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Gear For1

The document discusses a method for estimating crack size in gear teeth using pixel counts and a fusion approach of gear condition indices. Two tests were conducted to calibrate and validate the proposed algorithm, with a focus on improving the discriminatory power of the indices through scattering matrices and orthogonalization. The methodology aims to provide a more accurate physical size measurement of cracks compared to existing indices.

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pandunugraha04
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10 views2 pages

Gear For1

The document discusses a method for estimating crack size in gear teeth using pixel counts and a fusion approach of gear condition indices. Two tests were conducted to calibrate and validate the proposed algorithm, with a focus on improving the discriminatory power of the indices through scattering matrices and orthogonalization. The methodology aims to provide a more accurate physical size measurement of cracks compared to existing indices.

Uploaded by

pandunugraha04
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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extracts the pixel counts between the two marks, and between the crack tip and the

point of crack

initiation. The crack length is then calculated from the pixel counts and the distance between the

two marks.

Two gear tests were performed, with the same procedures and conditions, using two

notched pinions meshing with a normal gear. The first test (test 1) was used to obtain a data set for

calibrating the proposed algorithm and test 2 was used to test the proposed algorithm. Please refer

to Choi, 2001 for details of experimental setup and results.

4. Crack Size Estimation


The problem of the aforementioned gear condition indices is that none of them gives the

physical size of a crack and a near monotonic trend as evident from Figure 4 which shows the ten

gear condition indices over the course of test 1. This study takes a gear condition index fusion

approach to estimate the size of a tooth crack from a number of better indices in terms of

discriminating power (Li and Yoo, 1998). To select the better ones among the aforementioned gear

condition indices, scattering matrices and orthogonalization are used to rank them.

Scattering Matrices. Let’s say a sample vector is made of features as its components.

Suppose that one has a set of n sample vectors x1...xn that comes from c disjoint subsets (X1,...Xc).

Each subset is from a class, with samples in the same class being somehow more similar than

samples in different classes. Let ni be the number of samples in Xi.

Mean vector for the ith class:

1
mi =
ni
∑x
x ∈Xi
(11)

Total mean vector:

1
m= ∑x
n x∈X
(12)

9
Scatter matrix for the cluster:

1
Si =
ni
∑ (x − m )( x − m )
x ∈Xi
i i
t
(13)

Within-cluster scatter matrix:


c
ni
Sw = ∑ Si (14)
i =1 n

Between-cluster scatter matrix:


c
ni
Sb = ∑ ( mi − m)( mi − m )t (15)
i =1 n

The amount of discriminatory information, J ( x j ) , conveyed by the jth feature is then defined as

Sb ( j, j)
J(x j ) = (16)
Sw ( j, j)

where S b ( j, j ) and S w ( j, j ) are the jth diagonal elements of matrices S b and S w respectively. The

feature with the maximum J ( x j ) value is the one that provides maximum discriminatory

information.

Orthogonalization Although scattering matrices identify the best feature out of the set of

features that are ranked, it is frequently necessary to use more than one feature for a given task. If

this is the case, it is necessary to find out which one of the remaining features provides the most

discriminating power in addition to that of the best one. Frequently, features are not mutually

orthogonal to one another. Therefore, the remaining features must be made orthogonal to the best

one before the scattering matrices can be applied again to find out which one has the most

additional discriminatory information. This orthogonalization process can be performed with the

following equation which makes vector x i orthogonal to x j ,

10

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