Research on the Influence of Raceway Waviness and Groove Shape on the Vibration Performance of Angular-Contact Ball Bearings

Abstract

Raceway topography and lubrication jointly govern rolling-contact conditions and vibration transmission in angular-contact ball bearings. This study examined associations between eight inner- and outer-raceway descriptors—roundness, waviness, groove-profile deviation, and roughness—and four vibration responses of thirty production 7208 bearings from one batch. All bearings were tested as received with factory grease at 1800 r min⁻¹ and a radial load of 150 N. The responses comprised vibration-acceleration levels and vibration velocities in the 50–300, 300–1800, and 1800–10,000 Hz bands. Four grey relational schemes—initial-value-normalized, mean-value-normalized, relative, and absolute—were applied. A descriptor was retained when it ranked among the top three under at least two schemes. Inner-raceway roundness was retained for all four responses, while inner-raceway waviness was retained for acceleration and the low- and medium-frequency velocity responses. Inner-raceway roughness, outer-raceway groove-profile deviation, and outer-raceway roughness were each retained for two responses, whereas the remaining descriptors showed response-specific associations. Rankings varied with preprocessing and relational-degree formulation. The findings provide screening evidence for prioritizing raceway metrology and subsequent controlled experiments.

1. Introduction

With continuing advances in ultra-precision manufacturing and high-speed rotating machinery, increasingly stringent requirements are being imposed on the rotational accuracy, dynamic stability, and reliability of precision spindle systems. Angular-contact ball bearings can simultaneously support radial and axial loads and are therefore widely used in high-speed motorized spindles, aero-engines, and precision transmission systems. Their vibration behaviour affects spindle rotational accuracy, machined-surface quality, operating noise, and service life. It is therefore important to determine how raceway geometric deviations introduced during manufacturing are associated with the vibration performance of production bearings.

The dynamic response of an angular-contact ball bearing originates from time-varying and nonlinear interactions between the rolling elements, raceways, cage, and lubricant. It is also affected by the rotational speed, load, preload, contact angle, and structural deformation. Yi et al. developed an improved dynamic model for angular-contact ball bearings under constant preload [1], whereas Zhang et al. established a nonlinear rotor–bearing model incorporating preload and a varying bearing contact angle [2]. Bal et al. further examined the effect of preload on the vibration of elastohydrodynamically lubricated angular-contact ball bearings using theoretical and experimental approaches [3]. These studies demonstrate that bearing vibration should be interpreted with reference to both the internal contact state and the specified operating and lubrication conditions.

In manufactured bearings, deviations of raceway geometry occur over different spatial scales. Roundness errors represent the macroscopic departure of a circumferential profile from an ideal circle; waviness describes intermediate-scale, approximately periodic deviations; roughness characterizes shorter-wavelength surface irregularities; and groove-profile deviations describe the departure of the measured raceway cross-section from its nominal geometry. Raceway waviness can produce periodic variations in ball–raceway contact deformation and load as the rolling elements traverse the raceway. Adamczak and Zmarzły experimentally investigated the relationship between raceway waviness and rolling-bearing vibration levels [4]. Li et al. developed a nonlinear multi-degree-of-freedom model of an angular-contact ball bearing incorporating waviness [5], while Liu et al. analyzed the coupled effects of roundness and waviness errors [6]. Subsequent studies considered the combined influence of outer-race waviness and local defects, the effects of raceway waviness on cage dynamics, and nonuniform waviness in lubricated bearings [7,8,9].

Groove-profile deviation affects ball–raceway contact through a mechanism distinct from that of circumferential waviness. Changes in groove curvature and race conformity alter the contact geometry and may consequently modify the contact area, contact-pressure distribution, load sharing, and contact stiffness. Kwak and Kim analyzed the influence of shoulder height and bearing geometry on ball–raceway contact [10]. Xu et al. showed that inner-ring angular misalignment changes the contact characteristics and stiffness coefficients of duplex angular-contact ball bearings [11]. At the spindle-system level, bearing contact and geometric errors may contribute to dynamic rotational errors [12], and digital-twin-based bearing models have been developed for rotational-accuracy prediction [13]. These studies provide a contact-mechanics basis for investigating associations between groove-profile deviations and bearing vibrations, but they do not establish a universal quantitative relationship applicable to all bearing designs and operating conditions.

The effects of raceway waviness have also been investigated with respect to its location, spatial order, and representation. Aktürk analyzed vibrations associated with waviness on the inner race, outer race, and rolling elements of ball bearings [14]. Shah and Patel conducted theoretical and experimental vibration studies of lubricated deep-groove ball bearings with raceway waviness [15]. Xu et al. subsequently developed a Gaussian-filter-based waviness-characterization method for bearing vibration analysis [16], while Liu et al. investigated waviness-induced vibrations using signal-coherence analysis [17]. Sun et al. incorporated raceway waviness into the calculation of vibration-induced bearing noise [18]. Collectively, these studies indicate that the vibration response associated with waviness depends not only on its amplitude but also on its spatial order, location, bearing configuration, and operating condition.

Surface topography also interacts with the lubrication state of the rolling contact. Wang et al. developed a dynamic model showing that raceway roughness can affect rolling-element motion, lubrication performance, and subsurface stress in angular-contact ball bearings [19]. Their investigation of race conformity further demonstrated its influence on ball–raceway contact pressure and rolling-element skidding [20]. Liu et al. studied the vibration characteristics of a high-speed flexible angular-contact ball bearing while considering manufacturing errors [21], whereas Alfares et al. investigated the influence of bearing-component waviness on the dynamic performance of a grinding-machine spindle system [22]. Randall and Antoni reviewed vibration-based methods for rolling-element-bearing diagnostics, providing a general context for interpreting measured bearing vibration signatures [23]. These findings suggest that raceway topography may influence the measured response through geometric excitation, changes in contact conditions, and interactions with the lubricant film. However, lubricant-film thickness, traction, friction, and contact force were not measured in the present experiments. These mechanisms are therefore treated as physically plausible interpretations rather than experimentally verified causal pathways.

Grey relational analysis (GRA) provides a means of comparing a reference sequence with multiple comparison sequences after appropriate data preprocessing. Celik et al. applied GRA as a multi-criteria ranking method for lubricant selection and lubrication-interval evaluation, using normalized responses, grey relational coefficients, and grey relational grades [24]. Murugesan and Jung employed an experimental design and GRA for multi-response forming-process evaluation, including data normalization, deviation-sequence calculation, grey relational coefficient determination, weighting, and ranking [25]. These studies provide relevant methodological examples of GRA-based preprocessing and relational ranking. Nevertheless, their applications differ from the present bearing vibration problem and do not provide direct evidence of a physical mechanism linking raceway geometry to vibration. Accordingly, the grey relational grades calculated in this study are interpreted as measures of sequence association under the adopted formulations rather than as statistical significance tests or proof of causality.

Despite the progress described above, several issues remain insufficiently addressed for production bearings. First, many previous studies considered prescribed waviness patterns or a limited number of geometric variables, whereas roundness, waviness, groove-profile deviation, and roughness coexist on the inner and outer raceways of manufactured bearings. Second, systematic comparisons of the associations between these geometric descriptors, vibration acceleration, and low-, medium-, and high-frequency vibration velocities remain limited. Third, GRA rankings may vary with the normalization procedure and the definition of relational degree; conclusions based on one formulation may therefore be method-sensitive. In addition, the lubricant identity and properties were not included as comparison variables in the present dataset. The study can consequently evaluate raceway-geometry–vibration associations only under the reported factory-greased condition and cannot isolate lubricant effects or compare grease- and oil-lubricated responses.

To address these issues, thirty production 7208 single-row angular-contact ball bearings from one manufacturing batch were investigated. The roundness, waviness, groove-profile deviation, and roughness of the inner and outer raceways were measured and matched at the specimen level with the vibration acceleration and vibration velocity in three frequency bands. Four complementary GRA schemes—initial-value-normalized, mean-value-normalized, relative, and absolute grey relational analyses—were used to rank the associations between eight raceway descriptors and four vibration responses. The method-specific rankings were then compared using a predefined cross-method recurrence rule to identify descriptors that repeatedly occupied leading positions. The objective was to provide screening evidence for prioritizing raceway quality control and subsequent controlled validation while explicitly distinguishing relational ranking from experimentally demonstrated causality. The conclusions are limited to bearings and operating and lubrication conditions comparable to those examined in this study.

