Credibility in evaluating on-line wear-depth detection of self-lubricating spherical plain bearings

Abstract

The wear-depth detecting precision of self-lubricating spherical plain bearings directly affects the accuracy of evaluating on the tribological properties of bearings. However, because of the varying environmental factors, the detecting errors in the on-line experimental data of the wear depth affect the accuracy of the detecting data. In this article, we come up with the definition of the credibility of the on-line wear-depth detection of spherical plain bearings on the similarity between the on-line wear depth data and the conventional true value of the wear depth and present two evaluating indexes of the credibility of the on-line wear-depth detection according to the features of spherical plain bearing testers—the small sample, the dynamic measurement, and the complex operating conditions with much interference. The two indexes are the index I, the similarity between the on-line detecting curve and the off-line detecting curve using the algorithm of the discrete Fréchet distance, and the index II, the fluctuating amplitude of the on-line wear-depth detecting curve through adopting the computing method of the profile arithmetic average error of the surface roughness. Finally, we use the above evaluating indexes to analyze the comparative study of the detecting credibility for the same tester of spherical plain bearings before and after the thermal error compensation.

Introduction

The self-lubricating spherical plain bearings are special journal bearings which inlay or bond the self-lubricating solid materials between the inner and outer races. These bearings have some special characteristics, such as maintenance free, non-lubricating, and low coefficient of friction. Therefore, they are widely applied to aviation and aerospace fields.^1,2

In the process of operating spherical plain bearings, the safety of aircrafts is affected by the tribological properties and the life of bearings. So, the performance of spherical plain bearings should be evaluated by spherical plain bearing testers. Meanwhile, three performance parameters of the bearings (the wear depth, the friction torque, and the friction temperature) need to be detected in real time. Particularly, the wear depth is the most important parameter for the tribological properties and the life of self-lubricating spherical plain bearings.

Now, the on-line detecting method is used for the wear-depth detection of self-lubricating spherical plain bearings, which means the bearing testers constantly detect and record the varying wear depth in wear experiment. However, there are five principal error factors affecting the detection precision: the thermal errors, the loading errors, the fluctuating errors, the positioning errors, and the detecting interference errors. These factors have an effect on the on-line detecting precision of the wear depth.

At present, with regard to the detecting errors of commercialized and serialized measuring instrument, we use accuracy to define the closeness between the detecting value of measuring instruments and the true value and use different levels of accuracy to evaluate the detecting precision of measuring instruments. However, there are few research reports on the detecting errors of the small sample spherical plain bearing testers. Due to the existing detecting errors, we are not sure whether the wear-depth detecting data and curves are the real experimental results through the on-line detection of bearing testers. Meanwhile, there is no way to calculate the closeness between the wear-depth detecting data of bearing testers and the real one.

Based on the above-mentioned explanation, we put forward the definition of the credibility of the on-line wear-depth detection of spherical plain bearings. At the same time, according to the features of spherical plain bearing testers, we present two evaluating indexes of the credibility of the on-line wear-depth detection. Finally, we use the above evaluating indexes to analyze the comparative study of the detecting credibility for the same tester of spherical plain bearings before and after the thermal error compensation.

Credibility of the on-line wear-depth detection of spherical plain bearings

In the dictionary, the credibility means “the quantity that someone or something has that makes people believe or trust them” and also means the degree of uncertainty of the results. In the engineering, the credibility and its evaluating methods are widely used in the modeling and simulation (verification, validation, and accreditation (VV&A)),^3–5 the computer security authentication and access control, ⁶ the communication quality evaluation, ⁷ and so on. Essentially, the credibility is the closeness between the output values and the true values (or the standard values) in the engineering field. So, the definition of the credibility of the on-line wear-depth detection of spherical plain bearings is as follows.

The credibility of the on-line wear-depth detection of spherical plain bearings is the closeness between the displayed wear-depth data of the on-line detection of spherical plain bearing testers and the real one under the specified operating condition. Meanwhile, the credibility is a qualitative concept, and the on-line detecting credibility of bearing testers needs to be calculated upon the different degrees of credibility.

Evaluating methods of the credibility of the on-line wear-depth detection of spherical plain bearings

There are three features of spherical plain bearing testers: the small sample (the number of the testers is few), the dynamic measurement (the parameters of bearings are detected and recorded in real time), and the complex operating conditions with much interference (the experiments simulate the real operating condition, such as high- and low-temperature environment or the salt-spray environment, and they may take a long time). Therefore, the evaluating method of the accuracy of commercialized and serialized measuring instruments does not apply to the credibility evaluation of the on-line wear-depth detection of spherical plain bearings. In the following, we will present the evaluating indexes of the degree of credibility of the on-line wear-depth detecting data.

In the process of the tribological experiments of spherical plain bearings, the wear-depth data of the on-line detection on the spherical plain bearing tester are displayed in the form of the time-varying and fluctuant curve. The researchers could evaluate the tribological properties and the life of spherical plain bearings on the basis of the varying tendency and the range of this on-line detecting curve.

