## 4 Analysis of the Results  
In order to comprehensively analyze the relationship between journal movement and temperature, the decision was made to match the temperatures collected at the points and place them on the surface of the top foil. For this purpose, in the first step, the temperature values obtained at the highest speed were shown depending on the width of the foil in Figure 19a. Then, the temperature variation across the whole width of the foil for all columns of sensors (TC) was fitted using polynomial curves (in this case, second-order polynomials were used as they best reflected the nature of the distribution). This method followed the approach proposed in the article [20]. The curves along the length have been presented in a similar manner to Figure 19b, but in this case, the best fit guaranteed a fifth-order polynomial. As can be seen, the temperature values close to the lock (at the localization marked as 0 mm) were estimated to be too high, so they were discarded. Probably, if an additional column of thermocouples (placed closer to the foil lock) had been used, a more accurate fit would have been achieved. In order to find out about the fit of the curves to the point values, the statistical measure called R-squared was used to qualify the linear regression manifested by the percentage fit of the curves to the given values. R-squared is usually between 0 and 1, where 1 means a perfect representation. In our case, for all rotational speeds, the R-square coefficient was 1. In the literature, one can find the results of numerical calculations where the temperature is evenly distributed along the length of the top foil [21] and in some cases the temperature distribution of the top foil clearly reflects the shape of the bump foil. The peak temperature values on the top foil are at the apex of the convexity of the bump foil, while the temperature minima correspond to the segments between the bumps [15; 22]. The presented method overestimates the temperature values close to the lock, but it describes the temperature distribution between the measurement points with good accuracy. The next step was to match the two-dimensional temperature results obtained (Figure 19) to the surface of the top foil. In order to better reflect the temperature changes, an average value of the nearby measurement points (which are marked in yellow on the right side of Figure 19) was set in the foil lock (0 mm). This was based on the results described in the article [13], where the temperature close to the fixed end was close to the value at the free  
Figure 19: Temperature distribution curves fitted to temperature values obtained from measurement points distributed along: (**a**) the width of the top foil; (**b**) the length of the top foil. end of the top foil. A polynomial surface with a specified degree in x and y was used to illustrate the temperature distribution on the surface, according to the following formula:  
\[P_{i,j}(x,y)=c_{00}+c_{10}x+c_{01}y+c_{20}x^{2}+c_{11}xy+c_{02}y^{2}+c_{30}x^{3}+ c_{21}x^{2}y+c_{12}xy^{2} \tag{1}\]  
where 10 is the maximum value of \(i\) and \(j\); \(i\) and \(j\) stand for the foil width and length, respectively. In each case, the best fit was obtained using the polynomial P\({}_{2,7}\)(\(x\),\(y\)), and the R-squared coefficient ranged from 0.94 to 0.99. Below, examples of temperature distributions obtained at maximum (Figure 20a) and minimum (Figure 20b) rotational speed are shown. Different scales have been used to better illustrate the temperature distribution on the foil surface. The red dots indicate the measurement points enriched with temperature values at the edges of the foil, which were determined using a 2D method. In both cases of extreme speeds, the highest temperature occurs at the edges of the foil, close to the free end. This means that the lubrication gap was too small in this area, and this was caused, for example, by excessive pressure on the top foil exerted by the bump foil. Such a phenomenon could have caused local rubbing of the top foil against the journal.