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27.5 A Case Study

Back

This case study illustrates the application of the SBCN to an OH-58A main rotor gearbox (Figure 27.1;

Jammu et al., 1996). Experimental vibration data for the OH-58A gearbox were collected at the NASA

Lewis Research Center as part of a joint NASA/Navy/Army advanced lubricants program (Lewicki et al.,

1992). Various component failures in the OH-58A transmission were produced during accelerated

fatigue tests. The vibration signals were recorded by eight piezoelectric accelerometers (frequency

range of up to 10 kHz) using an FM tape recorder. The signals were recorded once every hour, for about

1 to 2 min per recording (using a bandwidth of 20 kHz). Two magnetic chip detectors were also used to

detect the debris caused by component failures. The location and orientation of the accelerometers are

shown in Figure 27.2. The OH-58A gearbox was run under a constant load, and was disassembled

and inspected periodically or when one of the chip detectors indicated a failure. A total of eleven failures

occurred during these tests. They consisted of three cases of planet bearing pitting fatigue, three cases of

sun gear pitting fatigue, two cases of top housing cover cracking, and one case each of spiral bevel pinion

(SBP) pitting fatigue, mast bearing micropitting, and planet gear pitting fatigue.

In order to extract the vibration features, the vibration signals from the gearbox were digitized and

processed by a commercially available signal analyzer (Stewart Hughes Ltd., 1986). Overall, 54 vibration

features were extracted from each accelerometer for the OH-58A gearbox. Out of these features, 35

features were gear-related features (7 for each of the five gears). The remaining 19 features were indicators

of either general faults (e.g., wear and out-of-balance), or general gear and bearing faults.

27.5.1 Structural and Featural Influences

The structural influences for the OH-58A gearbox were obtained through five primary vibration travel

paths: (1) duplex bearing-spiral bevel mesh-triplex bearing, (2) duplex bearing-sun-planet mesh-ring

gear, (3) mast roller bearing-main shaft-mast ball bearing, (4) ring gear-planet bearing-mast ball bearing,

and (5) duplex bearing-sun planet mesh-mast ball bearing. The first travel path was in connection to

accelerometers 4, 5, and 6, whereas all the other paths were connected to accelerometers 1, 2, 3, 6, 7, and 8.

The RMS values of vibration were then computed using the lumped-mass model of these paths with

excitation sources at each of the gearbox components. These RMS values were then used as the basis for

defining the fuzzy structural influences between each component – accelerometer pair (Table 27.3).

The structural influences in Table 27.3 indicate that all of the components in the gearbox are covered

by the accelerometers, and that some accelerometers have identical influences with respect to the

components. Although these influences are not completely accurate due to their neglect of the

orientation of accelerometers and various approximations (Jammu et al., 1998), they can still be used for

TABLE 27.3 Structural Influences between the Components of the OH-58A Gearbox and the Eight Accelerometers

Component/Accelerator 1 2 3 4 5 6 7 8

Triplex bearing — — — H H — — —

Spiral bevel pinion — — — H H — — —

Pinion roller bearing — — — H H — — —

Spiral bevel gear — — — H H — — —

Duplex bearing M M M H H M M M

Gear roller bearing M M M H H M M M

Mast roller bearing M M M — — M M M

Main shaft M M M — — M M M

Mast ball bearing M M M — — M M M

Sun gear H H H L M H H H

Planet bearing H H H L M H H H

Planet gear H H H L M H H H

Ring gear H H H L M H H H

The influences shown are: “—” nil, “L” low, “M” medium, and “H” high.

27-14 Vibration and Shock Handbook

an overall assessment of the effectiveness of various accelerometers and their redundancy. For example,

the influences in Table 27.3 are identical for accelerometers 1, 2, 3, 6, 7, and 8. This would indicate that

one or more of these accelerometers can be discarded without any drastic effect on fault diagnostic

effectiveness. However, it should be noted that the strategy used in SBCN relies on the averaging effect of

accelerometers, and that it will not function without a certain level of overlap in accelerometer coverage.

The featural influences were defined for individual vibration features according to the type of fault they

are supposed to represent. For example, bearing-related features such as envelope band and tone energy

were assigned high in association with gearbox bearings. Similarly, the signal averaged features for the five

gears in the OH-58A gearbox were assigned high with respect to these gears. Features that are indicators

of faults in all rotating elements in the gearbox were assigned an influence of medium for gears as well as

bearings.

