Modal analysis uses the overall stiffness, mass of a structure and constraints to determine the various frequencies at which it will vibrate. The mode shapes corresponding to those frequencies are also provided. Once we have these frequencies, we do our best to avoid exposing the structure to these frequencies in order to avoid damaging the structure. Here are two examples:
a) When designing a bridge in an earthquake zone, we will design the bridge to have natural frequencies of vibration that are different from the vibrational frequencies of the earthquakes.
b) Unbalanced forces from a motorcycle engine can damage parts mounted to the bike if the natural frequency of vibration of those components are the same as the frequency of vibration of the engine.
The equation used to model natural frequencies is [K - (ω^2 )M]X = 0 where:
K is the stiffness matrix
ω are the natural frequencies of the system (or eigenvalues)
M is the mass matrix
X is the systems eigenmode matrix