## 3 General Mathematical Model of the Static State of a Conventional Bearing
### System of Equations for the Balance of Forces and Air Flow Rate in the SECT  
The dimensionless system of equations describing the static equilibrium of the bearing includes one equation for forces and two equations for the balance of air flow rates in the SECT  
\[\left\{ \begin{array}{c}W=F,\\ Q_{p}-Q_{k}=0,\\ Q_{k}-Q_{h}=0.\end{array} \right. \tag{35}\]  
It is convenient to calculate the constant static pressure \(P_{k0}\) at the outlet of the damping feeders and the corresponding pressure \(P_{p0}\) in the inter-throttling chambers using the normalized coefficients [18; 21; 24]  
\[\chi=\frac{P_{k0}^{2}-1}{P_{s}^{2}-1}\in[0,1],\quad\varsigma=\frac{P_{p0}^{2} -P_{k0}^{2}}{P_{s}^{2}-P_{k0}^{2}}\in[0,1]. \tag{36}\]  
By setting the injection pressure \(P_{s},\chi\) and \(\varsigma\), we can find the dimensionless pressures of the unloaded bearing  
\[P_{k0}=\sqrt{1+\chi(P_{s}^{2}-1)},\quad P_{p0}=\sqrt{P_{k0}^{2}+\varsigma \big{(}P_{s}^{2}-P_{k0}^{2}\big{)}}. \tag{37}\]  
With the help of (27), it is now possible to calculate the flow rate in the bearing gap. Using (35)-(37) at \(\epsilon=0\), we find the criteria for the similarity of damping annular diaphragms and throttling simple diaphragms  
\[A_{k}=\frac{Q_{h0}}{\Pi\big{(}P_{p0},P_{k0}\big{)}},\quad A_{p}=\frac{Q_{h0}} {\Pi\big{(}P_{s},P_{p0}\big{)}}. \tag{38}\]  
After linearizing (33) and (34), we find the deviations of the flow rates through the diaphragms  
\[\Delta Q_{k}=A_{qkk}\Delta P_{k}+A_{qkp}\Delta P_{p}-A_{qkk}\Delta\epsilon, \tag{39}\]  
\[\Delta Q_{p}=A_{app}\Delta P_{p}, \tag{40}\]  
where  
\[A_{qkk}=\frac{\frac{\partial\;{\rm Brn}(P_{p0},P_{k0})}{\partial P_{k0}}}{ \partial P_{k0}},\quad A_{qkp}=\frac{\frac{\partial\;{\rm Brn}(P_{p0},P_{k0})} {\partial P_{k0}}}{\partial P_{k0}},\]  
\[A_{qke}=\;{\rm Brn}(P_{p0},P_{k0}),\;A_{qpp}=\frac{\frac{\partial\;{\rm Brn}( P_{s},P_{k0})}{\partial P_{k0}}}{\partial P_{k0}}.\]  
From (35), it follows that the equations of forces and flow rates for deviations can be written in the form  
\[\left\{ \begin{array}{c}\Delta W=\Delta F,\\ \Delta Q_{p}-\Delta Q_{k}=0,\\ \Delta Q_{k}-\Delta Q_{h}=0.\end{array} \right. \tag{41}\]  
Substituting (17), (27), (39), and (40) into (41), we obtain a system of linear equations, which can be written in matrix form as  
\[\left[ \begin{array}{ccc}0&A_{w}&0\\ A_{qke}&-A_{qkk}&\left(A_{qpp}-A_{qkp}\right)\\ \left(A_{qe}-A_{qke}\right)&\left(A_{qkk}-A_{qp}\right)&A_{qkp}\end{array}  \right]\left[ \begin{array}{c}K_{e}\\ K_{k}\\ K_{p}\end{array} \right]=\left[ \begin{array}{c}1\\ 0\\ 0\end{array} \right], \tag{42}\]where \(K_{e}=\frac{\Delta\varepsilon}{\Delta\Gamma},K_{k}=\frac{\Delta P_{p}}{\Delta \Gamma},K_{p}=\frac{\Delta P_{p}}{\Delta\Gamma}\) are the gains of the transfer functions, which are equal to the ratio of the output deviations \(\Delta\varepsilon,\Delta P_{k},\Delta P_{p}\) and the small input force deviation \(\Delta F\). The quantity \(K_{e}\) is the static compliance of the bearing. Having solved the system of (42), we find the static compliance of the bearing as  
\[K_{e}=\frac{1}{A_{w}}\frac{\left(A_{qp}-A_{qpk}\right)\left(A_{qpp}-A_{qkp} \right)-A_{qkk}A_{qkp}}{\left(A_{qq}-A_{qke}\right)\left(A_{qpp}-A_{qkp}\right) -A_{qke}A_{qkp}}. \tag{43}\]  
The resulting formula makes it possible to evaluate the effect of circumferential flows on bearing compliance at small shaft displacements. Figure 2 shows the dependences of the compliance \(K_{e}\) on the length \(L\) at various values of the relative elongation of the end length \(\lambda_{1}=L_{1}/L\), the injection pressure \(P_{s}=5\), and the recommended values of the SECT tuning factors \(\chi=0.45\), \(\varsigma=0.15\)[18; 21]. With an increase in \(L\), the part of the bearing surface lubricated by air increases should contribute to a decrease in the compliance \(K_{e}\). However, as the graphs show, a decrease in compliance occurs only up to a certain \(L_{opt}\), which corresponds to the minimum compliance.