27.2 Abnormality Scaling

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The abnormality-scaling module described here

relies solely on the value of each feature during

normal operation. Accordingly, it is referred to as

the single category-based classifier (SCBC) to signify

its independence from feature values associated

with faulty conditions (Jammu et al., 1996). In

order to perform abnormality scaling, the SCBC

compares features with their normal values, and if

they are sufficiently different, assigns values

between zero and one to characterize their degree

of deviation from their normal values. The

schematic of the SCBC, which is implemented by

a connectionist network, is shown in Figure 27.5.

The inputs to SCBC are the raw features from

signal processing, siðtÞ; i ¼ 1; …; n; and its outputs

are abnormality-scaled features, fiðtÞ; with values between zero and one. The value of zero indicates

normality, and the other extreme of one denotes complete abnormality. The individual weights of the

SCBC network, wi; represent the normal values of the features, which are initially set equal to the first

corresponding feature value supplied to the SCBC.

Classification in the SCBC is performed by first measuring the Euclidean distance of each feature, siðtÞ;

from its weight value, wi; and normalizing it into the range ½0; 1􀀉 using a matching factor, fi; defined as

(Figure 27.6)

fiðtÞ ¼ 1 2 exp

2ðsiðtÞ 2 wiÞ

wi

􀀏 􀀐2

ð27:1Þ

A fi value of zero indicates that the feature value matches the weight value precisely, and a value of one

denotes that it deviates from it considerably. Note that the exponential function used here is not unique,

and that other functions that can map the Euclidean distance into the range ½0; 1􀀉 can also be used for the

matching factor. Since during normal operation of the gearbox, noise in the features usually causes them

Normal Region

1

Matching

1 fmin

Abnormality-Scaling

FIGURE 27.6 Matching and abnormality-scaling in SCBC.

Abnormality-Scaled

Features

Vibration Features

Weight

Abnormality-Scaling

Matching

FIGURE 27.5 Schematic of the SCBC network.

Fault Diagnosis of Helicopter Gearboxes 27-5

© 2005 by Taylor & Francis Group, LLC

to deviate from their normal values, a threshold, u, is considered to account for deviations by noise. The

threshold, u, is used to hard-limit fiðtÞ in SCBC as

fiðtÞ ¼

0 ðnormal Þ if fiðtÞ , u

fiðtÞ ðhard-limited Þ otherwise

(

ð27:2Þ

In the above relationship, the threshold u is obtained as

u ¼

1

n

Xn

i¼1

1 2 exp

2½maxðsiÞ 2 mi􀀉2

m2i

􀁻 !

ð27:3Þ

where maxðsiÞ denotes the maximum value of the ith feature in a set of k samples of this feature recorded

during normal operation, and mi represents its mean, estimated as

mi ¼

1

k

Xk

t¼1

siðtÞ ð27:4Þ

The matching factor, defined by Equation 27.1, suppresses any positive value in ½0; 1􀀉 into the range

½0; 1􀀉: As such, only very large deviations in the feature values will be scaled to the value of one. Since

large deviations in feature values are uncommon for gearboxes, the value of matching factor is further

scaled to yield abnormality-scaled feature values fiðtÞ as (Figure 27.6)

fiðtÞ ¼ fmin þ expða p fiðtÞÞ ð27:5Þ

where fmin represents the minimum abnormality value assigned to any feature that violates the threshold,

u, and a controls the slope of the exponential curve. Since fiðtÞ is defined to have a value between zero and

one, it is set to one when fiðtÞ in Equation 27.5 exceeds the value of one.

After each round of classification of the vibration features, the weight values in the SCBC are updated

so as to cope with noise and small variations in the operating conditions. Adaptation is carried out in two

stages. In the first stage, called primary adaptation, a network weight is adapted if the feature associated

with it is classified as normal. In the second stage, referred to as contrast enhancement (CE) (Carpenter

and Grossberg, 1987), the rest of the weights are adapted to achieve homogeneity in the abnormalityscaled

values, thus increasing the likelihood of all of them being classified as normal or abnormal.