Meanwhile, there is a reference standard or the conventional true values for the credibility evaluation, namely, the real wear depth of the spherical plain bearing in the tribological experiment. The credibility evaluation of the on-line wear-depth detection of the spherical plain bearing is to compare the on-line displayed data with the conventional true values and distinguish the high from the low credibility of the on-line detecting data through the closeness of the above two sets.

In conclusion, on the basis of different degrees of credibility of the on-line wear-depth detecting data of spherical plain bearings, we present two evaluating indexes: the similarity between the on-line and the off-line detecting curves and the fluctuating amplitude of the on-line detecting curve. We will explain the two indexes in the following.

The index I—the similarity between the on-line and the off-line detecting curves

The on-line detecting curve is a fluctuant detecting curve showed by the tester in real time during the process of the wear experiment. This curve reflects the wear state of spherical plain bearings, so we could evaluate the tribological properties and the life of spherical plain bearings on the basis of the varying tendency and the range of this curve.

The off-line detecting curve—the real wear depth of spherical plain bearings—cannot be precisely measured at present during the process of the credibility evaluation comparing with the on-line detecting data. We will approximately get the conventional true value of the wear depth of the spherical plain bearing through measuring the radial clearance of it, that is, we use the variation of the radial clearance of the spherical plain bearing to replace the real wear depth of the bearing. Then, we continually collect the variation of the radial clearance of the spherical plain bearing and connect these variations according to the run time of the on-line detecting so as to get a smooth curve of the off-line wear-depth detection of the spherical plain bearing.

Now, we obtain two wear-depth curves: one is the on-line wear-depth detecting curve of spherical plain bearings and the other is the off-line wear-depth detecting curve of spherical plain bearings (the changing curve of the radial clearance of spherical plain bearings). Meanwhile, the principal error factors cause the deviation between the on-line and the off-line detecting curves of the wear depth, and then, the similarity degree of the two curves should be evaluated by the index I. The high similarity of these two curves means high closeness between the on-line and the off-line detecting curves and high degree of credibility of the on-line wear-depth detecting data and vice versa.

Form the above, the degree of credibility of the on-line wear-depth detecting data converts to the similarity between the on-line and the off-line detecting curves. At present, the common method of evaluating the similarity of two curves is the Fréchet distance.^8–10 The Fréchet distance is a distance measure for continuous shapes. ¹¹ Meanwhile, the higher the similarity of two curves, the smaller the Fréchet distance.

In this article, we use the discrete Fréchet distance to calculate the similarity of the two wear-depth curves. Intuitively, the discrete Fréchet distance between the two curves could be understood as the dog-leash distance, that is, a man with his dog walk without backtracking along their respective paths from one endpoint to the other. Every step of the man makes his path A, and every step of the dog forms its path B. So, the distance of the dog leash changes with the different positions between the man and the dog in the walking process. The discrete Fréchet distance is defined as the minimum length of a leash connecting the man and the dog, and meanwhile, the distance between the man and the dog is smaller than this minimum length as they move forward.^12–15

According to the above explanation, we can only get the discrete Fréchet distance through calculating the every discrete point one by one on the curves A and B, respectively. If the data of the experiment are large, the computing efficiency will be low. At the same time, we can express the feature and configuration of the curve by the characteristic points on the curve, such as the extreme points and the inflection points. Therefore, in order to improve the computing efficiency, we only need to calculate the important characteristic points on two curves without calculating every point one by one on two curves. In other words, we can evaluate the similarity of two curves so as to calculate the discrete Fréchet distance of characteristic points on two curves. In this article, we choose the maximum value points of the on-line detecting curve, the minimum value points of the on-line detecting curve, and the sampling points of the off-line detecting curve as the characteristic points of the discrete Fréchet distance calculation. Through calculating the discrete Fréchet distance of these three kinds of points, we can obtain ${\mathrm{d}}_1(\mathrm{A},\mathrm{B})$ —the discrete Fréchet distance of the maximum value points of the on-line detecting curve, ${\mathrm{d}}_2(\mathrm{A},\mathrm{B})$ —the discrete Fréchet distance of the minimum value points of the on-line detecting curve, and ${\mathrm{d}}_3(\mathrm{A},\mathrm{B})$ —the discrete Fréchet distance of the sampling points of the off-line detecting curve. Finally, we obtain the similarity of the curves A and B through calculating the average value of the above three discrete Fréchet distances, and the calculating formula is as follows

$$\mathrm{d}(\mathrm{A},\mathrm{B})=\frac{{\mathrm{d}}_1(\mathrm{A},\mathrm{B})+{\mathrm{d}}_2(\mathrm{A},\mathrm{B})+{\mathrm{d}}_3(\mathrm{A},\mathrm{B})}{3}$$

(1)

The index II—the fluctuating range of the on-line wear-depth detecting curve of spherical plain bearings

In the wear experiment of spherical plain bearings, the on-line wear-depth detecting curve of spherical plain bearings is a fluctuant and time-varying curve due to the electromagnetic interference, the unstable dynamic characteristics of the sensor, and the vibration of mechanical systems. If the fluctuating range of the on-line wear-depth curve is large, the experimental results will affect the accuracy of the displayed data and further influence the degree of credibility of the on-line wear-depth data.

