Three models—the viscous damping model, the new hysteretic damping model, and the Coulomb damping model—are studied in this paper. For the viscous damping and the Coulomb damping models, the free vibration problem is reviewed and demonstrated by animations. Regarding the new hysteretic damping model, the free vibration problem for the different range of parameters, namely 0<η<1, η=1, and η>1 are analytically derived and are also demonstrated by animations. In particular, the exact solutions of the latter two cases are derived for the first time. In animations, the trajectories for three damping models in the phase plane consist of straight lines, quarter ellipses, and hyperbolic curves. For the case of η≥1, it is interesting that permanent deformation may occur. In addition, the dead zone for the Coulomb damping model in the phase plane is also addressed. The envelope for the amplitude decay yields exponential, geometric, and linear curves for the viscous damping model, the new hysteretic damping model and the Coulomb damping model, respectively. It is also the primary focus that the same period and the same ratio of amplitude decay for the relation between the viscous coefficient and the hysteretic parameter are constructed. All animations are produced using the symbolic software Mathematica because it is easy for readers to understand the physical behavior of three damping models.
