/min, 600 rpm; (**b**) 50 L/min, 400 rpm; (**c**) 50 L/min, 600 rpm. Figure 15: Effect of feed pressure on drill string dynamics considering drill string internal fluid interaction. Regarding the annular fluid, Figure 16 illustrates the effect of varying rotational speeds on the drill string. An increase in rotational speed enhances the hydrodynamic pressure effect of the annular fluid on the drill string. The intensified hydrodynamic pressure reduces contact between the drill string and borehole wall, while the wall force exerted by the fluid on the drill string increases concurrently. Figure 17 shows the influence of annular fluid on the drill string under different feed pressures.