1. Introduction
Due to their properties and standardized nature, rolling bearings are used in almost all technical fields (automotive industry, machine industry, aviation industry, household goods industry, etc.) The structure of rolling ball bearings has not changed for many decades. The rolling element in rolling ball bearings is located between two rings. However, despite the standardized structure, rolling bearing manufacturers try to implement solutions to provide their products with a long service life [
1], low noise [
2], high stiffness [
3] and simultaneous minimization of manufacturing costs, which surely affects the products’ competitiveness.
Rolling bearings are the subject of many scientific and industrial research. The research mainly concerns the search for solutions that enable improving the rolling bearings’ operating parameters. Furthermore, there are many papers on diagnostic methods that evaluate the condition of new and operating bearings [
4,
5]. It is necessary to mention that rolling bearings in many cases are a critical element in a mechanism. In case of an unexpected bearing or bearing assembly failure, the entire machine is shutdown, which entails high costs.
One of the most important indicators determining the wear of rolling bearings is the vibration generated by them [
6]. During its operation, a bearing undergoes wear, which can lead to a gradual increase in generated vibration. The vibration diagnostics, which is a non-destructive procedure, allows for evaluating the tested bearing’s wear. Based on the vibration analysis, it is possible to introduce preventive treatment aimed at planned bearing replacement. This allows for the elimination of unplanned machine shutdown and minimizes repair costs. An analysis of the vibration generated by new rolling bearings also enables the detection of damage or factory defects. Moreover, excessive vibration generated by the bearing is an adverse phenomenon, because aside from a drastic reduction in the bearing’s service life, it has an adverse effect on the entire bearing assembly.
There are many factors that affect the vibration generated by rolling bearings. Despite the fact that there are many papers related to the analysis of the impact of specific factors on the vibration of rolling bearings, there is a lack in statistical evaluation of the significance of the factors’ impact.
Paper [
7] presents the mathematical model for the rolling ball bearings’ friction moment with consideration of the bearings’ imperfections, such as waviness. The model was verified experimentally. The mathematical models can also be used to predict the rolling bearings’ service life. In paper [
8], Wang et al. presented a mathematical model (matrix) in which the first dimension is the bearing’s service life given in hours and the other—performance. The mathematical model allowed for specifying the bearing’s condition after a specific operation cycle. The model was verified experimentally. On the other hand, paper [
9] presents the dynamic model of rolling bearing vibration generated due to the interaction between the rolling elements and the defect area. The paper proves that the raceway’s defect causes a double vibration impulse. This can be an indicator of raceway damage when analyzing the vibration of rolling bearings.
Vibration occurring during the operation of rolling bearings is a natural phenomenon and results directly from the rolling bearings’ structure. Natural vibration is generated during the turning of rolling elements. If it does not exceed the acceptable values, then such bearings are approved.
The testing of rolling bearing vibration can be classified into three groups. The first group concerns the evaluation of the vibration of new rolling bearings on testing rigs. The evaluation is aimed at detecting factory defects of new bearings and is commonly conducted by companies that manufacture rolling bearings by using industrial Anderon meters. This approach allows for streamlining the rolling bearings’ manufacturing and introducing a modification to their structure to meet the costumer’s requirements. The second testing group concerns the vibration analysis of rolling bearings operating in real application conditions, e.g., in electric engines [
10], etc. A certain inconvenience of this approach is the risk of transferring the vibration from the entire mechanism in which the bearing was used. In such a case, it is possible to obtain “false” vibration signals leading to erroneous conclusions. The third testing group concerns intentional induction of defects or damage in rolling bearing elements to determine their impact on the generated vibration. The testing is mainly aimed at developing new diagnostic methods for rolling bearings with the purpose of detecting specific defects.
The surface texture quality is one of the most important factors affecting the tribological performance and wear of mechanisms [
11,
12]. Paper [
13] presents the testing of the impact of 2D and 3D roughness parameters of rolling ball bearings’ inner ring raceway on the generated vibration. The raceway surface topography testing was conducted with the use of a focus microscope. The authors demonstrated a high correlation between the roughness parameters Ra and Sa, and the generated vibration. Similar testing was conducted by Adamczak et al. in paper [
14]. The evaluation of the impact of bearing raceway surface roughness on the generated vibration was conducted using 4 different roughness parameters (Ra (Sa), Rt (St), Rku (Sku) and Rsk (Ssk)). In the paper, the authors also obtained a high correlation between the inner raceway’s roughness parameters Ra and Sa, and the generated vibration. The testing allowed for the conclusion that in the case of stable rolling bearing manufacturing, it is sufficient to analyze the 2D raceway surface roughness parameters. On the other hand, in the case of distorted manufacturing or production of new bearing types, it is recommended to conduct a detailed analysis of the raceway’s topography.
