Sebastian Schwarz, M.Sc., Chair of Design Engineering (KTmfk), Friedrich-Alexander University Erlangen-Nuremberg (FAU), Erlangen; Dr. rer. nat. Hannes Grillenberger, Schaeffler Technologies AG & Co. KG, Herzogenaurach; Dr.-Ing. Stephan Tremmel, Chair of Design Engineering (KTmfk), Friedrich-Alexander University Erlangen-Nuremberg (FAU), Erlangen

Abstract
Today’s modern rolling bearings no longer simply have to meet high expectations with respect to service life and guiding accuracy; indeed, friction losses and noise must also be minimized. One type of noise is generated when, under certain operating conditions, the cage of a rolling bearing performs an overlapping motion – in addition to its rotation – that is of a very high frequency compared to its speed. This motion acts as a source of noise that may be accompanied by severe cage deformation. This phenomenon is typically referred to as “cage instability” or “cage rattle.”

Using numerous multi-body simulations, this paper will systematically analyze cage movement in various rolling bearings under different operating conditions. Basic, generalizable types of movement can be derived from this analysis, which yields a methodology that allows the cage movement, which has been calculated in a simulation or measured in an experiment, to be assigned automatically to the previously defined movement types with a high degree of reliability. Consequently, this methodology enables an objective assessment of the cage dynamics.

1. Introduction
In power train engineering and machine tool construction, low-vibration and low-noise behavior is considered to be one of the most important quality features. Accordingly, as power trains are increasingly being electrified, reducing noise in the drive train is currently of utmost importance in vehicle construction: Because an electric motor runs very quietly in comparison to an internal-combustion engine, the vibrations and noise caused by the other machine elements in the drive train – such as rolling bearings – are becoming increasingly apparent. At the same time, the operating conditions, especially for the bearings (such as high-speed characteristics, low-viscosity lubricants, etc.) are becoming more challenging.

A particularly important part of a rolling bearing is its cage, whose main functions include guiding the rolling elements and distributing them evenly over the circumference of the bearing. The cage’s design significantly influences the speed limit, the friction behavior as well as the temperature development and, last but not least, the vibration and noise behavior of the bearing. Under certain operating conditions, in addition to its rotation, the cage performs an overlapping motion that has a very high frequency relative to its speed. This cage’s resulting “unstable” motion causes high contact forces with the rolling elements due to its highly dynamic motion components. This can reduce the bearing’s performance. In addition – as described in numerous publications – a very disturbing noise can be detected [1–3]. In extreme cases, the unstable movement resulting from the impact forces and the associated increased stresses can cause the cage to break and, subsequently, immediate bearing failure [3].

It is now possible to examine cage dynamics with the help of special multi-body simulation programs [4; 5]. Of course, the quality of the conclusions that can be drawn from such simulations with regard to cage instabilities depends on having a very detailed representation of the individual contact points and the deformation behavior of the cage. One existing program that enables such specialized and comparatively sophisticated examinations is SCHAEFFLER’S CABA3D (Computer Aided Bearing Analyzer 3 Dimensional), which has already been used to perform dynamic simulations of rolling bearings for calculations in numerous publications [1; 6; 7]. Specialized expert knowledge is required to interpret the results of the transient simulations, in order to correctly classify the movement of the cage into critical or non-critical types of movement. If a large number of results is to be classified, however, a single expert will have trouble performing such an assessment manually. For this reason, a computer-aided method that represents the expert’s knowledge and allows an automated, objective and reliable assessment of the cage movement is needed.

2. Criteria for characterizing the cage dynamics
Numerous criteria have proposed in various technical articles to characterize the cage dynamics and to identify unstable cage behavior. These criteria are briefly presented below. The first, purely experimental investigations into unstable cage movements were carried out in 1965 by KINGSBURY for gyroscope bearings [2]. The unstable cage movement can therefore be identified on the basis of an unsteady course of the frictional torque. On the basis of the experimental results, various types of movement have been defined for the cage [2] (see Table 1).

GUPTA examined the cage instability using the multi-body simulation program ADORE (Advanced Dynamics of Rolling Elements) with regard to various influencing factors such as friction-slip curves [8] or the cage pocket play [9]. To assess the cage movement, GUPTA suggests the shape of the orbit plot – i.e., the trajectory of the cage’s center of gravity in the plane perpendicular to the bearing axis – in connection with the translational speed of the cage’s center of gravity [10]. Accordingly, in the case of unstable cage movement, a higherorder polygon or a very erratic and arbitrary trajectory of the cage’s center of gravity can be observed as an orbit plot (see Figure 1). This reveals a direct correlation between the shape of the orbit plot, the speed of the cage’s center of gravity and the development of noise (or “squeal”) – i.e., the unstable cage movement.

