Fault prediction method based on linear weighted summation

Abstract

A method to calculate the threshold of the wear particle concentration in lubricating oil was proposed, and it can also be used to predict the debris concentration in oil system. The concentration of the selected 6 elements was used as the monitoring object, and a linear weighted summation process was used to enhance numerical stability of the object value, the monitoring threshold was calculated using the Student distribution model. The computational process needs only the latest 7 $\sim$ 10 samples, so it is easy to update the monitoring threshold. The method was tested by some samples from aircraft engines and helicopter gear reducers. The test results show that the accuracy of the proposed prediction model is higher than traditional linear model. The proposed monitoring threshold algorithm can forecast wear out failure in advance.

Keywords

Wear particle concentration prediction threshold dynamic adjustment

1. Introduction

Many experimental results and published theoretical models indicate that wear and tear is a key factor that may lead to mechanical equipment failure. Since the lubricating state of mechanical system is closely related to the wear state, so if we can use the tribology system information contained in lubricating oil by analyzing the wear particles in lubricating oil, then it is feasible to evaluate the work condition of mechanical system and forecast failure [1, 2]. With the increase of operating time, the wear particles in the oil system can change gradually. In particular, before failure occurs, the wear rate often grows quickly, and wear particle concentration rises quickly. Therefore, by monitoring the change of debris concentration in lubricating oil system of a machine, it is possible to forecast abnormal wear failure in advance [3, 4]. The spectrometer analysis can easily and quickly acquire the concentration of wear elements, and this wear debris monitoring method is effective for revealing the wear condition of a machine without disassembly, and the method is suitable for monitoring the oil system of aero-engine and helicopter reducer.

Wear of rotating gear and bearings can generates metal debris in the oil system, the oil carries the information about the machine conditions as well as wear sources, rates and mechanisms. The oil monitoring method can be used to recognize the wear states of oil lubricated components when the machine is in operation, and the monitor method is effective to predit the mechanical faults in machine [5, 6, 7]. The oil system condition monitoring method can find the micro changes in the debris concentration and/or size. In practical application, it is discovered that, external environment and work condition may influence the test result of the Atomic emission spectrometer. The oil and the contacting surfaces in machinery components will also change over time. So, it is necessary to make dynamic adjustment of threshold value. Some meta-heuristic algorithms such as support vector machine, genetic algorithm and artificial neural network can train samples to automatically obtain the function relation contained in the sample data. Many scholars have applied it in the dynamic formulation of threshold value, and it can also realize failure forecast and classification [8, 9, 10]. The meta-heuristic algorithm needs to accumulate certain amount of sample data according to different failure types. The aero-engines are expensive and complex equipments, and the fault cases are rare, so it is difficult to find enough sample data for meta-heuristic algorithm. For example, it is costly to do wear and tear experiments in aero-engine, the meta-heuristic algorithm models are not suitable to compute threshold value for aero-engine fault monitoring. So it is difficult to use meta-heuristic algorithm model in the practical oil monitoring work.

In recent years, through an accurate measurement and statistics of the working state parameters and lubricating oil in the equipment, a variety of machine condition monitoring methods have been proposed [11, 12, 13, 14]. The test result shown, the wear particles concentration is a impormant indicator of the machine state monitoring, the concentration date can be used as the parameter about the wear extent and wear rate progression [15, 16]. However, the wear debris concentration in a practical lubrication system is influenced by many factors, such as the machine working time, the load and harsh working conditions, etc. This leads to some issues that the debris monitoring data is difficult to be explained, and the formula about the wear state of machine and the monitoring data is hard to be established.

Because the wear particles occurs between contact surfaces, and the wear particles numbers is a time-varying parameter, which can quantify the wear extent and wear state progression. So, the explicit expression for the wear particles number increase is the most important factor concering the wear rate of machines. Based on the models mentioned above, to overcome this drawback, the lognormal distribution model is used to formlate the relationship between the quantity of wear particles and the work time of lubricating oil. By analyzing the probability relation between particle quantity and time in the wear process, the method to calculate the wear particle concentration in lubricating oil used in the forecast equipment was proposed. In order to improve the stability and effectiveness of monitoring algorithm, taking aero-engine and helicopter reducer as an example, the spectral analysis results data of six key monitoring elements were selected for linear weighted calculation. According to the test results of new samples in the monitoring process of lubricating oil, the monitoring threshold value was dynamically adjusted, and Matlab platform was used to complete the programming test of algorithmic model. Finally, the trial run data of one type of aero-engine and helicopter reducer was used to verify the proposed computing method.