Homogeneity, however, needs to exist only within specific feature groups, because gearbox faults do not

necessarily cause abnormality in all of the features. For example, a gear fault will be reflected only by the

features related to the gear, and is not expected to cause abnormality in bearing features. In order to

preserve the functionality of individual feature groups (i.e., general features, gear features, and bearing

features), CE is performed exclusively for each feature group.

Adaptation in SCBC is performed as follows. Let wI represent the weight which is presently being

updated and wi the remaining weights in the group. In primary adaptation, the weight value wI is

modified according to the relationship

wI ¼ wI þ dwI ð27:6Þ

where

dwI ¼

h½sI ðtÞ 2 wI 􀀉 if fI ðtÞ ¼ 0

0 otherwise

(

ð27:7Þ

with the parameter h denoting the learning rate.

For CE, if the majority of features are classified as normal, then the weight values associated with the

features classified as abnormal will be adjusted such that the likelihood of all of the features being

27-6 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

classified as normal is increased for the same feature. CE is performed as

wi ¼ wi þ dwi for all i – I ð27:8Þ

where

dwi ¼

hL½siðtÞ 2 wi􀀉 if fI ðtÞ ¼ 0

2hL½siðtÞ 2 wi􀀉 otherwise

(

ð27:9Þ

In CE, the amount by which the weight values are adjusted is controlled by a neighborhood function, L

(Kohonen, 1989), which is assigned a value between zero and one. A value of zero is used for inputs with

no noise, and a value at the other extreme of one is used for unreliable features with large amounts of

noise. Usually, in practice, the value of L is set less than 0.5. For each round of primary adaptation

(Equation 27.6 and Equation 27.7), I is varied to include all the features in the group. If the jth group of

features contains mj features, then primary adaptation is applied by varying I from one to mj; to cover all

the weight values wI in the jth feature group. For each I, the remaining weight values, wi; in the group

(i ¼ 1 to mj and i – I) are adapted using CE according to Equation 27.8 and Equation 27.9.

The adaptation algorithm presented in Equation 27.6 to Equation 27.9 is biased towards the most

recent feature vector if only this vector were used for adaptation, whereas adaptation should be ideally

performed using all of the feature vectors that pertain to the current operating conditions. However, as

the number of available feature vectors for the operating condition progressively increases, adaptation

based on all of the features becomes computationally demanding. As a compromise, in SCBC only the

b most recent feature vectors are utilized for each adaptation sweep, such that adaptation is performed

iteratively over the b most recent feature vectors. The learning rate, h, is progressively reduced for each

adaptation iteration (Equation 27.7 and Equation 27.9). Abnormality scaling formulae are summarized

in Table 27.1.

TABLE 27.1 Summary of the Abnormality-Scaling Formulae

Classification in the single category-based classifier is performed by first measuring the Euclidean distance of each vibration

feature, si ðtÞ; from its weight value wi ; and normalizing it into the range ½0; 1􀀉 using a matching factor, fi ; defined as

fi ðtÞ ¼ 1 2 exp

2ðsi ðtÞ 2 wi Þ

wi

􀀏 􀀐2

The matching factor is then hard-limited by a threshold, u, as

fi ðtÞ ¼

0 ðnormalÞ if fi ðtÞ , u

fi ðtÞ ðhard-limitedÞ otherwise

(

In the above relationship, the threshold, u, is obtained as

u ¼

1

n

Xn

i¼1

1 2 exp

2½maxðsi Þ 2 mi 􀀉2

m2i

􀁻 !

where maxðsi Þ denotes the maximum value of the ith feature in a set of k samples of this feature, recorded during normal

operation, and mi represents its mean, estimated as

mi ¼

1

k

Xk

t¼1

si ðtÞ

The vibration feature is then abnormality-scaled as

fi ðtÞ ¼ fmin þ expða p fi ðtÞÞ

where fmin represents the minimum abnormality value assigned to any feature that violates the threshold, u, and a controls

the slope of the exponential curve.

Fault Diagnosis of Helicopter Gearboxes 27-7

© 2005 by Taylor & Francis Group, LLC

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