Perspective Chapter: On Rolling Bearing Fault Feature Extraction Based on Entropy Feature

Abstract

In large machinery, the most common element we can use is rolling bearing. When the rolling bearing fails, it is very likely to affect the normal operation of the equipment, or even cause danger. Therefore, it is necessary to monitor and diagnose the bearing fault in advance. The most important step in fault diagnosis is feature extraction. In this paper, the approximate entropy, the sample entropy, and the information entropy are analyzed, and the feature is extracted to diagnose the rolling bearing fault. Firstly, the concepts of approximate entropy, sample entropy, and information entropy are introduced briefly, and the approximate entropy, sample entropy and information entropy of rolling bearing vibration signals under different fault modes are calculated. The feasibility and shortcomings of the features extracted from these three entropy in the fault characteristics of rolling bearing are analyzed. In order to make up for their defects, a method of fault feature extraction based on approximate entropy, sample entropy, and information entropy is proposed, and its feasibility is verified. Simulation experiments are carried out to calculate the accuracy of fault feature extraction based on the joint analysis of approximate entropy, sample entropy, and information entropy.

1. Introduction

1.1 Research background and significance

In economic construction, as the main production equipment, it is inevitable that large-scale machinery will fail because of factors such as work difficulty or service time. However, the economic losses and casualties caused by the failure will make people lament [1]. In the 1990s, a power plant turbine unit caused great damage to the Korean high-speed rail derailment, causing many injuries and even deaths. Rolling bearing, as an important rotating part, has a long-term high-speed rotation, coupled with the precise and complex structure, and the failure may be quite high. Reliable data show that the motor faults caused by bearing faults are very frequent [2].

In the large or small machinery in various fields, the position of rolling bearings in it is indispensable and can be said to play an important role in production and life. Rolling bearing fault is the most common fault of rotating machinery, and once it happens, it will seriously affect the normal work of the whole machinery, so it is very important to study the fault diagnosis technology of rolling bearing [3, 4, 5, 6]. For such an important rolling bearing, its fault diagnosis must be valued. Specific to the rolling body, inner circle, or external circle failures, we need to make subsequent improvements. If only one standard is conducted to detect and repair, not only the accuracy is low but also the cost of manpower and material resources [7]. If the fault diagnosis can be accurately conducted on prevention and moderate maintenance to avoid adverse effects. It will inevitably play a very important role in promoting the economic and social aspects [8].

1.2 Current status of fault diagnosis technology

Research on fault diagnosis of rolling bearings started around 1960. Overall, there can be divided into five stages [9].

The first stage is spectral analysis in the 1950s. Spectrum analysis methods have attracted attention. However, due to the immature technology at that time, the spectrum analysis was not widely used in the field of bearing fault diagnosis technology because the results were widely affected by the interference noise, expensive price, and complex operation.

In the second stage of the 1960s, the impact impulse meter detection method appeared, and the effect was obviously better than the spectral analysis, which could directly save the complicated steps, and was still widely used in the fault diagnosis of roller capital bearings.

The third stage in the 1960s to 1980s, computers and signals in the trend of The Times, the more prominent is resonance demodulation technology, because of the advent of this technology, makes the rolling bearing fault diagnosis technology to a higher level, gradually from the beginning to maturity.

After the 1980s, the emergence of artificial intelligence provided a new soil for the rolling bearing fault diagnosis and the emergence of an intelligent diagnosis system greatly improved the accuracy of the fault diagnosis. Due to intelligence, the influence of human factors is greatly reduced, which has been applied in engineering practice.

The fifth stage is after the 21st century, that is, we are now, the rolling bearing fault diagnosis technology has taken an epoch-making step, more and more high-tech development, through the fault diagnosis of virtual instruments, has become a new pointing mark, has an important practical value.

At present, around the world, we are constantly studying the rolling bearing fault diagnosis, using a large number of different research fields. According to the most popular classification methods, it is divided into three categories, namely, model-based fault diagnosis technology, knowledge-based fault diagnosis technology, and data-based fault diagnosis technology.

Because of its national conditions, China began to study fault diagnosis much later than that in other countries. In the late 1970s, it was not first available in the early 1980s and formal research began. However, it is gratifying that under the hard work of Chinese researchers, in the 1990s, the field of fault research has been on the right track, and it has made great breakthroughs in both theory and practice and can be applied in production and life. But compared with other countries, China still has a long way to go.

1.3 Main work of the paper

In this paper, we study joint analysis of rolling bearings based on approximate entropy, sample, and information entropy.

It roughly describes the historical background and practical significance of fault diagnosis research in the 21st century today and briefly expounds on the current situation of the global research on fault diagnosis.

The universal structure of the rolling bearing is introduced and the relevant parameters are marked in the plane structure diagram. The most common form of rolling bearing failure is described, and the characteristics and hazards of the bearing are also mentioned.

The cause and mechanism of the bearing vibration are expounded in detail. A theoretical method for calculating the characteristic frequency of the rolling bearing is presented. Bearing fault diagnosis experimental equipment (Western Reserve University) is introduced, and its relevant basic parameters and bearing fault setting form are introduced.

