## 3 General Mathematical Model of the Static State of a Conventional Bearing
### Boundary Value Problems for a Dimensionless Function of Pressure in the Bearing Gap
\begin{array}{l}W=\frac{2\tau_{0}^{2}p_{a}}{2\pi\
u_{0}^{2}p_{a}}\int\limits_{0 }^{2\pi}\int\limits_{0}^{L}\left(\sqrt{\Psi_{0}+\Delta\Psi\cos\varphi}-1\right) \cos\varphi dZd\varphi=\\ =\frac{1}{\pi}\int\limits_{0}^{2\pi}\cos^{2}\varphi d\varphi\int\limits_{0}^{ L}\frac{\Delta\Psi}{2\sqrt{\Psi}_{0}}dZ=\frac{1}{2}\int\limits_{0}^{L}\frac{\Delta \Psi}{\sqrt{\Psi}_{0}}dZ.\end{array}  \tag{12}\]  
In Formula (12), we express \(\Psi\) in terms of \(P\). Since  
\[\Psi_{0}+\Delta\Psi\cos\varphi=\left(P_{0}+\Delta P\cos\varphi\right)^{2}=P_{0 }^{2}+2P_{0}\cos\varphi,\]  
then  
\[\Psi_{0}=P_{0}^{2},\Delta\Psi=2P_{0}\Delta P. \tag{13}\]  
Substituting (13) into (12), we obtain  
\[\Delta W=\int\limits_{0}^{L}\Delta PdZ. \tag{14}\]  
Replacing \(\Psi\) in (1) by \(P\), we write (10) in the form  
\[\Delta P(Z)=\left\{ \begin{array}{l}\Delta P_{k}\frac{P_{0}\sinh(L_{1}-Z)}{P _{0}(Z)\,\text{sh}L_{1}},\quad Z\leq L_{1},\\ \Delta P_{k}\frac{\text{sh}Z}{\text{sh}L_{2}},\quad Z\leq L_{2},\end{array} \right. \tag{15}\]  
where, taking into account (9)  
\[P_{0}(Z)=\sqrt{\Psi_{0}(Z)}=\left\{ \begin{array}{l}\sqrt{\left(P_{k0}^{2}-1 \right)\frac{L_{1}-Z}{L_{1}}+1},\quad 0\leq Z\leq L_{1},\\ P_{k0},\quad 0\leq Z\leq L_{2}.\end{array} \right. \tag{16}\]  
Substituting (15) and (16) in (14), we obtain the formula for determining the small deviation of the bearing capacity  
\[\Delta W=A_{w}\Delta P_{k}, \tag{17}\]  
where  
\[A_{w}=\text{th}L_{2}+\frac{P_{k0}}{\text{sh}L_{1}}\int\limits_{0}^{L_{1}} \frac{\text{sh}(L_{1}-Z)}{P_{0}(Z)}dZ.\]  
At the outlet of the pressurization line and the inlet into the bearing gap, the dimensionless gas flow rate \(Q_{h}\) is distributed in two directions: \(Q_{h2}\) into the inter-row zone and \(Q_{h1}\) into the end zone [26]  
\[Q_{h}=Q_{h2}+Q_{h1}, \tag{18}\]  
where  
\[Q_{h1}=-H^{3}\bigg{(}\frac{\partial\Psi}{\partial Z}\bigg{)}_{Z=L_{1}-0}, \tag{19}\]  
\[Q_{h2}=H^{3}\bigg{(}\frac{\partial\Psi}{\partial Z}\bigg{)}_{Z=L_{1}+0}. \tag{20}\]  
Taking into account (4) and (13) and performing linearization (19), we obtain  
\[Q_{h10}=-\bigg{(}\frac{dP_{0}^{2}}{dZ}\bigg{)}_{Z=0}, \tag{21}\]and  
\[\Delta Q_{h1}=\left[-2\frac{d(P_{0}\Delta P)}{dZ}+3\frac{dP_{0}^{2}}{dZ}\Delta \epsilon\right]_{Z=0}. \tag{22}\]  
Having performed the differentiation in (15) and (16), we describe the formula for the flow rate in the end zone with the central position of the shaft as  
\[Q_{h10}=\frac{P_{k0}^{2}-1}{L_{1}} \tag{23}\]  
and the formula for the small deviation of the flow rate in this part as  
\[\Delta Q_{h1}=A_{qp1}\Delta P_{k}-A_{qe1}\Delta\epsilon, \tag{24}\]  
where  
\[A_{qp1}=2P_{k0}\text{th}L_{1},A_{qe1}=3Q_{h10}.\]  
Similarly, for the inter-row zone, we obtain  
\[\left\{ \begin{array}{l}Q_{h20}=0,\\ \Delta Q_{h2}=A_{qp2}\Delta P_{k},\end{array} \right. \tag{25}\]  
where  
\[A_{qp2}=2P_{k0}\text{cth}L_{2}. \tag{26}\]  
Substituting (23)-(26) into (18), we find the final formulas for the separated components of the air flow rates in the bearing gap  
\[Q_{h0}=\frac{P_{k0}^{2}-1}{L_{1}}, \tag{27}\]  
\[\Delta Q_{h}=A_{qp}\Delta P_{k}-A_{qe}\Delta\epsilon, \tag{28}\]  
where  
\[A_{qp}=2P_{k0}(\text{cth}L_{1}+\text{th}L_{2}), \tag{29}\]  
\[A_{qe}=3Q_{h0}. \tag{30}\]  
To determine the mass flow rate of air through the diaphragms, we used the formula obtained by integrating the boundary value problem for the nonlinear Bernoulli Equation [26]  
\[q=\sqrt{\frac{2}{RT}\frac{\gamma}{\gamma-1}}s\cdot\text{Brn}(p_{1},p_{2}), \tag{31}\]  
where  
\[\text{Brn}(p_{1},p_{2})=p_{1}\left\{ \begin{array}{l}C_{\gamma},\quad\frac{P _{2}}{P_{1}}\leq C_{c},\\ \sqrt{\left(\frac{P_{2}}{P_{1}}\right)^{\frac{2}{\gamma}}-\left(\frac{P_{2}}{ P_{1}}\right)^{\frac{\gamma+1}{\gamma}}},\quad\frac{P_{2}}{p_{1}}>C_{c},\end{array} \right.