Miao Xuewen,Niu Cong,Yang Yun,Han Lei,Hong Jie

(1.Air Force Equipment Academy,Beijing,100076,P.R.China;2.School of Information and Electronics,Beijing Institute of Technology,Beijing,100081,P.R.China;3.School of Jet Propulsion,Beijing University of Aeronautics and Astronautics,Beijing,100083,P.R.China)

INTRODUCTION

Prognostics and health management(PHM)technology for potential applications on aircrafts is increasingly improved because of its support for safety,equipment reliability,and cost reduction.

Current state-of-the-art in aircraft preventive maintenance includes health and usage monitoring systems(HUMS)for AH-64,UH260 and CH247 helicopter,integrated vehicle health management(IVHM)for B22 Bomber, F-35 joint strike fighter (JSF) program, Boeing company′s airplane health management(AHM)for B777,B7472400, A320, A330 and A340 aircraft,aircraft condition analysis and management system (ACAMS) for B757 aircraft, and integrated systems health management(ISHM)for space shuttle,which incorporates PHM into its design,using sensors,advanced processing and reasoning,and a fully integrated system of information and supplies management[1-6].

The service life prediction of aircraft engines is vital for PHM technology.Currently,aircraft engine bearings highly rely on conservative life estimation to ensure that bearings can be replaced before failure.Due to the stochastic nature of a failure propagation process,the uncertain mission profile of aircrafts and uncertain factors in a bearing operating process,this method has two obvious deficiencies.First,the procedure reduces aircraft availability and leads to a significant labor cost to replace a bearing which is still quite useful. Second, although it has achieved extremely low failure rates,highly-conservative life estimation cannot avoid accidents due to those bearing failures outside scheduled maintenance hours under extreme and/or unpredictable circumstances. Therefore,the development of practical and verifiable prognostic models for the

service life of bearings plays a critical role in improving the reliability and safety of aircraft engines.

A new concept, grade-life(GL), is introduced to describe the service life of bearings.The entire service life is divided into four stages:good bearing condition(GBC),initial defect condition (IDC),damaged bearing condition(DBC)and failure coming condition(FCC).

In prognostic issues,mathematical models have a limited predictive capability.Mathematical models can not provide accurate GL prediction because of the highly variable fatigue-propagating process of bearings,the difference of mission profile and the various environmental effects.Therefore, a diagnostic estimation model is developed based on vibration theory and damage mechanics.

Finally,a GL prediction model of aircraft engine bearings is presented with the combination of the mathematical model and the diagnostic estimation model based on support vector machine(SVM)for the calculation of the physics-based GL (PGL)and the empirical GL (EGL).Experiments show that the methodology is scientific and reliable for predicting the lifetime of aircraft engine bearings.

1 GRADE-LIFE PROGNOSTIC MODEL

1.1 Mathematical model

The mathematical model is for rolling contact fatigue.In order to obtain the PGL value,the model utilizes life-limited parameters,including load, rotate speed, etc. to calculate the cumulative damage which the bearing suffers.The parameters are derived from the load spectrum of the bearing based on the statistic analysis of sensors data.

The PGL of bearings is defined as

where L is the cumulative damage of bearings,which can be acquired by Eq.(2),L 0 the minimum life,L 1 the life with 99% survival probability,L10 the rated life of bearings and L50 the life with 50% survival probability.

where LXis the bearing life at the load level X according to the dynamic capacity theory[7-8],tX the time under the load level X,which can be acquired by analyzing the sensors data[7].

Based on Tallian′s research[9],the minimum life parameter can be estimated by

According to ISO281/1.1977(E)[10],the life with 99% survival probability is

Assume m=1.5,which is the shape parameter of the Weibull probability distribution,and according to Tallian′s method,L 50 can be represented as

The relative consumption of lifeis defined as

Therefore,according to the survival probability,the entire service life is divided into four stages as shown in Fig.1.

