Quality-Achieving Reliability Assessment of Rolling Bearing Based on Bootstrap Maximum Entropy Method and Similarity Method

Abstract

Rolling bearing reliability is often affected by the processing quality, however the life reliability or operating reliability do not involve the study of rolling bearing processing quality assessment. In order to avoid bearing failure caused by bearing processing quality, it is necessary to evaluate the reliability of the bearing quality assurance capability. The difficulty in reliable evaluation of bearing's quality assurance capability is how to quantify the uncertain relationships between the bearing quality and its influence factors, and then determine the weight of different influencing factors. Therefore, in this paper, a novel method for determining the weight of bearing quality-influencing factors based on bootstrap maximum entropy method and similarity method is proposed, and then the quality-achieving reliability model is established. The experimental results show that the proposed method can effectively quantify the relationship between the bearing quality and its influencing factors, and accurately assess the bearing quality assurance capability and bearing quality processing level under the condition of a small number of experimental bearings. Compared with other quality-achieving reliability method, the proposed method is more effective.

Keywords

Tapered roller bearing quality-achieving reliability maximum entropy similarity influencing factor

Introduction

Rolling bearing is one of the key components of rotating machinery which sustain larger load,^1,2 and is also the main vibration sources of rotating machinery during operation. Therefore, the reliability of rolling bearing is directly related to the running performance of mechanical equipments,³ and it is necessary to establish an accurate model to evaluate the reliability of bearing.

Traditional reliability analysis usually refers to the ability or possibility of a product to perform tasks as required within a specified time, which can be generally divided into two main categories, that is, time-to-failure data-based methods and degradation data-based methods.^4–9 This traditional reliability calculation mainly focuses on life reliability analysis and operating reliability analysis. To date, many researches have been performed in this field. Gao¹⁰ proposed a prediction method of rolling bearing operational reliability based on isometric mapping and nonhomogeneous cuckoo search-least squares support vector machine. Wang¹¹ proposed a novel method based on kernel principal component analysis and Weibull proportional hazards model to assess the reliability of rolling bearings. Wang¹² proposed a method based on two-dimensional deep convolution neural network for rapidly evaluating bearing reliability and predicting remaining useful life.

However, the reliability of rolling bearings is often affected by their processing quality, and the life reliability or operating reliability do not involve the study of rolling bearing processing quality assessment. Therefore, in order to avoid bearing failure caused by bearing processing quality, it is necessary to evaluate the reliability of the bearing quality assurance capability.

The quality-achieving reliability refers to the probability at which the quality control level can be expected to reach a certain grade under the given conditions, which addresses the quality-achieving capability and probability from the perspective of weighted influence factors.¹³ The key step in evaluating the quality-achieving reliability is to analyze the quality-influencing factors of the bearing caused by the manufacturing process, and then determine the weight of different influencing factors. Bearing quality-influencing factors evaluation comes with two challenges. One challenge is how to quantify the uncertain relationship between the quality of the bearing and its influencing factors. Another challenge is that, due to economic reasons, the number of bearings in the experimental study is so small that it is difficult to obtain objective research results.

There are many contributions to the vibration response of rolling bearings for dynamic response analysis with different parameters considered, including surface roughness, waviness, localized and distributed defects, unbalance load, rub-impact, and misalignment condition.^14–19 Tadina²⁰ developed an improved bearing model to investigate the vibrations of a ball bearing during run-up. Guo²¹ determined the friction source of the spherical roller bearing by analyzing the relationship between the geometrical characteristics of the spherical roller bearing and the kinematics, and then obtained the main factors influencing the vibration and noise of the spherical roller bearing. Finally, the main factors were analyzed by orthogonal experiment. Aktürk²² studied the waviness of the contact surfaces of the inner and outer raceways and rolling elements of the bearing, and obtained the relationship between these factors and the vibration frequency of the bearing. Yan²³ analyzed the influence of geometric parameters, shape and position error, working condition parameters on the amplitude and frequency characteristics of ball bearing vibration according to the ball bearing vibration theory model, and used the mathematical statistics method to analyze the measurement data of the roundness, sparse wave waviness, roughness and shape error of the raceway of inner and outer ring channels of the low-noise bearing. Although the above research has obtained the analysis results of the quality-influencing factors by establishing the physical model of the bearing vibration and conducting comparative experiments, the complexity and inaccuracy of the physical model make them difficult to be used in engineering.

