This research offers a thorough analysis of the dynamic behavior of 1-, 2-, and 3-DoF mechanical systems under a sinusoidal force, examining both mechanical and dq coordinates. By utilizing standardized initial conditions, the 1-DoF system displays fascinating oscillatory patterns with dual frequency components, highlighting the significance of low damping. The adaptation to dq coordinates simplifies the analysis and highlights the system's nuanced behavior. In contrast, the 2-degree-offreedom system exhibits intricate interactions, oscillation phenomena, and multiple frequency components in mechanical coordinates. The contribution of masses that do not experience external forces in dq coordinates is minimal. On the other hand, the 3-degree-of-freedom system shows diverse interactions and frequency components that are different from the dq transformations. The observed dynamics not only enhance comprehension of these systems but also provide valuable insights for refining analytical approaches in the analysis of dynamic systems. This study sets the stage for future investigations and urges the development of streamlined analytical frameworks for a more focused exploration of externally influenced variables in dynamic mechanical systems.