## 1 Introduction
It was noted that this solution, like gas foil bearings, was the result of years of development of classical aerodynamic gas bearings. In recent publications on gas bearings, authors have focused, among other things, on design using artificial neural networks and genetic algorithms [13], modeling of bearings operating under atypical conditions [14], and the use of new design solutions for bearing sleeves [15]. Two basic types of instability can occur during gas bearing operation: forced resonance with damping and instability occurring at the so-called threshold speed caused by the loss of damping properties of the bearing [16; 17]. Resonance with damping, known as synchronous resonance, is related to natural frequencies and is caused by a force or moment due to unbalance, with the rotor vibrating at a frequency corresponding to the rotational speed of the shaft [18]. When passing through the critical speed, which corresponds to the natural frequency, there is an abrupt phase change, and as the speed continues to increase, the operation of the system stabilizes [19]. Once the speed, known as the stability limit, is exceeded, the damping capacity of the gas film disappears. A slight increase in rotational speed results in a rapid increase in vibration amplitude, leading to contact between the journal surface and the sleeve surface. This type of instability is characterized by the rotor axis spinning along a trajectory (circular or elliptical) at a frequency less than or equal to half the rotational frequency of the shaft. The loss of stability is caused by the effect of the rotational speed of the journal on the flow of the lubricating medium in the bearing [20; 21; 22]. The low stability of gas bearings at high speeds is a major limitation to their wider use. To improve the dynamic properties of gas bearings, in addition to self-acting bearings known as gas-dynamic (aerodynamic) bearings, which generate aerodynamic lift through the relative movement of two surfaces, externally pressurized bearings supplied with compressed gas (delivered through various types of supply orifices) are also used. The aerodynamic lift in externally pressurized bearings can be generated solely by high pressure from an external source, or by both an external source and the relative movement of two surfaces [5]. Various solutions have been proposed to increase the threshold speed in gas bearings. An example of such bearings is tilting pad journal bearings, the large-scale research and development of which began as early as the 1960s [23]. Since then, many researchers have endeavored to improve the dynamic properties of self-acting gas bearings. For example, in 2017, [PERSON] et al. [24] proposed modern solutions such as a novel squeeze-film air bearing with flexure pivot-tilting pads based on near-field acoustic levitation technology. Another approach to increasing the stability threshold of a gas bearing is to support the sleeve on a flexible pad, such as a rubber O-ring. As early as the 1960s, this solution was shown to significantly improve the stability range [25]. Unfortunately,it is not applicable at either low or high temperatures [1]. Research is still underway to find materials with appropriate damping properties and stiffness [26]. In the 1950s, the first papers on the aforementioned gas foil bearings appeared [27], in which segments of a flexible bump foil and top foil were placed between the journal and the sleeve of the bearing. Foil bearings operate as aerodynamic bearings, with the foil structure automatically adapting the lubrication gap geometry to the bearing's current operating conditions. Damping deficiencies in the flow part of the bearing can be partially compensated by damping in the structural part. This allows the application range of aerodynamic gas bearings to be extended to turbomachines operating in harsh environments [28]. The following section of the article discusses the modeling and analysis of aerodynamic gas bearings, which were carried out using two different numerical methods.