Keywords

1 Introduction

Rolling element bearings are vital parts in rotating machines, such as wind turbine, train and heavy equipment. Complex working conditions and harsh environment may cause faults on bearings, leading to fatal breakdowns, catastrophic accident and economic losses. Hence, for the reduction of faults and losses, many engineers and scholars have paid considerable attention on the health condition monitoring of rolling element bearings in nowadays [1].

Once a local defect, such as spall on raceways, appears on bearing, it will tempt an impact to the whole bearing vibration system. And then, resonant frequencies of the bearing parts and neighbouring components could be excited by the impulse [2]. Based on this, a large volume of research works have focused on fault detection through vibrational signal analysis with the acquired vibration data [3,4,5]. It has been proved to be a valid approach to realize fault diagnosis for bearings and most rotating machines. However, these signal processing methods are based on some foundational knowledge of mechanical structure and vibration mechanism, such as fault characteristic frequencies. On the other hand, mechanism analysis is an effective tool to reveal the mechanism behind the mechanical structure and obtain the representation form of fault. Usually, through establishing a dynamic model, characteristics of the system could be obtained [6, 7]. It is the basis of mechanical fault diagnosis, which provides foundation for data-driven methods. In this study, model-based method is utilized to explore the vibration characteristics of bearings influenced by bearing internal radial clearances.

Internal radial clearance means the clearance between the raceways and ball in a ball bearing. The bearing clearances have a critical impact on bearing operating temperature, system vibration and rotating life. Thus, it is a key factor for bearing selection. Investigations show that bearing usually go through various wear and tear during operating process, which results in the increase of clearance [8]. Besides, bearing clearance is an vital factor on bearing load distribution, which has a significant influence on the fatigue life of rolling bearings [9, 10]. Especially, investigation shows that wear and tear in the operating process shorten bearing rotating life by about 30% [11]. Consequently, it is urgent to study the mechanism of bearing internal clearances.

Bai et al. [12] presented a general rotor dynamic model for studying the dynamic properties considering the impact of bearing clearance and raceway waviness. Harsha [13] investigated the nonlinear characteristics of a balanced rotor with bearing internal radial clearance and detected the chaotic responses. Upadhyay et al. [14] studied the behavior of an unbalanced rotor affected by radial internal clearance of the supported bearings and high rotating speed. Zhuo et al. [15] utilized a vibration model to analyse the dynamic properties of the ball bearing as a result of applied load, waviness and internal clearances during the process of start and constant speed operation. Even a large number of literatures could be found focusing on bearing clearance through model-based methods, comparatively little scholars studied on different bearing clearances on vibrational responses. Nevertheless, sufficient acknowledge on vibration characteristics under different clearances will be benefit to account for various vibration phenomena and bearing clearances monitoring.

On mechanism of bearing clearance, Oswald [11] investigated the bearing load distribution and fatigue life under the effect of four different level of internal radial clearances on ball and roller bearings with radical load. Recently, Rehab et al. [16] presented a nonlinear dynamic model to investigate the diagnostic features under two group bearing clearances and point out that inconsistency on bearings with inner race defect according to characteristic frequency amplitude. More recently, the author Xu et al. [17] proposed a six degrees of freedom (DOF) dynamic model to study the vibration behavior influenced by continuously changing bearing clearances and working conditions, including external load and rotating speed. It is pointed out that ball pass frequency of outer race (BPFO) dominates the envelope spectrum for bearing before obvious local defect on raceways. On the other hand, to monitor the bearing clearances, Zmarzły [18] pointed out that medium frequency band, from 300 Hz to1800 Hz, has the biggest effect on bearing vibration level under different radial clearances through experiment study. Georgiadis et al. [19] predicted bearing clearance variations with spectrum kurtosis by test vibration data. Yakout et al. [10] investigated the relationship between the clearance and bearing system damping characteristics. Meier et al. [20] utilized kurtosis, recurrences and neural networks three methods to measure the clearances through experiment test and obtained a correlation between bearing clearance and vibration level. Recently, Wang et al. [21] utilized modulation signal bispectrum (MSB) to extract features and combined with Gini-index to monitor the variation of bearing clearances online, which show that useful information extracted from frequency domain is effective on detecting bearing clearances. Besides, in previous work (Ref. [17]), we have proposed a bearing vibration model to study the bearing vibration responses with different clearances and working conditions. Both RMS and spectral centroid increase with the radial clearance in general but some local fluctuations [17]. However, in some cases, there are local defects on bearings, such as outer race and inner race fault. When local defect appears, bearing will enter the period of failure rapidly and the vibration responses are different from bearing without faults, which brings much difficulty for online monitoring of bearing clearances.