2. Particle concentration forecasting model

2.1 Logarithmic model to compute particle content

The frictional wear process, due to environmental factors and the impact of different materials, demonstrates the complexity and diversity, and the most important factors are particle size, number and content. It is assumed that the object under analysis works in a steady state of friction and wear at a constant speed, and the particle content in lubricating oil at the initial moment is zero. After working for a certain period of time $t$ , the particle content in lubricating oil changes to $M(t)$ , so $M$ mainly depends on the particle number $n(t)$ and average particle size $\bar{m}(t)$ during this period of time. The formula is as below:

$\displaystyle M(t)=\sum\limits_{i=1}^{n(t)}m_{i}=n(t)\bar{m}(t)$ (1)

Suppose the probability that the particle number is increased by one unit amount in the period of time [ $t$ , $t+t$ ] is denoted by:

$\displaystyle Q(t,t+\Delta t)=P(t\leqslant T\leqslant t+\Delta t)=1-\frac{R(t+% \Delta t)}{R(t)}$ (2)

In which, the wear particle number $n$ in one period of time can be considered as a random variable of time $t$ , $R(t)$ represents the probability that the particle number does not change in the period of time $t$ , which distribution obeys the lognormal distribution. The formula is as below:

$\displaystyle R(t)=\exp\{-\alpha[\ln(1+t)]^{\beta}\}$ (3)

It is known from the Calculus theory that the growth rate of particle number at the moment $t$ can be denoted by:

$\displaystyle\lambda(t)=\mathop{\lim}\limits_{\Delta t\to 0}\frac{Q(t,t+\Delta t% )}{\Delta t}=-\frac{{R}^{\prime}(t)}{R(t)}$ (4)

The formula of increased particle number in the period of time $t$ is:

$\displaystyle n(t)=\int_{0}^{t}{\lambda(t)dt}=-\ln R(t)=\ln\frac{1}{R(t)}$ (5)

According to the friction and wear test and the online monitoring data of wear particles, for wear particles larger than 2 $\mu$ m, at the steady wear state, with the change of working time, the ratio of different sized wear particle numbers in total particle number is basically stable. So the average size of wear particles at the steady wear stage can be taken as a constant, then the size change function of wear particles is a constant function:

$\displaystyle m(t)=m_{k}$ (6)

As analysed above, the content of the wear particles in oil may increase over time, the content can be computed as the following formula:

$\displaystyle M(t)=n(t)m(t)=m_{k}\ln\frac{1}{R(t)}=m_{k}\alpha[\ln(1+t)]^{\beta}$ (7)

2.2 Algorithm model to predict wear particle concentration

In the normal work process, the lubricating oil amount $L$ changes very little. In order to simplify the calculation, $L$ is considered as a constant, $m_{k}\alpha$ and $\beta_{a}$ are constants. Let $\frac{m_{k}\alpha}{L}=a$ and $\beta=b$ , then the wear particle concentration can be denoted as:

$\displaystyle\hat{P}_{M}=a[\ln(1+t)]^{b}$ (8)

Because the iron is the most important element in wear particle, and the optical spectrum analyser (OSA) is used to detect the concentration of the iron element, so the concentration of element Fe can be used to compute unknown parameters in Eq. (8). Assume that $P(t)$ is the Fe concentration result at time $t$ , then the test result of lubricating oil collected in the working time period can be denoted as a vector below:

$\displaystyle P=[P(t_{1}),P(t_{2}),\ldots,P(t_{j})]$ (9)

The vector $t$ corresponding to the equipment working time can be denoted as:

$\displaystyle t=[t_{1},t_{2},\ldots,t_{j}]$ (10)

Take the logarithm at both sides of Eq. (8), then

$\displaystyle\ln\hat{P}(t)=\ln a+b\ln[\ln(1+t)]$ (11)

Suppose

$\displaystyle X=\left[\begin{array}[]{cc}1&\ln[\ln(1+t_{1})]\\ \vdots&\vdots\\ 1&\ln[\ln(1+t_{j})]\\ \end{array}\right],\ln a=b_{0},\ln[\ln(1+t)]=x,$

$\displaystyle b=b_{1},Y=[\ln P(t_{1}),\ldots,\ln P(t_{j})]^{T}$

then:

