For a linear hysteretic material (under predominantly uniaxial loading), as we have already seen, the loss factor is independent of the stress field and can be treated as a material constant. For such a material, we can construct a single degree- of-freedom model with equivalent viscous damping. In this lecture, we shall show, through a simplified analysis, how we can use these results to select a material for structural damping.
Let us consider a harmonic force excitation of the system shown in Fig. 10.1, where the linear hysteretic damping has been replaced by equivalent viscous damping.

Figure 10.1: Harmonic force excitation

For hysteretic damping, the energy dissipated per cycle is proportional to the square of the amplitude, i.e.,

with is a constant. Equating this with the expression of energy dissipition for equivalent damping (see Moduule 2, Lecture 8)

The equivalent viscous damping coefficient is

(10.1)

where is called the hysteretic damping coeffiient.