In this study, a nonlinear bearing vibration model is utilized to explore the vibration responses under various clearances with and without local defects on bearing raceways. Firstly, bearing dynamic equations are derived through is established. Then, simulation results are provided. Last, comprehensive analyses are carried out to display the difference between fault-free bearings and faulty bearings for bearing clearance monitoring.

2 Bearing Vibration Numerical Model

2.1 Bearing Clearances

Internal clearance is the geometrical clearance between the races and ball in a ball bearing. The diagram of bearing radial clearance is presented in Fig. 1.

According to the ISO standards and bearing manufacturer, the internal radial could be classified into five classes for deep groove ball bearing [9]. Table 1 represents the minimum and maximum of different level of clearance. It should be noted that the clearances shown in the table are the original values after manufacturing.

Fig. 1.
figure 1

Schematic diagram of (a) concentric arrangement, (b) initial contact

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Table 1. Clearance values of ball bearings under different classes
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2.2 Bearing Vibration Model

To monitor clearance, a vibration model is developed with six DOF, as shown in Fig. 2. In this shaft-house system, \(X\), \(\dot{X}\) and \(\ddot{X}\) represents the displacement, velocity and acceleration in \(X\) direction, respectively. \(s\), \(h\) and \(r\) denote shaft, housing and sensor in the bearing system.

The bearing dynamic model considering bearing clearances is given in Eqs. (1)–(6).

$$ M_{s} \ddot{X}_{s} + \sum\nolimits_{i = 1}^{{N_{b} }} {K\left[ {\delta_{i} } \right]^{3/2} cos\phi_{i} + } \sum\nolimits_{i = 1}^{{N_{b} }} {C[v_{i} ]cos\phi_{i} = F} , $$
(1)
$$ M_{s} \ddot{Y}_{s} + \sum\nolimits_{i = 1}^{{N_{b} }} {K\left[ {\delta_{i} } \right]^{3/2} sin\phi_{i} + } \sum\nolimits_{i = 1}^{{N_{b} }} {C[v_{i} ]sin\phi_{i} = 0} , $$
(2)
$$ M_{h} \ddot{X}_{h} + C_{h} \dot{X}_{h} + K_{h} X_{h} - \sum\nolimits_{i = 1}^{{N_{b} }} {K\left[ {\delta_{i} } \right]^{3/2} cos\phi_{i} } - \sum\nolimits_{i = 1}^{{N_{b} }} {C\left[ {v_{i} } \right]cos\phi_{i} = 0} , $$
(3)
$$ M_{h} \ddot{Y}_{h} + C_{h} \dot{Y}_{h} + K_{h} Y_{h} - \sum\nolimits_{i = 1}^{{N_{b} }} {K\left[ {\delta_{i} } \right]^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}} } sin\phi_{i} - \sum\nolimits_{i = 1}^{{N_{b} }} {C\left[ {v_{i} } \right]sin\phi_{i} = 0,} $$
(4)
$$ M_{r} \ddot{X}_{r} + C_{r} \left( {\dot{X}_{r} - \dot{X}_{h} } \right) + K_{r} \left( {X_{r} - X_{h} } \right) = 0, $$
(5)
$$ M_{r} \ddot{Y}_{r} + C_{r} \left( {\dot{Y}_{r} - \dot{Y}_{h} } \right) + K_{r} \left( {Y_{r} - Y_{h} } \right) = 0, $$
(6)

where \(M\), \(K\) and \(C\) denotes the mass, stiffness and damping, respectively. The subscript \(s\), \(h\) and \(r\) denote shaft, housing and sensor in the bearing system. \(\phi_{i}\) represents the angle position of the \(i^{th}\) ball. \(N_{b}\) is the total amount of rolling elements. \(\delta_{i}\) and \(v_{i}\), the nonlinear deformation and velocity of the \({\text{ith}}\) ball, for each rolling element can be obtained by Eq. (7) and (8)

$$ \delta_{i} = \left\{ {\begin{array}{*{20}c} {\left( {X_{s} - X_{h} } \right)cos\phi_{i} + \left( {Y_{s} - Y_{h} } \right)sin\phi_{i} - c/2 \times \left( {1 - cos\phi_{i} } \right)} & {\delta_{i} > 0,} \\ 0 & {\delta_{i} \le 0,} \\ \end{array} } \right. $$
(7)
$$ v_{i} = \left\{ {\begin{array}{*{20}c} {\left( {\dot{X}_{s} - \dot{X}_{h} } \right)cos\phi_{i} + \left( {\dot{X}_{s} - \dot{X}_{h} } \right)sin\phi_{i} } & {\delta_{i} > 0,} \\ 0 & {\delta_{i} \le 0.} \\ \end{array} } \right. $$
(8)

where \(c\) is the bearing internal radial clearance.