$\displaystyle\hat{y}=b_{0}+x\cdot b_{1}$ (12) $\displaystyle\left[\begin{array}[]{c}b_{0}\\ b_{1}\\ \end{array}\right]=(X^{T}X)^{-1}X^{T}Y$ (13)

After collecting lubricating oil samples at different moments, record the corresponding equipment working time and the tested wear particle concentration. According to Eq. (13), it is possible to solve the wear particle concentration computing coefficients $b_{0},b_{1}$ . Substituting these coefficients into Eq. (12), we can determine the concentration prediction function which takes working time as independent variable and wear particle concentration as dependent variable. In practical monitoring work, as the working time of monitoring equipment is lengthened, the sample data size will be continuously increased. In the analysis of test data, it is found that a stable concentration prediction function can be obtained by selecting 7 $\sim$ 10 sample data closest to current time.

3. Method for calculating oil monitoring threshold value

Now, the most common oil analysis equipment in domestic oil analysis labs is multielement atomic emission spectrometer, which can test more than 20 kinds of element concentration in one time. In order to forecast the abnormal wear fault of monitoring equipment, it is required to set relevant monitoring threshold value for different elements. Because the major chemical composition of the bearings and gears are metal elements, such as Cu and Fe, so in practical operation, the maintenance personnel focused on few major metal elements content in oil, while the test results of the concentration of other elements are not used effectively. The statistics of the test data indicate that the spectral analysis results of the major elements may change along with other elements. In order to promote the stability of calculation model and simplify the application of oil analysis results in the process of wear state monitoring, according to the relevancy analysis conclusions and the structural feature of rotating parts of 2 different aero-engine and helicopter reducer, totally six elements, i.e. Fe, Cu, Mg, Cr, Ni and Ti in the spectral analysis results are selected as the monitored elements, the linear weighted value of the test concentration of six elements are taken as the monitored targets, and the target function is computed according to Eq. (14). Considering that some random errors are inevitable in the process of lubricating oil detection, we suppose the random factor error of metal element $j$ content caused by error in the $j^{\rm th}$ sampling analysis process is $e_{ij}$ , its mean value is zero. Then, the predicted value corresponding to the concentration of six monitoring elements can be worked out according to Eq. (15).

$\displaystyle Y(x)=\sum\limits_{j=1}^{6}{C_{j}Y_{j}(x)}$ (14) $\displaystyle\hat{Y}_{j}(t_{i})=b_{0}+b_{1}\ln\left[\ln\left(\sum{\bar{N}g% \cdot t}\right)\right]+e_{ij}$ (15)

In which, $C_{j}$ is the corresponding weight factor of test results $j$ . When this factor is set as the optimal value, $Y$ is most sensitive to the wear state change of lubricating oil. Taking the signal-to-noise ratio of target function $Y$ as the target function, the weight factor is figured out according to Eq. (16).

$\displaystyle C_{j}=\frac{b_{0}+xb_{1}}{\sqrt{\textit{Var}Y_{j}(x)}}$ (16)

$\displaystyle\textit{Var}Y_{j}(x)=S_{j}^{2}\left[{\frac{N+1}{N}+\frac{\left({x% -\bar{x}}\right)^{2}}{\sum{\left({x_{i}-\bar{x}}\right)^{2}}}}\right]$ (17)

$\displaystyle S_{j}^{2}=\frac{S_{11}\cdot S_{22}-S_{12}\cdot S_{12}}{N(N-2)% \cdot S_{11}}$ (18)

$\displaystyle S_{11}=N\sum\limits_{i=1}^{N}{x_{i}^{2}}-\left({\sum\limits_{i=1% }^{N}{x_{i}}}\right)^{2}$ (19)

$\displaystyle S_{12}=N\sum\limits_{i=1}^{N}{y_{i}x_{i}}-\left({\sum\limits_{i=% 1}^{N}{x_{i}}}\right)^{2}$ (20)

In which: $S_{j}^{2}$ – the variance estimator of $e_{ij}$ ;

$N$ – the statistic test result sample number of lubricating oil;

$\overline{x}$ – the average of sample interval of statistic sample x.

