## 4 Nonlinear Mathematical Model of the Bearing Stationary State
### Analysis of the Static Characteristics of a Bearing with a Throttle SECT  
The calculations showed that the iterative process converges rapidly at small values of the eccentricity \(\varepsilon\). At medium \(\varepsilon\), the process slowed down, assumed an oscillatory character and diverged even at \(\varepsilon\) = 0.4. To improve the convergence, taking into account the oscillation of the process at each step, a new approximation for the vector was calculated by the formula  
\[\left(P_{k}^{j}\right)_{\alpha}=\left(P_{k}^{j}\right)_{\alpha-1}+\gamma\left[ \left(P_{k}^{j}\right)_{\alpha-1}+\left(P_{k}^{j}\right)_{\alpha-2}\right],j=0,1,\ldots,m,\]  
where \(\alpha\) is the iteration number, and \(\gamma\) is the coefficient.  
In addition, for medium and large \(\varepsilon\), a recursion was used, the meaning of which was that the pressure vector obtained after the end of the iterative process for the previous smaller \(\varepsilon\) was taken as the initial approximation for the new \(\varepsilon\). Thus, to obtain a solution to the problem at \(\varepsilon\) = 0.7, the problems for \(\varepsilon\) = 0.1, 0.2,..., 0.6 were successively solved. In this case, for the limiting \(\varepsilon\), it was necessary to decrease \(\gamma\) to 0.1. The maximum number of iterations recorded for the slowest process was \(\alpha\) = 980. The average number of iterations was \(\alpha\) = 20.  
An increase in the rate of convergence of the iterative process was also facilitated by the fact that matrices \(X\), \(V\) and \(A\) and vectors \(\overline{y},\overline{z},\overline{C}\) do not depend on pressures \(\overline{P}_{k}\), \(\overline{P}_{p}\),; therefore, for a single \(\varepsilon\), they would be calculated once. In this case, the speed of the iterative process would be determined by the rate of multiplication of the matrix \(A_{\gamma}\) by the vector \(\overline{d}_{k}\) and by the rate of solving \(m+1\) nonlinear equations when determining the vector \(\overline{P}_{p}\) by the bisection method. When solving the problem for a single \(\varepsilon\) on a regular PC, this required no more than 0.1 s of computation time.  
Figure 9 shows the load curves \(\varepsilon_{s}\) (\(F\)) of a bearing with a throttle SECT for different values of the elastic coefficient \(K_{e}\).  
The curve for \(i\) = 0 (\(K_{e}\) = 0) corresponds to a conventional bearing of positive compliance with a rigid suspension (4) of the sleeve (3). It can be seen from the graphs that, as in the case of the slotted SECT, the positive compliance of the bearing also remains in a rather  
Figure 9: Dependences of the total eccentricity \(\varepsilon_{s}\) of the throttle SECT on the external load \(F\) for different values of the coefficient of elasticity \(K_{e}\).  
wide range of loads and eccentricity. Similarly, in the range of eccentricity limit values at \(\varepsilon>0.85\), the bearing loses its rigidity and becomes statically unstable.  
The graphs show that the maximum bearing capacity of a throttle-type bearing is greater than one of a slotted bearing. Therefore, if for the latter, the maximum bearing capacity is \(W_{max}=0.93\), and then, for the first one, \(W_{max}=1.06\), that is \(14\%\) more. It is also seen that the first bearing has a more stable load characteristic. In this respect, the graphs in Figure 10 are meaningful, where the curves calculated according to the nonlinear theory and the straight lines obtained by Formula (65) for the \(K_{0}\) of the bearing with the throttle SECT are shown. The best stability of the characteristics is observed for all \(K_{0}\). Moreover, if the first has a stable load characteristic for \(F\leq 0.2\), then the second has this characteristic four times wider: \(F\leq 0.8\).  
The results obtained indicate that, even at arbitrary loads and eccentricities, a bearing with a throttling SECT is significantly better than a bearing with a slotted SECT.