Rotor machinery supports are susceptible to fracture arising from multi-period or irregular transient vibration and even the inconspicuous fatigue phenomenon. The novel dynamic model of a rotor mounted on the floating ring squeeze film damper (SFD) was developed. The proposed SFD implements low stiffness and small scale to overcome the deficiency. Based on the theory of hydrodynamic lubrication, the Reynolds equations on the working principle of the floating ring are established. Then, the dynamic characteristics of the rotor system during maneuvers, with the floating ring SFD supports, are subsequently examined by adopting the finite difference method. In addition, the oil film whirl mechanism of the floating ring SFD is demonstrated according to the transient analysis of the fluid–structure interaction model. The results of the SFD simulation reveal that, with increasing eccentricity ratio, both the inner and outer oil film pressure tend to be larger due to the shrinkage of the effective coverage of oil film. The maximum oil film pressure and bearing capacity increase nonlinearly within a certain eccentricity ratio range. Through the comparisons of the results, the vibration suppression effects of the proposed SFD are analyzed. This work will provide the practical reference for the dynamic design of the rotor support system.