It is necessary to add that the roughness profile was usually measured in the described scientific papers along a single raceway cross-section. On the other hand, due to the balls’ rolling motion on the raceway’s entire surface, it seems that a detailed analysis of the raceway’s surface is more reasonable, i.e., by evaluation the waviness and roundness deviations.
One of the dominant factors affecting the generated vibration is the raceway’s waviness. Paper [
15] features a simulation of the occurrence of waviness on rolling ball bearing rings and evaluation of its impact on the bearing’s behavior. It was demonstrated that the raceway’s waviness causes a non-linear vibration response. The paper lacked an experimental verification of the simulation tests. Similar testing was conducted by Wang et al. [
16]. In paper [
17], the authors applied the adopting signal coherence theory to evaluate the impact of the raceways surface waviness on the generated vibration. They demonstrated that the outer ring raceway’s waviness is more strongly correlated with the vibration analyzed in a low range of frequencies.
It is necessary to add that most papers only featured an analysis of the waviness’ impact, whereas a different type of deviations, especially long-wave deviations such as roundness is omitted. It is a certain simplification, because the ball does not only roll on the raceway’s roughness or waviness but also on “all deviations” [
18]. In such a case, it is necessary to apply filtration methods and measurement signal analyses [
19,
20]. Industrial practice usually features testing of the outer ring’s roundness, because its excess value contributes to issues with the bearings’ mounting. Viitala et al. [
21] analyzed the impact of the bearing’s inner ring roundness profile on the sub-critical vibrations in a flexible rotor. They demonstrated a significant correlation between the tested bearing’s inner ring roundness and the generated vibration. Minimization of the inner ring’s roundness deviation contributed to the reduction in the tested mechanism’s vibration. Interesting research was conducted in Japan [
22]. The method of mounting the bearing also affects the vibration generated by rolling bearings. One of the basic structural parameters of rolling ball bearings is the radial clearance. Paper [
23] presents an online system intended for evaluating the bearing clearance based on the vibration signal. In paper [
24], Zmarzły presented testing that demonstrated increased vibration as result of increasing the radial clearance.
Another parameter that describes the inner geometry of rolling ball bearings is the curvature ratio. Gloeckner [
25] tested the impact of the curvature ratio on the temperature and energy losses in bearings used in jet engines. However, the number of research papers related to the evaluation of the ratio’s impact on rolling bearing vibration is limited. Only the author of [
26] has conducted an analysis of the inner and outer raceway’s curvature ratio in relation to the vibration generated by the 6304 type bearings.
A novelty of the article is development multi-dimensional mathematical models used to quantitative analysis of the impact of selected factors on vibrations generated by rolling bearings made of AISI 52100 steel. A literature analysis demonstrated that the Authors are focusing mainly on testing the impact of single factors on the vibration of rolling bearings recorded in a full range of frequencies. It is a certain limitation, because the impact of some factors can be clearer in a narrowed range of frequencies. Due to the above, it was proposed in industrial practice to analyze vibration generated by bearings in 3 frequency ranges, i.e., low LB 50–300 Hz, medium 300–1000 Hz and high 1000–18,000 Hz. However, there is a shortage of papers that specify quantitatively the manner in which a given factor affects vibration recorded in a specific frequency range and whether this impact is relevant. It should be noted that it is an issue to clearly specify the factors that are dominant in their impact on the vibration generated by rolling bearings. Therefore, it is recommended to conduct a quantitative evaluation of the factors’ impact. Such evaluation can be conducted by using mathematical models developed on the basis of multiple regression. Furthermore, in most of the analyzed scientific papers, the testing is limited to solely computer simulation tests or experimental tests on a small group of rolling bearings from a single manufacturer. Despite the fact that the manufacturing of rolling bearings is a common and widely known process, bearing manufacturers apply various company secret processes to obtain products with satisfying quality. This is the reason for the large price and quality disproportion between bearings of the same type but from different manufacturers.
The main objective of the paper is a complex analysis of the impact of a selected factor on vibration level of 3 groups of rolling bearings offered by three different manufacturers and the development of mathematical models to quantitatively evaluate the significance of this impact. The models can be used to estimate the vibration generated by specific bearings based on the measured performance and structural parameters.
