A structure with several independent generalized coordinates is a multiple-degree-of-freedom (MDOF) system. In a linear undamped model,

$$M\ddot{\mathbf x}+K\mathbf x=\mathbf0,$$

where $M$ is the mass matrix and $K$ the stiffness matrix.

Coupled coordinates

The components of $\mathbf x$ need not oscillate independently. Motion of one coordinate can exert restoring forces on others through the off-diagonal couplings in the system matrices.

Normal modes

Seek harmonic motion of the form

$$\mathbf x(t)=\boldsymbol\phi e^{i\omega t}.$$

Substitution gives

$$(K-\omega^2M)\boldsymbol\phi=\mathbf0.$$

Nonzero mode shapes $\boldsymbol\phi$ exist only for particular natural frequencies $\omega$. Each pair of frequency and mode shape describes a normal mode.

Modal coordinates

Under suitable conditions, the normal modes provide a basis in which the coupled vibration problem can be decomposed into independent modal oscillators. A general response is then built by combining those modal contributions.

Why structures have many resonances

A beam, building or machine assembly can deform in many patterns. Each mode has its own natural frequency, so an MDOF structure can have many resonant regions rather than one.

The single-degree-of-freedom oscillator is therefore not a different theory: it is the simplest member of the same modal framework.