Fig. 2.
figure 2

Schematic diagram of a shaft-housing dynamic model

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Based on Hertzian contact theory [9, 17], the total stiffness between inner race and out race can be calculated through Eq. (9). In Eq. (9), \(K_{i} ,\) is the Hertzian contact stiffness between the inner raceway and rolling elements, while \(K_{o}\) is the stiffness between outer raceway and balls. In Eq. (10), \(\delta^{*}\) means the dimensionless contact deflection, \(\sum \rho\) denotes the curvature sum.

$$ K = \left[ {\frac{1}{{\left( {1/K_{i} } \right)^{{1/\left( {3/2} \right)}} + \left( {1/K_{o} } \right)^{{1/\left( {3/2} \right)}} }}} \right]^{3/2} $$
(9)
$$ K_{i,o} = \frac{{2\sqrt 2 \left( {\frac{E}{{1 - \nu^{2} }}} \right)}}{{3\left( {\sum \rho } \right)^{1/2} }}\left( {\frac{1}{{\delta^{*} }}} \right)^{3/2} $$
(10)

Besides, the damping between inner and outer raceway could be obtained through Eq. (11). In Eq. (11), \(\,\varsigma\) means the damping ratio, \(m_{b}\) stands for the mass of each ball.

$$ C = 2\,\varsigma \sqrt {m_{b} \times K} $$
(11)

2.3 Bearing Local Defects

Local defects on outer race and inner race are displayed in Fig. 3. In this study, the outer race defect is located at the bottom of the bearing, as shown in Fig. 3(a) and (b). The position of the inner race defect is rotating with the inner race and shaft. The initial position was shown in Fig. 3(c) and (d).

In this study, the defect width \(W\) and defect depth was set as 0.8 \({\text{mm}}\) and 0.1 \({\text{mm}}\), respectively. The additional deflection for outer race and inner race fault could be calculated in Eq. (12).

$$ \Delta = \frac{{d_{b} }}{2} - \frac{{d_{b} }}{2}\cos \left( {\frac{{\phi_{ball} }}{2}} \right) = \frac{{d_{b} }}{2} - \sqrt {\left( {\frac{{d_{b} }}{2}} \right)^{2} - \left( \frac{W}{2} \right)^{2} } $$
(12)
Fig. 3.
figure 3

Local defect: (a, b) outer race defect and (c, d) inner race defect

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3 Numerical Simulation Implementation

3.1 Assumptions and Considerations

In this study, to investigate the effect of clearance on vibration characteristics, a dynamic model was established. However, to simplify the question and speed up the numerical solution process, the model was established based on some assumptions and considerations [17], which are list as follow: (1) ideal bearing works under perfect operating conditions and ignore all manufacturing errors; (2) ignore the influence of temperature and lubrication is sufficient; (3) the contact between the balls and races are pure rolling, i.e., sliding and skidding are ignored in this model.

3.2 Numerical Simulation Implementation

In this study, the numerical simulation was carried out based on bearing 6206. In the dynamic model, the main geometry parameters and the model parameters adopted in numerical simulation are listed in following Tables 2 and 3.

Table 2. Main geometry parameters of 6206.
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Table 3. Physical properties for simulation study.
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Except the above parameters, the clearances in this study was set from 0 to 100 µm with the interval as 10  µm, duo to the wear and tear on bearing leads to increase of clearance, which is realized through the change of diameter of races and balls. In the simulation, the shaft rotating speed was arranged as 500 rpm, 1000 rpm and 1500 rpm, while the external load on the bearing were set from 800 to 4800 N. Besides, the numerical simulation was solved by sub-function of ode15s in MATLAB 2018a.

