## 1 Introduction  
Gas foil bearings (GFBs), i.e., bump-type hydrodynamic gas-lubricated bearings, exhibit a number of advantages over oil-lubricated bearings, such as fluid-film plain journal bearings or rolling element bearings. They do not require an oil lubrication system consisting of complex components that enable oil to circulate and lubricate bearings with low friction. Their outstanding characteristics include high-speed operations, extreme temperature capabilities, and reduced maintenance [1, 2]. In addition, their compliant structure consisting of bump and top foils increases their load-carrying capacity [3] and provides additional dry-friction damping [4, 5]. They also provide a variety of other advantages, such as simple design, light weight, high durability, and high-temperature operation. Although GFBs exhibit these advantages, they undergo subsynchronous motion with highly fluctuating amplitude in some cases. This is defined as a dynamic problem linked intrinsically to the behavior of the fluid film within a bearing referred to as oil whirl and oil whirl. This occurs typically when damped natural modes become unstable at the first critical speed [6, 7]. Subsynchronous motion is the primary cause of rotordynamic instability. In certain cases, the nonlinearity of the compliant structure of the GFBs can cause rotordynamic instability [8]. Many studies have been conducted and various designs of GFBs have been proposed to prevent and resolve this instability. Attempts have been made to modify the inner surface shape of the GFB to achieve improved performance. [PERSON] and [PERSON] [8] inserted three metal shims into bearing structures to introduce mechanical preloads. They examined the effects of the mechanical preloads on the rotordynamic behavior. It was revealed that preloads decrease the difference between two cross-coupled stiffness coefficients of the gas film, which results in reduced rotordynamic instability. Furthermore, the effects of the combination of mechanical preloads and bearing clearance were studied by [PERSON] et al. [9]. They demonstrated that an optimal combination of preload and clearance exists with regard to rotordynamic stability and bearing friction. Similarly, [PERSON] et al. [10] studied an optimum bearing-preload pattern and demonstrated that the mechanical preload should be designed based on GFB design variables. The principle underlying the mechanical preload is to develop a wedge shape between the bearing surface and a journal, and to enhance the hydrodynamic pressure at the preload locations. This results in increased direct stiffness and damping coefficients, and a variation in the journal equilibrium position (expressed as eccentricity and attitude angle). As a result, rotordynamic stability is improved. These solutions are simple and effective. However, they also exhibit limitations, such as reduced bearing performance at low rotor speeds [11; 12] owing to a decrease in film thickness, and the inability to respond instantaneously to variations in operation conditions such as rotor speed and external excitation. Recently, several studies addressing these issues have been performed by introducing piezoelectric (PZT) actuators in gas bearings, which are referred to as active or controllable gas bearings. [PERSON] and [PERSON] [13] proposed an active oil bearing with film thickness adjustment using a PZT actuator. Its bearing consisted of two horizontal segments. The lower segment was fixed, whereas the upper segment was movable in a vertical direction by the actuator, with a maximum stroke of 200 \(\upmu\)m. They controlled the actuator to obtain different mechanical preloads depending on the rotor speed and demonstrated that the control could effectively alter synchronous motions. Meanwhile, [PERSON] et al [14] used patch-type PZT actuators. These deform the supporting shells of the bearing substructure to control the radial clearance of GFBs.