In order to overcome the aforementioned drawbacks, this paper proposes a new method for determining the weight of bearing quality-influencing factors based on bootstrap maximum entropy method (BMEM) and similarity method, and then the quality-achieving reliability model is established.

In nature and social phenomena, different random variables obey different distributions, and different distributions have different probability density function (PDF). Even for the same distribution, the PDF will change as the parameters change. Therefore, the PDF can be used to describe the intrinsic characteristics of a variable.

Jianbo Yu²⁴ used PDF to describe the similarity of the metrics that characterize the rolling bearing at different stages of operation. Jianbo Liu²⁵ improved the accuracy of long-term prediction by comparing the signatures from any two degradation processes using measures of similarity that form a matching matrix. Liang Guo²⁶ calculated the similarity between the current spectrum and the initial spectrum to determine the degree of degradation of the rolling bearing, and the purpose of the remaining life prediction was achieved.

Therefore, PDF of the bearing quality-influencing factors and the bearing vibration acceleration are established by BMEM which is a novel method that combines the advantages of the bootstrap method²⁷ and the maximum entropy method.^28,29 The similarity method is used to obtain the weight of bearing quality-influencing factors by calculating the similarity between the PDF of the bearing quality-influencing factors and the PDF of the bearing vibration acceleration. The weight refers to the degree of overlap between the probability density curve of the vibration acceleration and the probability density curve of the quality-influencing factor, as shown in Figure 1.

Figure 1.

A schematic diagram of calculating the weight of bearing quality-influencing factor.

This paper is organized as follows. Section ‘Model’ establishes the quality-achieving reliability model of the rolling bearing, and introduces the BMEM and the similarity method in detail. Section ‘Experimental validation’ examines a case study. Section ‘Conclusions’ presents the conclusions.

Model

Performance data collection

According to the characteristics of the rolling bearing, the quality assessment indicator S₀ and its influence factors S_v can be established by an experimental investigation method.

The vibration performance of the rolling bearing is studied as the quality evaluation indicator, and the data sequence S₀ of the performance indicator can be obtained as

S_{0} = {s_{0} (1), s_{0} (2), \dots, s_{0} (t), \dots, s_{0} (q)}

(1)

Where, t is the data sequence number, t = 1, 2,…,q, q is the number of data in the sequence, s₀(t) is the tth datum in S₀.

Assuming that there are V factors influencing the vibration performance data sequence S₀, by measuring in advance, and the data sequence S_v of the vth influence factor is given by

S_{v} = {s_{v} (1), s_{v} (2), \dots, s_{v} (t), \dots, s_{v} (q)}

(2)

Where, s_v(t) is the tth datum of the vth influence factor, v = 1, 2,…,V.

Performance data classification

The quality of the bearing is divided into U grade, let H^u_v is used as the assessment standard value of the bearing quality or the influencing factors in grade u. If the measured value s_v(t) of a certain parameter satisfied

H_{v}^{u - 1} < s_{v} (t) \leq H_{v}^{u}

(3)

Where, v = 0, 1,…,V, u = 1, 2,…,U. In general, the quality grade number U = 4∼7, which may satisfy the different levels of rolling bearing quality control requirements.

Then the quality grade of the measured value s_v(t) is grade u.

According to the Equation (3), the quality grading of all measured performance data can be conducted, and then the frequency F_v^u of quality grade can be obtained, as shown in Table 1. The highest quality grade is 1 and the lowest quality grade is U. As the quality grade increases, the state of the bearing's quality influencing factors will improve and the bearing will be of higher quality.

Table 1.

Quality classifications of the measured performance data.

