Generally mathematical model of such damping is quite complicated and not suitable for vibration analysis.

Simplified mathematical model (such as viscous damping or dash-pot) have been developed which leads to simplified formulation.

A mathematical model of damping in which force is proportional to displacement i.e., F_d = cx is not possible because with cyclic motion this model will encounter an area of magnitude equal to zero as shown in
Figure 3.1(a). So dissipation of energy is not possible with this model.

The damping force (non-linearly related with displacement) versus displacement curve will enclose an area, it is referred as the hysteresis loop (Figure 3.1(b)), that is proportional to the energy lost per cycle.