This chapter explores the fundamental characteristics of vibratory systems and presents the general equation of motion for single-degree-of-freedom systems. The approach follows the engineering analysis cycle, beginning with the formulation of mathematical models, progressing to dynamic response analysis through computational simulation, and concluding with experimental validation. The activities outlined in this chapter can be easily reproducible by readers, offering visual aids that clarify abstract concepts like equilibrium points, stability, phase portraits, and the distinction between large and small vibrations. Four illustrative examples are provided to further demonstrate these concepts.