Quality Grade	Evaluation Standard Value	Frequency
1	H_v ¹	F_v ¹
2	H_v ²	F_v ²
…
u	H_v^u	F_v^u
…
U	H_v^U	F_v^U

From Table 1, the frequency sequence of rolling bearing quality achieving and its influencing factors can be expressed as follows

Z_{v}^{0} = {z_{v}^{0} (1), z_{v}^{0} (2), \dots, z_{v}^{0} (u), \dots, z_{v}^{0} (U)}

(4)

Where

z_{v}^{0} (u) = \frac{F_{v}^{u}}{q}; v = 0, 1, 2, \dots, V

(5)

The cumulative distribution sequence of rolling bearing quality achieving and its influencing factors can be obtained as

Z_{v} = {z_{v} (1), z_{v} (2), \dots, z_{v} (u), \dots, z_{v} (U)}

(6)

Where

z_{v} (u) = \sum_{j = 1}^{u} z_{v}^{0} (j); v = 0, 1, 2, \dots, V

(7)

Determining the PDF of bearing quality and its influencing factors based on BMEM

Let S_v be a real-valued observation sequence of rolling bearing performance, X is normalized values of S_v

X = {x_{1}, x_{2}, \dots, x_{q}}

(8)

A bootstrap sample, Y_b, can be obtained from X by equal probability sampling and can be expressed as follows:

Y_{b} = {y_{b} (1), y_{b} (2), \dots, y_{b} (k), \dots, y_{b} (K)}

(9)

Where K is the total number of the bootstrap sample, y_b(k) is the kth data of the bootstrap sample.

The mean of the bootstrap sample can be calculated by

\bar{Y_{b}} = \frac{1}{K} \sum_{k = 1}^{K} y_{b} (k)

(10)

Using the above-mentioned method, independently draw a large number of bootstrap samples and the sequence of the bootstrap sample can be expressed as follows:

Y_{B o o t s t r a p} = {\bar{Y_{1}}, \bar{Y_{2}}, \dots, \bar{Y_{b}}, \dots, \bar{Y_{B}}}, b = 1, 2, \dots, B

(11)

The MEM can be used to estimate the probability distribution of Y_Bootstrap with minimum deviation. For convenience of description, continuous variables x are used to represent discrete variables in the Y_Bootstrap sequence.

According to the MEP, the PDF with the least subjective bias should satisfy the condition in which the entropy reaches to the maximum, that is,

E (x) = - \int_{Ω} f (x) \ln f (x) d x \to max

(12)

Where E(x) is the information entropy, Ω is the feasible domain of the variable x, f(x) is the PDF of the variable x.

Equation (12) satisfying

{\begin{matrix} \int_{Ω} f (x) d x = 1 \\ \int_{Ω} x^{i} f (x) d x = m_{i} \end{matrix}

(13)

Where i is the i-order origin moments of x, i = 0, 1, 2, …,m; m is the order of origin moment and generally takes m = 3∼8; m₀ = 1.

In order to solve f(x), we apply the Lagrange multiplier method to Equation (12). The lagrange function L(x) is obtained as

L (x) = E (x) + (c_{0} + 1) [\int_{Ω} f (x) d x - 1] + \sum_{i = 1}^{m} c_{i} [\int_{Ω} x^{i} f (x) d x - m_{i}]

(14)

Where, c_i is the ith Lagrange multiplier.

Let $\frac{d L (x)}{d f (x)} = 0$ , then

- \int_{Ω} [\ln f (x) + 1] d x + (c_{0} + 1) \int_{Ω} d x + \sum_{i = 1}^{m} c_{i} (\int_{Ω} x^{i} d x) = 0

(15)

Therefore, the expression of the probability density function f(x) is obtained as

f (x) = \exp (c_{0} + \sum_{i = 1}^{m} c_{i} x^{i})

(16)

Substituting Equation (16) into Equation (13), we can have

\int_{Ω} \exp (c_{0} + \sum_{i = 1}^{m} c_{i} x^{i}) d x = 1

(17)

Then the first Lagrange multiplier, c₀, is obtained as

c_{0} = - \ln [\int_{Ω} \exp (\sum_{i = 1}^{m} c_{i} x^{i}) d x]

(18)

Other Lagrange multipliers should satisfy

g_{i} = g (c_{i}) = 1 - \frac{\int_{Ω} x^{i} \exp (\sum_{i = 1}^{m} c_{i} x^{i}) d x}{m_{i} \int_{Ω} \exp (\sum_{i = 1}^{m} c_{i} x^{i}) d x} = 0

(19)

Using vector to represent Equation (19)

G = G (c) = {g_{i}, i = 1, 2, \dots, m}^{T} = 0

(20)

And the vector of the Lagrange multiplier can be expresses as follows:

c = {c_{i}, i = 1, 2, \dots, m}^{T}

(21)

Solving vector of the Lagrange multiplier c by Newton Method

c^{j + 1} = c^{j} - G^{'} (c^{j})^{- 1} G (c^{j})

(22)

Where G′(c^j)⁻¹ is the Jacobian matrix that is iterated to step j, j = 0, 1, ….