2. Theoretical Analysis

2.1. Contact Mechanics Model Considering Geometric Defects

When establishing the contact mechanics model between the rolling elements and the raceways, it is necessary to comprehensively consider the influences of waviness and raceway-profile errors. According to the Hertz contact theory, the contact force between the rolling elements and the raceways can be expressed as:

$$Q_j=K_n\times {\delta }_j^{3/2}$$

(1)

where $K_n$ is the load-deformation constant, which depends on the bearing material and geometric dimensions.

Considering the contribution of all rolling elements, the expressions for the resultant force and resultant moment inside the bearing can be obtained. The fluctuation characteristics of these resultant forces and moments directly determine the vibration response of the bearing system. The additional deformation caused by waviness is a periodic function, whose frequency is related to the revolution speed of the rolling elements and the waviness order; while the additional deformation caused by raceway-profile errors is relatively stable, which mainly changes the static distribution of contact forces. When the two act in combination, the periodic excitation of waviness will be further amplified under the background of an uneven load caused by raceway-profile errors, resulting in complex vibration response characteristics.

2.2. Frequency-Domain Characteristics Analysis of Vibration Response

Based on the above-mentioned contact mechanics model, the frequency-domain characteristics of the vibration response can be further analyzed. The main frequency of the waviness excitation can be expressed as:

$$f_{wav}=l\times n_c/60$$

(2)

where $n_c$ is the cage rotational speed (unit: r/min). For waviness at different positions (inner ring, outer ring, or rolling elements), its excitation frequency has different characteristics, as the excitation frequency generated by outer-ring waviness is relatively fixed, since the outer ring is usually fixedly connected to the bearing housing; conversely, the excitation frequency generated by inner-ring waviness changes with the rotation of the inner ring.

Groove-shape errors mainly affect the vibration characteristics in the low-frequency band. By altering the load distribution of rolling elements, it generates a periodically varying restoring moment inside the bearing. When both groove-shape errors and waviness exist simultaneously, they produce a frequency modulation effect, forming side frequency bands in the vibration spectrum and distributing the vibration energy over a wider frequency range.

2.3. Theory of Poor-Information Systems and Correlation Degree Analysis Method

In practical engineering problems, the influencing factors of bearing vibration are complex and diverse, and there often exists a nonlinear relationship between each factor and the vibration response. Meanwhile, restricted by measurement conditions and sample size, it is difficult to obtain fully sufficient data and information. The theory of poor-information systems provides effective theoretical support for solving such problems. Its core idea is to excavate the key influencing factors through reasonable mathematical methods under the conditions of insufficient data information.

Grey relational analysis is an important tool of the poor-information system theory. It judges the correlation strength between influencing factors and vibration response by measuring the geometric similarity among data sequences. Compared with traditional statistical analysis methods, grey relational analysis does not require the assumption that data follow a specific probability distribution, and it is suitable for processing non-normal, nonlinear small-sample data. In this study, four grey relational schemes—initial-value-normalized, mean-value-normalized, relative, and absolute grey relational analyses—were therefore compared to evaluate the sensitivity of factor rankings to data preprocessing and relational-degree formulation.

We define the data sequence ${\mathit{X}}_i$ consisting of influencing factors as

$${\mathit{X}}_i=(x_i\left(1\right),x_i\left(2\right),\dots ,x_i(k),\dots ,x_i(n\left)\right)$$

(3)

In the formula, i denotes the number of influencing factors, where i = 1, 2,…, m; and k denotes the data serial number, where k = 1, 2,…, n.

The data sequence ${\mathit{Y}}_j$ consisting of bearing vibration measurements is defined as

$${\mathit{Y}}_j=(y_j\left(1\right),y_j\left(2\right),\dots ,y_j(k),\dots ,y_j(n))$$

(4)

In the formula, j represents the number of vibration value types, where (j = 1, 2, 3, 4).

2.3.1. Grey Relational Degree

After preprocessing the data sequences ${\mathit{X}}_i$ and ${\mathit{Y}}_j$ [20], the new data sequences can be obtained as:

$${\mathit{X}}_i^{\prime }=(x_i^{\prime }\left(1\right),x_i^{\prime }\left(2\right),\dots ,x_i^{\prime }(k),\dots ,x_i^{\prime }(n))$$

(5)

$${\mathit{Y}}_j^{\prime }=(y_j^{\prime }\left(1\right),y_j^{\prime }\left(2\right),\dots ,y_j^{\prime }(k),\dots ,y_j^{\prime }(n))$$

(6)

and

$$x_i^{\prime }(k)=x_i(k)\times K=x_i(k)/x_i\left(1\right)$$

(7)

$$y_j^{\prime }(k)=y_j(k)\times K=y_j(k)/y_j\left(1\right)$$

(8)

For initial-value processing,

$$K_1={\displaystyle \frac{1}{x_i\left(1\right)}}, K_2={\displaystyle \frac{1}{y_j\left(1\right)}}$$

(9)

For mean-value processing,

$$K_1={\displaystyle \frac{1}{\frac{1}{n}{\displaystyle {\sum }_{k=1}^n}{x}_i\left(k\right)}}, K_2={\displaystyle \frac{1}{\frac{1}{n}{\displaystyle {\sum }_{k=1}^n}{y}_j\left(k\right)}}$$

(10)

We define the absolute difference as

$${\Delta }_{\mathit{ij}}=\left|x_i^{\prime }(k)-y_j^{\prime }(k)\right|$$

(11)

The minimum absolute difference is defined as

$${\Delta }_{\mathit{min}}=\underset{i}{\mathit{min}} \underset{k}{\mathit{min}} |x_i^{\prime }(k)-y_j^{\prime }(k)|$$

(12)

The maximum absolute difference is defined as

$${\Delta }_{\mathit{max}}=\underset{i}{\mathit{max}} \underset{k}{\mathit{max}} |x_i^{\prime }(k)-y_j^{\prime }(k)|$$

(13)

The correlation coefficient between data sequences $x_j$ and $y_j$ is

$${\xi }_{\mathit{ij}}(k)={\displaystyle \frac{{\Delta }_{\min }+{\Delta }_{\max }\times \xi }{{\Delta }_{\mathit{ij}}+{\Delta }_{\max }\times \xi }}$$

(14)

In the formula, $\xi$ is the distinguishing coefficient, where ξ ∈ (0,1], and is generally taken as ξ ≤ 0.5. The larger $\xi$ is, the higher the resolution; the smaller $\xi$ is, the lower the resolution.

The grey relational degree between data sequences ${\mathit{X}}_i$ and ${\mathit{Y}}_j$ is

$${\gamma }_{\mathit{ji}},{\omega }_{\mathrm{ji}}=\gamma ({\mathit{X}}_i^{\prime },{\mathit{Y}}_j^{\prime })={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{k=1}^n}{\xi }_{\mathit{ij}}(k)$$

(15)

In the formula, ${\gamma }_{\mathit{ji}}$ is the initial-value grey relational degree; and ${\omega }_{\mathit{ji}}$ is the mean-value grey relational degree.

2.3.2. Relative and Absolute Relational Degrees

After the original data are processed by initialization, the sequences of ${\mathit{X}}_i$ and ${\mathit{Y}}_j$ are transformed into

$${\mathit{X}}_i^{\prime }={\displaystyle \frac{X_i}{X_i\left(1\right)}}=\left(x_i^{\prime }\left(1\right),x_i^{\prime }\left(2\right),\dots ,x_i^{\prime }(k),\dots ,x_i^{\prime }(n)\right)$$

(16)

$${\mathit{Y}}_j^{\prime }={\displaystyle \frac{Y_j}{Y_j\left(1\right)}}=(y_j^{\prime }\left(1\right),y_j^{\prime }\left(2\right),\dots ,y_j^{\prime }(k),\dots ,y_j^{\prime }(n))$$

(17)

Then, after performing zero-start processing, the data sequences become

$${\mathit{X}}_i^0=(x_i^0\left(1\right),x_i^0\left(2\right),\dots ,x_i^0(k),\dots ,x_i^0(n))$$

(18)

$${\mathit{Y}}_j^0=(y_j^0\left(1\right),y_j^0\left(2\right),\dots ,y_j^0(k),\dots ,y_j^0(n))$$

(19)

We define the relative relational degree of ${\mathit{X}}_i$ and ${\mathit{Y}}_j$ as

$${\epsilon }_{\mathit{ji}}={\displaystyle \frac{1+\left|s_j\right|+\left|s_i\right|}{1+\left|s_j\right|+\left|s_i\right|+\left|s_j-s_i\right|}}$$

(20)

where $s_i$ and $s_j$ are expressed accordingly as

$$\left|s_i\right|=\left|{\displaystyle {\sum }_{k=2}^{n-1}}{x}_i^0\left(k\right)+0.5x_i^0\left(n\right)\right|$$