In the practical statistic process, it is found that, as the working time of lubricating oil continues, the weight factor $C_{j}$ and monitoring value $Y$ will keep on rising. In order to reduce the influence of time factor on monitoring value $Y$ , the weight factor is normalized according to Eq. (21), and the processed statistic value $Y$ approximately obeys to $t$ distribution model. So we can work out the alarm threshold value and failure threshold value of monitoring value $Y$ according to $t$ distribution model. Take the confidence interval 90% and 99% of $t$ distribution respectively to compute the corresponding threshold values. The concentration prediction value of monitoring objects are calculated according to Eq. (22), the alarm threshold value is calculated according to Eq. (23), and the failure threshold value is calculated according to Eq. (24). So, in the analysis of the spectral test results of lubricating oil used in the monitored equipment, when the predicted value calculated according to Eq. (22) is larger than the alarm threshold value calculated according to Eq. (23), it is necessary to make a careful inspection on the oil filter, magnetic plug and other parts of the monitored equipment and determine whether the equipment could continue to be used. In case that the predicted value is larger than the failure threshold value calculated according to Eq. (24), it is required to conduct an overhaul on the key wear parts of the equipment, remove the fault and then put into use.

$\displaystyle C_{j}=\frac{c_{j}}{\sum\limits_{j=1}^{N=5}{c_{j}}}$ (21) $\displaystyle\hat{Y}(x)=\sum\limits_{j=1}^{7}{C_{j}\left({b_{0}+b_{1}\cdot t}% \right)}$ (22) $\displaystyle\textit{Th}_{W}=\hat{Y}(x)+t_{0.9}\cdot S_{j}=\hat{Y}(x)+1.6377% \cdot S_{j}$ (23) $\displaystyle\textit{Th}_{F}=\hat{Y}(x)+t_{0.99}\cdot S_{j}=\hat{Y}(x)+4.5407% \cdot S_{j}$ (24)

4. Experimental verification

4.1 Verification of wear particle concentration forecasting model

In order to verify the logarithmic function computation model of wear particle concentration and friction time, a test platform of the gearing of a type of engine accessories was modified, processed and newly equipped with friction gear and monitoring device. On the condition of maintaining a steady rotate speed and lubricating condition, a friction and wear test was performed for 200 hours. Lubricating oil sample was collected every 20 hours, and the metal element content of obtained 10 lubricating oil samples was inspected by MOAII type atomic emission spectrometer, see test results in Table 1. The first 5 sample data were used for data fitting according to the logarithmic function model of wear particle concentration and friction time, so as to determine the calculation parameters of wear particle concentration, and then the last 5 sample data were used for verification. The conventional linear polynominal model and the proposed logarithmic model were adopted to implement function fitting on the test data, and then the fitted models were used to compute the estimation value. The standard error, error sum and coefficient of correlation results of the linear model and the poposed model are shown in the Table 2.

Table 1
Spectral analysis results of the lubricating oil used in the transmission gear of an engine

S/N	Oil working	Added oil	6 key monitored metal element content [PPM]
	time [h]	amount [L]	Fe	Cu	Mg	Cr	Ti	Ni
1	25	0	3.8	3.5	0.4	0	0.1	0.1
2	50	0.4	4.8	4.5	0.6	0.2	0.2	0.1
3	70.5	0	4.7	4.2	0.5	0.3	0.2	0.2
4	101	0.3	4.6	5.2	0.6	0.5	0.3	0.2
5	153.5	0.5	5.2	5.9	0.3	0.4	0.3	0.2
6	221	0.2	5.5	7	0.5	0	0.4	0.2
7	240.5	0	5.7	7.4	0.4	0.5	0.2	0.3
8	280	0.2	6.6	8.6	0.8	0.8	0.2	0.2
9	20	0	4.5	1.8	0.5	0.8	0.4	0.1
10	50	0.6	5	2.1	0.5	0.1	0.5	0.4

Table 2

Comparison of prediction results of lineal model and logarithmic model

Function model	Std. deviation (RMSE)	Error sum (SSE)	Determination coefficient
			R-square	Adj R-q
$P=a+b\cdot t$	0.5673	1.9312	0.8835	0.8641
$\ln\hat{P}(t)=\ln a+b\ln[\ln(1+t)]$	0.1621	0.1577	0.9484	0.9398

Figure 1.

The curves of wear particle concentration-time of the aero-engine.

In Table 2, the error sum (SSE) and standard deviation (RMSE) of logarithmic model are much closer to zero, which indicates that using logarithmic model to conduct wear particle concentration prediction could get smaller error. The determination coefficient of logarithmic model is much closer to the optimal value 1.0 than linear model, which indicates that the correlation between concentration and time is more significant when logarithm is taken. Therefore, the proposed logarithmic model can forecast the wear particle’s concentration growth trend more objectively and accurately than traditional linear model.

