Scien Directvibration base fault dianogis of spur bevel gear box using fuzzy technique

This paper presents the use of decision tree for selecting best statistical features that will discriminate the fault conditions of the gear box from the signals extracted.. The statisti

Vibration-based fault diagnosis of spur bevel gear box

using fuzzy technique

Department of Mechanical Engineering, Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu 641105, India

Abstract

To determine the condition of an inaccessible gear in an operating machine the vibration signal of the machine can be continuously monitored by placing a sensor close to the source of the vibrations These signals can be further processed to extract the features and identify the status of the machine The vibration signal acquired from the operating machine has been used to effectively diagnose the condition of inaccessible moving components inside the machine Suitable sensors are kept at various locations to pick up the signals produced by machinery and these signals are very meaningful in condition diagnosis surveillance To determine the important charac-teristics and to unravel the significance of these signals, further analysis or processing is required

This paper presents the use of decision tree for selecting best statistical features that will discriminate the fault conditions of the gear box from the signals extracted These features are extracted from vibration signals A rule set is formed from the extracted features and fed to a fuzzy classifier The rule set necessary for building the fuzzy classifier is obtained largely by intuition and domain knowledge This paper also presents the usage of decision tree to generate the rules automatically from the feature set The vibration signal from a piezo-electric transducer is captured for the following conditions – good bevel gear, bevel gear with tooth breakage (GTB), bevel gear with crack at root of the tooth (GTC), and bevel gear with face wear of the teeth (TFW) for various loading and lubrication conditions The statistical features were extracted and good features that discriminate the different fault conditions of the gearbox were selected using decision tree The rule set for fuzzy classifier is obtained by once using the decision tree again A fuzzy classifier is built and tested with representative data The results are found to be encouraging

Ó 2008 Elsevier Ltd All rights reserved

Keywords: Feature selection; Statistical features; Decision tree; Gear box; Fuzzy; Fault detection

1 Introduction

A faulty gear system could result in serious damage if

defects occur to one of the gears during operation

condi-tion Early detection of the defects, therefore, is crucial to

prevent the system from malfunction that could cause

dam-age or entire system halt Diagnosing a gear system by

examining the vibration signals is the most commonly used

method for detecting gear failures The conventional

meth-ods for processing measured data contain the frequency

domain technique, time domain technique, and time– frequency domain technique These methods have been widely employed to detect gear failures The use of vibra-tion analysis for gear fault diagnosis and monitoring has been widely investigated and its application in industry is well established (Cameron & Stuckey, 1994; Gadd &

is particularly reflected in the aviation industry where the helicopter engine, drive trains and rotor systems are fitted with vibration sensors for component health monitoring The raw vibration signal in any mode from a single point

on a machine is not a good indicator of the health or con-dition of a machine Vibration is a vectorial parameter with three dimensions and requires to be measured at several carefully selected points

0957-4174/$ - see front matter Ó 2008 Elsevier Ltd All rights reserved.

doi:10.1016/j.eswa.2008.01.010

*

Corresponding author Tel.: +91 4222656422; fax: +91 4222656274.

E-mail addresses: n_saravanan@ettimadai.amrita.edu , nsaro_2000@

yahoo.com (N Saravanan).

www.elsevier.com/locate/eswa Expert Systems with Applications 36 (2009) 3119–3135

Expert Systems with Applications

Vibration analysis can be carried out using Fourier

transform techniques like Fourier series expansion (FSE),

Fourier integral transform (FIT) and discrete Fourier

transform (DFT) (Collacott, xxxx) After the development

of large-scale integration (LSI) and the associated

micro-processor technology, fast Fourier transform (FFT)

ana-lyzers became cost effective for general applications The

raw signatures acquired through a vibration sensor needed

further processing and classification of the data for any

meaningful surveillance of the condition of the system

being monitored

machine (SVM) and Fuzzy classifier are widely used as

1998; Jack & Nandi, 2000a; Nandi, 2000; Samanta &

Al-Baulshi, 2003; Samanta, Al-Al-Baulshi, & Al-Araimi, 2003;

