## 4 Calculations in the GAZBEAR Program  
In the GAZBEAR program, to calculate the aerodynamic lift of the bearing for a fixed journal position and, therefore, for a fixed shape of the lubrication gap, the pressure distribution acting on the sleeve surface is determined. The reaction force of the lubricating film acting on the bearing journal is then calculated based on this distribution. The pressure distribution in the bearing's lubrication film is determined by solving the Reynolds equation. The assumption of isothermal gas flow through the bearing allows the gas viscosity to be considered constant in the Reynolds equation, with its density replaced by pressure. Under these assumptions, for bearings that are not externally pressurized, Equation (3), the Reynolds equation, can be expressed in dimensionless form as follows:  
\[-\frac{\partial}{\partial\theta}\left(PH^{3}\frac{\partial P}{\partial\theta }\right)-\frac{\partial}{\partial\xi}\left(PH^{3}\frac{\partial P}{\partial \xi}\right)+\ \Lambda\frac{\partial}{\partial\theta}(PH)+\frac{\partial}{\partial\tau}\ PH=0, \tag{4}\]  
where  
\[ \begin{array}{l}\xi=z/R\\ P=p/p_{a}\\ H=h/c\\ \tau=\omega t/2\Lambda\end{array}  \tag{5}\]  
and the dimensionless constant \(\Lambda\), known as the bearing number or compressibility number, is defined by the following formula:  
\[\Lambda=\frac{6\mu\omega R^{2}}{p_{a}c^{2}} \tag{6}\]  
Furthermore, for a fixed journal position at the static equilibrium point, Equation (4) is assumed to take a steady-state form, implying that \(\frac{\partial H}{\partial\tau}=0\). A finite difference method was employed to discretize Equation (4), the Reynolds equation, where the computational mesh is defined by projecting the gas film onto the \(\theta\), \(\xi\) plane. In the circumferential direction \(\theta\), the mesh is assumed to have \(N+1\) nodes spaced \(\Delta\theta=2\pi/N\), and in the axial direction, M nodes spaced \(\Delta\xi=L/(R(\mathrm{M}-1))\). In the finite difference method, partial derivatives are replaced by finite differences, thereby reducing the Reynolds equation to a system of algebraic equations. Assuming parallelism of the sleeve and shaft axes, the dimensionless bearing gap is determined by the following formula:  
\[H(\theta)=1-\epsilon cos(\theta-\theta_{s}), \tag{7}\]  
where  
\[ \begin{array}{l}\varepsilon=\frac{1}{\zeta}\sqrt{\left(e_{x}\right)^{2}+ \left(e_{y}\right)^{2}}\\ \theta_{s}=arctg\frac{\zeta_{2}}{\varepsilon_{x}}\end{array} \]  
To solve Equation (4) and determine the pressure values at the mesh nodes, \(P2=Q\) is assumed, and the following boundary conditions are applied:  
* The gas pressure at the two edges of the bearing sleeve equals the ambient pressure. * The pressure distribution in the circumferential direction satisfies the periodicity condition. \[Q(\theta_{1},\xi)=Q(\theta_{1}+2\pi,\xi)\]  
Additionally, \(Q=1\) is taken as the initial condition for the square of the pressure. To obtain a solution, the alternating direction implicit (ADI) method is used, where calculations are performed at three time levels, \(n\), \(n+1\), and \(n+2\), separated by a time step \(\Delta\tau\), with \(\Delta\tau\leq 0.01\) assumed to ensure convergence of the method. One iteration cycle consists of solving a system of equations at time levels \(n+1\) and \(n+2\). For time level \(n+1\), the derivatives with respect to \(\xi\) are considered known from the previous iteration(at time level \(n\)), and a solution with respect to \(\theta\) is sought. In contrast, for time level \(n+2\), the derivatives with respect to \(\xi\) are considered unknowns, while the derivatives with respect to \(\theta\) are treated as known from time level \(n+1\). The results from time level \(n+2\) are then used as data for the next iteration (at time level \(n\)). In the first iteration, the pressure distribution is assumed to be known from the initial condition. The described calculation scheme is expressed in Equations (8) and (9). For time level \(n+1\):  
\[