Concept definitions and calculations for approximate entropy sample and information entropy are given. A single entropy feature of the rolling bearing vibration signal in different failure modes is extracted, combined with data to verify the feasibility of approximate entropy and sample entropy in terms of failure features. The three entropy features are jointly analyzed to distinguish the different fault modes of bearings.

In the end, the full research work is summarized, the deficiencies are proposed, and the future research is preliminary.

2. Basic nature of the rolling bearing

2.1 Structure and failure forms

2.1.1 Basic structure and parameters

The most basic structure of the rolling bearing is the outer ring in the outside, and the inner ring inside, and in the middle through the rolling body like the ball or the column, so as to transform the sliding friction into the rolling friction. In general, the outer ring is fixed, and the inner ring is connected with the matched axis, and the holder is also a common structure of the rolling bearing, which can avoid the friction seen by the rolling body.

The quality of the rolling body is quite critical, and a good rolling body can well reduce the friction coefficient, reduce the friction, loss, and the bearing rotation will become smooth and efficient. The smoother the inner circle is, the less the friction is in contact with the axis, and the smoother the rotation will be. The outer ring plays a supporting role, hardness must be high, otherwise easy to wear, cause bearing off. The role of the holder is to fix the rolling body; the size must fit with the rolling body, too large and too small will cause adverse effects.

As shown in Figure 1, we can clearly see the structure of the rolling bearing through the 3-dimensional diagram, and the main parameters are marked in the plane structure diagram, including: inner circle radius$r_1$ , outer circle radius$r_2$ , rolling body diameter d, bearing joint diameter D, rolling body number Z, bearing contact angle θ.

Figure 1.

Drawing and plane structure of the rolling bearing.

2.1.2 Common failure forms

Rolling bearing, as a component part of the mechanical equipment, will inevitably produce a loss. Even if their own quality, including material processing and other aspects are very excellent, and there is no working condition failure, after a period of use, there will still be fatigue and wear, and other conditions. Master the common failure form of bearings, and can repair and replace the bearing elements in advance to extend the service life [10].

1. Fatigue peeling

   Under the repeated action of various forces, rolling bearing outer ring and inner ring, the maximum force part appear crack, surface metal may show point peeling, in serious cases even sheet peeling, this phenomenon, we call fatigue peeling. Fatigue peeling is the most common without special circumstances.

2. Surface plastic deformation

   The surface of the rolling bearing will form mechanical damage because of the action of pressure, or hard objects involved in the rolling bearing, and the damage continues to expand, which will cause vibration, noise, etc. so that the damage speed of the bearing is greatly accelerated. This phenomenon is surface plastic deformation.

3. Corrosion

   When the rolling bearing is working, the surface and internal metal and the substances in the environment, such as acid and alkali, or the consumption phenomenon caused by chemical reaction into water, that is corrosion.

4. Wear and tear

   The relative movement of the two contact metal surfaces is inevitable. In relative motion, friction, metal consumption, deformation, change the size of the rolling bearing, and then cause a change in performance. This phenomenon is wearing.

5. Wriggle

   Rolling bearing in the work, affected by the load, the inner ring and axis in rotation, in the circular direction, relative movement, on the metal surface of friction, wear and other abnormal damage, this phenomenon is peristalsis.

6. Scaling loss

   In the process of use of the rolling bearing, due to the excessive bad lubrication load and other factors, itself is affected by high temperature, and not timely cooling will make the element surface burn, serious, the probability of the rolling bearing stuck, this phenomenon is burning loss.

2.1.3 Category of rolling bearings

The most used part in the machinery industry is bearings, and because bearings are often needed in all walks of life, the categories of bearings are also very diverse. The bearing can be divided into multiple categories according to the rolling body shape, column number, and outer diameter size of the bearing [11].

According to the shape of the rolling body, it can be divided into the ball bearing and the ball bearing, the ball bearing is well understood, that is, the rolling body is the ball bearing, and the rolling body in the roller bearing is generally other types of rolling body such as the cylindrical roller.

If the rolling bearings are classified according to the bearing load direction, they can be divided into thrust bearings, centripetal bearing, and centripetal thrust bearing. The load direction of the thrust bearing is from the axial direction, while the center bearing mainly bears the radial load. And the centripetal thrust bearing is more powerful, can bear the load formed by the axial and radial combination, but because of this reason, the life of the centripetal thrust bearings is often shorter than other bearings.

At the same time, it can also be classified according to the number of rolling body columns, there are single column bearings, double column bearings, and multiple column bearings, the literal meaning can be understood, and it will not repeat.

Finally, it can also be classified by the size of the bearing outer diameter, from micro bearings less than 26 mm in diameter to a major class bearing greater than 2000 mm in diameter, divided into bearings of various specifications.

2.2 Vibration mechanism

1. Vibration caused by the structural characteristics

   Inherent vibration of the collar, vibration caused by the elastic characteristics of the bearing, vibration when the rolling body passes through the bearing area.

