This study conducts a comparative analysis between black box and grey box models in handling torsional vibrations within drill strings, employing System Identification techniques such as NARX and ARX. The developed lumped parameter model, incorporating nonlinear friction torque, accurately replicates bit-rock interactions, effectively addressing torsional instability. Validation using field data sampled at 50 Hz demonstrates the model's robust fit to the interaction curve, closely mirroring real-world bit speeds. Although axial and lateral vibrations are excluded, the numerical model constructs a highly predictive torsional stability map. Results showcase the black box model with $R^2$ = 0.8897 and $RMSE$ = 30.1300, highlighting the superior performance of the grey box model with $R^2$ = 0.9287 and $RMSE$ = 24.2137 in dynamic system identification within drill string torsional vibration modeling.