We can conclude from the above that the stochastic fluctuation of the on-line wear-depth detecting curve affects the credibility of the on-line detection. So, we need to evaluate the fluctuating range of the on-line wear-depth detecting curve, so as to identify the influence of the fluctuant curve on the degree of credibility of the detecting data. On the basis of the fluctuant shape of the on-line detecting curve, the stochastic fluctuation of the on-line detecting curve is similar to the surface roughness of machine parts. At this moment, we take the concept of the surface roughness on the machine parts, namely, the microscopic surface irregularity of the workpiece. In addition, we introduce a characterization parameter of the surface roughness—the profile arithmetic average error ${\mathrm{R}}_{\mathrm{a}}$ . The formula of the profile arithmetic average error ${\mathrm{R}}_{\mathrm{a}}$ is ¹⁶

$${\mathrm{R}}_{\mathrm{a}}=\frac{1}{\mathrm{L}}\underset{0}{\overset{\mathrm{L}}{\int }}|\mathrm{z}-\mathrm{m}|\mathrm{dx}$$

(2)

where L is the sampling length, z is the distance between every point on the rough surface and the datum line, m is the average value of the distance between every point on the rough surface and the datum line, and its formula is

$$\mathrm{m}=\frac{1}{\mathrm{L}}\underset{0}{\overset{\mathrm{L}}{\int }}\mathrm{z}\mathrm{dx}$$

(3)

where z is also the distance between every point on the rough surface and the datum line.

On calculating the fluctuant range of the on-line wear-depth detecting curve, by aiming at the on-line wear-depth detecting curve, we first get a smooth curve through the moving average method in the middle of the fluctuating detecting curve, and based on this curve, we then calculate the average value of the absolute value of the distance between every point on the on-line detecting curve and the smooth curve. Finally, we take this average value as the evaluation index of the fluctuating range of the on-line wear-depth detecting curve.

From the above, we get two evaluating indexes of the degree of credibility of the on-line wear-depth detecting data of spherical plain bearing testers: the index I—the similarity between the on-line and the off-line detecting curves and the index II—the fluctuating range of the on-line wear-depth detecting curve of spherical plain bearings. In the following, we will evaluate the credibility of the on-line detection of testers and explain the computing process of these two indexes by the example.

Evaluating example of the degree of credibility of the on-line wear-depth detecting data

In order to improve the degree of credibility of the detecting data, we use the error compensating method to offset the errors during the experiment. In the following, we will demonstrate the differences of the detecting credibility before and after the thermal error compensation.

Of many causes triggering the detecting errors, the thermal errors caused by the varying environment temperature are the main factors. ¹⁷ How to reduce the thermal errors of the bearing tester has important implications for the evaluation of self-lubricating spherical plain bearings.

Based on the self-made modular spherical plain bearing tester, we studied the credibility of the on-line detection before and after the thermal error compensation. The picture of the spherical plain bearing tester is shown in Figure 1, and the experimental processes are introduced in the following.

Figure 1.

Picture of the modular self-lubricating spherical plain bearing tester.

The introduction of the experimental specimens and the detecting instrument: the experimental spherical plain bearings were provided by the Bearings Technology Institute of Shanghai in China, and the bearing designation was MS14102-9. The picture of the experimental spherical plain bearings is shown in Figure 2. Meanwhile, we used the self-made high-precise detecting instrument of the spherical plain bearing clearance to detect the bearing radial clearance. The picture of this self-made detecting instrument of the bearing clearance is shown in Figure 3.

Figure 2.

Picture of the experimental self-lubricating spherical plain bearings.

Figure 3.

Picture of the detecting instrument of the spherical plain bearing clearance.

Experimental processes of the credibility evaluation before the error compensation

• Step 1. Before the wear experiment of the self-lubricating spherical plain bearing, we needed to detect the original radial clearance at the selected location of the spherical plain bearing by the self-made detecting instrument of the bearing clearance. So, we obtained the original reference ${\delta }_0$ of the radial clearance.

• Step 2. We put the experimental spherical plain bearing into the bearing fixture module and ensured the selected location of the bearing in the previous step was as the wear area. Next, we conducted the first experiment under the low-speed oscillating and heavy-loading operating conditions (the testing load was 250 MPa, the oscillating frequency was 0.2 Hz, and the angle of oscillation was ±25°).

• Step 3. Through 1.5 h wear experiment, we collected the first set of the on-line wear-depth detecting data ${\theta }_1$ . Subsequently, we removed the experimental bearing and then detected the bearing radial clearance at the same selected location. And we obtained the first bearing radial clearance ${\delta }_1$ after the first wear experiment.

• Step 4. After detecting the first bearing radial clearance ${\delta }_1$ , we put the experimental bearing, within the same installation location, into the bearing fixture module again and conducted the second experiment under the low-speed oscillating and heavy-loading operating condition.