3. Results and Discussions
The experimental testing results were the basis for conducting statistical testing aimed at designating multi-dimensional regression models. Three regression models were developed for each vibration frequency band (low—LB, medium—MB and high—HB). The statistical calculations for a sample of 90 pieces of rolling bearings were conducted using the “multiple regression” package of the statistics software. The dependent variables (predictors) were as follows: inner ring’s roundness deviation (RONtPW(2–15)), outer ring’s roundness deviation (RONtPW:(16–50)), outer ring’s roundness deviation (RONtPZ:(2–15)), outer ring’s waviness deviation (RONtPZ:(16–50)), radial clearance (ΔR) and raceways’ total curvature ratio (ft).
Fisher’s variance test was conducted in order to evaluate the variance of the entire mathematical model. On the other hand, the student t-significance test was conducted for particular input variables.
Table 2 presents the analysis results for the low frequency band, i.e., 50–300 Hz, where b indicated regression coefficient, SE
b indicated estimation error for regression coefficient, t(83) present “t” statistic,
p—probability value and β—regression function’s structural parameter.
For the entire mathematical model designated for the low vibration frequency band (LB) the determination coefficient of R2 = 0.46 was obtained, whereas the statistics value is F(6.83) = 11.3777 > Fkr, which means that the model is statistically significant and was developed correctly. This is also confirmed by the value of coefficient p which amounted to p < 0.05. The standard estimation error for the entire model amounted to SEe = 1.197. The relatively small value of the determination coefficient R2 indicates that there may be other factors (not analyzed in the tests) affecting the vibration values recorded in this frequency band. It should be noted that that the value of vibrations generated in the low frequency range is influenced by difficult-to-measure factors, such as cage unbalance, contamination of the lubricant, etc.
Table 2 presents the statistically significant predictors marked in red. It is possible to see that in this case there are significant input factors that affect the vibration generated in the low frequency range. These are the inner ring’s waviness deviation, roundness deviation and the outer ring’s waviness deviation. The highest value of standardized regression coefficient β was obtained for the inner ring’s roundness deviation. Due to the above, it is possible to assume that the quality of the inner raceway’s surface layer has a substantial impact on the vibration generated in the low frequency band. A similarly high β value was obtained for the outer ring raceway’s waviness. Analyzing calculation result presented in
Table 2, it can be concluded that only three input factors among six have substantial impact at vibration generated in low frequency band. Therefore, this factors will be presented on the charts in detail. The impact of the inner raceway’s roundness and waviness deviations on the vibration generated in the low frequency range is presented in a single chart (
Figure 5), because they concern the same rings. On the other hand, the impact of the outer ring raceway’s waviness is presented on a separate chart. The vibration level is described by Anderon Unit (And).
When analyzing the chart presented in
Figure 5, it can be clearly stated that the increase in the inner ring race’s roundness and waviness deviation contributes to increased vibration tested in the frequency range of 50–300 Hz. Wherein, the waviness was higher than the roundness deviations measured for the inner ring’s race (see
Figure 5). A similar dependency can be read for the outer ring. Here, the increase in the waviness deviation RONt
PZ:(16–50) causes a moderate increase in the vibration recorded in the low frequency range (see
Figure 6). As shown in
Figure 1, waviness deviation is periodic or non-periodic irregularities appearing on bearing raceway profile. The largest waviness deviation indicated highest wave’s amplitude on the raceway profile. Therefore, during working, bearing balls generate higher vibration value, which is also recorded in the low frequency range.
When analyzing the mathematical model designated for the medium vibration frequency band (MB), the obtained determination coefficient amounted to R
2 = 0.44. On the other hand, the Fisher test result demonstrated that the model is statistically significant, because F(6.83) = 10.8 > F
kr. The value of coefficient p was substantially lower than 0.05. The estimation error for the entire model developed for the medium vibration frequency band amounted to SE
e = 1.832. When considering the results presented in
Table 3, it is possible to see only a single factor that substantially affects the vibration generated in the frequency range of 300–1800 Hz, i.e., the inner ring race’s waviness deviation (RONt
PW:(16–50)). This is demonstrated by the conducted student
t-tests (see
Table 3) and the high determination coefficient, which amounted to β = 0.7029.
When studying the data presented in the chart of
Figure 7, it is possible to see a concentration of results in two areas of axis x. One of them concerns waviness deviations in area
., while the other—in area
. This is related to the fact that the testing featured a group of 90 bearings of the same type but from different manufacturers. Due to the above, the chart includes visible disproportion between particular batches. The first concentration features the results of bearings from two manufacturers. The other concentration features bearings from the third manufacturer, which points to a lower quality of the outer raceway’s surface. This can derive from an incorrectly performed finishing process (excess vibration of the grinding wheel, incorrect processing parameters). However, when analyzing the trend line in the chart of
Figure 7 (red line), it is possible to see an upward trend in vibration along with the increase in the inner raceway’s waviness deviation. Similar results were obtained for the low vibration frequency range LB (
Figure 5 and
Figure 6).
