# Analysis of Dynamic Response of a Two Degrees of Freedom (2-DOF) Ball Bearing Nonlinear Model
## 6 Results Discussion and Summary  
This article reports the results obtained in the analysis of the dimensionless mathematical model for the single-row ball bearing by the recurrence-based methods. For the variable parameter, which is changing the bearing's dynamics, the shape error in form of the eccentricity was used. The model was studied in a wide range of the operating parameter showing the transition between the periodic and chaotic motion of the system. For studying the behavior of the bearing-shaft system dynamic, the recurrence-based methods were applied in the form of recurrence plots (RPs) and the Recurrence Quantification Analysis (RQA). The analysis aimed to find the transition between the periodic and chaotic response and how does it affect the topology of RPs and recurrence quantificators. The reconstructed recurrence plots by the constant value of the recurrence rate showed the general character of the system's dynamics. The periodic solutions are obtained for small values of the eccentricity and they are characterized by the diagonal structure without any disruptions. The square-like structures and isolated recurrence points are evidence of the chaotic motion of the bearing. The recurrence plots provide only the qualitative information on the system dynamics, so the recurrence quantificators from the Matlab CRP Toolbox were employed for the quantitative analysis. As the mathematical model is studied, the system is deterministic that is proved by the value of \(DET\) (Figure 8a). The character of bearing's response can be roughly determined by the diagonal- (Figure 8b-d) and vertical-based (Figure 8e-g) quantificators, however, the trapping time \(TT\) and the length of the longest vertical line \(V_{max}\) expresses the increasing nonlinearity in the clearest way for interpretation. The recurrence-time of the 2 nd type (Figure 8i) has a similar run to \(TT\) and \(V_{max}\) showing clearly the transition between periodic and chaotic motion by the drastic increase of the quantificator. For supplementary analysis, the deformation response was recalculated with the kurtosis. Within its course (Figure 9) the transition between periodic and chaotic solution is identifiable as in the case of vertical line based quantificators. The performed analysis of the nonlinear mathematical model of the ball bearing showed the usefulness of recurrence-based methods in the identification of its dynamic response. The transition between the periodic and chaotic motion in the variable eccentricity domain was detected with help of recurrence plots and recurrence quantificators. This is evidenced by the square-like structures observed at recurrence plots and radical increase or decrease of recurrence quantificators value in chaotic motion. As many other features are influencing the dynamic response of the ball bearings such as radial clearance or external load, the idea of the mathematical model's development is to analyze their influence on the system's dynamics in further steps applying input parameters with assumed uncertainty [89]. Moreover, the model validation is planned by the experimental verification and studying bearing's response from acceleration measurements. Figure 9: Kurtosis is calculated for the different values of the rotor’s eccentricity. **Author Contributions:** Conceptualization, B.A., G.L., A.G. ([PERSON]), N.M. and A.G. ([PERSON]); methodology, B.A., G.L., A.G. ([PERSON]) and N.M.; software, B.A. and G.L. ; validation, B.A. and G.L. ; formal analysis, B.A., G.L., A.G. ([PERSON]), N.M. and A.G. ([PERSON]); investigation, B.A. and G.L., resources, B.A., G.L. and A.G. ([PERSON]) data curation, B.A. and G.L. ; writing--original draft preparation, B.A. and G.L. ; writing--review and editing, G.L., A.G. ([PERSON]), N.M. and A.G. ([PERSON]); visualization, B.A. ; supervision, G.L. and A.G. ([PERSON]); project administration, B.A., G.L., A.G. ([PERSON]) and N.M.; funding acquisition, B.A. and G.L. All authors have read and agreed to the published version of the manuscript.