Fig.1 PGL of bearing

1.2 Diagnostic estimation model

The diagnostic estimation model is an intelligent health-signal prognostic one,in which time-domain statistical parameters of vibration acceleration signature are selected as the input of SVM to obtain the EGL value.

The service life of an aircraft engine bearing is based on the measured or calculated condition variables of the bearing and the limits of these variables. The condition variables include the measured vibration amplitude and frequency,the bearing temperature,acoustic emission,etc.

These variables are GL feature vectors.After normalized,the vector can be defined as

where xi is the value of the i th normalized condition variable.

The limits of these variables are bearing condition limits,which are represented as

The condition of a bearing depends on the GL feature vector rather than the service time.The bearing condition functionψcan be defined as

When a condition variable of the GL feature vector exceeds its bearing condition limits of the corresponding stage,the bearing EGL can be described as follows:

(1)0＜j＜1.5,if xi＜h1i for all condition variables,EGL=1.

(2)1.5＜j＜2.5,if xi＞ h1ifor one condition variable,EGL=2.

(3)2.5＜j＜ 3.5,if xi＞ h2i for one condition variable,EGL=3.

(4)j(t)＞ 3.5,if xi＞ h3i for one condition variable,EGL=4.

Fig.2 shows the process of the service life of a bearing, which represents the relationship between the GL of the bearing and the service time of the condition variable. The condition limits H j are difficult to be acquired practically,so the method to determine the thresholds of each condition limit is proposed as follows:

Fig.2 Illustration of GL of bearing

(1)The bearing health condition is good and EGL=1.

(2) The bearing appears defect characteristics,EGL=2.

(3)As shown in Fig.2,the phenomenon that the value of a condition variable increases drastically over a short time interval indicates that the bearing comes into damaged bearing condition,and EGL=3.

(4)That the value of the condition variable increases sharply indicates the failure of the bearing,and EGL=4.

EGL is the defect severity estimation of the bearing. The diagnostic estimation model,in essence,is to assess the condition function j with the GL feature vector.

1.2.1 Feature selection

Many challenges stand in the way of predicting the GL of a bearing efficiently,including how to choose the features vector to evaluate the condition degradation of the bearing.

In current researches,time features,such as RMS,Kurtosis,or Crest Factor,were often chosen.Ref.[11]adopted the RMS value and the Kurtosis Factor of the vibration.Ref.[12]used the average of the amplitudes of the defective frequency and its first six harmonics as the degradation index of thrush ball bearings.However,even though a large variety of features can be extracted to describe the characteristics of vibration signal from different aspects, the previous researches demonstrated that each feature could only affect a certain defect at a certain condition because of the highly stochastic nature of defect growth[13-14]. For instance,spikiness of the vibration signals denoted by Crest Factor and Kurtosis indicates incipient defects,the high energy level denoted by RMSindicates severe defects.Therefore,a good performance assessment method should take advantage of mutual information of multiple features for system degradation assessment.

According to Ref. [15], the selection criterion of the condition variables should include diagnostic ability,sensitivity,consistency and amount of calculation. Finally, Kurtosis,Skewness,Shape Factor and RMSare selected as the condition variables in this paper.

1.2.2 SVM model

SVM model is adopted to realize the mapping between the GL feature vector and the GL of rolling bearings.For the identification model,the run-to-failure data acquired from accelerated life tests of an aircraft bearing are used as learning samples.

A linearly separable binary classification problem can be represented as

where x i is a n-dimensional feature vector belonging to either of the two classes w1,w2,and yithe corresponding class indicator(+ 1 for w1,-1 for w2).

In order to achieve linear separation in classification,it is necessary to map the inputs x into a feature space H(x)by a mapping H(?).Therefore,with the maximum margin,SVM separates the points by an optimal hyper-plane where x is an input vector,w an adaptive weight vector and b a bias. The SVM searches the parameters w and b to maximize the geometric marginby solving the following optimization problem[16].

In general case,the two classes are not separable. Hence, the slack variables a i is introduced.The objective function is given as

where C is the penalty factor of training errors.