   As a mechanical element, vibration is inevitable, mainly reflected in the outer circle, and is determined by its own structural characteristics. When the rolling bearing is affected by the external force, the external circle vibration inevitably occurs.

   Rolling bearing in work, it is impossible to idling, must bear the load. Usually, the load is not small, which requires the rolling body to be very rigid. But large rigidity means that in special cases, the rolling experience produces a spring-like effect, producing vibration.

   The different number of rolling bodies serving as supports when passing through the carrying area also causes the inner circle ring to vibrate back and forth in the front and rear directions.

2. Vibration caused by the rolling bearing processing process

   In the processing process of the bearing production, due to the equipment accuracy and other problems, there will inevitably be an inner ring, and the outer ring will have a slight fluctuation. Only in the high-speed rotation, the effects of these fluctuations are also magnified, causing vibration.

   On the other hand, the uneven size of the rolling body is also one of the causes of the vibration, which will greatly reduce the service life of the bearing.

3. Vibration caused by component failure

   When the failure of the outer ring or the rolling body in the outer ring of the inner circle occurs, the vibration situation is also different. Different data can be obtained.

2.3 Feature frequency

To quantify the processing, the bearing fault characteristic frequency is given below. The rotation frequency of the ordering axis is$f_r$, the number of rolls is z, d is the diameter of ball, D is the diameter of rolling frame, and contact angle θ. When the outer ring of the rolling bearing is fixed, the theoretical calculation formula of the fault characteristic frequency of each element is as follows [12].

Inner ring fault frequency:

$$f_i=\frac{Z{f}_r}{2}\left(1-\frac{d}{D}cos\mathrm{\theta }\right)$$E1

Outer ring fault frequency:

$$f_o=\frac{Z{f}_r}{2}\left(1+\frac{d}{D}cos\mathrm{\theta }\right)$$E2

Ball failure frequency:

$$f_{\mathit{BS}}=\frac{D{f}_r}{2d}\left(1-{\left(\frac{d}{D}\right)}^2{cos}^2\mathrm{\theta }\right)$$E3

2.4 Diagnosis test of rolling bearing

2.4.1 Test device

Figure 2 shows the equipment used in the bearing data center of the rolling bearing data, with a 1kw motor on the left as the power provider, while in the center is the torque sensor, on the right is the power measuring motor, and the electronic control device.

Figure 2.

Experimental Equipment of Western Reserve University.

2.4.2 Basic parameters of rolling bearing fault

In this experiment, 6205-2RSJEMSKF bearing was used. The fault setting of the bearing was a single fault, and four damage degree faults were set on the inner ring, outer ring, and rolling body, respectively, namely 0.1778 mm, 0.3556 mm, 0.5334 mm, and 0.7112 mm. The rotational speed of the motor is 1797 r/min, 1772 r/min, 1750 r/min, and 1730 r/min, respectively.

This paper uses the SKF6205 bearing at a 12K sampling frequency, a fault diameter of 0.1778mm, a motor speed of 1750 r/min, and the vibration signal from the drive end bearing fault data of the acceleration sensor at the 6 o'clock position.

2.5 Influence and relationship between rolling bearing motion and chaos

2.5.1 Chaos phenomenon

With the advent of Lorentz nonlinear dynamical system, chaos has attracted more and more attention. As the representatives of nonlinear dynamical systems, Lorentz equation, and Lorentz-like equation have attracted the general attention of many scholars at home and abroad, and they have been deeply studied.

Lorentz system is a pioneer of chaos research. Chaos research based on Lorentz system can be divided into two independent methods, one is the study of the properties of equation solutions, numerical simulation by computer [13, 14, 15], and the other is chaotic water wheel physics experiment.

In Lorentz nonlinear dynamical system, the degree of randomness is generally determined by entropy. At present, in Lorentz nonlinear dynamical system, the determined entropy has no name, so we can call it Lorentz entropy for the time being.

The overall chaotic level of the system can be measured by the maximum Lyapunov exponent, which quantitatively describes the divergence rate of the phase volume exponent of the adjacent orbits of the system in the phase space. Although chaos is an irregular phenomenon, it comes from the deterministic system, so it is possible to predict it in a short term.

For Lorentz nonlinear dynamical systems, there is a relationship between Lorentz entropy and Lyapunov exponent. In chaotic one-dimensional mapping, a single Lyapunov exponent is consistent with Lorentz entropy.

An important quantitative method to judge whether the system is chaotic is whether there is a positive Lyapunov exponent. Lyapunov exponent is an important quantitative index reflecting the characteristics of dynamic system, which indicates the long-term average exponent of convergence or divergence between adjacent orbits of the system in phase space. For a time-delay dynamic system, its initial condition is a continuous function, so its Lyapunov exponent is related to the continuous function as the initial condition. The continuous function defined in the initial time period is uncountable, and so is the number of Lyapunov exponents of the system. Calculating Lyapunov exponents of time-delay dynamical systems is a complicated task. In the process of calculation, there may be some strange situations that make the results inaccurate. Therefore, judging whether the system is chaotic, as long as the maximum Lyapunov exponent is greater than zero, it can be used as a reliable basis for the existence of chaos.

