**1. Introduction**

The use of dynamics and noise behavior as criteria to assess the performance of a rolling bearing are coming into increasing focus besides the lifetime and energy efficiency. In addition to potentially negative health consequences of noise pollution [1], one reason for this is the increasing electrification of passenger cars and the associated sensitivity regarding disturbing and unpleasant noise of all machine elements contained in the technical system [2]. Besides unpleasant noise caused by bearing dynamics, in precision applications such as the bearing assembly of the main spindle of machine tools, vibration of the bearing can lead to a negative influence on manufacturing accuracy [3].

The vibrations emitted by a rolling bearing may have various causes. Due to the rotation of the rolling element set, the force transmitting points between the inner and outer ring differ. This leads to a changing stiffness and to unavoidable vibrations of the rolling bearing caused by the design itself and is known as variable compliance [4]. The characteristics of these vibrations differ depending on the rolling bearing type (geometry, number of rolling elements, and pitch diameter) and load conditions (operating contact

**Citation:** Schwarz, S.; Grillenberger, H.; Graf-Goller, O.; Bartz, M.; Tremmel, S.; Wartzack, S. Using Machine Learning Methods for Predicting Cage Performance Criteria in an Angular Contact Ball Bearing. *Lubricants* **2022**, *10*, 25. https:// doi.org/10.3390/lubricants10020025

Received: 14 December 2021 Accepted: 4 February 2022 Published: 11 February 2022

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angle and load zone). In addition to the geometry-related causes of vibrations in rolling bearings, production-related geometric deviations of the bearing rings or rolling elements, such as roughness, waviness, or surface damage (scratches and inclusions in the material), influence the radial displacement of the rings and can cause undesired vibrations [5]. Thus, depending on the frequency of vibration occurring, isolated surface deviations can be assigned to the inner or outer bearing part based on the respective ball-pass frequency [6].

The cage of the rolling bearing can also be a source of vibrations and noise. An example are highly dynamic cage movements, which are called "cage rattling" or "cage instability" in the literature and are associated with strong noise generation [7–9]. The normal and frictional forces at the guiding surfaces accelerate the cage, so that certain operating conditions lead to a high-frequency motion and severe deformation of the cage [10,11]. These cage dynamics lead to a sharp increase in frictional torque [8,9,12] and temperature in the rolling bearing [8] and can have a negative effect on cage life due to severe deformations and component stresses. Cage dynamics depend on many influencing factors; an overview of previous research papers is provided in Table 1.


**Table 1.** Influencing parameters on the cage dynamics that have been investigated in research papers.

The dynamic behavior of the cage depends on the bearing and cage properties as well as the operating conditions of the bearing. As the cage is (besides the rib contact) accelerated by the rolling elements contact, the dynamic behavior of the rolling elements has an influence on the cage motion. The kinematics of the rolling elements is affected by various factors, such as the bearing load and speed, the friction in the contact to the raceway, the rolling element geometry and the bearing clearance. However, these parameters are determined depending on the intended application with focus on bearing lifetime and accuracy of shaft guidance. The influence of the bearing design and load on the cage dynamics during the application is not usually in the focus in the bearing selection. Therefore, the cage dynamics must be adjusted by adapting the cage geometry in the available design space of the selected bearing. By varying the cage geometry, properties such as the pocket and guidance clearance, the mass inertia and stiffness, and the shape of the cage pocket are affected. By defining the cage properties, the dynamics can be adjusted, for example, to avoid unstable cage movements or to minimize the friction loss caused by the cage as well as the robustness against shock loads.

The influencing parameters on the resulting cage dynamics can be named in general, but the quantification of their effects is only partially known so far. There are two primary reasons for this. First, the calculation using numerical computer simulations or the measurement of the cage dynamics (motion, forces, or deformation) on a test rig are time consuming and complicated. In particular for experimental tests, the range of

influencing parameters that can be investigated is usually limited. Second, the interaction of the influencing variables is complex, so that it is not possible to determine the influence of the individual effects directly on the basis of the observed dynamics. Cage instability, for example, is caused by high frictional forces in the cage contacts and high rotational speeds of the bearing [14]. If one of the two parameters is low, the probability that highly dynamic movements will be excited is reduced. In addition to this example, other interactions can be found, making it more difficult to determine the cage dynamics depending of the influencing factors such as cage geometry and bearing load.

Machine learning methods are suitable for identifying complex patterns and relationships in the data provided. The application of machine learning algorithms in the field of tribological problems is increasing, especially in recent years. A comprehensive overview of the use of machine learning for tribological problems was provided by Marian and Tremmel [20]. Based on experimental test results, calculations, or information collected from the literature, regression methods are used to predict typical tribological behavior in the form of temperature, specific wear, or coefficient of friction. In addition to applications at the nano or micro scale, machine learning methods are also used at the macro scale, such as in bearing technology. Schwarz et al. used an ensemble classification model to determine the dependence between geometric parameters and load of a rolling bearing and the resulting dynamics of a cage. The result of the classification was one of the classes "unstable", "stable", or "circling" that were used to assess the qualitative behavior of the cage [21]. By extending this approach with a regression algorithm, not only the cage motion class but also the resulting forces on the cage or the acceleration of the cage can be estimated.

In previous research investigations [21], it was possible to quantify the dynamics of the cage for different operating conditions, but this was usually completed in isolated cases within the framework of complex numerical calculations or tests. A method for the time-efficient estimation of the quantitative dynamic behavior of rolling bearing cages for certain cage properties and rolling bearing loads is not yet available. The aim of this paper is to present a procedure for predicting the dynamics of a rolling bearing cage in an angular contact ball bearing using dynamics simulations and regression machine learning algorithms. This enables time-efficient estimation of the dynamics for the intended application during the development and selection of rolling bearing cages and also for operating conditions that are not directly included in the training data.