The structural and featural influences constitute the basis of the SBCN connection weights. As

discussed in Jammu et al. (1998), diagnosis in SBCN is performed hierarchically, first isolating the faulty

subsystems and then faulty components. As such, only the average of the structural influences associated

with each subsystem is used as the connection weights, vik, of SBCN to yield the fault possibility values,

pk(t), associated with each subsystem, according to Jammu et al. (1998):

pkðtÞ ¼

i¼1

fiðtÞvik ð27:26Þ

In subsystem diagnosis, fi(t) denotes the average abnormality-scaled value of general features. In the

second stage of fault diagnosis, at the component level, the combination of structural and featural

influences are used as vik, and fi(t) consist of individual abnormality-scaled features.

27.5.2 Evaluation of Influences

As explained in Jammu et al. (1998), the structural influences representing the proximity effect of

component faults on accelerometer readings were obtained from the RMS value of the frequency

response of the lumped-mass model of the gearbox. In this case, however, the actual RMS values of the

vibration were available at several fault instances for the OH-58A gearbox, which could also be used to

yield a set of experimentally obtained structural influences. In order to evaluate the modeled influences, a

comparison between these two sets of structural influences was conducted. The experimental influences

were obtained by normalizing the experimental RMS values when a faulty component had been

detected and using them as the basis for assigning the level of fuzzy influences (Table 27.4). The results in

Table 27.4 indicate mixed agreement between experimental and structural influences. For example, the

structural influences of sun gear, planet bearing, and planet gear on accelerometers 1, 2, and 3 are close to

the experimentally obtained influences, but the influence of mast ball bearing on accelerometers 2 and 3

TABLE 27.4 Influences between Accelerometers and Gearbox Components Obtained from Experimental RMS

Values of Vibration for the OH-58A Gearbox

Influences from RMS Values of Vibration

Accelerator/Parts Spiral Bevel Pinion Mast Ball Bearing Sun Gear Planet Bearing Planet Gear

1 —( –) —( –) H(H) M (H) M (H)

2 M( –) —(H) M (H) M (H) M (M)

3 M( –) —(H) M (H) H (H) M (M)

4 L(H) — (– ) H (L) — ( –) M ( – )

5 H(H) — (– ) H (M) H (M) — (M)

6 L( –) —( –) L(H) M (H) L (H)

7 —( –) —( –) H(H) M (H) L (H)

8 —( –) —( –) M(H) M (H) M (H)

For comparison, the influences from the lumped-mass model are shown inside parentheses.

Fault Diagnosis of Helicopter Gearboxes 27-15

do not match. In this case, the mismatch between the two sets of influences may be due to (1) the

limitation of the RMS value in reflecting the change in vibration as a result of various component faults

(e.g., mast ball bearing micropitting), (2) variation in the level of change of the RMS values as a function

of the type and size of the fault, and (3) lack of faults in every component of the OH-58A gearbox, which

limits the ability to determine the influences for every gearbox component.

Defining influences based on an approximate model of the gearbox was motivated by the need to avoid

supervised training of the SBCN. However, given that experimental data were available for the OH-58A

gearbox, an evaluation of the structural influences could be performed by comparing them with the

weights of a connectionist network structurally similar to SBCN, but trained by supervised learning. For

this purpose, the OH-58A gearbox was divided into three subsystems (Figure 27.1), and the supervised

network, having three output units for the three subsystems and eight input units for the eight

accelerometers, was trained using least-mean-square (LMS) learning. The weights of this network were

trained until the number of false alarms and misdiagnoses were reduced to zero. The trained weights were

then normalized and converted into fuzzy variables for comparison with the structural influences

obtained from the lumped-mass model of the gearbox. Table 27.5 includes the influences of the two

networks, where the modeled influences (inside parentheses) represent the average of component

influences within each subsystem (Table 27.3). The results in Table 27.3 indicate general agreement

between the trained weights and modeled influences. Some of the trained weights had negative values

(indicated by “ p ”), which is inevitable due to the use of LMS learning. However, all of these negative

weights were quite small in magnitude, which makes them consistent with their modeled nil counterparts

(denoted by “ – ”). As in the case of influences from the RMS values (Table 27.4), some of the mismatches

in Table 27.5 are expected to be due to the limited number of faults represented in the experimental data.

For example, the considerably different influence of subsystem 2 on accelerometer 1, or subsystem 3 on

accelerometer 1, is attributed to the lack of specific faults in these subsystems that would lead to a more

accurate influence on accelerometer 1.