generalization of the results in models that can over fit

the data (Samanta et al., 2003) SVM has high classification

accuracy and good generalization capabilities for crisp data

the problem at hand, the nature of the fault itself is fuzzy

in nature Fuzzy classifier models the physical problem

under study more closely The flow chart of the fault

diag-nostic system is shown inFig 1

1.1 Different phases of present work

The signals obtained are processed further for machine

condition diagnosis as explained in the flow chartFig 1

2 Experimental studies

DC motor (0.5 hp) with speed up to 3000 rpm is the basic

drive A short shaft of 30 mm diameter is attached to the shaft of the motor through a flexible coupling; this is to minimize effects of misalignment and transmission of vibra-tion from the motor The shaft is supported at its ends through two roller bearings From this shaft the drive is transmitted to the bevel gear box by means of a belt drive

and the full lubrication level is 110 mm and half lubrication level is 60 mm

SAE 40 oil was used as a lubricant An electromagnetic spring-loaded disc brake was used to load the gear wheel A torque level of 8 N-m was applied at the full-load condi-tion The various defects are created in the pinion wheels and the mating gear wheel is not disturbed With the sensor mounted on top of the gear box vibrations signals are obtained for various conditions The selected area on the top of the gearbox for mounting the sensor is made flat and smooth to ensure effective coupling between the sensor and the gearbox The sensor used is a piezoelectric acceler-ometer (Dytran model) which is mounted on the flat

Vibration Signals

Feature Selection Using J 48 Algorithm

Rule Generation

Test data set

Modeling Fuzzy

system

Fuzzy inference engine

Fuzzy output

Machine Condition Diagnosis

Fig 1 Flowchart for bevel gear box health diagnosis.

Bevel

Fig 2 Fault simulator setup.

Pinion Wheel Gear

Wheel

Electromagnetic spring loaded disc brake

Fig 3 Inner view of the bevel gear box.

surface using direct adhesive mounting technique The

accelerometer is connected to the signal-conditioning unit

(DACTRAN FFT analyzer), where the signal goes through

the charge amplifier and an analogue-to-digital converter

(ADC) The vibration signal in digital form is fed to the

computer through a USB port The software RT Pro-series

that accompanies the signal-conditioning unit is used for

recording the signals directly in the computer’s secondary memory The signal is then read from the memory and pro-cessed to extract different features

2.1 Experimental procedure

In the present study, four pinion wheels whose details

wheel and was assumed to be free from defects In the other three pinion wheels, defects were created using electron dis-charge machine (EDM) in order to keep the size of the defect under control The details of the various defects are depicted inTable 2and its views are shown inFig 4 The size of the defects is in-line with work reported in literature (Gadd & Mitchell, 1984) The vibration signal from the piezoelectric pickup mounted on the test bearing was taken, after allowing initial running of the bearing for sometime The sampling frequency was 12,000 Hz and sample length was 8192 for all speeds and all conditions The sample length was chosen arbitrarily, however, the fol-lowing points were considered Statistical measures are more meaningful, when the number of samples is more

On the other hand, as the number of samples increases the computation time increases To strike a balance, sample length of around 10000 was chosen In some feature extrac-tion techniques, which will be used with the same data, the number of samples is to be 2n The nearest 2n–10,000 is

8192 and hence, it was taken as sample length Many trials were taken at the set speed and vibration signal was stored

Table 1

Details of faults under investigation

Gears Fault description Dimension (mm)

G2 Gear tooth breakage (GTB) 8

G3 Gear with crack at root (GTC) 0.8  0.5

Table 2

Gear wheel and pinion details

Chordal tooth thickness (mm) 3.93 0.150 3.92 0.110

Chordal tooth height (mm) 2.53 2.55

c

Fig 4 (a)View of good pinion wheel (b) View of pinion wheel with face wear (GFW) (c) View of pinion wheel with tooth breakage (GTB).

