## 4 Nonlinear Mathematical Model of the Bearing Stationary State
### Analysis of the Static Characteristics of a Bearing with a Slotted SECT  
The static characteristics calculated according to the formulas in Section 3.1 correspond to the values of the parameters used for plotting the graphs in Section 3.5.2. The characteristics were calculated point-by-point for different values of the eccentricity \(\varepsilon\in(0,1)\) with a small step \(\varepsilon_{h}=0.05\). The dependences \(\varepsilon_{s}(F)\) were determined by calculating the parametric functions \(\varepsilon_{s}(\varepsilon)\) and \(W(\varepsilon)\), since \(F=W\). The calculations showed that, sufficient for practice accuracy, \(|W_{n}-W_{2n}|<0.001\), \(|W_{m}-W_{2m}|<0.001\) is achieved when dividing the above integration intervals into \(n=40\) and \(m=16\) parts.  
In Figure 7, the load curves \(\varepsilon_{s}(F)\) for different values of the coefficient of elasticity \(K_{e}\) are shown. The curve for \(i=0\) (\(K_{e}=0\)) corresponds to a conventional bearing of positive compliance with a slotted SECT and rigid suspension (4) of the sleeve (3). It can be seen from the graphs that the positive compliance of the bearing remains in a fairly wide range of loads and eccentricity. However, in the area of extreme values of eccentricity at \(\varepsilon>0.85\), the bearing loses its rigidity and even acquires negative compliance. This area of theoretical performance is unrealizable in practice, since it corresponds to a statically unstable bearing.  
At \(i=1\) (\(K_{e}=K_{e0}\)), the bearing at \(\varepsilon=0\) has zero compliance (\(K=0\)). With an increase in load \(F\), the curve deviates into the region of negative eccentricities, while the nature of the curve changes; now, the eccentricity does not increase but, on the contrary, decreases. The same applies to the regimes of negative compliance \(i>1\) (\(K_{e}>K_{e0}\)) at the initial point of the curves. As in a bearing with a rigid suspension in the area of loads at which a conventional bearing loses its static stability, this bearing also loses stability at zero and negative compliance. The difference is only in the sign of the curvature of the characteristics.  
Figure 8 shows the characteristics of nonlinear and linear models.  
Figure 8: Comparative nonlinear (solid curves) and linear (dashed straight lines) dependences of the total eccentricity \(\varepsilon_{s}\) of the slotted SECT on load \(F\) for different values of the elasticity \(K_{c}\).  
Figure 7: Dependences of the total eccentricity \(\varepsilon_{s}\) of the slotted SECT on the external load \(F\) for different values of the elasticity coefficient \(K_{c}\).  
The first (solid curves) are calculated by the formulas of Section 4.1, and the second (dashed lines) are calculated by the formula  
\[e_{s}=K_{0}F, \tag{65}\]  
where \(K_{0}\) is the bearing compliance calculated by Formula (33).  
In the region of low loads, the graphs show complete coincidence of the characteristics, which confirms the correctness of the formulas derived in Section 4.1 for calculating nonlinear dependencies.  
At \(K_{e}=0\), the compliance is stable up to \(F=0.4\). With the increasing \(K_{e}\), the stability of the characteristics deteriorates. Thus, at \(K_{e}=2K_{e0}\), with negative compliance, a noticeable divergence of lines is observed even at a load of \(F=0.2\). At \(F>0.7\), the discrepancy becomes significant.  
The stability of compliance is important in the sense that the deformations of the technological system of the machine, as a rule, are linear functions of the load; therefore, those bearings of negative compliance are the most effective, the load characteristics of which are close to linear ones. As can be seen from the graph in Figure 8, bearings with slotted SECT meet this criterion only in the areas of low and moderate loads. Earlier, it was shown that bearings with throttle SECT have better characteristics. It should be expected that they will be the best under arbitrary loads and eccentricities.