The comparison between the modeled structural influences and those obtained from the supervised

neural network indicates that the modeled influences are in good agreement with the trained influences,

and that the lumped-mass modeling used here provides an acceptable set of structural influences for

the SBCN.

27.5.3 Fault Detection Results

The fault detection network (FDN) in the proposed diagnostic system (Figure 27.4) is used first to identify

the presence of faults in the gearbox. Fault diagnosis is then performed when the presence of a fault is

prompted. A total of eight FDNs, one for each of the eight OH-58A accelerometers, were used. The inputs

to each FDN were the 19 general features not specific to any particular gear or bearing. The initial weight

TABLE 27.5 Normalized Weight Values of the

Supervised Connectionist Network

Accelerator Subsystem

1 2 3

1 p ( – ) p (M) p (H)

2 p ( – ) H (M) H (H)

3 p ( – ) p (M) L (H)

4 H (H) p ( – ) p (L)

5 M (H) p ( – ) H (M)

6 M (M) p (M) H (H)

7 M ( – ) L (M) H (H)

8 p ( – ) H (M) M (H)

For comparison, the subsystem influences of the SBCN

are included inside parentheses. A “p” indicates a negative

weight value.

27-16 Vibration and Shock Handbook

values of the FDNs were set as the values of the first set of features for each of the five tests, and were

subsequently adapted using 50 adaptation sweeps for each training batch. The occurrence of a fault was

prompted when any of the FDNs indicated a fault. The detection results for individual test sets obtained

from the FDNs are shown in Table 27.6. A “ – ” in this table indicates normal conditions, whereas a “1”

indicates the presence of a fault. The expected detection results are indicated inside parentheses. The results

for test 1 indicate that the presence of faults was detected on days 5, 7, and 8, while faults were expected to be

present from days 5 to 9. Of course, it should be noted that the gearbox was not inspected on a daily basis, so

the actual condition of the gearbox is unknown for each day of the tests. In test 1, which was run for nine

days, a fault was actually observed only on day 9 during a routine inspection of the gearbox (indicated by

1p). However, based on an inspection of the vibration features, it was estimated that the fault could have

been present as early as day 5. For the other tests, the days when the faults were actually observed are also

indicated by 1p. While it is discouraging to note that day 6 of test 1 was classified as normal though a fault

was present on day 5, the results are in agreement with observations by experts who believe that sometimes

increased noise levels immediately after the occurrence of faults mask the effect of faults on vibration

features. For the other tests, the results indicate that except for an undetected fault in test 3 and a false alarm

in test 4, excellent fault detection was obtained. It should also be noted that the fault on day 9 of test 3 was a

hairline crack, which was perhaps undetectable through vibration monitoring. The quality of fault

detection in the proposed system is particularly important to the overall diagnostic results, since it is only

after a fault is detected that SBCN is engaged in diagnosis.

In summary, the detection results obtained (Table 27.6) indicate that the occurrences of most of the

faults were identified. This provides assurance that the later stage of diagnostics would not be hampered

by the detection phase.

27.5.4 Fault Diagnostic Results

In the proposed system, fault diagnosis is performed by the SBCN only after the presence of a fault is

detected. In this system, fault diagnosis is performed in two hierarchies so as to take full advantage of the

separation of the structural and featural influences. In the first hierarchy the gearbox is divided into

subsystems (Figure 27.1) and the faults in individual subsystems are isolated by the SBCN based on the

structural influences alone. For each subsystem, the weights of the SBCN are set equal to the average of

the structural influences of the components within that subsystem (Table 27.5). The inputs to the SBCN

TABLE 27.6 Fault Detection Results for the OH-58A Gearbox

Day Fault Detection Network: Predicted and Actual Failures

Test1 Test2 Test3 Test4 Test 5

1 — ( – ) — (– ) — ( –) — ( – ) — ( – )

2 — ( – ) — (– ) — ( –) — ( – ) — ( – )

3 — ( – ) — (– ) 1 (1) — ( –) — ( – )

4 — ( – ) — (– ) 1 (1p) — ( – ) — ( – )

5 1 (1) — (– ) — ( –) — ( – ) — ( – )

6 — (1) — (– ) — ( –) — ( – ) — ( – )

7 1 (1) — (– ) — ( –) — ( – ) — ( – )

8 1 (1) — (– ) — ( –) — ( – ) — (1)

9 — (1p) —( –) — (1p) — ( – ) 1 (1)

10 — ( –) 1 (1) — (1)

11 1 (1) 1 (1) — (1p)

12 1 (1) 1 (1p)

13 — (1p) 1 ( – )

14 1 (1)

15 1 (1p)

A “—” indicates normality and a “1” represents the presence of a fault. For reference, the

expected faults determined by an expert are included inside the parentheses, with “ p ”

indicating actual observation of the fault.