4 Results and Analysis

4.1 Results

Taking the rotational speed of 1500 rpm, the radial load of 1600 N and the internal radial clearance of 20 \(\upmu {\text{m}}\) as an instance, Fig. 4 depicts the vibration waveform, frequency spectrum and envelope spectrum of the \(X_{r}\). As can be seen, both for bearing with and without local defects, the vibration accelerate displays impulse periodically, but with quite different amplitude. The amplitude of normal bearing is the lowest, while that of bearing with inner race fault is the highest. In addition, from the spectrum, resonant frequency bands could be clearly recognized. As can be seen from envelope spectrum, for bearings both without local defects and with outer race defect, it shows that BPFO and its harmonics are key modulation component. However, for bearing with inner race defect, ball pass frequency of inner race (BPFI), BPFO and rotating frequency (\(f_{r}\)) have the comparatively higher amplitude. The modulation is more complex than normal and outer race defect cases. It may be due to the different location of the inner race defect with the rotation. When the defect is in the load region of the bearing, the defect interacts with the balls and BPFI is significant. While in non-load region, the influence of BPFO exceeds BPFI. Under different rotational speed, the characteristic frequencies, including BPFO, BPFI and \(f_{r}\) of the bearing are shown in Table 4.

Table 4. Characteristic frequency of bearing 6206 under different speed.
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Fig. 4.
figure 4

Results of vibration acceleration (a, b, c), spectrum (d, e, f) and envelope spectrum (g, h, i) of the sensor in \(x\) direction.

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4.2 Comprehensive Analysis

In this section, comprehensive analysis is displayed from time domain and frequency domain. Firstly, statistical indicators from the time domain, such as RMS, kurtosis and entropy, under different radial clearance and eternal load are described in Fig. 5.

Fig. 5.
figure 5

RMS (a, b, c), kurtosis (d, e, f) and entropy (g, h, i) values under the case of normal (a, d, g), outer race defect (b. e, h) and inner race defect (c, f, i).

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Figure 5 depicts the effect of clearances and load on vibration responses in frequency domain. As can be seen, with the increase of clearances and load, the vibration RMS value shows an uptrend. However, for the normal case, there are several fluctuations from the RMS on both directions of clearances and loads. When there are local defects on outer race, the RMS presents better linear growth trend with the increase of clearances and loads, while RMS values change slightly with clearances under different loads. Besides, kurtosis and entropy fluctuate too much, which brings much difficulty for clearance monitoring of rolling element bearings.

On the other hand, FFT spectrum and envelope spectrum are effective tools from frequency domain to reveal the frequency components. These two spectra were carried out and shown in Fig. 6. It is noted that ‘800-0’ in Fig. 6 stands for 800 N and 0 μm.

Fig. 6.
figure 6

Spectrum and envelope spectrum of normal (a, d), outer (b, e) and inner (c, f) race defect

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From the envelope spectrum, characteristic frequencies of normal bearing and bearing with local defect was extracted, as shown in Fig. 7. As can be seen, when there is no local defect on races, even BPFO is dominant in envelope spectrum, there is no obvious law could be found based on the amplitude of BPFO. However, for bearing with outer race and inner race defect, BPFO and BPFI increases with clearances in general. Especially, BPFO shows an uptrend with the increase of loads in outer race defect case.

Fig. 7.
figure 7

Characteristic frequency of (a) normal, (b) outer race defect and (c) inner race defect

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But for inner race fault, there are many fluctuations form the BPFI amplitude, which brings difficulties for clearance monitoring and needs special care in detection. In the previous work (Ref. [17]), spectral centroid [22] was utilized to monitor the change of the clearances. The calculation of the spectral centroid is shown in Eq. (13). Based on the spectrum, the spectral centroid values are shown in Fig. 8. As can be seen, the uptrends are more clear than statistical indicators in time domain, especially for bearing with local defect, i.e. outer race and inner race defect.

$$ Centroid = \frac{{\mathop \sum \nolimits_{n = 0}^{N - 1} f\left( n \right)x\left( n \right)}}{{\mathop \sum \nolimits_{n = 0}^{N - 1} x\left( n \right)}} $$
(13)
Fig. 8.
figure 8

Spectral centroid of bearing (a) normal, (b) outer race defect and (c) inner race defect

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5 Conclusion

To investigate the influence of bearing radical clearance when there are local defects on bearing races, this study proposes a nonlinear dynamic model with six DOF, taking into continuously changing clearances and external load. The numerical results show that indicators in frequency domain outperforms statistical indicators in time domain, such as RMS, kurtosis and entropy. Especially, for bearing with local defect on outer and inner races, BPFO and BPFI shows an uptrend with the increase of bearing clearances in general, respectively. However, BPFI of inner race fault fluctuates throughout the clearance, which needs special care in monitoring. Besides, spectral centroid shows good performance on describing the change of bearing clearances except fluctuations in bearing without local defect. All these findings provide theory and data support for bearing clearances monitoring and fault diagnosis at an early stage.