(21)

$$\left|s_j\right|=\left|{\displaystyle {\sum }_{k=2}^{n-1}}{y}_j^0\left(k\right)+0.5y_j^0\left(n\right)\right|$$

(22)

$$\left|s_j-s_i\right|=\left|{\displaystyle {\sum }_{k=2}^{n-1}}(y_j^0\left(k\right)-x_i^0\left(k\right))+0.5(y_j^0\left(n\right)-x_i^0\left(n\right))\right|$$

(23)

The calculation process of the absolute relational degree does not require initialization; direct zero-start processing is performed instead, and the remaining calculation steps are consistent with those of the relative relational degree. Therefore, the absolute relational degree can be expressed as

$$e_{\mathit{ji}}={\displaystyle \frac{1+\left|s_j\right|+\left|s_i\right|}{1+\left|s_j\right|+\left|s_i\right|+\left|s_j-s_i\right|}}$$

(24)

2.3.3. Relational Order

In the research process, the specific values of relational degrees are usually not focused on; rather, the relational order comprising each relational degree is the focus. We suppose that sequence Z consisting of the initial grey relational degrees between ${\mathit{X}}_i$ and ${\mathit{Y}}_j$ is

$$\mathit{Z}=\left({\gamma }_{j1},{\gamma }_{j2},\dots ,{\gamma }_{\mathit{ji}},\dots ,{\gamma }_{\mathit{jm}}\right)$$

(25)

By sorting the data values in Z in descending order, the grey relational order can be obtained. In the grey relational order, elements with higher grey relational-degree values (i.e., larger ${\gamma }_{\mathit{ji}}$ values and more left positions in the ranking) indicate a stronger correlation and deeper influence degree between ${\mathit{X}}_i$ and ${\mathit{Y}}_j$; conversely, the correlation is weaker and the influence degree is slighter. The mean-value grey relational degree ${\omega }_{\mathit{ji}}$, relative relational degree ${\epsilon }_{\mathit{ji}}$, and absolute relational degree $e_{\mathit{ji}}$ also follow this relational order rule. Therefore, the ranking order of the influence degrees of different influencing factors on the vibration value can be determined according to the relational orders obtained by different methods.

2.3.4. Application Method of Qualitative-Fusion Principle

As the system information is extremely complex and diverse, different research methods may yield different results, and sometimes even completely opposite conclusions. Therefore, it is crucial to screen for valuable information. If all possible solutions are regarded as a set H, the process of qualitative fusion is to select a subset h₀ with certain consistent characteristics from set H under the guidance of specific criteria as the final solution of the system.

Given the number of factors selected that may affect a certain vibration value of the rolling bearing as m, the symbol set of the influencing factors $\mathit{X}$ can be obtained as

$$\mathit{X}=(X_1, X_2,\dots , X_i, \dots , X_m)$$

(26)

By analyzing the experimental data using q relational-degree methods, q sequences of the influence degree ranking of factors can be obtained as

$$X_l^p>X_{l-1}^p\hspace{1em}l=2,3,\dots ,m\hspace{1em}p=1,2,\dots ,q$$

(27)

In the formula, $\succ$ denotes “is superior to”.

Then, the set of ranking sequences is expressed as

$$X_i^p={\left[X^1,X^2,\dots ,X^p,\dots ,X^q\right]}^T$$

(28)

$$X^p=\left(X_1^p,X_2^p,\dots ,X_i^p,\dots ,X_m^p\right)$$

(29)

From the set of ranking sequences, we select n ≤ m symbols of influencing factors with top positions as the solution set H, namely

$$\mathit{H}={\left[h_1,h_2,\dots ,h_p,\dots ,h_q\right]}^T$$

(30)

Here, $h_p$ is

$$h_p=\left(X_1^p,X_2^p,\dots ,X_i^p,\dots ,X_n^p\right)$$

(31)

Then, the final solution $h_0$ is the set comprising the factors in H, namely

$$h_0={\displaystyle {\bigcap }_{k=1}^4}{h}_k=h_1 \bigcap h_2 \bigcap h_3 \bigcap h_4$$

(32)

If h₀ ≠ Φ, then the qualitative fusion has a unique solution.

In the final solution h₀, the order of the z influencing factors is irrelevant; all of them are key factors affecting bearing vibration. After re-numbering and sorting, we can obtain:

$$h_0=\left(x_1,x_2,\dots ,x_i,\dots x_z\right) \text{⊂} X$$

(33)

where 1 ≤ z ≤ n and n ∈ [0.3m,m]. It can be concluded that there is no new information in the final solution.

In this study, 7208 angular-contact ball bearings produced by Harbin Bearing Co., Ltd. (Harbin, China) were selected as the research objects. Multiple parameters of the inner and outer rings of the bearings, as well as various vibration value parameters, were measured through the experiments. A variety of methods were adopted to explore the correlations between these parameters and different vibration values of the bearings, and the key factors affecting each vibration value were determined through qualitative analysis. Subsequently, further comprehensive analysis was conducted on these key factors to determine the final influencing factors.

3. Experimental Scheme Design

3.1. Experimental Objects and Equipment

In this experimental study, 7208 angular-contact ball bearings produced by Harbin Bearing Co., Ltd. were taken as the research objects. The structure of the angular-contact ball bearing is shown in Figure 1.

The geometrical parameters of the 7208 angular-contact ball bearings are shown in

Table 1. Geometrical parameters of the 7208 angular-contact ball bearings.

Parameter	Symbol	Value	Description
Bore diameter	d	40 mm	Nominal inner diameter
Outside diameter	D	80 mm	Nominal outer diameter
Bearing width	B	18 mm	Overall bearing width
Inner-ring shoulder diameter	d1	53.3 mm	Diameter of the inner-ring shoulder
Inner-ring recess diameter	d2	53.3 mm	Diameter associated with the inner-ring recess
Outer-ring shoulder diameter	D1	66.7 mm	Diameter of the outer-ring shoulder
Minimum inner-ring chamfer dimension	r1,2 min	1.1 mm	Minimum chamfer dimension at the inner-ring edges
Minimum outer-ring chamfer dimension	r3,4 min	0.6 mm	Minimum chamfer dimension at the outer-ring edges
Load-centre distance	a	17.1 mm	Axial distance from the bearing-side face to the load centre

.

The measurement method for vibration acceleration of 7208 angular-contact ball bearings is shown in Figure 2.

Throughout the experiments, a Talyrond 365 roundness measuring instrument (Taylor Hobson Ltd., Leicester, UK) was used to measure various parameters of the inner and outer rings of the bearings, and a BVT-5 bearing vibration measuring instrument (Hangzhou Bearing Research Institute, Hangzhou, China) was used to measure various vibration value parameters, as shown in Figure 3.

The vibration tests were performed at an inner-ring speed of 1800 r/min with a radial load of 150 N applied to the outer ring. During each measurement, the permitted control bands were 1800 ± 30 r/min for speed, 150 ± 5 N for load, and 20 ± 2 °C for ambient temperature; relative humidity remained below 70%. The vibration-velocity bands were 50–300 Hz, 300–1800 Hz, and 1800–10,000 Hz. All 30 bearings were tested in the as-received factory-greased condition, and the lubricant type was not treated as an independent experimental variable. The grease-related information required for reproducible reporting and the nominal operating settings with their specified control limits are summarized in

Table 2. Grease-related information required for reproducible reporting of the bearing vibration tests.

Category	Grease
Identity	Manufacturer, product name, batch; base-oil type; thickener type; NLGI grade
Viscosity	Base-oil kinematic viscosity at 40 and 100 °C (mm2/s); viscosity index
Density	Density and reference temperature
Pressure–viscosity coefficient	α at stated temperature and pressure range
Additives	EP/AW, antioxidant, corrosion inhibitor, solid additives
Application	Fill mass (g), fill fraction (% free volume), placement, re-greasing/cleaning protocol
Conditioning	Run-in duration/speed/load; time to thermal and vibration steady state

and

Table 3. Nominal operating conditions and specified control limits for the bearing vibration tests.

Quantity	Nominal Setting	Specified Control Range	Maximum Permitted Deviation
Rotational speed	1800 r/min	1770–1830 r/min	±30 r/min
Applied radial load	150 N	145–155 N	±5 N
Ambient temperature	20 °C	18–22 °C	±2 °C
Relative humidity	<70% RH	<70% RH	Upper-limit criterion only

, respectively.