The following diagram shows the transformation of the maximum Lyapunov exponent with parameters in the chaotic waterwheel experiment of Lorenz typical system. It can be seen that when the parameter reaches 15, the maximum Lyapunov exponent is positive, resulting in chaos.

2.5.2 Brief introduction of chaotic waterwheel device

As shown in Figure 3, the chaotic waterwheel device is similar to the ancient waterwheel, with a constant water flow at the top of the waterwheel injected into the water cup hanging on the edge of the wheel. There is a small hole at the bottom of each cup that can constantly discharge water. If the water flow speed on the top is very slow, the water in the top cup is small, so the friction force of the axle cannot be overcome, and the water wheel will not rotate; If the water flow speeds up, with the increase of water in the top cup, the water wheel will start to rotate at a constant speed; As the water flow continues to increase, the rotation will be chaotic, and the direction and speed of rotation will have complex motion characteristics due to the inherent nonlinearity of the system (Figure 4).

Figure 3.

Maximum Lyapnov exponential.

Figure 4.

Schematic diagram of chaotic water wheel experimental device.

2.5.3 Influence and relationship between rolling bearing motion and chaos

Rolling bearing is similar to chaotic water wheel device. The fault location will affect the rotation of rolling bearing, and the rotation will be chaotic. The direction and speed of rotation will have complex but regular motion characteristics due to the inherent nonlinearity of the system. Because of the different fault locations, the vibration signals presented by the rolling bearing rotation are different.

2.5.4 Entropy and chaos phenomenon

Entropy is a measure of disorder, in a chaotic system. The greater the entropy, the more chaotic the chaotic system is. The smaller the entropy, the more orderly the chaotic system is. Approximation entropy, sample entropy, and information entropy are all entropy, and as characteristics, they also affect the chaotic degree of chaotic phenomena.

2.6 Summary

The basic structure of the rolling bearing and the relevant parameters are introduced briefly. Six common rolling bearing failures are subsequently described. The cause of bearing vibration is clearly explained and the method of the characteristic frequency of inner ring, outer ring, and rolling body is given. We briefly introduce the western storage university bearing fault diagnosis equipment and state the data used in this paper.

3. Analysis of rolling bearing fault features based on approximate entropy, sample entropy, and information entropy

3.1 Nonlinear dynamic analysis of vibration signal

3.1.1 Time domain analysis

Figures 5–8 are the time domain signals of four working conditions of rolling bearing in turn.

Figure 5.

The normal bearing signal.

Figure 6.

The inner ring fault signal.

Figure 7.

The outer ring fault signal.

Figure 8.

The rolling body fault signal.

From the time domain image of the vibration signal of the rolling bearing, it can be seen that the vibration signals generated by the bearing under different working conditions are different, although there are differences, but they are not obvious. No matter whether the human eye or the computer can make an accurate judgment only by the vibration signal, the error is great. Therefore, it is necessary to dig deep into the vibration signal data and find out the characteristics of each working condition.

By calculating the entropy of the vibration signal of rolling bearing, its characteristics can be extracted well.

3.1.2 Fast Fourier Transform of power density spectrum

Fast Fourier transform (FFT), that is, the general name of the efficient and fast computing method of computing discrete Fourier transform (DFT) by computer, is abbreviated as FFT. Fast Fourier transform was proposed by J.W. Cooley and T.W. Tuki in 1965. By using this algorithm, the number of multiplications required by computer to calculate discrete Fourier transform can be greatly reduced. Especially, the more sampling points N are transformed, the more significant the saving of FFT algorithm is.

Fast Fourier transform (FFT) is a method to quickly calculate the discrete Fourier transform or its inverse transform of a sequence. Fourier analysis can convert the signal from the original domain (usually time or space) to the frequency domain for representation. For sequence $x\left(n\right)=\left\{x_0,x_1,\dots ,x_{N-1}\right\},0\le n<N$, the discrete Fourier transform expression is:

$$\widehat{x}\left(k\right)=\sum _{n=0}^{N-1}x\left(n\right)e^{-i\frac{2\mathrm{\pi }}{N}\mathit{nk}}$$E4

Figures 9–12 are the Fast Fourier Transform of vibration signals of the rolling bearing in four working conditions.

Figure 9.

Normal Working Condition Fast Fourier Transform.

Figure 10.

Inner Ring Fault Fast Fourier Transform.

Figure 11.

Rolling Element Fault Fast Fourier Transform.

Figure 12.

Outer Ring Fault Fast Fourier Transform.

It can be seen that the normal working conditions have obvious peaks at frequencies of 0,500 and 1000Hz, which are very regular. When the inner ring fails, the peak value is concentrated at 1000∼2000Hz. When the rolling element fails, the peak value is concentrated at 1500∼1800Hz. When the outer ring fails, the peak values are concentrated in 800∼1500Hz and 1700∼1800Hz. Therefore, FFT algorithm can not only diagnose signals without noise, but also apply to fault signals with noise.