The norm criterion for iterative convergence is given by

‖ G^{j + 1} - G^{j} ‖_{1} \leq ε

(23)

Where ε is convergence accuracy and generally takes ε = 10⁻¹².

In order to make the solution to converge, the sample data is arranged in increasing order and divided into Q-2 group, draw a histogram, then the group median(ζ_q) and the frequency (F_q, where q = 0, 1, …,Q, and let F₁ = F_q = 0.) of each group are obtained.

Mapping the raw data sequences x (i) dimensionlessly into interval [ − e, e], let

x = a t + b

(24)

Where a and b are the mapping parameters, t∈[ − e, e].

Then, we have

t = x / a - b / a

(25)

Since

d t = d x / a

(26)

Thus

{\begin{matrix} a = \frac{2 e}{ξ_{Q} - ξ_{1}} \\ b = e - a ξ_{Q} \end{matrix}

(27)

And the i-order origin moment (m_i) is given by

m_{i} = \sum_{q = 1}^{Q} i_{q}^{i} F_{q}

(28)

Thus, the PDF based on the BMEM can be expresses as follows:

f (t) = \exp [c_{0} + \sum_{i = 1}^{m} c_{i} {(a t + b)}^{i}]

(29)

Where t∈ [ − e, e].

Determining the weight of bearing quality-influencing factors based on the similarity method

Let the PDF of the bearing vibration performance sequence be

f_{0} (t) = \exp [c_{0} + \sum_{i = 1}^{m} c_{i} {(a t + b)}^{i}]

(30)

Let the PDF of vibration-influencing factors sequence be

f_{v} (t) = \exp [c_{0} + \sum_{i = 1}^{m} c_{i} {(a t + b)}^{i}]

(31)

Where, v = 1, 2,…,V, and V is the total number of the vibration-influencing factors.

The similarity can be used to evaluate the level of the vibration-influencing factors by overlapping the maximum entropy distribution curve of the vibration performance with the maximum entropy distribution curve corresponding to vibration-influencing factors. The similarity between the vibration performance and the vibration-influencing is defined by

I_{v} = \int_{Θ} f_{0} (t) \cup f_{v} (t) d t

(32)

Where, Θ is the feasible domain of the variable t.

The value of I_v can be from 0 to 1: the closer the value of I_v is to 1, the greater similarity; the closer the value of I_v is to 0, the smaller the similarity.

The weight of the influence factor S_v is given by

w_{v} = \frac{I_{v}}{\sum_{v = 1}^{V} I_{v}}

(33)

Quality-achieving reliability model

As we all know, the factors that affect bearing’s quality-achieving reliability are complicated. According to factor space theory, it is usually difficult to forecast the status of a complex factor.¹³ The most effective approach is to decompose the complex factor into a series of simple facilitating factors and then to use these simple factors to determine the status of the complex factor.³⁰ Therefore, it is feasible to decompose the quality-achieving reliability of rolling bearing into a series of simple factors whose statuses are easily examined. Then, the factor status of rolling bearing can be synthesized by examining the statuses of all of the simple factors.

According to the factor synthesis method, the weight can be used to synthesize the status of the influencing factors in each grade. Therefore, the status composite value o_u is given by

o_{u} = \sum_{v = 1}^{V} w_{v} z_{v} (u)

(34)

According to the theory of organizational reliability,^30,31 the quality-achieving reliability function of mechanical products can be given by

R_{u} = \frac{1}{1 - \exp (- a)} [1 - \exp (- a o_{u}^{b})]

(35)

Where, R_u is the quality-achieving reliability when the quality of the bearing is in the u grade, a and b are quality influence coefficients.