Different methods were adopted to study the relationships between various parameters and different vibration values of the bearings. Qualitative fusion was carried out to identify the main influencing factors affecting each vibration value, and then secondary fusion was performed on each main influencing factor to obtain the final influencing factors.

Since both the inner and outer rings of angular-contact ball bearings have raceways, which allow relative movement along the bearing axis, such bearings are particularly suitable for bearing combined loads, i.e., loads acting radially and axially simultaneously. Their axial load-carrying capacity increases with the increase in the contact angle α.

In this experiment, a special experimental platform was constructed to accurately investigate the vibration of angular-contact ball bearings. The device integrates the main components, such as a motor, a loading device, a supporting structure, and a vibration detector. Vibration velocity and vibration acceleration were measured using the BVT-5 bearing vibration measuring instrument developed by Hangzhou Bearing Research Institute, which conforms to international bearing vibration testing standards and features high measurement accuracy and good stability. A roundness measuring instrument was used to measure the geometric parameters of the bearing raceways, whose core components include a high-precision rotating spindle, measuring elements, and a data processing module.

3.2. Experimental Methods and Steps

Thirty 7208 single-row angular-contact ball bearings were randomly selected from the same production batch. Each bearing was assigned a unique identification number, which was retained throughout vibration measurement, disassembly, raceway measurement, and data processing. This coding procedure ensured that the geometric parameters of each inner and outer raceway could be matched with the vibration results obtained from the same bearing. The experimental procedure comprised the following five steps.

Step 1: Bearing vibration measurement. Each bearing was mounted on a BVT-5 bearing vibration tester. During measurement, the outer ring remained stationary, while the inner ring was rotated at a nominal speed of 1800 r min⁻¹. A nominal radial load of 150 N was applied to the outer ring. The nominal ambient temperature was 20 °C, and the relative humidity was maintained below 70% RH. The lubricant information and the nominal operating conditions with their specified control limits are summarized in Table 2 and Table 3, respectively.

After the bearing had been mounted and the instrument reading had stabilized, the vibration-acceleration levels and vibration velocities were recorded. The overall vibration-acceleration level (Y1) was measured directly in decibels using the BVT-5 tester. In accordance with the convention for rolling-bearing vibration-acceleration measurements, the reference root-mean-square acceleration corresponding to 0 dB was 0.00981 m/s².

The root-mean-square vibration velocities were measured in three frequency bands. The low-frequency vibration velocity (Y₂) covered 50–300 Hz; the medium-frequency vibration velocity (Y₃) covered 300–1800 Hz; and the high-frequency vibration velocity (Y₄) covered 1800–10,000 Hz. Thus, one vibration-acceleration indicator and three band-limited vibration-velocity indicators were obtained for each bearing.

Step 2: Bearing disassembly and component identification. After completion of the vibration measurements, each bearing was carefully removed from the tester and disassembled. The inner ring, outer ring, rolling elements, and cage were marked using the identification number assigned to the complete bearing. Components belonging to different bearings were stored separately to prevent specimen mismatch.

During disassembly, direct mechanical contact with the functional raceway surfaces was avoided. The components were protected against scratching, contamination, and corrosion during handling and storage. No grinding, polishing, or other surface-modification operations were performed before the raceway measurements. This procedure preserved the manufactured surface conditions and ensured that the measured raceway parameters corresponded to the bearing conditions during the preceding vibration tests.

Step 3: Raceway roundness and waviness measurement. The circumferential profiles of the inner and outer raceways were measured using the contact-type roundness and waviness measurement system shown in Figure 4. Before each measurement, the bearing ring was mounted on the rotary table. The ring was carefully centred and levelled to minimize eccentricity and inclination between the ring axis and the rotational axis of the instrument.

The contact stylus was positioned on the functional raceway track corresponding approximately to the ball–raceway contact path. The rotary table was then driven through one complete revolution to acquire a closed circumferential profile. The instrument’s analysis system removed the mounting-related eccentricity and inclination components and evaluated the circumferential form of the measured raceway.

Three repeated measurements were conducted for each reported raceway parameter, and the arithmetic mean of the three readings was used as the representative value. The inner-raceway roundness and waviness were denoted by (X₁) and (X₂), respectively. The outer-raceway roundness and waviness were denoted by (X₅) and (X₆), respectively. All four quantities were expressed in micrometres.

Step 4: Raceway groove-profile and surface-roughness measurement. The cross-sectional groove profiles and surface roughness of the inner and outer raceways were measured using the contact-type profile measurement system shown in Figure 5. Before measurement, each ring was securely positioned on the instrument stage, and the stylus was aligned with the functional raceway surface.

For groove-profile measurement, the stylus traversed the raceway in the axial direction to acquire a cross-sectional trace of the complete groove. The measured trace was evaluated relative to the reference groove geometry implemented in the instrument analysis system. The resulting inner- and outer-raceway groove-profile deviations were denoted by (X₃) and (X₇), respectively.

Surface roughness was subsequently measured on the functional surface of each raceway using the contact-stylus method. Three repeated measurements were performed for each raceway, and their arithmetic mean was used in the subsequent analysis. The inner- and outer-raceway roughness values were denoted by (X₄) and (X₈), respectively. The groove-profile deviations and surface-roughness values were expressed in micrometres.

Step 5: Data verification and specimen-level matching. After completion of the measurements, the data were checked for specimen identification, completeness, unit consistency, and obvious acquisition errors. When an incomplete profile or an instrument acquisition failure occurred, the corresponding measurement was repeated rather than removed on the basis of its numerical magnitude. No bearing was excluded merely because it exhibited a relatively high or low geometric or vibration value.

For each bearing, the arithmetic means of the three repeated measurements of (X₁)–(X₈) were used as the representative raceway parameters. These eight geometric parameters were then matched with the four vibration indicators (Y₁)–(Y₄) using the unique bearing identification number. All 30 bearings had complete matched records and were retained in the final dataset. The resulting specimen-level dataset contained 30 observations for each of the eight raceway parameters and four vibration indicators and was subsequently used in the grey relational analysis.

During the measurement process, each parameter was measured three times repeatedly, and the average value was taken to reduce the impact of random errors. At the same time, a validity test was performed on the measurement data to eliminate abnormal data and ensure the rationality and usability of the data.

3.3. Parameter Definition and Data Acquisition

Eight bearing ring parameters were selected as the research objects, including four inner-ring parameters and four outer-ring parameters. For the convenience of research, the symbols and their meanings used in this experiment are listed in

Table 4. Symbols and their definitions.

Symbol	Definition	Component	Symbol	Definition	Component
X1/μm	Roundness of inner raceway	Inner ring	X7/μm	Groove shape of outer raceway	Outer ring
X2/μm	Waviness of inner raceway	Inner ring	X8/μm	Roughness of outer raceway	Outer ring
X3/μm	Groove shape of inner raceway	Inner ring	Y1/dB	RMS of vibration acceleration	Whole bearing
X4/μm	Roughness of inner raceway	Inner ring	Y2/(μm/s)	RMS of low-frequency vibration velocity	Whole bearing
X5/μm	Roundness of outer raceway	Outer ring	Y3/(μm/s)	RMS of medium-frequency vibration velocity	Whole bearing
X6/μm	Waviness of outer raceway	Outer ring	Y4/(μm/s)	RMS of high-frequency vibration velocity	Whole bearing

.

The measured values of various bearing parameters are shown in

Table 5. Measured values of bearing parameters.