A stability metric for differentiating the cage movements as a function of several indicators (see Table 2) is suggested by BOESINGER [11]. These investigations included both experimental measurements as well as calculations. By comparing a movement that is to be evaluated with a reference movement, the relative behavior and, therefore, a greater or lesser tendency towards instability can be determined. However, an absolute classification of both movements is not possible – accordingly, this metric cannot be used to determine whether the movement used as a basis for comparison is stable or unstable.

V. AUL uses dimensionless metrics to evaluate cage vortex movements [5] (see Table 3). These were derived from the results of numerous simulations. An example is used to illustrate that limits can be defined on the basis of these metrics in order to be able to differentiate between stable and unstable cage behavior.

A significant extension of the aforementioned simulation options for analyzing cage instability has been introduced by GRILLENBEGER using macro-elastic modeling of the cage with the aid of rigid cage-, spring- and damper elements [1]. It was determined that with the unstable movement of the cage, its rigid body movement is overlapped by a strong elastic deformation [1]. Using simulations and experiments, the influence of the coefficient of friction in the contact area between the cage and rolling elements as well as the bearing load on the cage movement were investigated in greater detail [4]. The various types of cage movements are differentiated with the help of the coordinates of the cage center of gravity, the comparison of the rotational frequency of the cage’s center of gravity with that of the rolling element set, and the deformation of the cage. [1]

In summary, it is now possible to map the movement behavior of cages using special multibody simulation programs. For the most part, the simulation results correspond with the experimental investigations [4], although the latter tend to be very complex and can, therefore, only be performed occasionally. In addition, the technical literature on this subject offers a large number of metrics that serve to identify unstable cage movements. Extensive, systematic simulation studies performed by the authors over the past two years show, however, that tests for cage instability produce false-positive or positive-false results rather quickly for all known metrics. Consequently, a reliable, objective and automated assessment of cage movements has not yet been possible – until now.

A methodology that makes this possible for the first time is presented below. Based on a large number of multi-body simulations for different bearings and operating situations, it enables the automatic classification of cage movements into characteristic movement types with a high degree of reliability. In addition, all types of elastically modeled cages can be considered.

3. Description of the simulation models
The multi-body simulations are carried out using CABA3D software. Modeling of the cages is fully elastic, whereby the degrees of freedom are reduced according to the method described by CRAIG and BAMPTON[12]. Details with respect to implementation can be found at HAHN [6]. Accordingly, the cage’s deformation can be mapped in high resolution based on the nodal displacements (see Figure 2 as well as [7]).

In order to identify characteristic types of movement, several bearings are examined under different load conditions as well as initial- and boundary conditions. The variable parameters are the coefficient of friction in the contact between cage and rolling element as well as the ring speeds and the bearing forces.

In this case, deep groove and angular contact ball bearings, tapered and cylindrical rolling bearings as well as spindle bearings with polyamide, laminated fabric and sheet steel cages were analyzed. The examined bearings’ bore diameter ranges from 65 mm to 90 mm, while the cage pocket play varies between 0.18 mm and 1.05 mm. By selecting and examining the above-listed different bearings and cage types, it is possible to make a general statement about the cage movement and its applicability to types of rolling bearings that were not considered in this article.

4. Deriving properties of the cage movements for classification purposes
The results of the simulations mentioned above support the conclusion that cage movements can be divided into three types of movement: stable, unstable and circling. These three types 7 of movement exhibit certain properties; on the basis of these characteristics, virtually any cage movement can be classified into one of the three types of movement.

The first property used is the temporal course of the coordinates of the cage’s center of gravity for the duration of one revolution of the rolling element set with respect to the stationary, inertial coordinate system (see Figure 3). The trajectory of the stable cage movement illustrates that the cage’s center of gravity moves slightly. The other two types of movement show circular movement courses. In addition, the trajectory of the unstable cage movement makes it clear that the cage’s center of gravity rotates more frequently around the origin of the coordinate system than the rolling element set. Consequently, the speed of the cage’s center of gravity is significantly higher than the speed of the rolling element set. By relating the coordinates to the guiding clearance, it is possible to compare cages with different guiding clearance (see Figure 3). The standard deviation of these normalized coordinates for the duration of one cage revolution is used as an indicator; this is primarily done in order to distinguish the “stable” type of movement from the other two movement types.

In addition to its coordinates, the speed of the cage’s center of gravity also indicates that it belongs to one of the three movement types. The whirl ratio (per GUPTA [10]) is defined as the quotient of the speed of the cage’s center of gravity and the speed of the rolling element set. Figure 4 illustrates characteristic whirl conditions for the three types of movement. The stable cage movement shows values fluctuating around zero. Accordingly, the cage’s center of gravity rotates alternately in the same and in the opposite direction as compared to the rolling element set. In the case of the rotational cage movement, the values fluctuate slightly around the number 1; that is, the speed of the center of gravity roughly corresponds to that of the rolling element set. With the unstable cage movement, comparatively high as well as strongly fluctuating values can be observed for the whirl ratio’s value. The arithmetic mean and the standard deviation for the time range of a cage revolution are determined as scalar properties for classifying the movements.