The maximum likelihood method is used to solve the quality influencing coefficient. Derivation of R_u to obtain the quality-achieving density function of bearing

r_{u} = \frac{d R_{u}}{d o_{u}} = \frac{a b}{1 - \exp (- a)} o_{u}^{b - 1} \exp (- a o_{u}^{b})

(36)

The maximum likelihood function is given by

Γ (a, b) = {[\frac{a b}{1 - \exp (- a)}]}^{U} (\prod_{u = 1}^{U} o_{u}^{b - 1}) \exp (- a \sum_{u = 1}^{U} o_{u}^{b})

(37)

The maximum likelihood estimation equation of the parameter a and b is

\begin{aligned} \frac{\partial \ln Γ}{\partial a} = & \frac{U}{a} - \frac{U \cdot \exp (- a)}{1 - \exp (- a)} - \sum_{u = 1}^{U} o_{u}^{b} = 0 \\ \frac{\partial \ln Γ}{\partial b} = & \frac{U}{b} + \sum_{u = 1}^{U} \ln o_{u} - a \sum_{u = 1}^{U} (o_{u}^{b} \cdot \ln o_{u}) \end{aligned}

(38)

The influencing coefficients a and b can be solved by Equation (38).

Experimental validation

Experimental setup and performance data collection

Tapered roller bearings 30306X2B are taken as the research object and the bearing vibration measuring equipment is the bearing vibration tester: BVT-6, as show in Figure 2. The detailed parameters of the bearing vibration tester are shown in Table 2. Depending on the type and size of the bearing, a finely filtered lubricating oil with a nominal kinematic viscosity between 10 and 100 mm²/s is used to lubricate the bearings, and carrying out a test run during the lubrication process to distribute the lubricant evenly. An axial load of 110N is added to the bearing, the rotation speed is kept constant at 900 r/min during the experiment. The loading position is the end face of the bearing outer ring. After applying an axial load to the bearing, start the test after a delay of more than 0.5 s, and the test time is between 10 and 60 s. The ambient temperature is 20 degrees Celsius and the relative humidity of air is less than 70%. The basic layout of the test rig is shown in Figure 3.

Figure 2.

Bearing vibration tester: BVT-6.

Figure 3.

Tapered roller bearing vibration measurements rig.

Table 2.

Parameters of bearing vibration tester.

Parameters	Numerical value
Measured bearing size range/mm	65≤inner diameter≤120outer diameter≤280
Dynamic range/(μm/s)	0– 10,000
Frequency range/Hz	low frequency band 50–300mid frequency band 300–1800high frequency band 1800– 10,000
Spindle speed range/(r/min)	900–1800
Axial load range/N	0–500
Radial load range/N	0–600
Spindle motor	380 V, 50 Hz, three phase, 370 w
Oil pump motor	380 V, 50 Hz, three phase, 180 W
Measuring amplifier	220 V ± 10%, 50 Hz
Power consumption/W	600
Ambient temperature/°C	5–40
Total mass/kg	620
Overall size/mm	600 × 1200 × 1100

The vibration acceleration is selected as the physical quantity to characterize the bearing vibration and the vibration acceleration level can be expressed as follows:

Ψ = 20 \log \frac{χ}{g}

(39)

Where, Ψ is the vibration acceleration level which unit is dB; χ is the root mean square value of the vibration acceleration or the peak value of the vibration acceleration of the bearing in a certain frequency range which unit is m·s⁻²; g is the reference acceleration which value is 9.81 × 10⁻³ m·s⁻².

In this study, 30 sets of tapered roller bearings are selected and each bearing is installed (separately) on the bearing vibration tester (BVT-6) to collect the vibration data, and then each bearing is disassembled to measure the parameters of the inner ring, outer ring and rolling element, respectively. The measuring parameters and measuring instruments are shown in Table 3. Some measuring instruments are shown in Figure 4.

Figure 4.

Instruments used in the measurement process. (a) Wall thickness difference measuring instrument: H903. (b) Inner raceway measuring instrument: D712. (c) Shape measuring instrument: XZ-150G. (d) Roundness measuring instrument: Y9025C. (e) Roughness profiler: PGI1220. (f) Bearing width measuring instrument: G904. (g) Perpendicularity measuring instrument: C923. (h) Roller measuring instrument: D744.

