Many vibrating mechanical systems can be approximated by concentrating their inertia into one generalized mass and their elastic behavior into one generalized stiffness. This is a single-degree-of-freedom (SDOF) model.

A basic mass-spring system obeys

$$m\ddot x+kx=0.$$

One generalized coordinate

The phrase “one degree of freedom” means one independent coordinate $x(t)$ is sufficient to describe the modeled motion. A real structure may deform in many places while still being approximated by one dominant coordinate for a particular vibration problem.

Natural frequency

The undamped natural angular frequency is

$$\omega_n=\sqrt{\frac{k}{m}}.$$

Increasing stiffness raises the natural frequency; increasing mass lowers it.

Lumped models are approximations

A point mass and ideal spring are not claims that the real object literally has those components. They are a reduced model that preserves the inertia and restoring behavior relevant to the motion being studied.

Why start simple

The SDOF model makes the relationships among mass, stiffness, damping, forcing and resonance explicit before those ideas are extended to structures with many coupled coordinates.