## 1 Introduction  
The working speed of modern aeroengines is generally above 10,000 rpm, while that of some small engines is as high as 40,000-50,000 rpm. Operation at high speeds induces severe dynamic loading with large amplitude journal motions at bearing supports [1]. Simultaneously, there are some large-amplitude response issues, which may cause serious accidents such as friction between the rotors and rotor fatigue cracks [2]. In order to meet the utmost desire of adequate critical-speed margin and high stable rotor speed in modern small aeroengines, squeeze film damper (SFD) bearings are an effective solution to add viscous damping that aids to attenuate vibration and also serve to isolate the rotors from the aeroengine frame [3]. Their simplicity of construction and consequent robustness means that they are commonplace in modern aircraft gas turbine engines [4, 5]. Although SFDs have been frequently used in the turbo-machinery industry, the non-synchronous vibrations can result in some problems of fatigue [6, 7]. Thus, it is a tremendous requirement to investigate and improve the ring SFD-rotor system dynamics for vibration attenuation of high-speed small rotating machinery. The typical and more practical approach in SFD modeling is based on the incompressible Reynolds lubrication equation to describe the relationship between the pressure in a fluid film and the journal motion. [PERSON] et al. [8] carried out the bifurcation behaviors of a rigid rotor-squeeze film damper considering the effect of fluid inertia. By changingthe value of the Reynolds number that reflects the fluid inertia of SFD, the hysteresis set is moved obviously. [PERSON] [9] proposed the pressure distribution model of SFD with finite length, and the generalized Reynolds equation of hydrodynamic pressure distribution considering that the inertial effect is obtained. According to the simulation results, the pressure distribution and fluid film reaction forces are significantly influenced by fluid inertia effects even at small and moderate Reynolds numbers. As in a much earlier work [10], the pressure distribution in the full oil film was then described by a modified Reynolds equation. In Ref. [11], modified Reynold equation with short damper approximation was used to derive the SFD forces for 2\(\pi\)-film. [PERSON] [12] proposed the use of Chebyshev polynomial fits to identify a finite difference (FD) solution of the incompressible Reynolds equation. Based on the complex fluid-structure interaction phenomenon inside the ring SFD, the pressure distribution of the damper oil film and the phenomenon of oil film whirl are worthy of further study for ring SFDs. To improve the performance of SFDs, various new design concepts had been proposed. In Ref. [13], [PERSON] proposed a calculation procedure to investigate the dynamic characteristics and response of an ERSFD-supported rotor system. The study reveals that three effects of elastic cause the better dynamical performance of ERSFD than SFD and rigid ring squeeze film damper. [PERSON] et al. [14; 15; 16; 17; 18] analyzed the influence of SFD oil film damping, oil film stiffness, and elastic ring stiffness on the rotor system through theoretical modeling, numerical calculation, finite element analysis, and so on. [PERSON] [19] studied the response of a rigid rotor system supported by SFD under the short-bearing approximation theory. The study showed that the oil film of SFD can withstand a certain negative pressure. [PERSON] [20] studied the damping mechanism of SFD and showed that through the elastic ring deformation, the nonlinearity of SFD can be effectively lightened. [PERSON] [21] established a rotor system model considering the nonlinear characteristics of SFD and analyzed the influence of SFD on the bistable response and non-coordinated precession response of the rotor system. [PERSON] [22; 23] used a rotor tester with an SFD to study the influence of the static eccentricity of SFD on the rotor vibration response and combined the design of SFD with the rotor dynamics design.