Number	Measured Values/μm
	X1	X2	X3	X4	X5	X6	X7	X8
1	0.18	0.06	0.66	0.04	1.15	0.08	0.90	0.04
2	0.21	0.06	0.78	0.03	0.87	0.07	0.82	0.04
3	0.24	0.06	0.67	0.03	0.48	0.07	0.67	0.03
4	0.27	0.06	0.88	0.03	1.10	0.08	0.73	0.03
5	0.17	0.06	0.71	0.04	0.71	0.07	0.73	0.03
6	0.28	0.05	0.85	0.03	0.71	0.07	1.00	0.03
7	0.34	0.07	0.78	0.03	0.44	0.07	0.90	0.03
8	0.17	0.06	0.68	0.03	0.61	0.07	1.07	0.01
9	0.14	0.05	0.69	0.03	0.63	0.06	1.04	0.04
10	0.18	0.06	0.93	0.03	0.56	0.07	0.71	0.03
11	0.14	0.05	0.92	0.03	0.66	0.08	0.70	0.04
12	0.18	0.06	0.78	0.02	0.90	0.06	0.69	0.03
13	0.59	0.10	0.73	0.03	0.54	0.06	0.71	0.03
14	0.25	0.06	0.74	0.03	0.30	0.05	0.82	0.03
15	0.18	0.06	0.53	0.03	0.30	0.08	0.67	0.03
16	0.36	0.06	0.94	0.03	0.48	0.05	0.65	0.03
17	0.29	0.08	0.82	0.03	0.74	0.10	0.51	0.03
18	0.23	0.08	0.91	0.03	0.67	0.08	0.69	0.03
19	0.46	0.07	0.87	0.04	0.75	0.08	0.49	0.03
20	0.36	0.08	0.74	0.03	0.34	0.05	0.67	0.02
21	0.26	0.08	0.76	0.03	0.46	0.06	0.88	0.03
22	0.18	0.05	0.70	0.03	0.89	0.08	0.70	0.03
23	0.21	0.06	0.68	0.03	0.51	0.06	0.38	0.03
24	0.14	0.08	0.87	0.03	0.67	0.05	0.72	0.03
25	0.13	0.05	0.85	0.03	0.56	0.07	0.85	0.04
26	0.20	0.05	0.68	0.03	0.85	0.10	0.68	0.03
27	0.21	0.05	0.92	0.02	0.97	0.07	0.73	0.03
28	0.33	0.08	0.73	0.03	0.86	0.08	0.53	0.03
29	0.21	0.06	0.93	0.02	0.71	0.08	0.62	0.03
30	0.17	0.06	0.87	0.03	0.36	0.06	0.79	0.03

.

The measured values of various influencing factor parameters of the bearing inner and outer rings are shown in Figure 6.

It can be seen intuitively from Figure 6 that there are obvious differences in various parameters among different bearings, which provides a sound data foundation for the subsequent correlation analysis.

The measured values of various bearing vibration data are shown in

Table 6. Measured values of various vibration data of bearings.

Number	Measured Values				Number	Measured Values
	Y1/dB	Y2/(μm/s)	Y3/(μm/s)	Y4/(μm/s)		Y1/dB	Y2/(μm/s)	Y3/(μm/s)	Y4/(μm/s)
1	40	110	20	50	16	39	30	50	45
2	37	120	20	20	17	39	50	20	30
3	38	110	20	40	18	40	150	20	50
4	40	60	20	40	19	42	30	20	50
5	41	100	20	40	20	38	90	30	30
6	39	100	20	30	21	39	120	20	30
7	39	50	30	40	22	38	20	20	30
8	40	50	20	50	23	39	110	20	50
9	39	100	20	30	24	38	110	20	40
10	39	50	20	30	25	38	80	20	40
11	37	100	20	30	26	40	50	20	50
12	38	100	40	30	27	40	4	18	50
13	38	40	10	40	28	37	110	20	40
14	39	90	20	30	29	39	110	20	40
15	37	50	50	40	30	39	20	20	40

.

It can be seen from the table that the vibration values of different bearings exhibit large discreteness, which reflects the significant influence of manufacturing errors on vibration performance.

4. Experimental Analysis

4.1. Correlation Analysis of Influencing Factors on Vibration Acceleration

To investigate the influence degrees of each geometric parameter on vibration acceleration, four correlation methods were first used to process the measured data. Each point in Figure 7, Figure 8, Figure 9 and Figure 10 represents a single grey relational grade calculated from the complete 30-bearing dataset rather than the mean of independently replicated grade estimates; therefore, conventional standard-deviation error bars are not applicable.

4.1.1. Initialization Grey Relational Analysis

The original data were initialized, and the results are shown in

Table 7. Initialization data.

Number	Initialization Data
	ξ11	ξ21	ξ31	ξ41	ξ51	ξ61	ξ71	ξ81
1	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
2	0.83	0.83	0.47	0.66	0.68	0.78	0.94	0.95
3	0.75	0.88	0.78	0.56	0.40	0.70	0.58	0.69
4	0.70	1.00	0.41	0.52	0.89	1.00	0.59	0.57
5	0.94	0.93	0.81	0.85	0.47	0.54	0.56	0.63
6	0.67	0.72	0.42	0.52	0.50	0.64	0.69	0.60
7	0.56	0.65	0.53	0.45	0.38	0.64	0.95	0.60
8	0.95	1.00	0.84	0.42	0.43	0.58	0.61	0.33
9	0.86	0.72	0.76	0.56	0.46	0.44	0.61	0.88
10	0.98	0.93	0.35	0.68	0.42	0.64	0.59	0.66
11	0.89	0.80	0.33	0.73	0.50	0.70	0.65	0.95
12	0.96	0.88	0.50	0.39	0.68	0.47	0.60	0.81
13	0.33	0.33	0.59	0.48	0.43	0.47	0.63	0.72
14	0.74	0.93	0.62	0.98	0.33	0.33	0.80	0.69
15	0.94	0.83	0.66	0.73	0.35	0.70	0.61	0.76
16	0.53	0.93	0.34	0.85	0.39	0.33	0.52	0.66
17	0.65	0.50	0.46	0.45	0.52	0.39	0.40	0.73
18	0.81	0.52	0.38	0.52	0.46	1.00	0.54	0.60
19	0.44	0.75	0.47	0.51	0.47	0.78	0.35	0.60
20	0.53	0.48	0.58	0.74	0.35	0.35	0.58	0.53
21	0.71	0.50	0.56	0.45	0.38	0.44	1.00	0.73
22	0.96	0.75	0.67	0.61	0.67	0.78	0.61	0.69
23	0.86	0.93	0.78	0.61	0.40	0.44	0.33	0.55
24	0.87	0.48	0.38	0.67	0.49	0.35	0.64	0.92
25	0.84	0.75	0.41	0.74	0.44	0.70	0.98	0.83
26	0.91	0.68	0.87	0.42	0.58	0.41	0.53	0.69
27	0.87	0.68	0.37	0.35	0.70	0.58	0.59	0.55
28	0.56	0.47	0.55	0.73	0.67	0.70	0.45	0.98
29	0.86	0.93	0.35	0.33	0.50	0.88	0.49	0.60
30	0.97	0.93	0.40	0.56	0.35	0.44	0.73	0.73

.

Taking the resolution coefficient ξ = 0.5, the initialization grey relational degrees between each influencing factor parameter and vibration-acceleration value are shown in Figure 7.

Figure 7 presents the initial-value-normalized grey relational grades. The three leading descriptors are X1, X2, and X8 for Y1; X1, X2, and X4 for Y2; X6, X2, and X1 for Y3; and X1, X8, and X6 for Y4. X1 ranks among the top three for all four responses, while the other leading descriptors vary with responses. These rankings represent method-specific sequence associations; the final retained descriptors are determined by the subsequent qualitative-fusion analysis.

4.1.2. Calculation Results of Mean Grey Relational Degree

The original data were normalized by their mean values, and the results are shown in

Table 8. Mean-value data.

Number	Mean-Value
	ξ11	ξ21	ξ31	ξ41	ξ51	ξ61	ξ71	ξ81
1	0.73	0.80	0.45	0.57	0.35	0.68	0.55	0.62
2	0.91	1.00	0.84	0.89	0.51	0.86	0.60	0.61
3	1.00	0.92	0.56	0.96	0.61	0.96	0.80	1.00
4	0.91	0.80	0.67	0.89	0.37	0.68	0.87	0.75
5	0.69	0.75	0.52	0.51	0.98	0.81	0.79	0.89
6	0.84	0.59	0.69	0.86	0.86	1.00	0.41	0.79
7	0.66	0.78	1.00	0.67	0.53	1.00	0.53	0.79
8	0.70	0.80	0.51	0.62	0.80	0.89	0.36	0.34
9	0.64	0.59	0.56	1.00	0.91	0.60	0.37	0.56
10	0.75	0.86	0.47	0.83	0.72	1.00	0.87	0.93
11	0.67	0.66	0.43	0.77	0.91	0.55	1.00	0.61
12	0.77	0.92	0.98	0.54	0.50	0.65	0.88	0.79
13	0.34	0.34	0.81	0.73	0.72	0.65	0.96	0.93
14	0.98	0.86	0.73	0.59	0.41	0.43	0.69	1.00
15	0.79	1.00	0.36	0.77	0.43	0.55	0.88	0.87
16	0.61	0.86	0.46	0.65	0.59	0.43	0.67	0.93
17	0.80	0.56	0.83	0.67	0.78	0.35	0.44	0.91
18	0.92	0.58	0.58	0.89	1.00	0.68	0.74	0.80
19	0.48	0.97	0.94	0.35	0.89	0.82	0.37	0.83
20	0.60	0.53	0.83	0.75	0.45	0.46	0.81	0.63
21	0.93	0.56	0.86	0.67	0.56	0.60	0.55	0.91
22	0.77	0.63	0.66	1.00	0.51	0.59	0.89	1.00
23	0.86	0.86	0.55	0.96	0.63	0.60	0.33	0.69
24	0.66	0.53	0.56	0.86	0.94	0.46	0.99	0.68
25	0.64	0.63	0.63	0.75	0.76	0.96	0.57	0.54
26	0.80	0.57	0.50	0.62	0.60	0.36	0.70	0.98
27	0.84	0.57	0.54	0.47	0.46	0.89	0.88	0.70
28	0.65	0.51	0.96	0.77	0.52	0.55	0.51	0.65
29	0.86	0.86	0.47	0.43	0.86	0.63	0.60	0.79
30	0.72	0.86	0.62	1.00	0.45	0.60	0.78	0.91

.