Table 3.

Measured parameters and measuring instruments.

Symbol	Measuring parameters	Measuring position	Measuring instruments
S ₀	Vibration acceleration	Bearing	Vibration tester: BVT-6
S ₁	Wall thickness difference of inner raceway	Inner ring	Wall thickness difference measuring instrument: H903
S ₂	Vertical difference of inner raceway	Inner ring	Inner raceway measuring instrument: D712
S ₃	Angle difference of inner raceway	Inner ring	Inner raceway measuring instrument: D712
S ₄	Straightness of inner raceway	Inner ring	Shape measuring instrument: XZ-150G
S ₅	Roundness of raceway	Inner ring	Roundness measuring instrument: Y9025C
S ₆	Waviness of raceway	Inner ring	Roundness measuring instrument: Y9025C
S ₇	Roughness of raceway	Inner ring	Roughness profiler: PGI1220
S ₈	Thickness variation of guide flange	Inner ring	Bearing width measuring instrument: G904
S ₉	Roughness of guide flange	Inner ring	Roughness profiler: PGI1220
S ₁₀	S_d	Inner ring	Perpendicularity measuring instrument: C923
S ₁₁	Wall thickness difference of outer raceway	Outer ring	Wall thickness difference measuring instrument: H903
S ₁₂	Vertical difference of outer raceway	Outer ring	Outer ring measuring instrument: D712
S ₁₃	Angle difference of outer raceway	Outer ring	Outer ring measuring instrument: D712
S ₁₄	Straightness of outer raceway	Outer ring	Shape measuring instrument: XZ-150G
S ₁₅	Roundness of raceway	Outer ring	Roundness measuring instrument: Y9025C
S ₁₆	Waviness of raceway	Outer ring	Roundness measuring instrument: Y9025C
S ₁₇	Roughness of raceway	Outer ring	Roughness profiler: PGI1220
S ₁₈	S_d	Outer ring	Perpendicularity measuring instrument: C923
S ₁₉	Base surface run-out	Roller	Roller profile measuring instrument: R943
S ₂₀	Surface roughness	Roller	Roughness profiler: PGI1220
S ₂₁	Diameter variation	Roller	Roller measuring instrument: D744
S ₂₂	Cone angle error	Roller	Roller measuring instrument: D744
S ₂₃	Ball base roughness	Roller	Roughness profiler: PGI1220
S ₂₄	Roundness	Roller	Roundness measuring instrument: Y9025C
S ₂₅	Waviness	Roller	Roundness measuring instrument: Y9025C
S ₂₆	Straightness	Roller	Roller convexity measuring instrument

H903 is a comparative mechanical measuring instrument, which is mainly used for the measurement of the thickness variation of the bearing inner ring. There are special positioning bolts on the instrument plate, the position of the positioning bolts can be adjusted according to the diameter of the tested ferrule, the positioning bolts are equipped with a disc-shaped positioning piece, and the height can be adjusted according to the requirements of the tested ferrule. D712 is also a comparative mechanical measuring instrument, which is mainly used for the measurement of the diameter and angle of the outer ring raceway of tapered roller bearings.

XZ-150G is mainly used to measure the shape of the element line and the profile of the section of various mechanical parts. In the bearing industry, it can be used to measure the element lines of various rolling elements and raceways, such as: protrusion, contact angle, radius of curvature of curved surface, and angle of reference parts. The instrument is controlled by a microcomputer, which can automatically realize the measurement cycle, automatically eliminate the installation error, and directly display the shape and parameters of the measured parts. The instrument workbench adopts closed air flotation guide rail.

Y9025C is mainly used to measure the roundness, waviness, coaxiality, perpendicularity, parallelism, flatness and other parameters of the inner and outer ring raceways and rolling elements of the bearing, and can obtain the eccentricity and phase of the parts. The instrument is equipped with high-precision air flotation spindle and air filtration system. It adopts Windows system operation platform, which can perform spectrum analysis on the surface state of parts. It is equipped with two types of sensors to meet the measurement of various diameters and apertures.