• Step 5. By repeating the above experimental process, we conducted 12 wear experiments together and detected 12 bearing radial clearance detections. So, we obtained 12 sets of the on-line detecting data ( ${\theta }_1$ , ${\theta }_2$ , …, ${\theta }_{12}$ ) and 12 sets of the bearing radial clearance ( ${\delta }_1$ , ${\delta }_2$ , …, ${\delta }_{12}$ ). The original reference ${\delta }_0$ and the 12 wear experimental data of the radial clearance are shown in Table 1.

Table 1.

Original reference ${\delta }_0$ and the 12 wear experimental data of the radial clearance before the thermal error compensation.

Time	${\delta }_0$	${\delta }_1$	${\delta }_2$	${\delta }_3$	${\delta }_4$	${\delta }_5$	${\delta }_6$	${\delta }_7$	${\delta }_8$	${\delta }_9$	${\delta }_{10}$	${\delta }_{11}$	${\delta }_{12}$
Clearance (mm)	0.093	0.116	0.1 35	0.144	0.165	0.173	0.188	0.193	0.199	0.205	0.207	0.209	0.215

Connecting preprocessing of the experimental data

After 12 on-line and off-line detections of the wear depth, we obtained 12-set detecting data, respectively. In the following, we separately connected the 12-set on-line and the 12-set off-line detecting data and obtained the on-line wear-depth detecting curve and the off-line wear-depth detecting curve of the spherical plain bearing.

The preprocessing of connecting the on-line wear-depth detecting data: since the self-lubricating spherical plain bearing were marked at the selected location, the 12-group wear experiments were all conducted at the same wear zone, hence the wear depth of the selected location was the accumulated value of every experimental data. So, the total on-line wear depth ${\theta }_{\mathrm{T}}$ after the 12-group wear experiments was the superposition of the relative variation of the wear depth in every wear experiment. The on-line total wear depth ${\theta }_{\mathrm{T}}$ of the 12-group on-line wear experiments was

$${\theta }_T={\theta }_1+{\theta }_2+{\theta }_3+\cdots +{\theta }_{12}$$

(4)

The preprocessing of connecting the off-line wear-depth detecting data: from the above, the 12 radial clearance detections all aimed at the selected wear zone of the spherical plain bearing. Before the wear experiment, we detected the original radial clearance ${\delta }_0$ of the spherical plain bearing. After the first wear experiment, we detected the radial clearance ${\delta }_1$ of the bearing at the same location, so we obtained the relative variation of the radial clearance $\mathrm{\Delta }{\delta }_1$ of the spherical plain bearing at the selected location, namely, $\mathrm{\Delta }{\delta }_1={\delta }_1-{\delta }_0$ . From the above, we used the relative variation of the radial clearance to replace the real wear depth of the spherical plain bearing and thus obtained the conventional true value of the wear depth of the bearing after the first wear experiment which was $\mathrm{\Delta }{\delta }_1={\delta }_1-{\delta }_0$ . Similarly, after 12 wear experiments, the conventional true value of every wear experiment was the difference between the radial clearance of the spherical plain bearing in this experiment and the previous one, that is, $\mathrm{\Delta }{\delta }_{\mathrm{i}}={\delta }_{\mathrm{i}}-{\delta }_{\mathrm{i}-1}$ (i = 1, …, 12). Therefore, the total off-line wear depth ${\delta }_T$ after 12 wear experiments was the superposition of the relative variation of the radial clearance in every radial clearance detecting. The total wear depth ${\delta }_T$ of 12 off-line wear experiments was

$${\delta }_T=\mathrm{\Delta }{\delta }_1+\mathrm{\Delta }{\delta }_2+\mathrm{\Delta }{\delta }_3+\cdots +\mathrm{\Delta }{\delta }_{12}$$

(5)

In conclusion, we obtained a fluctuant on-line detecting curve and a smooth off-line detecting curve after connecting the 12-group on-line wear-depth detecting data and the 12-set off-line wear-depth detecting data. As a result, under the low-speed oscillating and heavy-loading operating condition, the on-line and the off-line wear-depth detecting curves without the thermal error compensation are shown in Figure 4.

Figure 4.

Curves of the on-line wear depth and the off-line wear depth before compensation.

Experimental processes of the credibility evaluation after the error compensation

In order to improve detecting credibility of the spherical plain bearing tester, we adopted the mechanical compensating method to offset the thermal errors of the wear-depth detecting system. ¹⁸ The evaluating experiment of the detecting credibility of the bearing tester after the error compensation, such as the experimental spherical plain bearing, the operating condition, the experimental procedure (conducting 13 on-line and off-line detections) and the preprocessing of connecting the experimental data, was the same with that before the error compensation. The original reference ${\delta }_0$ and the 13 wear experimental data of the radial clearance are shown in Table 2. Under the low-speed oscillating and heavy-loading operating condition, the on-line and the off-line wear-depth detecting curves with the thermal error compensation are shown in Figure 5.

Table 2.

Original reference ${\delta }_0$ and the 13 wear experimental data of the radial clearance after the thermal error compensation.

