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The simplest vibratory system can be described by a single mass connected to a spring (and possibly a dashpot). The mass is allowed to travel only along the spring elongation direction. Such systems are called Single Degree-of-Freedom (SDOF) systems and are shown in the following figure,

Equation of Motion for SDOF Systems

SDOF vibration can be analyzed by Newton's second law of motion, F = m*a. The analysis can be easily visualized with the aid of a free body diagram,

The resulting equation of motion is a second order, non-homogeneous, ordinary differential equation:

with the initial conditions,

The solution to the general SDOF equation of motion is shown in the damped SDOF discussion.