# Analysis of Dynamic Response of a Two Degrees of Freedom (2-DOF) Ball Bearing Nonlinear Model
## 1 Introduction
The significant increase of the radial clearance during bearing operation is an undesirable phenomenon from the exploitation point of view and it. On its variability, the following factors can have an influence: fitting on the shaft, the thermal expansion, subjected loads, shape errors, and the radial run-out. The first three factors are related to earlier determined operating conditions and they can be more or less predicted. The shape errors in form of roundness or waviness of the rolling surface can be observed in form of numerous and small amplitude frequency peaks. This effect is only measurable during the ball bearing's assembly process or its disassembly. Another factor is the eccentricity related to the shaft's manufacturing imperfections, external loads on the rotating shaft, or its improper connection with the motor by the clutch. In our research, we will present the effect of variable eccentricity on the dynamic response of the rolling element bearing. As the eccentricity introduces additional excitation into the system nonlinearities, such as contact loss in multi-body interactions present in the system play a more important role. Consequently, we expect the appearance of evolution and bifurcations of periodic solutions with the change of the eccentricity. Finally, various instabilities [30] occur in the system. In the specific conditions, chaotic motion [31] can develop, those effects can be studied by different methods [32], such as the Maximal Lyapunov Exponents, Test 0-1 [33] or Multi-Scale Entropy [34]. One of the promising methods to analyze the chaotic response of ball bearings is the recurrence analysis [35; 36]. According to [PERSON]'s work, the dynamic system comes back to the initial state or its close neighborhood in the phase space after some characteristic time forming so-called Poincare sections [37]. [PERSON] et al. [38] proposed the tool for visualization of the small parts of time series in the form of a recurrence plot (RP), however, to obtain the quantitative information on the considered state, the Recurrence Quantification Analysis has to be conducted [39]. The mentioned method has been already applied to characterize nonlinear dynamics in the variable dynamics of the mechanical systems [40; 41; 42; 43; 44] and manufacturing processes [45; 46]. The advantage of the recurrence analysis is that it detects the natural behavior of mechanical systems, i.e., occurrence of the same state after some time. That is why, with help of recurrences, it is possible to recognize defects, rapidly changing vibrations as chatters in milling, or varying misalignments or clearance in time. Ball bearings and other rotational systems generate rapidly changing vibrations in time and minor quantitative or qualitative changes in the dynamical response can be studied in short-time intervals. In this paper, it is used for the analysis of ball bearing's nonlinear dynamics. The rest of the paper is arranged in the following way. In Section 2, the derived dimensionless mathematical model of rolling-element bearing with its assumptions is discussed. Next, simulation results are presented for different values of eccentricity through FFT, phase plots, and orbit diagrams. In Section 4, the brief theory on the recurrence analysis is presented, and obtained recurrence plots and recurrence quantificators are discussed. Section 5 contains the discussion on results obtained and the next challenges in the mathematical model are mentioned. The last section summarizes the paper.