## 4 Nonlinear Mathematical Model of the Bearing Stationary State
### Bearing Model with Throttle SECT  
This model is based on a slotted bearing model. The only difference is that, instead of dependences (9) that are linear with respect to the square of the pressure, nonlinear dependences (10) are used. This difference is reflected in the entire system of equations, which also will be nonlinear for this model. From the total number of \(n\) (\(m+1\)) equations with respect to the value of the squared pressure at the nodes of the finite-difference mesh, \(2\) (\(m+1\)) equations will be nonlinear (these equations correspond to the injection line \(i=k\)), and the rest of the equations remain linear. Using the fact that, for \(i\
eq k\), the system (37) is linear, we apply the counter-sweep method, which will allow excluding \((n-2)\) (\(m+1\)) linear equations from this system. In this case, only nonlinear equations will remain, to the solution of which, the Seidel iterative method will be applied. To eliminate linear equations, we use the opposite methods of left and right sweep. For the left sweep, we will use the recurrent Formula (55) for \(i=1,2,\,\ldots\,,\,k\). For the right sweep, we use the recurrent formula  
\[\overline{\mathbf{v}}_{i+1}=V_{i}\overline{\mathbf{v}}_{i}+\Xi_{i}, \tag{66}\]  
where \(V_{i},\Xi_{i}\) are matrices of size \((m+1)\times(m+1)\) and vectors of size \(m+1\), \(i=n-1,2\), \(\ldots\,,\,k\). If the left straight sweep starts from the left edge of the segment \(Z\in[0,\,L]\), then the right straight sweep will start from the right edge of this segment, for which the initial value of the unknown function is determined by Formula (39). Writing (66) for \(i=n-1\), and comparing the formula with (39), we find the initial sweep coefficients of the right straight sweep  
\[V_{n-1}=0,\Xi_{n-1}=\overline{\mathbb{T}}. \tag{67}\]  
Performing straight left and right sweeps with Formulas (55) and (67), we find sweep matrices and vectors \(X_{i},\overline{y}_{i}\) for \(i=1,2,\,\ldots\,,\,k\) and for \(i=n-1,2,\,\ldots\,,\,k\). For the flow rate of the bearing gap, we will use Formula (46). Similar to (47), the formulas (10) for the flow rates through the throttles take the form  
\[\left\{ \begin{array}{c}Q_{k,j}=A_{k}H_{j}\Pi\Big{(}P_{p,j},P_{k,j}\Big{)},\\ Q_{p,j}=A_{p}\Pi\big{(}P_{s},P_{p,j}\Big{)}.\end{array} \right. \tag{68}\]Substituting (46) and (68) into the balance equations for the flow rate (7), we obtain a nonlinear system of finite-difference equations  
\[\left\{ \begin{array}{c}\frac{H_{j}^{3}}{2z_{z}}\Big{(}\overline{\mathbf{v}}_{k+2 }^{j}-4\overline{\mathbf{v}}_{k+1}^{j}+6\overline{\mathbf{v}}_{k}^{j}-4 \overline{\mathbf{v}}_{k-1}^{j}+\overline{\mathbf{v}}_{k-2}^{j}\Big{)}=\\ =A_{k}H_{j}\Pi\big{(}P_{p,j},P_{kj}\big{)},\\ A_{k}H_{j}\Pi\big{(}P_{p,j},P_{kj}\big{)}=A_{p}\Pi\big{(}P_{s},P_{p,j}\big{)}. \end{array} \right. \tag{69}\]  
After performing the transformations, we write System (69) in a matrix form  
\[\left\{ \begin{array}{c}\overline{\mathbf{v}}_{k+2}-4\overline{\mathbf{v}}_{k +1}+6\overline{\mathbf{v}}_{k}-4\overline{\mathbf{v}}_{k-1}+\overline{\mathbf{ v}}_{k-2}=\overline{d}_{k},\\ \overline{d}_{k}=\overline{d}_{p},\end{array} \right. \tag{70}\]  
where \(d_{k,j}=\frac{2z_{z}A_{k}}{H_{j}^{3}}\Pi\big{(}P_{p,j},P_{kj}\big{)}A_{p,j}= \frac{2v_{z}A_{p}}{H_{j}^{3}}\Pi\big{(}P_{s},P_{p,j}\big{)}.\)  
For \(i=k-1\) for the left sweep formula and \(i=k+1\) for the right sweep formula, we have a sweep system of matrix equations:  
\[\left\{ \begin{array}{c}\overline{\mathbf{v}}_{k-2}=X_{k-1}\overline{\mathbf{ v}}_{k-1}+\overline{y}_{k-1},\\ \overline{\mathbf{v}}_{k+2}=V_{k+1}\overline{\mathbf{v}}_{k+1}+\overline{z}_{k +1}.\end{array} \right. \tag{71}\]  
Substituting (71) into the first equation of (70), we obtain  
\[A_{5}\overline{\mathbf{v}}_{k+1}+6\overline{\mathbf{v}}_{k}+A_{6}\overline{ \mathbf{v}}_{k-1}=\overline{d}_{k}-\overline{C}_{4}, \tag{72}\]  
where \(A_{5}=(V_{k+1}-4E)\), \(A_{6}=(X_{k-1}-4E)\), \(\overline{C}_{4}=\overline{y}_{k-1}+\overline{z}_{k+1}\).