## 4 Calculations in the GAZBEAR Program
\begin{array}{c}2P_{ij,j}^{n}\frac{H_{ij,j}^{n}-H_{ij,j}^{n-1}}{\Delta\tau}+ \frac{H_{ij}^{n}}{P_{ij}^{n}}\frac{Q_{ij}^{n+1}-Q_{ij}^{n}}{\Delta\tau}+2 \Lambda\frac{Q_{ij}^{n+1}}{P_{ij}^{n}}\frac{H_{ij,i+1}^{n}-H_{ij,-1}^{n-1}}{2 \Delta\theta}+\Lambda\frac{H_{ij}^{n}}{P_{ij}^{n}}\frac{Q_{ij}^{n+1}-Q_{ij}^{n +1}}{2\Delta\theta}+\\ -3\left(H_{ij}^{n}\right)^{2}\frac{\left(H_{ij,j+1}^{n}-H_{ij,-1}^{n}\right) \left(Q_{ij,j+1}^{n+1}-Q_{ij,j+1}^{n+1}\right)}{4\left(\Delta\theta\right)^{2} }-3\left(H_{ij}^{n}\right)^{2}\frac{\left(H_{ij,i+1}^{n}-H_{ij,-1}^{n}\right) \left(Q_{ij,j+1}^{n}-Q_{ij,-1}^{n}\right)}{4\left(\Delta\xi\right)^{2}}+\\ -\left(H_{ij,j}^{n}\right)^{3}\frac{Q_{ij,j+1}^{n+2}-Q_{ij,j}^{n+1}+Q_{ij,-1}^ {n+1}}{\Delta\theta^{2}}-\left(H_{ij,j}^{n}\right)^{3}\frac{Q_{ij,j}^{n}-2Q_{ ij}^{n+2}+Q_{ij,-j}^{n+2}}{\Delta\xi^{2}}=0\end{array}  \tag{8}\]  
For time level \(n+2\):  
\[ \begin{array}{c}P_{ij,j}^{n}\frac{H_{ij,j}^{n}-H_{ij,j}^{n-1}}{\Delta\tau}+ \frac{H_{ij}^{n}}{P_{ij,j}^{n}}\frac{Q_{ij}^{n+2}-Q_{ij,j}^{n+1}}{\Delta\tau} +2\Lambda\frac{Q_{ij}^{n+1}}{P_{ij,j}^{n}}\frac{H_{ij,j+1}^{n}-H_{ij,-1}^{n-1 }}{2\Delta\theta}+\Lambda\frac{H_{ij}^{n}}{P_{ij,j}^{n}}\frac{Q_{ij,j+1}^{n+1} -Q_{ij,j-1}^{n+1}}{2\Delta\theta}+\\ -3\left(H_{ij}^{n}\right)^{2}\frac{\left(H_{ij,j+1}^{n}-H_{ij,-1}^{n}\right) \left(Q_{ij,j+1}^{n+1}-Q_{ij,j-1}^{n+1}\right)}{4\left(\Delta\theta\right)^{2 }}-3\left(H_{ij}^{n}\right)^{2}\frac{\left(H_{ij,i+1}^{n}-H_{ij,-1}^{n}\right) \left(Q_{ij,j+1}^{n+1}-Q_{ij,j}^{n+2}\right)}{4\left(\Delta\xi\right)^{2}}+\\ -\left(H_{ij,j}^{n}\right)^{3}\frac{Q_{ij,j+1}^{n+1}-2Q_{ij,j}^{n+1}+Q_{ij,j-1 }^{n+1}}{\Delta\theta^{2}}-\left(H_{ij,j}^{n}\right)^{3}\frac{Q_{ij,j}^{n+2}-2 Q_{ij}^{n+2}+Q_{ij,-j}^{n+2}}{\Delta\xi^{2}}=0\end{array}  \tag{9}\]  
By grouping the components calculated at the appropriate nodes of the mesh, Equations (8) and (9) are transformed to form the following:  
\[A_{ij}^{n}Q_{i,j-1}^{n+1}+B_{ij}^{n}Q_{i,j}^{n+1}+C_{ij}^{n}Q_{i,j+1}^{n+1}=f_ {i,j}^{n} \tag{10}\]  
\[D_{ij}^{n}Q_{i-1,j}^{n+2}+E_{ij}^{n}Q_{i,j}^{n+2}+F_{ij}^{n}Q_{i+1,j}^{n+2}=g_ {i,j}^{n+1} \tag{11}\]  
and are then solved using the iterative method of tridiagonal elimination. Knowing the pressure distribution in the gas film, it is possible to determine the components of the dimensionless forces acting on the journal using the following formulas:  
\[F_{x}=-\int\limits_{0}^{2\pi\,L/R}\int\limits_{0}^{R}P(\xi,\theta)cos\theta d \xi d\theta \tag{12}\]  
\[F_{y}=-\int\limits_{0}^{2\pi\,L/R}\int\limits_{0}^{L/R}P(\xi,\theta)sin\theta d \xi d\theta \tag{13}\]  
The GAZBEAR program was used to carry out a series of calculations in which, for a given bearing geometry and operating conditions, the aerodynamic force of the bearing was determined as a function of the relative eccentricity of the journal position. In subsequent calculations, the eccentricity value ranged from 0.01 to 0.99 with a step of 0.01. Following indications from the literature [33] and our own experience, a mesh with 17 nodes in the axial direction and 48 nodes in the circumferential direction was used for the calculations. Results were obtained for a dimensionless eccentricity ranging from 0.01 to 0.89. The coordinate system in which the calculation results were presented corresponded to that shown in Figure 1. Figure 2 presents the calculated changes in aerodynamic lift and the angular position of the journal center as functions of the dimensionless eccentricity in the gas-dynamic bearing. In the range tested, as the eccentricity increased, the angular position of the journal center decreased from approximately 338\({}^{\circ}\) to approximately 278\({}^{\circ}\). Over the entire range tested, the load capacity of the bearing increased as the center of the journal moved away from the center of the sleeve, ranging from approximately 1.1 N to more than 350 N. The nominal value of the aerodynamic lift (18 N) was exceeded at a relative eccentricity of 0.16 and an angular position of the journal center of 336\({}^{\circ}\).