## 1 Introduction  
The bearing system of any rotating machine is crucial in terms of the permissible speed and mechanical and thermal loads. Bearings also often determine the service life and reliability of a device. Bearing properties, therefore, have a decisive influence on the operating parameters of the entire machine. That is why much attention should be paid to these components of a rotating machine during the design, construction and operation stages. Numerous types of bearings are currently available on the market, which can be customised to match the specific operating parameters of a rotating machine. With increasing target speed and temperature, the number of commercially available bearing systems decreases. Currently, the most used high-speed bearings are gas bearings [1], magnetic bearings [2, 3] and high-precision rolling bearings [4, 5]. Each bearing type has its own properties that make it suitable for specific applications. Unfortunately, at very high speeds (approx. 100,000 rpm) and high temperatures, the selection of ready-to-use bearings is very limited. Foil bearings, predominantly classified as gas bearings, perform excellently under such conditions. While they have been known for many decades, their rapid advancement in recent years has been made feasible by developing novel materials and unique structures. The history of foil bearings is also linked to the growing popularity of high-speed rotating machines such as microturbines and turboexpanders [6, 7, 8]. A lot of research and development work is currently being carried out on the application of foil bearings in aviation [9, 10],small-scale cogeneration [11; 12; 13] and the automotive industry [14; 15; 16]. Well-designed foil bearings have many advantages, such as the absence of external lubrication, low friction losses, low noise and vibration levels, long service life, and the ability to operate at high speeds and temperatures [17]. Foil bearings typically function as aerodynamic bearings, but less often as hydrodynamic bearings, using the viscosity of the surrounding fluid to form a lubricating film, eliminating the need for additional complex lubrication systems. The design of foil bearings incorporates the use of both the top and bump foils, in addition to the journal and sleeve [18]. Such a foil bearing structure is shown in Figure 1. By employing compliant foils, the dynamic stability of rotors supported by foil bearings experiences a significant improvement. This is because the shape of the lubrication space adapts to the current operating conditions. At the same time, the poor damping properties of the lubrication film are compensated by the flexible foil structure (i.e., top foil and bump foils) [19; 20; 21]. Many inventors have proposed numerous modifications of foil bearings, replacing standard bump foils with other structures with favourable static and dynamic properties. These include metal mesh foil bearings and hybrid bump-metal mesh foil bearings [22], gas foil bearings with metal shims between the bumps and the sleeve [23], foil bearings with an embedded compression spring [24] or foil bearings with embedded smart materials [25]. Hybrid (hydrodynamic and hydrostatic) foil bearings [26] and multi-layer protuberant foil bearings [27] have also been developed. Certain arrangements of foil bearings employ diverse cooling methods, including cooling fans [28], side-feed pressurisation [8] or the utilisation of thermocouples [29]. Foil bearings are typically gas lubricated to operate at very high rotational speeds with low frictional losses. A less popular method of lubricating foil bearings is the use of liquids. Bearings lubricated in this manner do not reach as high rotational speeds as gas foil bearings, but they have many other advantages. Lubricating the bearing with a liquid increases the load capacity. It allows direct use of process media, which in some machines (such as pumps) are present in liquid form, broadening the range of applications.