## 1 Introduction
### Foil Bearings  
A foil bearing is the type of bearing that is relatively new but is becoming increasingly common around the world. The first papers about foil bearings were published in the 1950s. Since then, the design and materials of foil bearings have changed significantly, but the principle of their operation has remained unchanged. A key element of a bump-type foil bearing is a set of compliant foils, usually made of thin metal sheets. So far, many experimental, analytical and numerical techniques for predicting the performance of foil bearings have been developed. Such parameters as load capacity, power loss, the temperature of the bearing parts as well as static and dynamic stiffness or damping can be thoroughly analysed experimentally. Different approaches are used to research the compliant bearing structure. Most of them are a variety of numerical methods. Modellingof the dynamic behaviour of the foil bearing structure, due to its mechanical complexity, remains a challenge for scientists and engineers from all over the world. Foil bearings operate with a relatively thin hydrodynamic lubricating film, which changes within a narrow range. Therefore, the overall stiffness and damping of a foil bearing depend mainly on the compliant supporting structure, rather than on a very stiff gas film. The first dynamic models of the foil bearing structure were most often based on static models [1]. The model, which was developed by [PERSON] and [PERSON], considered the elastic deformation of the bumps, the friction between the corrugated foil, the top foil and the sleeve, the interaction between the bumps as well as the curvature of the sleeve. The dynamic stiffness and damping of the entire bearing were obtained using a single-DOF (degree of freedom) model. [PERSON] and [PERSON] [2] provided test results for a bump-type foil bearing structure as well as stiffness and damping (Coulomb- and viscous-type) coefficients. The dynamic tests were conducted on a stiff non-rotating shaft. In 2010, [PERSON] et al. [3] presented an experimental evaluation of the steady-state (load capacity) dynamic performance of first- and second-generation foil bearings of a relatively large size (with a nominal diameter of 70 mm) under various controlled operating conditions. These tests aimed at collecting high-fidelity data needed to validate the theoretical model of a foil bearing. [PERSON] et al. [4] presented the results of experimental research carried out on foil bearings operating under elevated temperatures. The dynamic properties of the bearings were also investigated when a cooling system was turned on. This article presents various methods for cooling foil bearings and discusses the impact of these methods on dynamic parameters of the rotating system equipped with such bearings. The authors of the article measured the temperature of the top foil using thermocouples. They showed that their measurement method did not exert any significant impact on the functioning of the rotor-bearings system. [PERSON] et al. [5] wrote an article in which they discussed experimental research and simulation tests of foil bearing prototypes. All these tests were carried out on a special test stand under different operating conditions (including a broad speed range). This study shows that adverse operating conditions caused the bearing damage after its short operation, accompanied by a significant increase in temperature. The main factors that affected the durability of foil bearings were the material used for coatings of the mating surfaces, bearing geometry, way of assembling the bearing, speed, and load. To better understand the physical phenomena that occur in foil bearings, a numerical model has been developed which enabled performing thermal analyses. The results showed that the design process of a new foil bearing could be a challenging task.