## 1 Introduction  
Gas foil bearings (GFBs) have been considered as a core lubrication technology for oil-free high-speed rotation machinery [1]. Bump-type GFBs have compliant sub-structures including a bump and top foil, which improve the load-carrying capacity and rotordynamic stability [1, 2]. The use of GFBs reduces system complexity and maintenance costs for their simple design [3, 4]. Major applications of GFBs include high-speed turbomachinery and electro-mechanical machinery such as turbo-compressors, turbo-blower, micro gas turbines, and electric permanent magnet motors/generators. Specifically, GFBs have the following advantages as compared to conventional rigid hydrodynamic gas bearings [5]:  
High Load Capacity: A compliant bump foil structure increases the minimum thickness region in the gas film, leading to a capability to support higher loads. Low Power Losses: The prevalence of gas-film thickness results in lower viscous friction losses of the bearing. High-Speed Operation: Improved damping characteristics caused by the sliding motion between the bump foil and top foil/bearing housing makes the journal rotate stably at high speed. Self-Alignment: The flexibility of the bearing structure accommodates misalignment of assembled system and vibration/shock, or any other unexpected maneuver. Figure 1 presents the configuration of a one-pad journal GFB and gas foil-polymer bearing (GFPB) proposed by [PERSON] and [PERSON] [3]. For the GFB, the top and bump foils are located between the journal and bearing housings. The top foil provides a smooth bearing surface and assists in generating a hydrodynamic film pressure between the journal and top foil. The top foil is supported on a series of bump foils, which provide structural compliance and energy dissipation, acting as springs and dampers. The GFPB consists of laminated top, bump, and bottom foils made of Inconel X-750. An additional polymer layer made of nitrile butadiene rubber is filled in the circumferential direction. The polymer layer leads to radial motions of the sub-structure, providing additional structural compliance and hysteretic damping. One end of the top, bump, and bottom foils is spot-welded to the bearing key, whereas the other end is free. The static and dynamic performances of the GFBs mainly depend on the mechanical properties of the sub-structures, i.e., the material properties and geometrical configuration of the bump and top foils, particularly for heavily loaded GFBs at high speeds [2]. To predict the exact rotordynamic performance of the rotor supported on GFBs, the precise estimation of the mechanical properties of the sub-structures is important [3; 6]. Thus, studies on the mechanical properties of sub-structures have been one of the main research topics in GFBs. [PERSON] et al. [7; 8] proposed the first simple elastic foundation model of a bump foil. The bump foils were modeled as uniform springs with structural compliance, which was determined by the material and geometric properties of the bump foil. [PERSON] [9] presented another simple elastic foundation model of a bump foil under either free-free end or fixed-free end conditions for a single bump. Specifically, this model included the dry-friction coefficients of the sub-structures. Subsequently, [PERSON] et al. [1] proposed finite element (FE) models that coupled the elastic deformations of a two-dimensional (2D) shell or one-dimensional (1D) beam-like top foil with the bump foil deflections. The structural stiffness of the bump foil was estimated from a simple elastic foundation model [9]. In contrast, [PERSON] et al. [10] introduced a fully coupled FE model of a bump and top foil. This model included the moment, tension, curvature, and strain expressions for the cylindrical shell deflections of the foil structures. The bump foil was also considered as a continuous elastic foundation with coupled radial and circumferential deflections of the foil.