Time	${\delta }_0$	${\delta }_1$	${\delta }_2$	${\delta }_3$	${\delta }_4$	${\delta }_5$	${\delta }_6$	${\delta }_7$	${\delta }_8$	${\delta }_9$	${\delta }_{10}$	${\delta }_{11}$	${\delta }_{12}$	${\delta }_{13}$
Clearance (mm)	0.069	0.078	0.089	0.096	0.109	0.11	0.115	0.119	0.123	0.128	0.13	0.133	0.137	0.14

Figure 5.

Curves of the on-line wear depth and the off-line wear depth after compensation.

Analysis of the experimental error

The error analysis of the on-line detection: on the basis of the above experimental process, we collected the on-line wear-depth data through the discontinuous methods. Whether the wear zone of the spherical plain bearing was the same in each experiment was the key to guaranteeing the precision of the on-line detection. The spherical plain bearing and the T-type plate were marked before the wear experiment so that the mutual positions of the spherical plain bearing and the T-type plate were kept as the same in the installation process of each wear experiment as before. The marks of the spherical plain bearing and the T-type plate are shown in Figure 6. The wear zones of the spherical plain bearing were kept the same during the on-line detection in this way, which lay the foundation for connecting preprocessing of the experimental data.

Figure 6.

Marks of the spherical plain bearing and the T-type plate.

The error analysis of the off-line detection: there were two influence factors for the off-line detecting errors and they are given as follows:

1. The errors caused by the variation in the installation position in each off-line detection. Similar with the solution of the on-line detection, in order to ensure the detecting positions were the same in each off-line detection, we marked the relevant parts of the detecting instrument of the bearing clearance, so as to make the bearing installing position the same on the detecting instrument in each off-line detection. The picture of the installing position of the spherical plain bearing and the detecting instrument of the bearing clearance is shown in Figure 7.

2. The errors caused by the different loadings between the on-line detection and the off-line detection. The research object of this article example was the woven fabric self-lubricating spherical plain bearing, and there was the self-lubricating fabric liner between the outer ring and inner ring of the spherical plain bearing. Meanwhile, the fabric liner was weaved with the polytetrafluoroethylene (PTFE) fiber and Kevlar, and it was the fabric elastic body. According to the above experimental process, the testing load was 250 MPa during the on-line detection, so the liner was compressed during the on-line experiment. The wear-depth data of the on-line detection were the thickness variation of the liner which was in compression state. However, the detecting loadings of the off-line detection were only the measuring pressure of the detecting instrument of the bearing clearance, and the liner compression amount of the bearing detecting zone was small. Comparing to the on-line detecting data, the detecting data of all the off-line detections contained the liner uncompressed amount. Meanwhile, according to the experimental data preprocessing, the conventional true value of every wear experiment was the difference between the radial clearance of the spherical plain bearing and the previous one in this experiment, that is, $\mathrm{\Delta }{\delta }_{\mathrm{i}}={\delta }_{\mathrm{i}}-{\delta }_{\mathrm{i}-1}$ (i = 1, …, 13). The liner uncompressed amount of the two adjacent off-line detection offset each other in the computing process of the off-line detecting wear depth, and the conventional true value of the wear depth was only the relative variation of the liner thickness. So, the errors caused by the different loadings between the on-line detection and the off-line detection affected the detecting results slightly.

Figure 7.

Picture of the installing position of the spherical plain bearing and the detecting instrument of the bearing clearance.

Credibility evaluation of the on-line wear-depth detection

According to the above two evaluating indexes of the on-line wear-depth detecting data of the spherical plain bearing tester, we evaluated the credibility of the on-line wear-depth detection of the spherical plain bearing tester before and after the thermal error compensation.

The credibility evaluation of the on-line wear-depth detection before the thermal error compensation

The index I—the similarity between the on-line and the off-line detecting curves $\gamma$

• Step 1. The denoising and smoothing processes of the on-line detecting curve

In order to simplify calculation, we needed to smooth the fluctuant curve of the on-line detection before evaluating the similarity between the on-line and the off-line detecting curves and obtained a smooth, time-varying curve which was similar to the off-line detecting curve. First, we eliminated the singular points on the on-line detecting curve by MATLAB software and then smoothed the on-line detecting curve through the moving average method. So, we obtained two smooth on-line and off-line curves depending on Figure 4. The processed curves are shown in Figure 8.

Figure 8.

Curves of the on-line wear depth and the off-line wear depth after curve smoothing.

• Step 2. The extreme point calculation of the on-line wear-depth detecting curve

In the above algorithm introduction of the discrete Fréchet distance, we selected the maximum and minimum value points of the on-line detecting curve and the sampling points of the off-line detecting as the characteristic points of the discrete Fréchet distance calculation. Then, we extracted 101 maximum value points and the 100 minimum value points on the smooth on-line wear-depth detecting curve by MATLAB software. The distribution diagrams of the maximum and the minimum value points are shown in Figure 9.

Figure 9.

Distribution diagrams of (a) the maximum points and (b) the minimum points.

