## 5 Mathematical Model of the Unsteady State of the Bearing
### Bearing Dynamic Characteristics and Their Discussion
For them, there are also optimal values of the parameter \(V_{p}\) in terms of the speed  
Figure 14: Dependences of the criterion of damping of the oscillations for the period \(\xi\), of the bearing with the throttle SECT on the squeezing number \(\sigma\) and volume \(V_{p}\), where \(\varsigma=0.15\). Figure 13: Dependences of the degree of stability \(\eta\) of a bearing with a throttle SECT on the squeezing number \(\sigma\). \(V_{p\ opt}\) at which the function \(\eta(V_{p})\) reaches its maximum. At \(V_{p}<V_{p\ opt}\), the \(\xi\) criterion indicates the oscillatory nature of the transient processes (\(\xi<100\%\)), while, at \(V_{p}>V_{p\ opt}\), the transients become aperiodic (\(\xi=100\%\)). It is also seen that, with an increase in \(V_{p}\), the region of aperiodicity of the transient processes expands--that is, an increase in the volume \(V_{p}\) contributes to an increase in the stability margin of the bearing as a dynamic system. The analysis of the dependences for the performance criteria \(\eta\) and the oscillation of transient processes \(\xi\) shows that, in the space of changes in the parameters \(\xi\), \(\sigma\) and \(V_{p}\), for each of which, the criterion \(\eta\) has an optimal value, and the criterion \(\xi\) reaches its maximum \(\xi=100\%\); with other fixed parameters, there is the global maximum of the speed function \(\eta_{max}=\eta(\xi,\sigma,V_{p})\). The optimization of the \(\eta\) criterion showed that, for these parameters, \(\eta_{max}=0.71\) at \(\xi_{max}=0.149\), \(\sigma_{max}=3.48\) and \(V_{p}\) max = 139. In practical terms, when designing a bearing, it is recommended to choose slightly larger values: \(\varsigma=0.15\), \(\sigma=3.5\) and \(V_{p}=140\), since, at these values, the speed will be close to optimal, and the transient response will be aperiodic, providing a sufficient stability margin. Figure 16: Dependences of the criterion of damping of oscillations for the period \(\xi\), of the bearing with a throttle SECT on the volume \(V_{p}\) and squeezing number \(\sigma\), where \(\varsigma=0.15\). Figure 15: Dependences of the degree of stability \(\eta\) of a bearing with a throttle SECT on the volume \(V_{p}\) and the squeezing number \(\sigma\), where \(\varsigma=0.15\). Figure 17 shows the amplitude-frequency characteristics of the bearing with the throttle SECT. As can be seen from the graph, an increase in \(V_{p}\) contributes to a noticeable decrease in the resonance peaks \(A(\Omega)\), which also indicates that the \(V_{p}\) parameter is an important factor in increasing the stability margin of a bearing with a throttle SECT. The parameter \(\sigma\) in the region of optimality of the criteria \(\eta\) and \(\xi\) barely affects the frequency characteristics of the bearing. An Example of Calculating the Dimensional Characteristics of a Bearing and their Experimental Validation  
Let us give an example of calculating the dimensional characteristics of a bearing with a throttle SECT for \(\varsigma=0.15\), \(\sigma=3.5\) and \(V_{p}=140\). For the calculation and subsequent experimental verification, we take the bearing radius \(r_{0}=15\cdot 10^{-3}\) m and the ambient pressure \(p_{a}=0.1\cdot 10^{6}\) Pa used in [40].