27.4 Sensor Location Selection

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Suitable mounting locations for accelerometers on the housing are generally those that provide both

close proximity to gearbox components and ease of mounting (also see Chapter 15). However,

considering that only a limited number of accelerometers can be used, the candidate locations need to be

selected so as to provide a comprehensive coverage of the components. The method presented here relies

on indices to quantify the suitability of various mounting locations. It uses a coverage index to define the

reach of each accelerometer, and an overlap index to represent the overlap of the locations in coverage of

the gearbox components. The monitoring effectiveness of various combinations of accelerometer

locations can then be estimated as a function of the coverage and overlap indices of the accelerometer

locations (Wang et al., 1999).

27.4.1 Coverage Index

The coverage index denotes the reach of each accelerometer in monitoring various components of the

gearbox. It can be computed based on an influence matrix such as that defined in the SBCN. It can be

defined, for example, as the sum of the corresponding row of the influence matrix, as

Cj ¼

1

N

XN

i¼1

uij þ lij

2 ð27:20Þ

Fault Diagnosis of Helicopter Gearboxes 27-11

© 2005 by Taylor & Francis Group, LLC

where uij and lij represent the upper and lower limits, respectively, of the fuzzy influence coefficients

between the gearbox component, i, and accelerometer, j, and N denotes the total number of gearbox

components. According to this index, an accelerometer location with a higher coverage index should be

considered a better candidate.

27.4.2 Overlap Index

The coverage index can be used to rank accelerometers individually. However, for a complex gearbox

where more than one accelerometer is needed for monitoring, the sum of the coverage indices associated

with all of the accelerometers in the suite will not provide a correct representation of their overall

effectiveness because it will ignore the overlap among them.

Although the overlap between accelerometer pairs may be defined according to the influence

coefficients, that definition will only consider the distance of the accelerometers from the components,

and will ignore factors such as accelerometer orientation and gearbox size. In order to consider these

other factors, the overlap index may be defined independent of the influence coefficients, based on an

overlap matrix that represents the level of cocoverage between all of the accelerometer pairs. The

individual components Ojk in this matrix can be defined as

Ojk ¼

Ejk

eaDjk ð1 þ SjkÞ ð27:21Þ

where Ojk denotes the overlap between the accelerometers, j and k, Ejk represents the similarity of

orientation between the two accelerometers, a denotes a constant to account for the size of the gearbox,

Djk accounts for the physical distance between the two accelerometers, and Sjk represents a symmetry

factor between the two accelerometers. The formulation of Equation 27.21 is based on the following

understanding. The overlap in coverage of two accelerometers depends mainly on their orientation and

location, that is, the more identical their orientation is and the closer they are mounted to each other, the

higher is their expected level of overlap. In the above equation, the orientation factor, Ejk [ ½0; 1􀀉; is set to

one when the accelerometers, j and k, have identical orientation. The distance factor, Djk [ ½0; 1􀀉; is

defined as the normalized shortest geometrical distance between the two accelerometers along the casing,

where the term eaDjk approximates the attenuation of vibration due to distance (Lindsay, 1960) and the

constant a, with values ranging between zero and two, accounts for the effect of gearbox size on vibration

attenuation. Although the main factors in the coverage overlap of two accelerometers are orientation and

distance, the accelerometers that are symmetrically positioned are believed to have a larger overlap with

each other. In order to account for this factor, a symmetry factor, Sjk [ ½0; 1􀀉; is also included in Equation

27.21, which is set to one when two accelerometers are perfectly symmetrical with respect to the housing.

Based on Equation 27.21, the values of Ojk would ordinarily fall in the range ½0; 1􀀉; where the value of one

indicates that the two accelerometers, j and k, have 100% overlap with each other and that one of them

can be removed. Similarly, when Ojk is zero then the two accelerometers are assumed to be covering

completely different components. It should be noted, of course, that some overlap between

accelerometers is considered useful and is often factored in when selecting accelerometer locations.