• Step 3. The calculation of the discrete Fréchet distance

From the above calculation of the extreme points, we extracted the maximum and the minimum value points on the on-line detecting curve and the sampling points on the off-line detecting curve. Next, we calculated the discrete Fréchet distance of the three kinds of characteristic points.

In calculating this example, we further simplified the calculation method of the discrete Fréchet distance, that is, we compressed the searching range of characteristic points on two curves. ¹⁹ So, we set up the following initial conditions—the characteristic points on the curve A only corresponded to the characteristic points ${\mathrm{b}}_{\mathrm{i}-1}$ , ${\mathrm{b}}_{\mathrm{i}}$ , and ${\mathrm{b}}_{\mathrm{i}+1}$ on the curve B, thus there were seven matching situations about the characteristic points between two curves

$${\mathrm{A}}_{\mathrm{i}}\to {\mathrm{a}}_{\mathrm{i}},{\mathrm{B}}_{\mathrm{i}}\to \{\begin{array}{l}0\\ {\mathrm{b}}_{\mathrm{i}-1}\\ {\mathrm{b}}_i\\ {\mathrm{b}}_{\mathrm{i}+1}\\ {\mathrm{b}}_{\mathrm{i}-1},{\mathrm{b}}_{\mathrm{i}}\\ {\mathrm{b}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}+1}\\ {\mathrm{b}}_{\mathrm{i}-1},{\mathrm{b}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}+1}\end{array}\mathrm{i}\ge 1$$

After selecting the initial condition, the next characteristic point ${\mathrm{a}}_{\mathrm{i}+1}$ on the curve A affected the selection of the corresponding point on the curve B. In the initial conditions, the possible situations about the next matching characteristic points between two curves were shown as follows

$$\begin{array}{c}\mathrm{If}{\mathrm{A}}_{\mathrm{i}}\to {\mathrm{a}}_{\mathrm{i}},{\mathrm{B}}_{\mathrm{i}}\to 0\Rightarrow {\mathrm{A}}_{\mathrm{i}+1}\to {\mathrm{a}}_{\mathrm{i}+1},{\mathrm{B}}_{\mathrm{i}+1}\to \{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}}}\\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}+1}}\\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}+1},{\mathrm{b}}_{\mathrm{i}+2}}\end{array}\mathrm{i}\ge 1\hfill \\ \mathrm{If}{\mathrm{A}}_{\mathrm{i}}\to {\mathrm{a}}_{\mathrm{i}},{\mathrm{B}}_{\mathrm{i}}\to {\mathrm{b}}_{\mathrm{i}-1}\Rightarrow {\mathrm{A}}_{\mathrm{i}+1}\to {\mathrm{a}}_{\mathrm{i}+1},{\mathrm{B}}_{\mathrm{i}+1}\to \{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}}}\\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}+1}}\\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}+1},{\mathrm{b}}_{\mathrm{i}+2}}\end{array}\mathrm{i}\ge 2\hfill \\ \mathrm{If}{\mathrm{A}}_{\mathrm{i}}\to {\mathrm{a}}_{\mathrm{i}},{\mathrm{B}}_{\mathrm{i}}\to {\mathrm{b}}_{\mathrm{i}-1},{\mathrm{b}}_{\mathrm{i}}\Rightarrow {\mathrm{A}}_{\mathrm{i}+1}\to {\mathrm{a}}_{\mathrm{i}+1},{\mathrm{B}}_{\mathrm{i}+1}\to \{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}}}\\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}+1}}\\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}+1},{\mathrm{b}}_{\mathrm{i}+2}}\end{array}\mathrm{i}\ge 2\hfill \\ \mathrm{If}{\mathrm{A}}_{\mathrm{i}}\to {\mathrm{a}}_{\mathrm{i}},{\mathrm{B}}_{\mathrm{i}}\to {\mathrm{b}}_{\mathrm{i}-1},{\mathrm{b}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}+1}\Rightarrow {\mathrm{A}}_{\mathrm{i}+1}\to {\mathrm{a}}_{\mathrm{i}+1},{\mathrm{B}}_{\mathrm{i}+1}\to \{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}+1}}\\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}+1},{\mathrm{b}}_{\mathrm{i}+2}}\end{array}\mathrm{i}\ge 2\hfill \\ \mathrm{If}{\mathrm{A}}_{\mathrm{i}}\to {\mathrm{a}}_{\mathrm{i}},{\mathrm{B}}_{\mathrm{i}}\to {\mathrm{b}}_{\mathrm{i}}\Rightarrow {\mathrm{A}}_{\mathrm{i}+1}\to {\mathrm{a}}_{\mathrm{i}+1},{\mathrm{B}}_{\mathrm{i}+1}\to \{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}}}\\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}+1}}\\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}+1},{\mathrm{b}}_{\mathrm{i}+2}}\end{array}\mathrm{i}\ge 1\hfill \\ \mathrm{If}{\mathrm{A}}_{\mathrm{i}}\to {\mathrm{a}}_{\mathrm{i}},{\mathrm{B}}_{\mathrm{i}}\to {\mathrm{b}}_{\mathrm{i}-1},{\mathrm{b}}_{\mathrm{i}}\Rightarrow {\mathrm{A}}_{\mathrm{i}+1}\to {\mathrm{a}}_{\mathrm{i}+1},{\mathrm{B}}_{\mathrm{i}+1}\to \{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}+1}}\\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}+1},{\mathrm{b}}_{\mathrm{i}+2}}\end{array}\mathrm{i}\ge 1\hfill \\ \mathrm{If}{\mathrm{A}}_{\mathrm{i}}\to {\mathrm{a}}_{\mathrm{i}},{\mathrm{B}}_{\mathrm{i}}\to {\mathrm{b}}_{\mathrm{i}+1}\Rightarrow {\mathrm{A}}_{\mathrm{i}+1}\to {\mathrm{a}}_{\mathrm{i}+1},{\mathrm{B}}_{\mathrm{i}+1}\to \{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}+1}}\\ \multicolumn{1}{c}{{\mathrm{b}}_{\mathrm{i}+1},{\mathrm{b}}_{\mathrm{i}+2}}\end{array}\mathrm{i}\ge 1\hfill \end{array}$$