in the data file The raw vibration signals acquired for

var-ious experimental conditions form the gearbox using FFT

are shown inFig 5

3 Feature extraction

Statistical analysis of vibration signals yields different

reported (James Li & Wu, 1989) use these in combinations

to elicit information regarding bearing faults Such

proce-dures use allied logic often based on physical

Consider-ations A fairly wide set of these parameters is selected as

a basis for our study, as detailed below

a Mean b Standard error c Median d Standard

devi-ation e Sample variance f Kurtosis g Skewness h Range

i Minimum j Maximum k Sum

All the above mentioned statistical features were

extracted for the vibration signals obtained for various

conditions and fed as an input to J 48 algorithm for select-ing the best features which classify the different fault conditions

4 Descriptive statistics The statistical features are explained below

4.1 Standard deviation This is a measure of the effective energy or power con-tent of the vibration signal and clearly indicates deteriora-tion in the bearing condideteriora-tion The following formula was used for computation of standard deviation

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

nPx2ðPxÞ2

nðn  1Þ

s

:

-0.4

-0.2

0

0.2

Sample No.

Good-Dry-Unload

-0.4 -0.2 0 0.2

Good-Dry-FullLoad

Sample No.

-0.4

-0.2

0

0.2

Good-HalfLub-Unload

Sample No.

-0.2 0 0.2

Good-HalfLub-FullLoad

Sample No.

-0.2

0

0.2

Good-FullLub-Unload

Sample No.

-0.2 0 0.2

Good-Full-FullLoad

Sample No.

-0.4 -0.2 0 0.2

Sample No.

GTB-Dry-Unload

-0.4 -0.2 0 0.2

GTB-Dry-FullLoad

Sample No.

-0.4 -0.2 0 0.2

GTB-HalfLub-Unload

Sample No.

-0.4 -0.2 0 0.2

GTB-HalfLub-FullLoad

Sample No.

-0.4 -0.2 0 0.2

GTB-FullLub-Unload

Sample No.

-0.4 -0.2 0 0.2

GTB-FullLub-FullLoad

Sample No.

-0.1

0

0.1

Sample No.

GTC-Dry-Unload

-0.1 0 0.1

GTC-Dry-fullLoad

Sample No.

-0.1

0

0.1

GTC-HalfLub-Unload

Sample No.

-0.1 0 0.1

GTC-HalfLub-FullLoad

Sample No.

-0.1

0

0.1

GTC-FullLub-Unload

Sample No.

-0.1 0 0.1

GTC-FullLub-FullLoad

Sample No.

-0.2 0 0.2

Sample

TFW-Dry-Unload

-0.2 0 0.2

TFW-Dry-FullLoad

Sample

-0.2 0 0.2

TFW-HalfLub-Unload

Sample

-0.2 0 0.2

TFW-HalfLub-FullLoad

Sample

-0.2 0 0.2

TFW-FullLub-Unload

Sample

-0.2 0 0.2

TFW-FullLub-FullLoad

Sample

Fig 5 (a) Vibration signal for good pinion wheel under different lubrication and loading conditions (b) Vibration signal for pinion wheel with teeth breakage under different lubrication and loading conditions (c) Vibration signal for pinion wheel with crack at root under different lubrication and loading conditions (d) Vibration signals for pinion wheel with teeth face wear under different lubrication and loading conditions.