Figures 13–16 are the calculation of the power spectrum function of vibration signals of the rolling bearing in four working conditions.

Figure 13.

Normal Working Condition Power Spectral Density Estimate.

Figure 14.

Inner Ring Fault Power Spectral Density Estimate.

Figure 15.

Rolling Element Fault Power Spectral Density Estimate.

Figure 16.

Outer Ring Fault Power Spectral Density Estimate.

Through the power density spectrum, it can be seen that the power density distribution range of vibration signals of rolling bearings in different working conditions is different, which means that the chaotic degree of this nonlinear system is different, and the entropy is naturally different, so it can be extracted as a feature.

The collected vibration signal is greatly influenced by noise, so the time domain signal cannot be directly extracted. From the analysis of the frequency domain curve, it can be seen that there are peaks at the characteristic frequencies of each working condition. Under normal working conditions, the peak amplitude obtained by fast Fourier transform is concentrated at a certain frequency. However, the vibration data of the inner ring fault, outer ring fault, and rolling element fault, and the amplitude peaks obtained by fast Fourier transform are scattered at various frequencies, showing different chaotic phenomena and different chaotic degrees.

However, the frequency components such as frequency conversion and frequency doubling are not obvious in the frequency domain curve. If the fault degree is light or the fault mode is complex, the characteristic frequency peak of each working condition is likely to be submerged in the noise, and it cannot be identified by the frequency domain diagram alone. However, different chaotic degrees mean different entropy, so entropy can be used as the fault characteristics of rolling bearings.

3.2 Approximate entropy

3.2.1 The concept of the approximate entropy

Approximate entropy ApEn is a non-linear kinetic parameter of sequence proposed by Pincus in 1991. ApEn reflects the degree of self-similarity of the sequence in the pattern [16].

The larger the ApEn value means that it is a complex sequence, and the less likely the system will be able to predict it. It gives cases where the incidence of new patterns increases with dimension, thus reflecting the structural complexity of the data.

From the above, we can know that the rolling bearing produces vibration, and the vibration signal are different in different failure modes. Depending on the physical meaning of ApEn, different signals imply different complexities that can be used as features for rolling bearing fault diagnosis.

3.2.2 Fast algorithm for the approximate entropy

When calculating approximate entropy, too much extra calculation is a waste of time. A fast algorithm to approximate the entropy is presented, exactly as follows:

Let the original sequence be {u (i), i = 0,1, …, N}, r = 0.1∼0.25SD (u) (SD indicates the standard deviation of the sequence {u (i)}), then the approximate entropy is more reasonable, select m = 2, N = 500–1000.

Calculated distance matrix D of $N\times N$. The element in line i and column j of D is marked as $d_{\mathit{ij}}$.

$$d_{\mathit{ij}}=\left\{\begin{array}{c}1\left|u\left(i\right)-u\left(j\right)\right|<r\\ 0\left|u\left(i\right)-u\left(j\right)\right|\ge r\end{array}\right.\mathrm{i}=1\sim \mathrm{N},\mathrm{j}=1\sim \mathrm{N},\mathrm{i}\ne \mathrm{j}$$E5

Using the elements in D, the and $C_i^2\left(r\right)C_i^3\left(r\right)$.

$$C_i^2\left(r\right)=\sum _{j=1}^{N-1}{d}_{\mathit{ij}}\cap d_{\left(i+1\right)\left(j+1\right)}$$E6

$$C_i^3\left(r\right)=\sum _{j=1}^{N-2}{d}_{\mathit{ij}}\cap d_{\left(i+1\right)\left(j+1\right)}\cap d_{\left(i+1\right)\left(j+2\right)}$$E7

Calculate $H^2\left(r\right)$ and$H^3\left(r\right)$ from $C_i^2\left(r\right)$ and $C_i^3\left(r\right)$, and finally, calculate approximate entropy.

$$\text{ApEn}\left(\mathrm{m},\mathrm{r},\mathrm{N}\right)=H^n\left(r\right)-H^{n+1}\left(r\right)$$E8

The approximate entropy of the sequence can be calculated from the above calculations.

3.2.3 Application of approximate entropy in mechanical fault diagnosis

Here, take six hundred vibration signals as a group and calculate ten groups of approximate entropy.

From Figure 17, it is not difficult to see that when the rolling bearing is normal, the value of the approximate entropy is not large, because, under normal circumstances, the generated signal is relatively single. When the rolling bearing fails, a lot of complicated information is generated, increasing the approximate entropy value. However, under the rolling body failure and normal working conditions, the two approximate entropy value is very similar, can not easily distinguish the two. If only a single entropy feature is used, it is easy to misjudge.

Figure 17.

Approximates the entropy curve.