PGI1220 is a stylus optical profiler that can be used to measure the roughness and shape of plastic lenses, spherical, aspherical, free-form surfaces, infrared glass, crystals, and diffractive surfaces up to 300 mm in diameter. PGI (Phase Grating Interferometry) technology utilizes a short stylus to measure large lenses, enabling high-precision measurements.

G904A is mainly used to measure the height and parallel difference of bearing rings. C923 is a comparative mechanical measuring instrument, which is mainly used to measure the perpendicularity of the center line of the inner diameter to the reference end face. D744 is a comparative mechanical measuring instrument, which is mainly used to measure rolling element diameter and related parameters.

Vibration acceleration of the 30 sets of bearing is shown in Figure 5. From Figure 5, we can see that under the same working condition the vibration acceleration value of tapered roller bearings which belong to the same type are different. Figures 6–8 show the measuring data of inner ring, outer ring and roller, respectively. It can be seen from the Figures 6–8, although all the tapered roller bearings belong to same type, the measuring values are different for a specific parameter which influences the vibration. It can be inferred that all the measuring parameters may have an effect on the vibration of the tapered roller bearings to varying degrees.

Figure 5.

Vibration acceleration data of tapered roller bearing.

Figure 6.

Measuring data of the inner ring.

Figure 7.

Measuring data of the outer ring.

Figure 8.

Measuring data of the roller.

Performance data classification

According to Table 1, the vibration acceleration data of the test bearing is classified. Take the number of bearing quality grades U = 6, that is, the bearing vibration acceleration is divided into six grades. The tapered roller bearing vibration acceleration quality classification results are shown in Table 4.

Table 4.

Quality classifications of the vibration acceleration.

Quality Grade, u	Evaluation Standard Value, H₀ ^u /dB	Frequency, F₀ ^u	Cumulative Distribution, z₀(u)
1	53	3	3/30
2	54	6	9/30
3	55	6	15/30
4	56	8	23/30
5	57	4	27/30
6	58	3	30/30

Similarly, the influence factors data can be classified. And the cumulative distributions of the quality grade of the inner ring (S₁, S₂, …, S₁₀), the outer ring (S₁₁, S₁₂, …, S₁₈), and the roller (S₁₉, S₂₀, …, S₂₆) is obtained as shows in Table 5.

Table 5.

Cumulative distributions of the influence factors quality grade.

Symbol	QualityGrade
	1	2	3	4	5	6
Z ₁	0	0.133	0.3	0.7	0.9	1
Z ₂	0	0.067	0.767	0.933	1	1
Z ₃	0.3	0.733	0.867	0.933	0.967	1
Z ₄	0.033	0.033	0.1	0.567	1	1
Z ₅	0	0	0.3	0.533	1	1
Z ₆	0	0	0.033	0.133	0.867	0.967
Z ₇	0	0	0	0.167	0.933	1
Z ₈	0.067	0.3	0.6	0.967	1	1
Z ₉	0	0	0.067	0.367	0.633	1
Z ₁₀	0	0.033	0.3	0.5	0.967	1
Z ₁₁	0	0.133	0.4	0.667	0.933	1
Z ₁₂	0	0.167	0.4	0.667	0.933	0.967
Z ₁₃	0.3	0.533	0.733	0.833	0.867	0.967
Z ₁₄	0.033	0.067	0.333	0.7	0.867	0.967
Z ₁₅	0	0.033	0.1	0.433	0.9	0.967
Z ₁₆	0	0.067	0.6	0.867	1	1
Z ₁₇	0	0	0.333	0.6	1	1
Z ₁₈	0	0.2	0.667	0.867	0.933	0.967
Z ₁₉	0	0	0	0.3	0.767	1
Z ₂₀	0	0	0	0.467	1	1
Z ₂₁	0	0	0	0	0.133	1
Z ₂₂	0.1	0.367	0.6	0.8	0.9	0.967
Z ₂₃	0	0	0.3	0.9	0.967	0.967
Z ₂₄	0	0	0	0.2	1	1
Z ₂₅	0	0	0	0.967	1	1
Z ₂₆	0.033	0.267	0.867	1	1	1

Determining the weight of bearing quality-influencing factors

The probability density curves of the vibration acceleration and all the measuring parameters can be obtained based on BMEM, as shown in Figures 9–11. It can be seen from these three Figures that all the probability density curves have difference in shape. From equation (32) and equation (33), the weights of the influencing factors can be obtained, as shown in Figure 12. From equation (34), the status composite value of the influencing factors can be obtained, as summarized in Table 6. From equation (32), the influencing coefficients can be calculated, that is a = −0.9831, and b = 0.6757. Substituting the quality influencing coefficients into the bearing quality-achieving reliability model, and then the quality-achieving reliability values of the vibration acceleration of the tapered roller bearing at different quality grades can be obtained, as shown in the Table 7.