Based on the algorithm, we calculated the discrete Fréchet distance of these three kinds of characteristic points by MATLAB software and obtained the discrete Fréchet distance of the maximum value points of the on-line detecting curve— ${\mathrm{d}}_1(\mathrm{A},\mathrm{B})=0.417\mathrm{mm}$ , the discrete Fréchet distance of the minimum value points of the on-line detecting curve— ${\mathrm{d}}_2(\mathrm{A},\mathrm{B})=0.355\mathrm{mm}$ , and the discrete Fréchet distance of the sampling points of the off-line detecting curve— ${\mathrm{d}}_3(\mathrm{A},\mathrm{B})=1.557\mathrm{mm}$ . So, the similarity between the on-line and the off-line detecting curves was

$${\gamma }_1=\mathrm{d}(\mathrm{A},\mathrm{B})=\frac{{\mathrm{d}}_1(\mathrm{A},\mathrm{B})+{\mathrm{d}}_2(\mathrm{A},\mathrm{B})+{\mathrm{d}}_3(\mathrm{A},\mathrm{B})}{3}=0.776\mathrm{mm}$$

The index II—the fluctuating range of the on-line wear-depth detecting curve of the spherical plain bearing $\lambda$

In calculating the process of the index I, we already obtained the datum line of the fluctuant on-line wear-depth detecting curve through the moving average method. In the following, we calculated the arithmetic average value of the absolute value of the distance between every point on the on-line curve and the datum line. The fluctuant on-line detecting curve and the datum line are shown in Figure 10.

Figure 10.

Fluctuating curve and the datum line of the on-line wear-depth detection.

According to formula (2), the formula of the fluctuating range of the on-line wear-depth detecting curve was

$$\lambda =\frac{1}{\mathrm{S}}\sum _0^{\mathrm{S}}|{\mathrm{y}}_{\mathrm{i}}-{\overline{\mathrm{y}}}_{\mathrm{i}}|,\mathrm{i}=0,\dots ,\mathrm{S}$$

(6)

where S is the number of the sampling points on the on-line detecting curve, ${\mathrm{y}}_{\mathrm{i}}$ is the ordinate value of the fluctuant on-line detecting curve, and ${\overline{\mathrm{y}}}_{\mathrm{i}}$ is the ordinate value corresponding to ${\mathrm{y}}_{\mathrm{i}}$ on the datum line.

As shown in Figure 10, the numbers of the sampling points on the on-line detecting curve were 10645, and the sum of absolute values of the distance between every point on the fluctuant on-line detecting curve and the datum line was 8.504 mm. So, the index II—the fluctuating range of the on-line wear-depth detecting curve ${\lambda }_1=0.799\mu \mathrm{m}$ .

In conclusion, before the thermal error compensation for the spherical plain bearing tester, we obtained the similarity between the on-line and the off-line detecting curves ${\gamma }_1=0.776\mathrm{mm}$ and the fluctuating range of the on-line wear-depth detecting curve ${\lambda }_1=0.799\mu \mathrm{m}$ .

The credibility evaluation of the on-line wear-depth detection after the thermal error compensation

The index I—the similarity between the on-line and the off-line detecting curves $\gamma$

• Step 1. The denoising and smoothing processes of the on-line detecting curve

With the data processing procedure of the on-line detecting curve before the thermal error compensation, first, we eliminated the singular points on the on-line detecting curve and then smoothed the on-line detecting curve through the moving average method. So, we obtained two smooth on-line and off-line curves depending on Figure 5. The processed curves are shown in Figure 11.

Figure 11.

Curves of the on-line wear depth and the off-line wear depth after curve smoothing.

• Step 2. The extreme point calculation of the on-line wear-depth detecting curve

With the data processing procedure of the on-line detecting curve before the thermal error compensation, we extracted 105 maximum value points and 104 minimum value points on the smooth on-line wear-depth detecting curve. The distribution diagrams of the maximum and the minimum value points are shown in Figure 12.