By using Eq.(11),Lagrangian optimization function can be obtained, where Lagrangian multiplier is represented as T=[T1,T2,…,T n]T.The function is differentiated with respect to w,T,a to obtain the Karush-Kuhn-Tucker condition.Substituting the condition for the above equation,and holding k(x,y)= ＜H(x),H(y)＞ ,the dual optimi-sation function is given as

The solution T*is guaranteed to be the global minima.These inputs x i at this T*ι≠0 are called the support vector(SV).Based on SVs,the SVM classification can be carried out as

The typical kernel functions are polynomials,radial basic and hyperbolic tangent ones.In the study,the radial basic function is adopted.

The SVM model input is the GL feature vector which is constructed by the four condition variables (Kurtosis,Skewness,Shape Factor and RMS).The model output is the value of condition function j. The EGL is acquired by discriminating the condition functions.

The implementation process of diagnostic estimate model based on SVM is as follows.

(1)Determinate the GL set,that is,status={GBC,IDC,DBC,FCC},whose corresponding value is f s={1,2,3,4}.Obtain the run-to-failure test data from the experiment.

(2)Preprocess the run-to-failure test data and compute the GL feature vector x,which consists of Kurtosis, Skewness,Shape Factor and RMSof the signal.

(3)Train SVM with the input parameter as the GL feature vector and the output as the corresponding value f s.

(4)Obtain the EGL of the bearing when the feature vector x of the vibration signals of a bearing is input to the SVM model.

2 GL FUZZY INFERENCE FUSION PROGNOSTIC MODEL

″GL″itself is fuzzy in nature.The fuzzy logic inference method[17]is adopted to fuse the mathematical model and diagnostic estimation model.The final GL of bearings is the fusion result of PGL and EGL. It can reduce uncertainties of the mathematical model. The architecture is shown in Fig.3.

Fig.3 Architecture of prognostic model

2.1 Fuzzy rule generation

The designed rules are based on the expert knowledge.The process consists of two parts:

(1) Knowledge acquisition and rule formation.

(2)Combination of rules.

Rule 1:If EGL is 1 and PGL is 1,GL is 1

Rule 2:If EGL is 1 and PGL is 2,GL is 1

Rule 3:If EGL is 2 and PGL is 1,GL is 2

Rule 4:If EGL is 2 and PGL is 2,GL is 2

Rule 5:If EGL is 2 and PGL is 3,GL is 2

Rule 6:If EGL is 1 and PGL is 3,GL is 2

Rule 7:If EGL is 2 and PGL is 4,GL is 3

Rule 8:If EGL is 1 and PGL is 4,GL is 3

Rule 9:If EGL is 3 and PGL is 1,GL is 3

Rule 10:If EGL is 3 and PGL is 2,GL is 3

Rule 11:If EGL is 3 and PGL is 3,GL is 3

Rule 12:If EGL is 3 and PGL is 4,GL is 4

Rule 13:If EGL is 4 and PGL is 1,GL is 4

Rule 14:If EGL is 4 and PGL is 2,GL is 4

Rule 15:If EGL is 4 and PGL is 3,GL is 4

Rule 16:If EGL is 4 and PGL is 4,GL is 4

2.2 Fuzzy inference parameter selection

2.2.1 Selection of quantization factor and fuzzy subset

The fuzzifica tion of fuzzy variables consists of two processes: (1) Quantization factor selection in which the basic range is transformed into fuzzy range effectively,(2)Fuzzy variables fuzzifica tion based on fuzzy subset and membership function.

The selection of quantization factor and fuzzy subset of PGL,EGL and GL is shown in Table 1-3 respectively.

Table 1 Quantization f actor and fuzzy subset of PGL

Table 2 Fuzzy subset of EGL

Table 3 Fuzzy subset of GL

2.2.2 Membership function

A membership function(M F)is a curve that defines how each point in the input or output space is mapped to membership value(or degree of membership)between 0 and 1.

The PGL,EGL and GL of bearings have four possible outcomes from a fuzzy set:GBC,IDC,DBC and FCC,which are defined and shown in Fig.4.

