The vibratory behaviour of materials has traditionally been modelled by a linear equation of second order where the most significant parameters are the mass (m), the rigidity (k) and the damping (c).
Model mass-shock absorber-spring of one degree of freedom: Assuming that this system has a degree of freedom, we can write the equation of the movement in the following way: x(t) represents the temporary law of that degree of freedom, and f(t) represents the temporary law of the cause of that movement x(t).
This behaviour defines frequencies for which the amplitude of the answer is the maximum. These frequencies are known as the resonances of the system. For that reason, the system is adapted and simplified to work in the theoretical scope of the frequency. A tool that allows us to transfer the temporary domain of the frequency domain is the transformed of Fourier.
The most commonly used parameters are density (distributed mass by volume unit), the Young's modulus (associate to the rigidity of the material) and the loss coefficient (associate with the behaviour of a material).
These laws are obeyed by the materials under certain hypotheses, such as small deformations and linearity between the force and the resulting displacement. When these hypotheses are not fulfilled, we speak of models with non linear behaviour.