Figure 12.

Distribution diagrams of (a) the maximum points and (b) the minimum points.

• Step 3. The calculation of the discrete Fréchet distance

With the data processing procedure of the on-line detecting curve before the thermal error compensation, we calculated the discrete Fréchet distance of the three kinds of characteristic points and obtained the discrete Fréchet distance of the maximum value points of the on-line detecting curve— ${\mathrm{d}}_1(\mathrm{A},\mathrm{B})=0.301\mathrm{mm}$ , the discrete Fréchet distance of the minimum value points of the on-line detecting curve— ${\mathrm{d}}_2(\mathrm{A},\mathrm{B})=0.274\mathrm{mm}$ , and the discrete Fréchet distance of the sampling points of the off-line detecting curve— ${\mathrm{d}}_3(\mathrm{A},\mathrm{B})=1.302\mathrm{mm}$ , respectively. So, the similarity between the on-line detecting curve and the off-line detecting curve was

$${\gamma }_2=\mathrm{d}(\mathrm{A},\mathrm{B})=\frac{{\mathrm{d}}_1(\mathrm{A},\mathrm{B})+{\mathrm{d}}_2(\mathrm{A},\mathrm{B})+{\mathrm{d}}_3(\mathrm{A},\mathrm{B})}{3}=0.626\mathrm{mm}$$

The index II—the fluctuating range of the on-line wear-depth detecting curve of the spherical plain bearing $\lambda$

In the data processing procedure of the on-line detecting curve before the thermal error compensation, the fluctuant on-line detecting curve and the datum line are shown in Figure 13.

Figure 13.

Fluctuating curve and the datum line of the on-line wear-depth detection.

As shown in Figure 13, in formula (6) of the fluctuating range of the on-line wear-depth detecting curve, the numbers of the sampling points on the on-line detecting curve were 11907, and the sum of absolute values of the distance between every point on the fluctuant on-line detecting curve and the datum line was 8.995 mm. So, the index II was ${\lambda }_2=0.755\mu \mathrm{m}$ .

In conclusion, after the thermal error compensation for the spherical plain bearing tester, we obtained the similarity between the on-line and the off-line detecting curves ${\gamma }_2=0.626\mathrm{mm}$ and the fluctuating range of the on-line wear-depth detecting curve ${\lambda }_2=0.755\mu \mathrm{m}$ .

From the former calculation and analyses, aiming at the first index of the degree of credibility of the on-line wear-depth detecting data, the similarity between the on-line and the off-line detecting curves, we found that the similarity before the thermal error compensation was ${\gamma }_1=0.776\mathrm{mm}$ and the similarity after the thermal error compensation was ${\gamma }_2=0.626\mathrm{mm}$ , namely, ${\gamma }_1>{\gamma }_2$ . This showed that the discrete Fréchet distance of the on-line and the off-line detecting curves after the thermal error compensation was less than that before the thermal error compensation, and the two curves after the thermal error compensation had a high similarity. So, after the thermal error compensation, the on-line show values of the bearing tester were close to the conventional true values of the wear depth. Meanwhile, the fluctuating range of the on-line wear-depth detecting curve before the thermal error compensation was ${\lambda }_1=0.799\mu \mathrm{m}$ and that after the thermal error compensation was ${\lambda }_2=0.755\mu \mathrm{m}$ , namely, ${\lambda }_1>{\lambda }_2$ . This showed that the fluctuation of the on-line wear-depth detecting curve after the thermal error compensation was less than that before the thermal error compensation. In conclusion, from the above results of the two indexes of the credibility evaluation, the degree of credibility of the experimental data of the spherical plain bearing tester after the thermal error compensation was higher than that before the thermal error compensation.

Conclusion

In this article, on the basis of the similar degree between the on-line wear-depth data and the conventional true value of the wear depth, the definition of the credibility of the on-line wear-depth detection of spherical plain bearings was set forth. Meanwhile, two evaluating indexes of the credibility of the on-line wear-depth detection were presented: the index I—the similarity between the on-line and the off-line detecting curves obtained using the algorithm of the discrete Fréchet distance and the index II— the fluctuating range of the on-line wear-depth detecting curve obtained by adopting the computing method of the profile arithmetic average error ${\mathrm{R}}_{\mathrm{a}}$ of the surface roughness.

Finally, we used the above evaluating indexes to analyze the comparative study of the detecting credibility for the same tester of spherical plain bearings before and after the thermal error compensation, whose results showed that the index I and the index II of the credibility after the thermal error compensation were less than those before the thermal error compensation ( ${\gamma }_1=0.776\mathrm{mm}$ , ${\gamma }_2=0.626\mathrm{mm}$ ; ${\lambda }_1=0.799\mu \mathrm{m}$ , ${\lambda }_2=0.755\mu \mathrm{m}$ ). So, the degree of credibility of the experimental data of the spherical plain bearing tester after the thermal error compensation was higher than that before the thermal error compensation, that is, the detecting precision of the tester after the thermal error compensation was improved.