The overlap coefficients only represent the level of overlap between pairs of accelerometers and not the

specific components covered jointly by each accelerometer pair. As such, defining the level of overlap

between several accelerometers is not as straightforward. At one extreme, one can assume that the

coverage overlap of each accelerometer pair in the suite does not coincide with the coverage overlap of the

other accelerometer pairs, so the total overlap, Os, for the accelerometer suite can be computed as

Os ¼

Xm

j¼1

min

Xm

k¼jþ1

Ojk

Cj þ Ck

2

; Cj

0

@

1

A ð27:22Þ

where m denotes the number of accelerometers in the suite, and the min function ensures that the

computed overlap will not exceed the total coverage of accelerometer j. At the other extreme, it can

27-12 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

be assumed that the overlaps of all of the accelerometer pairs coincide, and that the total overlap of

the accelerometer suite can be defined as

Os ¼

Xm

j¼1

max Ojk

Cj þ Ck

2

for all k . j

􀀏 􀀐

ð27:23Þ

The latter formulation only considers the largest overlap between accelerometer j and the other accelerometers

in the suite. It is also possible to use the average of the above two extremes as a compromise:

Os ¼

Xm

j¼1

1

2

min

Xm

k¼jþ1

Ojk

Cj þ Ck

2

; Cj

0

@

1

A þ max Ojk

Cj þ Ck

2

for all k . j

0 􀀏 􀀏 􀀐􀀐

@

1

A

2

4

3

5 ð27:24Þ

It should be noted that, unlike the coverage index, the overlap index is suite related because the overlap of

an accelerometer depends on the other accelerometers in the suite.

27.4.3 Monitoring Effectiveness

As mentioned earlier, the coverage index can be used to evaluate the effectiveness of individual

accelerometers in monitoring, but its sum cannot be a sole measure of effectiveness of suites of

accelerometers. For instance, a suite consisting of the top three accelerometer locations in terms of

coverage may be inferior to another suite of three accelerometer locations with a lower total level of

coverage but less overlap among the accelerometers. The best set of accelerometer locations is, therefore,

one which provides the highest coverage of the gearbox components and the least overlap among the

accelerometers. As such, the monitoring effectiveness (ME) of a suite can be expressed as

MEs ¼

Xm

j¼1

Cj

0

@

1

A 2 Os ð27:25Þ

where m represents the number of accelerometers in the suite, Ci denotes the coverage of individual

accelerometers in the suite, and Os represents the total overlap among the accelerometers in the suite

(Equation 27.24), estimated from Equation 27.22, Equation 27.23, or Equation 27.24. The indices used in

sensor location are summarized in Table 27.2.

TABLE 27.2 Summary of Indices Used for Sensor Location Selection

Sensor location is performed according to the following indices:

Coverage index

Cj ¼

1

N

XN

i¼1

uij þ lij

2

where uij and lij represent the upper and lower limits of the fuzzy influence coefficients between the gearbox component, I,

and accelerometer, j, and N denotes the total number of gearbox components.

Overlap index

Ojk ¼

Ejk

eaDjk ð1 þ Sjk Þ

where Ojk denotes the overlap between the accelerometers, j and k, Ejk represents the similarity of orientation between the

two accelerometers, a denotes a constant to account for the size of the gearbox, Djk accounts for the physical distance

between the two accelerometers, and Sjk represents a symmetry factor between the two accelerometers.

According to the above indices, the monitoring effectiveness of an accelerometer suite is defined as

MEs ¼

Xm

j¼1

Cj

0

@

1

A 2 Os

where m represents the number of accelerometers in the suite, Cj denotes the coverage of individual accelerometers in the

suite and Os represents the total overlap among the accelerometers in the suite.

Fault Diagnosis of Helicopter Gearboxes 27-13

© 2005 by Taylor & Francis Group, LLC

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