3.3 Sample entropy

3.3.1 The notion of sample entropy

In 2000, the concept of sample entropy was first proposed by Richman et al., a similar but more robust time-series complexity metric to approximate entropy, with greater resistance to interference and noise compared to approximate entropy [17].

Sample entropy improves the algorithm of approximate entropy and can reduce the error of approximate entropy when calculating. It is an algorithm similar to the approximate entropy but with superior computational accuracy.

Sample entropy has a better agreement. That is, if a time series has higher values than another time series, it also has higher values for other m and r values.

3.3.2 Algorithm for sample entropy

Generally, r takes 0.1∼0.25SD (SD is the standard deviation of raw data), this paper$r=0.15\mathit{SD}$, When m = 2 is selected, N = 500–1000.

Assuming the data is $\left\{X_i\right\}=\left\{x_{1,}{x}_2,\dots ,x_N\right\}$, with its length N, a m-dimensional vector is reconstructed from the original signal:

$$x_i=\left[x_i,x_{i+1},\dots ,x_{i+m-1}\right],\mathrm{i}=1,2,\dots ,\mathrm{N}-\mathrm{m}$$E9

Defines the distance between $x_i$ and $x_j$:

$$d_{\mathit{ij}}{d}_{\mathit{ij}}=d\left[x\left(i\right),x\left(j\right)\right]=$$

$$\underset{k\in \left[0,m-1\right]}{max}\left[\left|x\left(i+k\right)-x\left(j+k\right)\right|\right]i,j=1,2,\dots ,N-m,i\ne j$$E10

Then find the average value of $B_i^m\left(r\right)$

$$B^m\left(r\right)=\frac{1}{N-m}\sum _{i=1}^{N-m}{B}_i^m\left(r\right)$$E11

Similarly, again for the dimension m + 1, repeat the above steps, to obtain, and further obtain the final definition of the sample entropy, when N is a finite number

Similarly, repeat the above steps for dimension m+1 to obtain $B_i^{m+1}\left(r\right)$, and further obtain the final definition of sample entropy of $B^{m+1}\left(r\right)$. When n is finite,

$$\mathit{\text{SampEn}}\left(m,r,N\right)=ln{B}^m\left(r\right)-ln{B}^{m+1}\left(r\right)$$E12

The sample entropy of the sequence data is obtained from the above calculations.

3.3.3 Application of sample entropy in mechanical fault diagnosis

The first 6,000 data were taken in 600 sets to calculate the sample entropy, as shown in Figure 18.

Figure 18.

Sample entropy curves.

From Figure 18, it is not difficult to see that the inner circle fault, the roll body fault, and the normal working conditions are very difficult to distinguish, but the outer circle fault can be distinguished.

3.4 Information entropy

3.4.1 The concept of information entropy

Information entropy is mostly used as a quantitative indicator of the information content of a system. The information entropy can be further used as a criterion for the optimization of the system equations [18].

3.4.2 Algorithm for information entropy

$$H\left(X\right)=-\sum p\left(\mathit{xi}\right)log\left(p\left(\mathit{xi}\right)\right)\left(i=1,2,\dots ,n\right)$$E13

X represents the random variable$\left(x1,x2,\dots \mathit{xn}\right)$, the value of which is, and p (xi) indicates the probability of an event xi.

3.4.3 Application of information entropy in mechanical fault diagnosis

The first 6000 data were taken in 600 sets and the information entropy was calculated in Figure 19.

Figure 19.

Information entropy curve.

As we can see from Figure 19 the rolling body fault and the normal working condition intersect, and there are several very close places between the inner circle fault and the outer circle fault. If the information entropy is not processed, it is difficult to distinguish the rolling body fault and the normal working condition, and it is also easy to misjudge the inner circle fault and the outer ring fault.

3.5 Approximate entropy, sample entropy, information entropy, and maximum Lyapunov exponent

Taking 6000 data from each of four working conditions randomly, and taking 600 data as a group, the approximate entropy, sample entropy, information entropy, and maximum Lyapunov exponent are calculated. See Tables 1–4 for specific results.

	Approximate entropy	Sample entropy	Information entropy	Maximum Lyapunov exponent
1 group	0.0145	1.5295	2.7995	0.8572
2 group	0.0122	1.6364	2.944	1.1640
3 group	0.0122	1.6746	2.9223	1.0358
4 group	0.0099	1.5549	2.6987	0.8828
5 group	0.0284	1.5904	2.9571	1.0570
6 group	0.0168	1.6752	2.6942	0.8198
7 group	0.0122	1.5137	2.7286	0.8005
8 group	0.0214	1.6044	2.7931	0.9247
9 group	0.0214	1.6455	2.6883	0.4819
10 group	0.0145	1.6316	2.8451	0.9472

Table 1.

Normal working condition.