Figure 9.

The PDF of the inner ring parameters and vibration acceleration.

Figure 10.

The PDF of the outer ring parameters and vibration acceleration.

Figure 11.

The PDF of the roller parameters and vibration acceleration.

Figure 12.

The weights of the quality influence factors.

Table 6.

The status composite value of the influencing factors.

o ₁	o ₂	o ₃	o ₄	o ₅	o ₆
0.0357	0.1263	0.3345	0.6173	0.9105	0.9910

Table 7.

Tapered roller bearing quality-achieving reliability values.

	1	2	3	4	5	6
R_u	0.0652	0.1643	0.3578	0.6177	0.9064	0.9904

It can be seen from Table 7 that, when the quality grade is 1, the quality-achieving reliability of the tested bearing is the lowest, and as the quality grade increases, the quality-achieving reliability gradually increases. When the quality grade is 5, the quality-achieving reliability can reach about 90.64%. Therefore, under the current processing level, the vibration acceleration value of the tapered roller bearing used in the research is generally maintained at the fifth quality grade, that is, the vibration acceleration value is 57 dB (57 dB is obtained according to Table 4.), and the corresponding bearing quality-achieving reliability is 90.64%.

In order to verify the effectiveness of the proposed method, the method in Literature 13 is used to calculate the quality-achieving reliability, as shown in Figure 13. The comparison results are in Figure 14 and Table 8. As shown, the quality-achieving reliability calculated by the method in Literature 13 differ greatly from the experimental results, the maximum of relative error is nearly 90%, mean relative error is up to 44.26%, and the relative error of the proposed method is considerably lower than the method in Literature 13. According to the results, it can be clearly noticed that the proposed method is more accurate than the method in Literature 13.

Figure 13.

Tapered roller bearing quality-achieving reliability using different methods.

Figure 14.

The comparison results of quality-achieving reliability of different methods.

Table 8.

The quantitative evaluations of different methods.

Index	Proposed method	Method in Literature 13
Mean Relative Error	7.36%	44.26%
Maximum of relative Error	15.70%	88.95%

Since the bearing quality-achieving reliability is affected by many factors, and the mechanism of rolling bearing vibration is also very complex. There are many factors affecting bearing vibration, including ring raceway waviness, rolling element size difference, roughness, structure type of bearing itself, assembly clearance, lubrication conditions, installation conditions and work conditions, etc.. Only the main influencing factors are considered in the quality-achieving reliability model, so there is a certain difference between the theoretical calculation results and the experimental results.

Conclusions

In this paper, a novel method for determining the weight of bearing quality-influencing factors based on bootstrap maximum entropy method and similarity method is proposed, and then the quality-achieving reliability model is established. The proposed method is applied to investigate the experimental tapered roller bearings. The results show that the proposed method can be employed to quantify the uncertain relationship between the bearing quality and its influencing factors, and to assess the bearing quality assurance capability and bearing quality processing level under the condition of a small number of experimental bearings. The results also confirm that the proposed method is more effective than other quality-achieving reliability method.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Li Cheng

Author biographies

Li Cheng was born in 1990. His research interests related to data driven rolling bearing reliability.

Wensuo Ma was born in 1969. He works as Professor in Henan University of Science and Technology, Luoyang, China. His research interests include basic and applied basic research in mechanical engineering and cross-material science.

Xintao Xia was born in 1957. He works as Professor in Henan University of Science and Technology, Luoyang, China. His research interests include the poor information theory. He has 12 published monographs and more than 200 research papers published.

Liangwen Wang is a professor in School of Mechanical and Electrical Engineering, Zhengzhou University of Light Industry, Henan Provincial Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, China. His research interests are in robotics and mechanism design.

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