When considering the mathematical model calculated for the high frequency band (HB), it can be stated that the highest determination coefficient among all tested models was obtained, i.e., R
2 = 0.69. This means that 69% of the dependent variables were explained in this model. The good matching of the mathematical model and its statistical significance is also confirmed by the Fisher test, where F(6.83) = 30.66 > F
kr. As with the model obtained for the medium vibration frequency band, the coefficient p was lower than 0.05. The estimation error for the entire model amounted to SE
e = 0.396. When analyzing the student
t-test results and the coefficient p presented in
Table 4, it is possible to see a statistical correlation of 4 input factors (RONt
PZ:(2–15), RONt
PW:(16–50), ΔR, f
t) on the generated vibration recorded in the frequency range 1800 Hz–10,000 Hz. Due to the above, they will be analyzed in detail in the charts.
When analyzing the results presented in
Figure 8, it is possible to see a small diversification between the measured outer ring’s roundness deviations. Most results do not exceed 2 µm, which indicates a stable manufacturing process maintained in all of the tested rolling bearing groups. There are only two exceptions where RONt
PZ:(2–15) = 2.04 µm and RONt
PZ:(2–15) = 5.86 µm were obtained. This may result from the analyzed ring’s factory defects. When analyzing all test results, it is possible to see discrepancies in the vibration characteristics despite the small variation of the inner ring’s roundness deviations. This indicates a dominant impact of other factors that can affect the generated vibration. The trend line presented in the chart points to a slight upward trend in the vibration recorded in the high frequency range due to the increase in the tested deviation.
When considering the results presented in
Figure 9, it is possible to see a clear increase in the vibration generated in the high frequency band due to the increase in the inner ring’s waviness deviation (see the trend’s red line). Similarly, to
Figure 7, it is possible to see a concentration of results in two intervals, which also results from the analysis of rolling bearings from different manufacturers. However, in the case of the high vibration frequency band, the trend line is more vertical, which points to a clearer impact of the inner ring’s waviness deviation on the generated vibration.
When testing the impact of radial clearance on the rolling bearings’ operation, it can be clearly stated that the increase in radial clearance increases the vibration generated in the frequency band of 1800–10,000 Hz. It is necessary to add that the testing featured bearings of the same type (6304ZZ) with a standard radial clearance. According to the ISO 5753-1:2009 standard [
30], this type of bearings should have a radial clearance in the range of 0.005 mm–0.020 mm. It is possible to see that some bearings have a greater radial clearance, which allows for classifying them to the bearing group with the C3 clearance, i.e., bearings with greater radial clearance. It is necessary to note that the bearings with a greater radial clearance can operate at higher revolving speeds and have a lower friction moment. However, this has an adverse effect on the generated vibration, because excessive radial clearance allows the rolling elements to generate additional vibration during motion, which is visible in the chart of
Figure 10. Due to the above, it is a significant factor visible in the high vibration frequency band.
When analyzing the results of testing the impact of the total curvature ratio on the vibration generated in the high frequency band presented in
Figure 11, it is possible to see three measurement result groups, which is also related to the testing of bearings from three manufacturers. As can be seen from Dependency (14), the total curvature ratios derive from the inner and outer ring raceway’s radii and from the rolling balls’ diameter. In most cases, it is a parameter that is not standardized nor provided to users by the manufacturers. Hence the disproportions between the total curvature ratios obtained for different manufacturers. However, when analyzing the trend line, it can be stated that the increase in the total curvature ratio causes a moderate increase in the vibration generated in the frequency range of 1800–10,000 Hz. This is confirmed by the test presented in paper [
26].
The test results in
Table 5 were used to present the mathematical models specifying the impact of significant factors on the generated vibration. Only statistically significant factors were taken into consideration in these models.
When analyzing the mathematical models presented in
Table 5, it can be stated that the most statistically significant predictors exists for the model developed for the high vibration frequency band. Due to the above, in terms of analyzing the impact of geometry on the quality of the rolling bearing race’s surface layer, this range should especially subjected to analysis. This was also confirmed by the statistical significance testing of the developed models, where the determination coefficient of R
2 = 0.69 and estimation error of SE
e = 0.396 were obtained for HB. This points to the mathematical model’s good quality.
The innovation of mathematical models presented in
Table 5 is the possibility of predicting the values of vibrations generated in specific frequencies based on the measured deviations and bearing parameters. Moreover, the developed models allow the producers of rolling bearings to indicate factors that may significantly affect the vibrations generated by the bearing type 6304zz. Then, it is possible to carry out a correction to the production process in order to obtain bearings of satisfactory quality.