4.2 Verification of the method for calculating lubricating oil threshold value

In order to verify the proposed method for calculating lubricating oil monitoring threshold value, the spectral analysis data of two lubricating oil samples collected from the helicopter main reducing gear box and aero-engine were taken as data source, and the Matlab platform was used to program the data processing and curve plotting software. The software was used to conduct fitting calculation of 10 sample data of an aero-engine collected during the trial run time of 50 h $\sim$ 200 h according to Eq. (22). The sample data is analysed by the psoposed method and calculation results is show in Table 3. The results show that the model obtained by using logarithmic weighted model fitting is significantly improved in accuracy than linear function model.

Table 3
Comparison of fitting results of linear weighted model and logarithmic weighted model

Function model	Std. deviation (RMSE)	Error sum (SSE)	Determination coefficient
			R-square	Adj R-q
$P=a+b\cdot t$	0.07071	0.015	0.9879	0.9839
$\hat{Y}(x)=\sum\limits_{j=1}^{7}{C_{j}\left[{b_{0j}+b_{1j}\ln(\ln t)}\right]}$	0.03269	0.004328	0.9989	0.9901

Figure 2.

The picture of the wear granule when the engine has worked 220 hours.

Figure 3.

The picture of the wear granule when the engine has worked 280 hours.

In order to test the practicability of the proposed method, the software was used to analyse the data sampled from an aero-engine in service. The lubricating oil samples were collected from the aero-engine every 20 hours during its continuous trial run period and the concentration of the most important six elements are selected as the monitoring objects. When calculating the threshold value, the eight data closest to end time were taken as samples, and the alarm threshold value and failure threshold value were calculated according to Eqs (23) and (24) respectively. The curves of analysis results are shown in Fig. 1. When this engine works for 220 hours, the test results exceed the alarm threshold value. When it works for 360 hours, the test results exceed the failure threshold value. When it works for 370 hours, the engine is heard abnormal. The T2FM analyzing iron spectrum is used to analyzed the wear granule, the picture of the wear granule extracted by iron spectrum are shown in Figs 2–4, the number of wear particle in Fig. 2 is small when the engine worked no more than 220 hours, the number of wear particle grows quickly after then engine worked more than 280 hours as shown in Figs 3 and 4. After halting and disassembling the equipment, it is found that the main drive gear of accessory drive casing of this engine is seriously worn, resulting in the fracture of gear tooth. During this period, the spectral analysis results of lubricating oil show that the concentration of Fe, Cu and all other signal elements do not exceed the alarm threshold value specified in the traditional method. So, by using the proposed threshold value calculation model, the dynamic adjustment of lubricating oil monitoring threshold value can be realized, which helps to promote the fault forecasting accuracy.

Figure 4.

The picture of the wear granule when the engine has worked 300 hours.

5. Summary

Through an analysis of the probability relationship between wear particle number and time in the wear process, a calculation method for forecasting the wear particle concentration of lubricating oil used in the equipment was proposed. The calculation results indicate that this prediction model has been substantially improved in accuracy than traditional linear model.

Through the linear weighting on the spectral analysis results of multiple monitoring elements, the monitoring value was computed according to the t distribution model, effectively improving the stability and accuracy of monitoring threshold value.

The test results show that, using the spectral analysis results of 7 $\sim$ 10 lubricating oil samples closest to current analysis time to work out the monitoring threshold value could achieve the dynamic adjustment of monitoring threshold value and improve fault prediction accuracy.

The most common lubricating oil spectrometer could not effectively test the wear particles larger than 20 $\mu$ m in lubricating oil. When there are many large-size wear particles in lubricating oil, the test method of different wear particles and their variation trend will be further studied.

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Fault prediction method based on linear weighted summation

Abstract

Keywords

1. Introduction

2. Particle concentration forecasting model

2.1 Logarithmic model to compute particle content

4.1 Verification of wear particle concentration forecasting model

Table 1 Spectral analysis results of the lubricating oil used in the transmission gear of an engine

Table 3 Comparison of fitting results of linear weighted model and logarithmic weighted model

References

Table 1
Spectral analysis results of the lubricating oil used in the transmission gear of an engine

Table 3
Comparison of fitting results of linear weighted model and logarithmic weighted model