Taking the resolution coefficient ξ = 0.5, the mean grey relational degrees between each influencing factor parameter and vibration-acceleration value are shown in Figure 8.

Figure 8 presents the mean-value-normalized grey relational grades. The three leading descriptors are X8, X1, and X4 for Y1; X1, X2, and X5 for Y2; X1, X3, and X2 for Y3; and X6, X1, and X8 for Y4. Compared with Figure 7, changes in the leading descriptors and their rankings demonstrate the sensitivity of the method-specific sequence associations to data preprocessing.

4.1.3. Calculation Results of Relative Relational Degree

The data of relative relational degrees are shown in

Table 9. Relative relational-degree data.

Number	Value
	y1 − x1	y1 − x2	y1 − x3	y1 − x4	y1 − x5	y1 − x6	y1 − x7	y1 − x8
1	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
2	−0.24	−0.08	−0.27	0.10	0.17	0.05	0.02	−0.02
3	−0.38	−0.05	−0.07	0.15	0.53	0.08	0.21	0.17
4	−0.50	0.00	−0.34	0.17	0.04	0.00	0.19	0.27
5	0.08	0.03	−0.06	−0.03	0.41	0.15	0.22	0.21
6	−0.58	0.14	−0.33	0.18	0.36	0.10	−0.13	0.25
7	−0.91	−0.19	−0.22	0.23	0.59	0.10	−0.02	0.25
8	0.06	0.00	−0.04	0.26	0.47	0.13	−0.18	0.73
9	0.20	0.14	−0.07	0.15	0.43	0.23	−0.18	−0.05
10	−0.03	−0.03	−0.45	0.09	0.49	0.10	0.19	0.19
11	0.15	0.09	−0.48	0.07	0.35	−0.08	0.15	−0.02
12	−0.05	−0.05	−0.24	0.29	0.17	0.20	0.18	0.09
13	−2.33	−0.72	−0.17	0.21	0.48	0.20	0.17	0.14
14	−0.41	−0.03	−0.15	0.00	0.71	0.35	0.07	0.16
15	−0.08	−0.08	0.12	0.07	0.66	−0.08	0.18	0.11
16	−1.03	−0.03	−0.46	0.03	0.56	0.35	0.26	0.19
17	−0.64	−0.36	−0.28	0.23	0.33	−0.28	0.42	0.14
18	−0.28	−0.33	−0.38	0.17	0.42	0.00	0.24	0.24
19	−1.51	−0.12	−0.27	−0.18	0.40	0.05	0.51	0.24
20	−1.05	−0.38	−0.17	0.06	0.65	0.33	0.20	0.33
21	−0.47	−0.36	−0.19	0.23	0.58	0.23	0.00	0.14
22	−0.05	0.12	−0.12	0.12	0.18	−0.05	0.18	0.17
23	−0.19	−0.03	−0.07	0.12	0.53	0.23	0.56	0.30
24	0.17	−0.38	−0.38	0.09	0.37	0.33	0.16	0.03
25	0.23	0.12	−0.35	0.06	0.46	0.08	0.01	−0.08
26	−0.11	0.17	−0.04	0.26	0.26	−0.25	0.25	0.16
27	−0.17	0.17	−0.41	0.34	0.16	0.13	0.19	0.30
28	−0.91	−0.41	−0.20	0.07	0.18	−0.08	0.34	0.01
29	−0.19	−0.03	−0.45	0.38	0.36	−0.03	0.29	0.25
30	0.03	−0.03	−0.35	0.15	0.66	0.23	0.10	0.14

.

The relative relational degrees between each influencing factor parameter and vibration-acceleration value are shown in Figure 9.

Figure 9 presents the relative grey relational grades between the eight raceway surface descriptors and the four vibration responses. The three leading descriptors are X2, X6, and X4 for Y1; X8, X7, and X4 for both Y2 and Y4; and X3, X2, and X1 for Y3. These rankings indicate that the descriptors showing the closest relative sequence-change patterns vary among the vibration responses, although Y2 and Y4 exhibit the same leading order.

4.1.4. Calculation Results of Absolute Relational Degree

The data of absolute relational degrees are shown in

Table 10. Absolute relational-degree data.

Number	Value
	y1 − x1	y1 − x2	y1 − x3	y1 − x4	y1 − x5	y1 − x6	y1 − x7	y1 − x8
1	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
2	36.79	36.94	36.22	36.97	36.13	36.93	36.18	36.97
3	37.76	37.94	37.33	37.97	37.52	37.93	37.33	37.97
4	39.73	39.94	39.12	39.97	38.90	39.92	39.27	39.97
5	40.83	40.94	40.29	40.96	40.29	40.93	40.27	40.97
6	38.72	38.95	38.15	38.97	38.29	38.93	38.00	38.97
7	38.66	38.93	38.22	38.97	38.56	38.93	38.11	38.97
8	39.83	39.94	39.32	39.97	39.39	39.93	38.93	39.99
9	38.86	38.95	38.31	38.97	38.37	38.94	37.96	38.96
10	38.82	38.94	38.07	38.97	38.44	38.93	38.29	38.97
11	36.86	36.95	36.08	36.97	36.34	36.92	36.30	36.97
12	37.82	37.94	37.22	37.98	37.10	37.94	37.31	37.97
13	37.41	37.90	37.27	37.97	37.46	37.94	37.29	37.97
14	38.75	38.94	38.26	38.97	38.70	38.95	38.18	38.97
15	36.82	36.94	36.47	36.97	36.70	36.92	36.33	36.97
16	38.64	38.94	38.06	38.97	38.52	38.95	38.35	38.97
17	38.71	38.92	38.18	38.97	38.26	38.90	38.49	38.97
18	39.77	39.92	39.09	39.97	39.33	39.92	39.31	39.97
19	41.54	41.93	41.13	41.96	41.25	41.92	41.51	41.97
20	37.64	37.92	37.27	37.97	37.66	37.95	37.33	37.98
21	38.74	38.92	38.24	38.97	38.54	38.94	38.12	38.97
22	37.82	37.95	37.30	37.97	37.11	37.92	37.30	37.97
23	38.79	38.94	38.32	38.97	38.49	38.94	38.62	38.98
24	37.86	37.92	37.13	37.97	37.33	37.95	37.29	37.97
25	37.87	37.95	37.15	37.97	37.44	37.93	37.15	37.96
26	39.80	39.95	39.32	39.97	39.15	39.90	39.32	39.97
27	39.79	39.95	39.08	39.98	39.03	39.93	39.27	39.97
28	36.67	36.92	36.27	36.97	36.14	36.92	36.48	36.97
29	38.79	38.94	38.07	38.98	38.29	38.92	38.39	38.97
30	38.83	38.94	38.13	38.97	38.64	38.94	38.21	38.97

.

The absolute relational degrees between each influencing factor parameter and vibration-acceleration value are shown in Figure 10.

Figure 10 presents the absolute grey relational grades between the eight raceway surface descriptors and the four vibration responses. All four responses yield the same descending order: X3, X7, X5, X1, X6, X2, X4, and X8. The grades are closely clustered near 0.5 and are sensitive to the numerical scale and overall level of the original sequences. Therefore, the absolute-grade rankings should be interpreted together with the results obtained using the other three GRA schemes.

4.2. Ranking and Comparative Analysis of Correlation Degrees by Multiple Methods

The influencing factors of vibration acceleration obtained by various methods are ranked according to different correlation degrees, and four sequences with correlation from strong to weak are obtained, as shown in

Table 11. Influencing factors of vibration-acceleration values.

Method	Influencing Factors
Initial-value grey relational degree	X1, X2, X8, X7, X6, X4, X3, X5
Mean-value grey relational degree	X8, X1, X4, X2, X7, X6, X5, X3
Relative relational degree	X2, X6, X4, X7, X8, X3, X1, X5
Absolute relational degree	X3, X7, X5, X1, X6, X2, X4, X8

.