4.2 Skewness

Skewness characterizes the degree of asymmetry of a

distribution around its mean The below shown expression

was used to calculate the skewness, where ‘n’ is the sample

size and ‘s’ is the sample standard deviation

ðn  1Þðn  2Þ

s

:

4.3 Kurtosis

Kurtosis indicates the flatness or the spikiness of the

sig-nal Its value is very low for good bevel gearbox and high

for faulty gearbox due to the spiky nature of the signal

ðn  1Þðn  2Þðn  3Þ

s

2

ðn  2Þðn  3Þ:

where ‘s’ is the sample standard deviation

4.4 Standard error

Standard error is a measure of the amount of error in

the prediction of y for an individual x in the regression,

where x and y are the sample means and ‘n’ is the sample

size

Standard error of the predicted

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

ðn  2Þ

X

ðy  yÞ2

Pðx  xÞðy  yÞ

Pðx  xÞ2

v

u

:

4.5 Sample variance

It is variance of the signal points and the following

for-mula was used for computation of standard variance

4.6 Range

It refers to the difference in maximum and minimum

sig-nal point values for a given sigsig-nal

4.7 Minimum value

It refers to the minimum signal point value in a given

signal As the gear parts (crack, breakage, face wear) get

degraded, the vibration levels seem to go high Therefore,

it can be used to detect faulty gears

4.8 Maximum value

It refers to the maximum signal point value in a given signal

4.9 Sum

It is the sum of all signal point values in a given signal

5 Using J 48 algorithm in the present work

A standard tree induced with c5.0 (or possibly ID3 or c4.5) consists of a number of branches, one root, a number

of nodes and a number of leaves One branch is a chain of nodes from root to a leaf; and each node involves one attri-bute The occurrence of an attribute in a tree provides the information about the importance of the associated

represen-tation methodology used to represent classification rules J48 algorithm (A WEKA implementation of c4.5 Algo-rithm) is a widely used one to construct Decision Trees

The Decision Tree algorithm has been applied to the problem under discussion Input to the algorithm is set of statistical features of vibration signatures It is clear that the top node is the best node for classification The other features in the nodes of Decision Tree appear in descending order of importance It is to be stressed here that only fea-tures that contribute to the classification appear in the Decision Tree and others do not Features, which have less discriminating capability, can be consciously discarded by deciding on the threshold This concept is made use for selecting good features The algorithm identifies the good features for the purpose of classification from the given training data set, and thus reduces the domain knowledge required to select good features for pattern classification

condi-tions of different faults compared with good condicondi-tions of the pinion gear wheel

Based on the output of J 48 algorithm, the decision tree various statistical parameters are selected for the various conditions of the gearbox The values appearing between various nodes in the decision tree are used for generating the fuzzy rules to classify the various conditions of the gearbox under study

5.1 Application of decision tree for feature selection The algorithm has been applied to the problem under discussion for feature selection Input to the algorithm is

extracted from raw vibration signatures, the output is the Decision Tree It is clear there from that the top node is the best node for classification The other features appear

in the nodes in Decision Tree in descending order of

impor-tance It is to be stressed here that only features that

con-tribute to the classification appear in the Decision Tree

and others do not The level of contribution is not same

and all statistical features are not equally important The

level of contribution by individual feature is given by a

sta-tistical measure within the parenthesis in the Decision Tree

The first number in the parenthesis indicates the number of

data points that can be classified using that feature set The

second number indicates the number of samples against

this action If the first number is very small compared to

the total number of samples, then the corresponding

fea-tures can be considered as outliers and hence ignored

Fea-tures that have less discriminating capability can be

consciously discarded by deciding on the threshold This

concept is made use of in selecting good features The

algo-rithm identifies the good features for the purpose of

classi-fication from the given training data set and thus reduces

the domain knowledge required to select good features

for pattern classification problem

6 Methodology adopted for fuzzy classification

7 Fuzzy logic (classifier)

Fuzzy Logic provides a precise approach for dealing

with uncertainty Fuzzy inference is a method that

inter-prets the values in the input vector and, based on some

set of rules, assigns values to the output vector The

point of fuzzy logic is to map an input space to an

out-put space, and the primary mechanism for doing this is a

list of ‘if-then’ statements called rules Rules are the

inputs for building a fuzzy inference engine The

method-ology adopted for fuzzy classification is shown in Fig 6

All rules are evaluated in parallel, and the order of the

rules is unimportant The real world data do not have

sharply defined boundaries where information is often

incomplete or sometimes unreliable In quest for

preci-sion, scientists have generally attempted to manipulate

the real world into artificial mathematical models that

make no provision for gradation Because Fuzzy Logic

provides the tools to classify information into broad,

coarse categorizations or groupings, it has infinite

possi-bilities for application which have proven to be much cheaper, simpler and more effective than other systems

