The drilling industry faces a complex challenge in modeling dynamic vibrations during drilling because it directly affects drilling efficiency, tool longevity, and overall operational safety. The emphasis on creating reliable control systems that can either completely eliminate or significantly reduce these vibrations’ negative effects has increased as a result of the necessity to understand and manage them. Thus, developing a thorough grasp of the drill bit’s response to these dynamic forces is an essential first step in creating efficient control schemes. Numerous studies in the literature have previously examined the intricacies of the drill string’s behavior when subjected to axial-torsional vibrations. Numerous models have been put forth in these studies to capture the complex interaction between axial and torsional forces. But incorporating such control mechanisms calls for a more thorough investigation of the ways in which the drill string’s dynamic behavior and the control systems interact, especially in relation to the axial-torsional effects. Within this framework, our work aims to advance current understanding by providing a thorough model of our system that takes into account the particular subtleties of torsional and axial vibrations. Our main goal is to understand the complex interrelationship between these two vibration types and how they affect the drilling process as a whole. By dissecting this relationship, we hope to reduce the harmful effects of the coupled vibrations and increase the control system’s robustness and effectiveness. We think that this research has a lot of potential to improve our knowledge of and ability to control dynamic vibrations that occur during drilling operations, which will ultimately increase the overall effectiveness and safety of drilling procedures.