Fig.4 Membership functions of variables

Trapezoidal, bell and gaussi an curve membership functions are used.To some extent,the selection of this membership function is arbitrary.Other membership functions may yield better results.This problem needs to be further studied in future.

2.3 Fuzzy inference engine

After defining membership functions and generating the″if-then″rules,the fuzzy inference engine is built.Each rule is taken at a time,and the rules are employed using membership functions and fuzzy operators. For instance,when rules are applied or fired,the membership function outputs are shown in Fig.5. The numbers on the left hand of the figure refer to the Rule 1 to Rule 16 listed in Section 2.1.The input values and membership functions for PGL and EGL are listed in the first two columns(from the left to the right).Based on the inputs for each feature,the fired rules are shaded. The last column shows the output membership functions(GL)and the fired rules are shaded. For example,if EGL is 2.56 and PGL is 3.14,GL is 2.98.It shows the final GL of the bearing is 3.

Fig.5 Test results for bearing GL

3 EXPERIMENTAL VERIFICATION

3.1 Setup

The experimental equipment is constructed to perform the accelerated life test on bearings for the model verification.Test rig layout is shown in Fig.6.

Fig.6 Test rig layout

The setup has three sub-systems: a test housing system,an oil circulation system and a data acquisition sub-system.

The test housing system consists of a test housing,a hydraulic loading mechanism and a drive mechanism. The test housing loads and spins four bearings,in which,two roller bearings NU1010 are used as auxiliary load bearings.During the experiment,continuous lubrication of the testing bearings is provided by the oil circulation system.The type of the test bearing used in this research is a 6008 bearing,which is a single-row deep-groove ball bearing of the Conrad type assembly.The dynamic load is 17.0 k N.

Some undamaged 6008 test bearings run at a constant rotational speed(3 310 r/min)with an equivalent dynamical load of 8 kN to failure.The degradation databases are divided into two parts,a training set and a validation set.

An accelerometer is attached to the housing of the bearings. The vibration signals are acquired by the DASP and then input into a computer.RMS,Kurtosis,Skewness and Shape Factor can be captured as the condition variables by analyzing the vibration signals, and the samples are divided into four categories.

3.2 Result

The accelerated life test of the massive 6008 bearing is carried on the test platform. Three bearings are used.They are the number 1#,2#and 3#.Bearing 1# and 2# are designated as validation bearings.Bearing 3# has a ″seeded″damage.According to the load level,PGL of the test bearings can be calculated by mathematical model,shown in Table 4.

Table 4 PGL of test bearings

The analysis results of the 1#,2# and 3#bearings are shown in Figs.7-9 respectively.In these figures,once the load level is confirmed,the PGL curve is a deterministic curve.The EGL curve represents the results acquired by diagnostic estimation model. It indicates that bearing 1# comes into stageⅡ(EGL=2)in 2 420 min,stageⅢ (EGL=3)in 3 450 min and stageⅣ(EGL=4)in 3 782 min,and that bearing 2#comes into stageⅡ(EGL=2)in 2 722 min,stageⅢ (EGL=3)in 4 885 min and stageⅣ (EGL=4)in 10 250 min.The GL curve is generated by fuzzy logic inference.It reduces the affection of the uncertainty of the mathematical model.In Figs.(7-9),points A,B and C are the″critical point″of the GL stages.

Fig.7 PGL,EGL and GL of test bearing 1#

Fig.8 PGL,EGL and GL of test bearing 2#

Fig.9 PGL,EGL and GL of test bearing 3#

It should be noted that bearing 3# with a″seeded″damage does not experience the GL stage 1,which illustrates that the assessment based on vibration features reduces the uncertainty′s affection towards prediction results.

4 CONCLUSIONS

(1)GL model is better than traditional life model when describing the service life.Since the service life of bearings is vastly different,the traditional life definition based on time,obviously,is not suitable for prediction of an individual bearing life.

(2)GL model fuses the mathematical model and diagnostic estimation model to provide reliable lifetime prediction of bearings.

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