For the problem at hand, the condition of the gearbox, good or faulty is basically fuzzy in nature All the faults do not occur in the gearbox instantly It comes gradually In that case, there is no threshold value (crisp data) based

on which the decision on the condition of the gearbox can be taken (Whether gearbox is now good or faulty) The problems of this kind can be modeled using fuzzy logic

8 Membership function

A membership function (MF) is a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1 Observ-ing the values of the feature, based on which the branches

of the Decision Tree are created for different conditions of the gearbox, the membership functions for the correspond-ing features are defined There are four possible outcomes from a fuzzy classifier, namely: good bevel gear, bevel gear with tooth breakage (GTB), bevel gear with crack at root

of the tooth (GTC), and bevel gear with face wear of the teeth (TFW) for various loading and lubrication condi-tions Hence, four membership functions are defined with equal range for the output

9 Rule generation from decision tree Artificial neural network and support vector machine are used to generate rule for classification problems

xxxx) In this study, Decision Tree is used for that pur-pose Decision Tree shows the relation between features and the condition of the gearbox Tracing a branch from the root node leads to a condition of the gearbox (Refer

informa-tion available in a branch in the form of ‘if-then’ state-ment gives the rules for classification using fuzzy for

Dry/Half/Full

GOOD/GTC/

GTB/TFW Fig 6 Methodology of classification using Fuzzy Fig 7 Decision tree from J 48 Algorithm for dry-lub no-load condition.

various conditions of the gearbox Hence the usefulness of

the decision tree in forming the rules for fuzzy

classifica-tion is established

10 Generation of rules for various gearbox conditions and

discussions

The preceding section describes how the classification

has been carried out using fuzzy technique

10.1 Dry-lubrication and no-load condition

variance play a decisive role in classifying the various gear-box faults under dry lubrication and no-load condition This output of the decision tree is used to design the mem-bership function for fuzzy classifier as shown inFig 8–10

A membership function (MF) is a curve that defines how each point in the input space is mapped to a membership

Fig 8 Membership function for ‘‘standard error”.

Fig 9 Membership function for ‘‘kurtosis”.

Fig 10 Membership function for ‘‘variance”.

value (or degree of membership) between 0 and 1 In the

present study, trapezoidal membership function is used

The selection of this membership function is to some extent

arbitrary However, the following points were considered

while selecting membership function The Decision Tree

for the selected three features is shown inFig 7 Observing

the values of the feature, based on which the branches of

the Decision Tree is created, the membership functions

for all three features are defined for standard error,

kurto-sis and variance, respectively

10.1.1 Rules designed for the dry-lubrication and no-load

condition

1 If (stderr is not stderr) then (Output1 is GTC)

2 If (stderr is stderr) and (kurtosis is Kur) then (Output1 is

GOOD)

3 If (stderr is stderr) and (kurtosis is not Kur) and

(vari-ance is Var) then (Output1 is GTB)

4 If (stderr is stderr) and (kurtosis is not Kur) and

(vari-ance is not Var) then (Output1 is TFW)

