# Analysis of Dynamic Response of a Two Degrees of Freedom (2-DOF) Ball Bearing Nonlinear Model
## 1 Introduction  
Ball bearings are one of the main components in mechanical systems dealing with transferring the rotational movement and carrying loads, simultaneously assuring high reliability of the structure [1, 2]. Years of ball bearings' development brought the high-precision rolling element bearings in application to a more demanding environment such as space crafts, high-speed rails, or in machines for the semiconductor industry [3], where the reduced friction, vibrations, and heat generation is required. Growing needs and demands require the application of advanced signal processing techniques [4, 5] or mathematical description of dynamical phenomena in ball bearings [6, 7]. Rotating rolling element bearing generates vibrations related to the parametrical excitation called varying compliance (VC) [8] and characteristic frequencies referring to the specific bearing element [9]. Varying compliance vibrations are related to the number of carrying load rolling elements resulting in rapidly changing stiffness [10, 11]. The VC characteristic frequency peak independently from the quality of rolling surfaces or faults and they are related to one of the characteristic ball bearing's frequencies, i.e., fundamental train frequency (FTF). Other characteristic frequencies refer to the faults of inner ring (BPFI--Ball Passage Frequency Inner), outer ring (BPFO--Ball Passage Frequency Outer), or rolling element (BSF--Ball Spin Frequency) [12, 13]. Many internal and external phenomena affect the dynamics of rotational mechanical systems and it is the same in the case of ball bearings. Except for varying compliance vibrations and characteristic frequencies corresponding to the bearing's design, the following features are the source of nonlinearity in rolling element bearings:  
* variable stiffness in time, related to Hertzian contact theory [14; 15],
* shape errors and evident faults [16; 17],
* friction torque [18],
* radial internal clearance [19],
* subjected external forces [20],
* eccentricity [21],
* thermal expansion influencing the internal geometry [22; 23]. In most of the papers related to the mathematical modeling of ball bearings, analyses of response frequencies or statistics-based approaches were proposed to identify faults, while the optimal working conditions and factors inducing vibrations are crucial for future design developments. One of the most important operational parameters is the radial internal clearance denoting the total distance in the radial direction that the inner ring and outer ring can be displaced in relation to each other. The selection of the radial clearance depends on the operating conditions, both too small and too big will result in premature fatigue and short bearing life [24]. Over the years, the influence of the radial clearance was the subject of the research and was discussed by the analysis of the mathematical model and the experiment. [PERSON] et al. [25] studied the influence of the radial clearance on the rotor's dynamics. His analysis brought the information on appearing super- and sub-harmonics in the frequency spectra depending on the clearance value. [PERSON] et al. [26] found the correlation between the value of radial clearance and the amplitude of vibrations, which has a linear character. [PERSON] et al. provided the source of knowledge on radial clearance influence on the ball bearing service life [27] and performed an analysis of grease contamination influence on the RIC by the thermographic inspection [28]. [PERSON] et al. [29] analyzed the bearing response with RMS indicator by variable internal clearance and load. Based on the above-cited literature, the bearing clearance is a subject of study over the years. However, in most of the papers, the case study bearings with defined parameters in the model are discussed. The alternative is to derive the dimensionless model for a better understanding of terms applied into the mathematical model on each other and characterize the ball bearing dynamic response.