	Approximate entropy	Sample entropy	Information entropy	Maximum Lyapunov exponent
1 group	0.3906	1.8951	2.2813	1.1505
2 group	0.3550	1.8330	2.3621	1.1941
3 group	0.3812	1.7537	2.2053	1.1570
4 group	0.3512	1.8701	2.2987	0.9440
5 group	0.3144	1.8343	2.1981	0.8531
6 group	0.3608	1.9440	2.2503	0.9038
7 group	0.3259	1.8736	2.3636	0.9706
8 group	0.3903	1.8948	2.1622	0.8296
9 group	0.3767	1.8621	2.2511	0.9782
10 group	0.3590	1.6874	2.2865	0.7633

Table 2.

Inner ring fault.

	Approximate entropy	Sample entropy	Information entropy	Maximum Lyapunov exponent
1 group	0.0400	1.9635	2.9744	1.3315
2 group	0.0649	1.8153	2.8584	1.2093
3 group	0.0449	1.9966	2.8589	1.5054
4 group	0.0644	1.9115	2.7397	1.3533
5 group	0.0547	1.7711	3.0546	1.4797
6 group	0.0487	1.7368	2.7809	1.2739
7 group	0.0796	1.6927	2.9196	1.8167
8 group	0.0519	2.0092	2.7850	1.4544
9 group	0.0547	2.0074	2.9782	1.2152
10 group	0.0492	2.0193	2.7702	1.3233

Table 3.

Rolling element fault.

	Approximate entropy	Sample entropy	Information entropy	Maximum Lyapunov exponent
1 group	0.9484	1.1286	1.9842	1.0986
2 group	0.8777	1.1193	1.9447	0.8310
3 group	0.8442	0.9298	2.1366	0.5168
4 group	0.8804	1.0431	2.0030	0.8996
5 group	0.9411	1.1166	1.9253	0.9002
6 group	0.9405	1.1085	1.9136	0.6443
7 group	0.9071	1.0950	2.0648	0.7009
8 group	0.8832	1.1136	2.0648	0.5603
9 group	0.9296	1.0335	1.8442	0.9588
10 group	0.8982	0.9782	2.0407	0.9048

Table 4.

Outer ring fault.

It can be seen from the above data that there is a certain correlation between the maximum Lyapunov exponent and entropy, but it can only show that the rolling bearings are in different degrees of chaotic systems under various working conditions. The maximum Lyapunov exponent has little discrimination in various working conditions, so it is still difficult to diagnose the fault of rolling bearings.

At the same time, we can see that the entropy with the highest correlation with the maximum Lyapunov exponent is also different under different working conditions of rolling bearings. For example, when the outer ring fails, the sample entropy can better improve the chaos phenomenon, but it is quite different from the maximum Lyapunov exponent under normal conditions.

In order to improve the chaotic phenomenon and diagnose the fault of rolling bearing, this paper does not use single entropy as the feature but uses approximate entropy, sample entropy, and information entropy as the fault feature to diagnose the fault of rolling bearing.

3.6 Joint analysis

3.6.1 Feature extraction

The three entropy have considerable disadvantages under independent judgment, and it is easy to appear misjudgment, which needs to be further handled. So I began to extract the fault features of rolling bearings by jointly analyzing the approximate entropy, sample entropy, and information entropy.

The data are taken from the Western Reserve University, with a vibration acceleration signal in four different modes under a load of 2 horsepower, a fault diameter of 0.1778 mm, and a rotational speed of 1750 r/min. A total of four groups correspond to different patterns, with 60000 data, 10 segments, 6000 data per segment, with 10 segments, and 600 data per segment.

Each section finds an approximate entropy, sample entropy, information entropy, ten entropy values for each group, you can obtain ten approximate entropy mean, sample entropy mean, and information entropy mean. See Figures 20–22.

Figure 20.

Approximates the entropy mean.

Figure 21.

Mean sample entropy.

Figure 22.

Information entropy mean.

From the above figure, it is not difficult to see that some working conditions are still difficult to distinguish if you want to pass a single entropy feature. Approximate entropy has certain errors in the diagnosis of normal working condition and outer working condition, sample entropy has certain errors in the diagnosis of inner ring fault and rolling element fault, and information entropy has certain errors in the diagnosis of normal working condition and rolling element fault.

Therefore, as long as the three entropy means are extracted in each fault mode, four sets of column vectors are formed, such as Table 5, and each column corresponds to the entropy feature vector under the working condition.

	Normal working condition	Outer ring fault	Inner ring fault	Ball fault
Approximate entropy mean	0.0164	0.9050	0.3605	0.0553
Sample entropy mean	1.6056	1.0666	1.8448	1.8923
Information entropy	2.8071	1.9922	2.2659	2.8720

Table 5.

Entropy feature matrices.

The same entropy feature column vectors obtained from the same test data form the entropy feature matrix of the test data.

Based on the average entropy feature vector, we compare it with the test data entropy feature vector, that is, we take the absolute value after making the difference to obtain four new vectors.

The four new vectors are formed into an entropy feature matrix, and the minimum of each row in the matrix is taken. The largest number of columns corresponding to the minimum is the judged test data closest to the failure mode.

However, in rare cases, three entropy features will determine the three failure modes. At this time, the failure mode corresponding to the approximate entropy mean with the greatest discrimination will be taken as the final failure mode of the test data.