The membership value of the condition being GTC is

when the standard error value is less than or equal to

0.000175 (fromFig 7) which is the threshold value Hence,

up to this threshold value the membership function

gener-ates the value ‘0’ and afterwards it increases linearly

(assumption) The trapezoidal membership function suits

this phenomenon and hence it was selected to map each

point in the input space to a membership value To review,

the threshold values are given by decision tree and the slope

is defined by the user through heuristics The threshold

value (0.000175) is defined based on the representative

training dataset If standard error value is less than or

equal to 0.000175, a membership function which is defined

on a 0–1 scale gives a value of 0 which means that it is not a

standard error If threshold value is greater than 0.000175,

the membership function generates a value of 1 Similarly

membership functions for other features are designed

accordingly and shown inFigs 9 and 10

There are four possible outcomes from a fuzzy classifier, namely: Good, GTC, GTB and TFW Hence, four member-ship functions are defined with equal range and shown in

10.1.2 Fuzzy inference engine After defining membership functions and generating the

‘if-then’ rules, the next step is to build the fuzzy inference engine The fuzzy toolbox available in MATLAB 7 was used for building fuzzy inference engine Each rule was taken at a time and using membership functions and fuzzy operators the rules were entered The rules were obtained from a training data set (150 trials in each condition) For testing the built model a portion of the data (100 trials

in each condition) called testing data was kept aside Using the testing data, the fuzzy inference engine was evaluated and its performance was presented as confusion matrix in

table (3) show the number of correctly classified instances

In the first row, the first element shows the number of data points belonging to ‘good’ class and classified by fuzzy logic as ‘good’ The second element shows the number of data points belonging to ‘GTC’ class and classified by fuzzy logic as ‘GTC’ The third element shows the number of data points belonging to ‘GTB’ class and classified by fuzzy logic as ‘GTB’ The fourth element shows the number of data points belonging to ‘TFW’ class and classified by fuzzy logic as ‘TFW’.Table 3illustrates the powerfulness

of the fuzzy rules designed with the aid of the decision trees

by the authors

Here each row corresponds to each rule as discussed in

Fig 11 Membership functions for condition (output).

Table 3

section 6.4.2 The first three blocks in rows represents the

membership function of standard error, kurtosis, variance,

respectively The fourth block corresponds to the

help of sample inputs for standard error, kurtosis and

var-iance the rules are tested as follows, for a sample input of

standard error as 0.0005, kurtosis as 10 and variance as

0.005 which satisfies the second rule completely and the

corresponding output condition is GOOD, which is shown

in the output block of the second row in the rule viewer

shown inFig 12

10.1.3 Confusion matrix

In lieu with the above discussions the fuzzy rules,

mem-bership functions, confusion matrix and rule viewer are

shown in Sections10.2–10.5 and 10.6

10.2 Dry-lubrication and full-load condition

10.2.1 Rules designed for the dry-lubrication and full-load condition

1 If (stderr is stderr) then (output1 is GTC)

2 If (stderr is not stderr) and (variance is var3) then (out-put1 is GTB)

3 If (kurtosis is kur) and (variance is var2) then (output1 is GOOD)

4 If (kurtosis is not kur) and (variance is var1) then (out-put1 is TFW)

Here there are three membership functions to represent three threshold values of variance in the decision tree

stderr = 0.0005, kurtosis = 10, Variance = 0.005, then the output is 6.25, i.e., the condition is GTB

10.2.2 Confusion matrix

Fig 12 Rule viewer for one of the test data.

Fig 13 Decision tree from J 48 Algorithm for dry-lub full-load condition.

10.3 Half-lubrication and no-load condition

10.3.1 Rules designed for half -lubrication and no-load

condition

1 If (stderr is stderr1) then (output1 is GTC)

2 If (stderr is not stderr2) then (output1 is GOOD)

3 If (stderr is stderr2) and (kurtosis is kur) then (output1 is

GTB)

4 If (stderr is stderr2) and (kurtosis is not kur) then (out-put1 is TFW)

10.3.2 Confusion matrix

10.4 Half-lubrication and full-load condition

10.4.1 Rules designed for half-lubrication and full-load condition

Fig 14 Membership function for ‘‘stderr”.

Fig 15 Membership function for ‘‘kurtosis”.

Fig 16 Membership functions for ‘‘variance”.