Table 11 summarizes the method-specific rankings of the eight raceway surface descriptors for vibration-acceleration level (Y1). The three leading descriptors are X1, X2, and X8 under the initial-value-normalized scheme; X8, X1, and X4 under the mean-value-normalized scheme; X2, X6, and X4 under the relative scheme; and X3, X7, and X5 under the absolute scheme. According to the predefined qualitative-fusion rule, which retains a descriptor when it ranks among the top three under at least two schemes, X1, X2, X4, and X8 are retained for Y1. The differences among the method-specific rankings demonstrate sensitivity to data preprocessing and relational-degree formulation.

4.3. Analysis of Influencing Factors of Vibration Velocity in Different Frequency Bands

The same method was adopted to analyze the influencing factors of vibration velocity in the low-, medium-, and high- frequency bands, and the correlation degree rankings of each frequency band were obtained, which are shown in

Table 12. Influencing factors of low-frequency vibration velocity.

Method	Influencing Factors
Initial-value grey relational degree	X1, X2, X4, X3, X6, X8, X5, X7
Mean-value grey relational degree	X1, X2, X5, X7, X3, X4, X8, X6
Relative grey relational degree	X8, X7, X4, X5, X6, X2, X3, X1
Absolute grey relational degree	X3, X7, X5, X1, X6, X2, X4, X8

,

Table 13. Influencing factors of medium-frequency vibration velocity.

Method	Influencing Factors
Initial-value grey relational degree	X6, X2, X1, X8, X4, X3, X7, X5
Mean-value grey relational degree	X1, X3, X2, X6, X7, X8, X4, X5
Relative grey relational degree	X3, X2, X1, X6, X4, X7, X8, X5
Absolute grey relational degree	X3, X7, X5, X1, X6, X2, X4, X8

, and

Table 14. Influencing factors of high-frequency vibration velocity.

Method	Influencing Factors
Initial-value grey relational degree	X1, X8, X6, X7, X2, X4, X5, X3
Mean-value grey relational degree	X6, X1, X8, X7, X4, X3, X5, X2
Relative grey relational degree	X8, X7, X4, X6, X5, X2, X3, X1
Absolute grey relational degree	X3, X7, X5, X1, X6, X2, X4, X8

respectively.

Table 12 summarizes the method-specific rankings of the eight raceway surface descriptors for low-frequency vibration velocity (Y2). The three leading descriptors are X1, X2, and X4 under the initial-value-normalized scheme; X1, X2, and X5 under the mean-value-normalized scheme; X8, X7, and X4 under the relative scheme; and X3, X7, and X5 under the absolute scheme. According to the predefined qualitative-fusion rule, X1 and X2 are retained through the initial-value-normalized and mean-value-normalized rankings, X4 through the initial-value-normalized and relative rankings, X5 through the mean-value-normalized and absolute rankings, and X7 through the relative and absolute rankings. Therefore, X1, X2, X4, X5, and X7 are retained for Y2. The differences among the rankings demonstrate the sensitivity of the method-specific sequence associations to data preprocessing and relational-degree formulation.

Table 13 summarizes the method-specific rankings of the eight raceway surface descriptors for medium-frequency vibration velocity (Y3). The three leading descriptors are X6, X2, and X1 under the initial-value-normalized scheme; X1, X3, and X2 under the mean-value-normalized scheme; X3, X2, and X1 under the relative scheme; and X3, X7, and X5 under the absolute scheme. According to the predefined qualitative-fusion rule, X1 and X2 are each retained through the initial-value-normalized, mean-value-normalized, and relative rankings, while X3 is retained through the mean-value-normalized, relative, and absolute rankings. Therefore, X1, X2, and X3 are retained for Y3. Although X6 ranks first under the initial-value-normalized scheme, it does not enter the top three under any other scheme and is therefore not retained for this response.

Table 14 summarizes the method-specific rankings of the eight raceway surface descriptors for high-frequency vibration velocity (Y4). The three leading descriptors are X1, X8, and X6 under the initial-value-normalized scheme; X6, X1, and X8 under the mean-value-normalized scheme; X8, X7, and X4 under the relative scheme; and X3, X7, and X5 under the absolute scheme. According to the predefined qualitative-fusion rule, X1 and X6 are retained through the initial-value-normalized and mean-value-normalized rankings, X7 through the relative and absolute rankings, and X8 through the initial-value-normalized, mean-value-normalized, and relative rankings. Therefore, X1, X6, X7, and X8 are retained for Y4. The differences among the method-specific rankings demonstrate sensitivity to data preprocessing and relational-degree formulation.

4.4. Identification of Key Factors Based on Qualitative Fusion

To reduce reliance on any single GRA scheme, a predefined qualitative-fusion rule was applied: a surface descriptor was retained for a given vibration response when it ranked among the top three under at least two of the four schemes. The resulting retained descriptor sets are summarized in

Table 15. Influencing factors of different vibration velocities and accelerations.

Type of Vibration Value	Influencing Factors
Vibration acceleration	X1, X2, X4, X8
Low-frequency vibration velocity	X1, X2, X4, X5, X7
Medium-frequency vibration velocity	X1, X2, X3
High-frequency vibration velocity	X1, X6, X7, X8

.

As summarized in Table 15, the retained descriptor sets differ among the four vibration responses, indicating response- and frequency-dependent sequence associations between the measured raceway characteristics and bearing vibration. Inner-raceway roundness (X1) is retained for all four responses and therefore exhibits the broadest cross-response recurrence under the adopted rule. Inner-raceway waviness (X2) is retained for vibration acceleration and for the low- and medium-frequency vibration velocities.

The three vibration-velocity bands also exhibit different retained descriptor combinations. The low-frequency response is associated with both inner- and outer-raceway descriptors (X1, X2, X4, X5, and X7); the medium-frequency response is associated only with inner-raceway descriptors (X1, X2, and X3); and the high-frequency response is associated with one inner-raceway descriptor and three outer-raceway descriptors (X1, X6, X7, and X8). These differences describe the composition of the retained sets and should not be interpreted as evidence of causal influence or effect magnitude.

Because the retained descriptor sets differ among the three vibration-velocity frequency bands, their cross-band recurrence was compared. The number of frequency bands for which each descriptor was retained is summarized in

Table 16. Number of vibration velocity and acceleration items affected by each factor.

No.	Influencing Factor	Number of Affected Frequency Bands
1	X1	3
2	X2, X7	2
3	X3, X4, X5, X6, X8	1

.

Table 16 shows that inner-raceway roundness (X1) is retained for all three vibration-velocity frequency bands and therefore exhibits the broadest cross-band recurrence. Inner-raceway waviness (X2) and outer-raceway groove-profile deviation (X7) are each retained for two bands. Inner-raceway groove-profile deviation (X3), inner-raceway roughness (X4), outer-raceway roundness (X5), outer-raceway waviness (X6), and outer-raceway roughness (X8) are each retained for one band, indicating more response-specific associations. These counts represent recurrence across the investigated frequency bands rather than the magnitude or causal effect of any descriptor.

4.5. Comprehensive Analysis of Influencing Factors

Because the retained descriptor sets differ among the four vibration responses, their cross-response recurrence was further summarized to evaluate the breadth of their associations. The results are presented in

Table 17. Fusion results of main influencing factors for four vibration values.

No.	Influencing Factor	Number of Affected Vibration Values	Influence Degree
1	Inner-raceway roundness	4	Most important
2	Inner-raceway waviness	3	Second-most important
3	Inner-raceway roughness, outer-raceway groove- profile deviation, outer-raceway roughness	2	Third-most important
4	Inner-raceway groove-profile deviation, outer-raceway roundness, outer-raceway waviness	1	Fourth-most important

.

Table 17 shows that inner-raceway roundness (X1) is retained for all four vibration responses and therefore exhibits the broadest cross-response recurrence under the adopted qualitative-fusion rule. Inner-raceway waviness (X2) is retained for three responses. Inner-raceway roughness (X4), outer-raceway groove-profile deviation (X7), and outer-raceway roughness (X8) are each retained for two responses, whereas inner-raceway groove-profile deviation (X3), outer-raceway roundness (X5), and outer-raceway waviness (X6) are each retained for one response and therefore exhibit greater response specificity.