In addition to the translational speed, the rotational speeds of the cage are also accounted for by the speed ratio for the analysis. Speed ratio is defined as the quotient of the Euclidean norm of all cage rotations and the linear trend of this norm within the time range that is to be classified – i.e., one cage rotation. For this purpose, the Euclidean norm of the rotational speeds of the cage around its center of gravity axis system is calculated. Furthermore, the linear trend for this norm’s chronological sequence is determined for the period of one cage rotation. This means that in a case where the speed is constant, the linear trend corresponds to the arithmetic mean. The change in this property over time indicates how much the cage’s speed varies (see Figure 5). The unstable cage displays a very volatile sequence for this characteristic; this is because high frequency tilting often occurs in addition to the rotation around the bearing’s axis.

Since unstable cage movements are often accompanied by a strong elastic deformation of the cage (see Figure 2 and [4]), the displacements of the nodes of the FE model are an integral part of the classification. Nodal displacements or cage deformations that occur at particularly high frequencies and vary greatly are indicators of unstable movement, while small, steadystate displacements are characteristic of the rotational and stable form of movement (see Figure 6). The extent of the fluctuations is represented for each spatial direction by the standard deviation.

To summarize, it can be said that the three types of movement

Stable, Rotational and Unstable

can be identified by the following properties (in combination):

Standard deviation of the normalized coordinates of the cage’s center of gravity Standard deviation of the speed ratio Arithmetic mean and standard deviation of the whirl ratio Standard deviation of the nodal displacements

These values, calculated for the period of one cage rotation, are referred to as CDI (Cage Dynamics Indicator), which forms the initial basis for the classification. The components of the CDI were deliberately chosen to ensure that it can be determined not only from simulations but also from optical measurements.

5. Discriminant analysis to classify the cage dynamics
Quadratic discriminant analysis allows the cage movement to be classified as stable, unstable or rotational by evaluating the stability metric CDI, which is composed of the characteristics that were described above. The defined metric CDI from Section 4 (see Figure 7) is calculated for the period of one cage rotation.

The discriminant analysis requires training data that represent the classification problem – that is, the classification of the CDI for a specific calculation case into the stable, unstable or rotational types of movement. The training data contain the results from 402 multi-body simulations and 4,788 feature vectors. On the basis of the training data, multivariate normal distributions are determined for each movement type in order to determine the class membership for the CDI that is to be classified. To obtain the desired function requires multiplying this probability by a function that represents the costs of a misclassification. Finally, the CDI is assigned to the class that causes the lowest costs resulting from a potential misclassification.

As an example of the distribution of two characteristics, Figure 8 shows the whirl ratio’s arithmetic mean and the standard deviation. Although for certain simulations a classification into the three movement types can be made based on this property alone, there are often overlapping or transitional areas between the movement types. Using the CDI, these areas can be relatively precisely separated from each other.

With a five-fold cross validation, the discriminant analysis has a 93.3%prediction accuracy, which can be considered very reliable in practice. The incorrectly classified cage movements are largely due to CDIs that do not have a clear class affiliation; consequently, they cannot be clearly classified – even with the help of manual classification performed by an expert.

6. Summary and outlook
This paper describes a procedure for the objective and reliable automated classification of rolling bearing cage movements into the three characteristic movement types: stable, rotational and unstable. This procedure is able to take into account elastic deformations of the cage. The basis for this classification procedure is the Cage Dynamics Indicator (CDI), which subsumes several properties. A total of 402 simulations and 4,788 CDIs from different bearing and cage types are used to train the quadratic discriminant analysis to classify the cage movement. After a five-fold cross-validation, the algorithm has a prediction accuracy of 93.9%. The majority of the incorrectly assigned cage movements do not show any clear movement behavior that is characteristic of a certain class.

When the training data was created, special attention was paid to make sure that this classification can also be used for other rolling bearing and cage types that are unknown to the algorithm. This ensures a broad scope of application. In addition, the procedure described above also enables the results of large-scale tests to be automated and objectively classified into the three movement types (stable, unstable and circling), so that the cage dynamics can be examined with regard to a variety of influences, such as a change in the geometry or the load on the rolling bearing.

Acknowledgements
The findings and conclusions published in this paper are the result of the project “KILL VIB – Reduzierung von Vibrationen und Geräuschemissionen infolge von Käfiginstabilitäten in Wälzlagerungen (AZ-1233-16)” (Reduction of vibrations and noise emissions as a result of cage instabilities in rolling bearings), funded by the Bavarian Research Foundation. The authors gratefully acknowledge the Bavarian Research Foundation’s funding and support of their work.

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