3.6.2 Simulation experiment

After the early extraction of the rolling bearing fault characteristics, we have extracted the approximate entropy mean, sample entropy mean and information entropy mean in the four-fault modes of the rolling bearing. We can effectively distinguish the four-fault modes by comparing the feature vectors.

A matrix with a data quantity of 6000 was randomly generated using the data, and the fault features were extracted and determined to be a working condition. Later, the results were compared with the previous established standard, each group was tested 500 times, at least 10 groups, the test accuracy rate was based on the above description, and the simulation experiment began.

Also taken from Western Reserve University, the vibration acceleration signal in four different modes under a load of 2 HP, fault diameter of 0.1778 mm, and rotational speed of 1750 r/min.

The data of 6000 outer circle faults from the last 60000 data were divided into ten sections and 600 data for each section to calculate approximate entropy, sample entropy and fuzzy entropy mean to form a column vector.

The specific process is shown in Figure 23.

Figure 23.

Test process.

Choose any one of the four random working conditions as the test condition, extract 6000 vibration signals, and calculate the approximate entropy, sample entropy, and information entropy in groups of 600, and calculate the average value. The entropy mean is arranged as a test vector. The test vector is different from the corresponding components in the four working condition feature vectors, and the absolute value is taken. Then the test condition is the same as the condition corresponding to the feature vector with the smallest absolute value difference of each component. For the convenience of understanding and observation, the difference between the test vector and the feature vector group is visualized, resulting in Figure 24.

Figure 24.

Entropy difference 1.

It can be clearly seen from the figure that the absolute value of the difference between the yellow line characteristic vector corresponding to the inner ring fault and the test vector is the smallest, so the test condition is the inner ring fault (Figure 25).

Figure 25.

Entropy difference 2.

It can be clearly seen from Figure 16 that the absolute value of the difference between the purple line feature vector corresponding to the rolling element fault and the test vector is the smallest, so the test condition is the rolling element fault.

3.6.3 Comparative Experiment

In order to further prove the effectiveness of this method, a comparative experiment with the single entropy feature was carried out. The accuracy of the method using triple entropy combined features is 98.28%. The accuracy of the method characterized by single approximate entropy is 89.64%. The accuracy of the method characterized by single sample entropy is 79.80%. The accuracy of the method characterized by single information entropy is 71.96%. The four methods were repeated ten times respectively. Table 6 shows the accuracy and average results of the experiment.

	Approximate entropy	Sample entropy	Information entropy	Present method
1	90.80%	80.00%	71.60%	99.00%
2	89.60%	80.60%	75.80%	97.40%
3	91.00%	79.20%	70.00%	98.80%
4	86.80%	77.80%	74.20%	99.00%
5	89.60%	81.40%	71.20%	97.80%
6	91.20%	79.80%	70.60%	98.60%
7	90.00%	79.00%	72.40%	98.80%
8	89.40%	76.20%	69.00%	98.00%
9	89.20%	82.40%	71.60%	97.60%
10	88.80%	81.60%	71.20%	97.80%
Mean value	89.64%	79.80%	71.76%	98.28%

Table 6.

Comparison of accuracy of four methods.

The accuracy of the triple entropy combination feature method is 8.64% higher than that of the single approximate entropy method, 18.48% higher than that of the single sample entropy method, and 98.28% higher than that of the single information entropy method. It can be seen that the method of triple entropy combination features can effectively diagnose the rolling bearing faults.

3.7 Test results

Through many experiments, the accuracy rate of this method is above 97%, which is higher than other methods of the same type. The joint analysis of rolling bearing fault feature extraction based on approximate entropy, sample entropy, and information entropy is effective.

4. Conclusion

The chaotic system of rolling bearing is discussed, and the vibration signal of rolling bearing is changed by fast Fourier. It is found that different chaotic phenomena appear in different working conditions. Therefore, a fault diagnosis method of rolling bearings based on entropy features is proposed, and the relationship between approximate entropy, sample entropy, information entropy, and the maximum Lyapunov exponent is studied. It is concluded that single entropy can’t improve the chaotic phenomenon. Furthermore, a fault extraction method of rolling bearing combining approximate entropy, characteristic entropy, and information entropy is proposed. The feasibility of this method is proved by the comparative experiment of single entropy and approximate entropy sample entropy information entropy joint analysis. In the final test, the classification accuracy reached 98.28%. The feasibility of this method is proved, and four working conditions of rolling bearings are diagnosed with high precision.

Nomenclature

$c$	Radius of rolling bearing inner ring
$r_2$	Radius of rolling bearing outer ring
D	Rolling element diameter of bearing
D	bearing joint diameter
Z	Number of rolling bodies
Θ	bearing contact angle
$f_r$	Rotation frequency of rolling bearing
$f_i$	Inner ring fault frequency
$f_o$	Outer ring fault frequency
$f_{\mathit{BS}}$	Ball failure frequency
E	Base of natural logarithm
I	Imaginary unit