These results provide a dataset-specific basis for prioritizing X1 and X2 in subsequent raceway-metrology and controlled-validation studies, while the other descriptors should be considered according to the vibration response of interest. The recurrence categories describe the breadth and consistency of association across the four responses; they do not represent effect magnitude, statistical significance, causal importance, or a universal hierarchy of manufacturing priorities.

4.6. Physical Mechanism Explanation of the Experimental Results

The GRA and qualitative-fusion results reveal response-dependent associations between the measured raceway characteristics and bearing vibration. These associations can be interpreted from the perspectives of rolling-contact mechanics and tribology. However, GRA quantifies similarity between data sequences and does not establish causality. Moreover, contact force, contact stiffness, friction, lubricant-film thickness, and traction were not measured directly. The following interpretations should therefore be regarded as physically plausible explanations of the observed associations rather than as experimentally verified mechanisms.

Inner-raceway roundness and waviness (X1 and X2): Inner-raceway roundness (X1) was retained for all four vibration responses and therefore exhibited the broadest cross-response association under the adopted qualitative-fusion rule. Inner-raceway waviness (X2) was retained for vibration-acceleration level (Y1) and for the low- and medium-frequency vibration velocities (Y2 and Y3). Because the inner ring rotated during the tests, its circumferential form deviations moved relative to the loaded contact region. Roundness and waviness deviations may therefore introduce periodic variations in ball–raceway contact geometry, contact position, and load distribution. For waviness, the resulting excitation characteristics may also depend on waviness order and bearing kinematics. These effects provide a plausible explanation for the recurrence of X1 and X2 across multiple vibration responses, although the corresponding excitation orders and contact-force variations were not measured directly.

Groove-profile deviations (X3 and X7): Inner-raceway groove-profile deviation (X3) was retained only for the medium-frequency vibration velocity (Y3), whereas outer-raceway groove-profile deviation (X7) was retained for the low- and high-frequency vibration velocities (Y2 and Y4). Groove-profile deviations may alter ball–raceway conformity, local contact geometry, and the effective contact angle. These changes may redistribute the contact load and modify the effective contact-stiffness distribution, thereby affecting vibration transmission. The different response associations of X3 and X7 may reflect differences between the kinematic roles of the rotating inner raceway and stationary outer raceway. Nevertheless, the contact angle, contact force, conformity, and contact stiffness were not measured directly, and the present results do not establish a unique frequency-domain mechanism.

Raceway roughness (X4 and X8): Inner-raceway roughness (X4) was retained for vibration-acceleration level (Y1) and low-frequency vibration velocity (Y2), whereas outer-raceway roughness (X8) was retained for vibration-acceleration level (Y1) and high-frequency vibration velocity (Y4). Raceway roughness represents relatively short-wavelength surface irregularities and may influence local asperity interactions, interfacial friction, grease-film behaviour, and local fluctuations in rolling-contact conditions. These effects provide plausible pathways through which roughness may be associated with bearing vibration. However, the lubrication regime was not determined from direct film-thickness or traction measurements, and the present data do not demonstrate why the inner- and outer-raceway roughness descriptors were retained for different velocity-frequency bands.

Outer-raceway roundness and waviness (X5 and X6): Outer-raceway roundness (X5) and outer-raceway waviness (X6) showed response-specific associations with low-frequency vibration velocity (Y2) and high-frequency vibration velocity (Y4), respectively. Because the outer ring remained stationary during the tests, its circumferential form deviations constituted spatial variations in raceway geometry. As the rolling elements successively traversed these circumferential positions, the deviations may have produced variations in contact geometry and local load distribution. Such spatially repeated contact variations provide a plausible explanation for the observed associations. Their frequency characteristics may depend on the spatial order of the deviations and bearing kinematics, but these relationships were not evaluated directly in the present study.

Overall, the results indicate that different raceway characteristics exhibit distinct response- and frequency-dependent association patterns. Roundness and waviness deviations may affect circumferential contact geometry, load distribution, and periodic variations in rolling-contact conditions; groove-profile deviations may modify ball–raceway conformity and contact-stiffness distribution; and roughness may influence asperity interactions, interfacial friction, and grease-film behaviour. These interpretations are physically compatible with the GRA rankings and qualitative-fusion results but should not be regarded as direct evidence of causality. Controlled experiments that independently vary individual raceway.

4.7. Limitations

This study is limited to one bearing type, one production batch, one nominal speed–load condition, and as-received factory-greased conditions. The lubricant formulation was not varied as an experimental factor. Consequently, the present results cannot be generalized directly to other grease formulations or lubrication conditions. The modest sample size and observational manufacturing variability also mean that the rankings should be interpreted as screening evidence rather than causal effect estimates. Future work should use fully characterized greases, directly measure lubricant-film thickness, and validate the findings under different speeds, loads, temperatures, and preload conditions.

5. Conclusions

This study investigated the associations between eight inner- and outer-raceway geometric and surface descriptors and four vibration responses of thirty production 7208 angular-contact ball bearings from one manufacturing batch. All bearings were tested in the as-received, factory-greased condition at a nominal rotational speed of 1800 r min⁻¹ with an applied radial load of 150 N. Four grey relational analysis (GRA) schemes—initial-value-normalized, mean-value-normalized, relative, and absolute grey relational degrees—were applied to assess the sensitivity of descriptor rankings to data preprocessing and relational-degree formulation. Under the qualitative-fusion rule adopted in this study, a descriptor was retained for a given response when it ranked among the top three under at least two of the four schemes. Based on the method-specific rankings and qualitative-fusion results presented in Table 11, Table 12, Table 13, Table 14, Table 15, Table 16 and Table 17, the following conclusions are drawn:

(1)

For the vibration-acceleration level (Y1), inner-raceway roundness (X1), inner-raceway waviness (X2), inner-raceway roughness (X4), and outer-raceway roughness (X8) were retained. Their recurrence among the leading descriptors under multiple GRA schemes indicates comparatively consistent sequence associations with the vibration-acceleration response under the tested operating and lubrication conditions. This result represents screening evidence and does not establish statistical significance, causal effects, or universal manufacturing-tolerance limits.

(2)

The retained descriptors differed among the three vibration-velocity frequency bands. For the low-frequency vibration velocity (Y2, 50–300 Hz), X1, X2, X4, outer-raceway roundness (X5), and outer-raceway groove-profile deviation (X7) were retained. For the medium-frequency vibration velocity (Y3, 300–1800 Hz), X1, X2, and inner-raceway groove-profile deviation (X3) were retained. For the high-frequency vibration velocity (Y4, 1800–10,000 Hz), X1, outer-raceway waviness (X6), X7, and X8 were retained. These response-specific combinations indicate that the associations between raceway characteristics and vibration velocity depend on the frequency band considered and cannot be represented adequately by a single overall ranking.

(3)

Across the four vibration responses, X1 was the only descriptor retained for all four responses, whereas X2 was retained for three. X4, X7, and X8 were each retained for two responses, while X3, X5, and X6 were each retained for one. Thus, inner-raceway roundness exhibited the broadest cross-response recurrence under the adopted fusion rule, followed by inner-raceway waviness, whereas the other descriptors showed greater response specificity. These recurrence counts describe the breadth and consistency of association across the investigated responses rather than the magnitude of influence on any individual response.

(4)

The four GRA schemes produced different descriptor rankings, demonstrating sensitivity to data preprocessing and relational-degree formulation. The predefined qualitative-fusion rule reduced reliance on any single scheme by retaining descriptors that repeatedly appeared among the leading positions. Nevertheless, cross-method recurrence represents consistency among the method-specific rankings and does not constitute a test of statistical significance, an estimate of effect magnitude, or independent causal validation.

Overall, inner-raceway roundness (X1) showed the most consistent cross-response association with the measured vibration responses, followed by inner-raceway waviness (X2), whereas the remaining descriptors exhibited stronger response- or frequency-specific associations. The observed relationships are compatible with plausible rolling-contact and tribological interpretations: roundness and waviness deviations may alter circumferential contact geometry, load distribution, and periodic rolling-contact conditions; groove-profile deviations may modify ball–raceway conformity and contact-stiffness distribution; and roughness may affect asperity interactions, interfacial friction, and lubricant-film behaviour. Because contact force, contact stiffness, friction, and lubricant-film behaviour were not measured directly, these interpretations remain as hypotheses requiring controlled validation. The results provide dataset-specific screening evidence for prioritizing raceway metrology and selecting candidate parameters for subsequent controlled studies rather than establishing causal relationships or universal manufacturing limits. Future work should include additional production batches and operating conditions, repeated vibration measurements with uncertainty evaluation, fully characterized grease properties, controlled variation in individual raceway descriptors, and uncertainty-aware statistical or resampling analyses.