Fault Diagnosis of Helicopter Gearboxes 27-17

for this stage of the diagnosis consist of the averaged values of abnormality-scaled features from each

accelerometer, and its outputs are the fault possibility values for each subsystem. In the second hierarchy,

the faulty components within each subsystem are isolated. The inputs to the SBCN for this stage of the

diagnosis are the abnormality-scaled vibration features, and the weights are the product of featural

influences and structural influences of the subsystem containing each component. The averaging of

fuzzy influences here was done by taking the average of upper bounds and lower bounds of individual

influences separately, and then defining the fuzzy variable that would match the range. The product of

fuzzy influences was determined by multiplying the upper bounds and lower bounds of fuzzy variables

separately, and then defining the fuzzy variable for the resultant range.

27.5.4.1 Faulty Subsystem Isolation

The fault possibility values for the three subsystems of the OH-58A gearbox are shown in Table 27.7. The

results in this table represent the hard-limited fault possibility values (threshold of 0.5) and include, for

comparison, the actual condition of the gearbox reported from routine inspection inside parentheses.

As before, a “ p” indicates actual observation of the fault during inspection of the gearbox. The results

in Table 27.7 indicate that in test 1, faults in subsystems 1 and 3 were correctly identified on days 5, 7, and

8. In test 3, the faults in subsystem 3 on days 3 and 4 were correctly identified, along with a possible fault

in subsystem 1. The housing crack on day 9 of this test was left unidentified because it was never

prompted during the detection phase. In any case, this particular fault (a housing crack) could not be

isolated by the current SBCN due to absence of features that reflect this fault. Also for this test, faults in

subsystems 2 and 3 were correctly identified on days 11 and 12. In Test 4, the fault in subsystem 3 was

correctly diagnosed on days 10, 11, 12, 14, and 15. Moreover, on day 13 of test 4, even though the gearbox

was supposed to be normal, the SBCN indicated faults in subsystem 3. This was due to the replacement of

the three-planet assembly with a four-planet assembly, which changed the vibration characteristic of

subsystem 3. In test 5, the fault in subsystem 3 was correctly identified on day 9. There was also a

misdiagnosis in subsystem 1.

In summary, the diagnostic results from the gearbox subsystems indicate that all of the eight subsystem

faults were correctly identified in the OH-58A gearbox and that four faults were misdiagnosed.

Considering that these results were obtained by using structural influences alone as the connection

weights of the SBCN, the results validate the utility of these influences and of lumped-mass modeling as a

means of representing the vibration travel path of gearboxes for their model-based fault diagnosis.

TABLE 27.7 Faulty Subsystem Isolation Results for the OH-58A Gearbox

Day Faulty Subsystems Isolation for OH-58A

Test1 Test2 Test3 Test4 Test 5

1 — ( – ) — (– ) — ( –) — ( – ) — ( – )

2 — ( – ) — (– ) — ( –) — ( – ) — ( – )

3 — ( – ) — (– ) 1, 3 (3) — ( –) — ( – )

4 — ( – ) — (– ) 1, 3 (3p) — ( – ) — ( – )

5 1, 3 (1, 3) — (– ) — ( –) — ( – ) — ( – )

6 — (1, 3) — (– ) — ( –) — ( – ) — ( – )

7 1, 3 (1, 3) — (– ) — ( –) — ( – ) — ( – )

8 1, 3 (1, 3) — (– ) — ( –) — ( – ) — (3)

9 — (1p, 3p) —( –) — (3) — ( –) 1, 3 (3)

10 — ( –) 3 (3) — (3)

11 2, 3 (2, 3) 3 (3) — (3p)

12 2, 3 (2, 3) 3 (3p)

13 — (2p, 3p) 3 ( – )

14 3 (3)

15 3 (3p)

The three subsystems in the table are the input subsystem (1), the output subsystem (2), and the transmission subsystem (3).

For comparison, the actual faults are included inside parentheses with “ p ” indicating the observed faults.

27-18 Vibration and Shock Handbook

27.5.4.2 Faulty Component Isolation

Fault possibility values associated with the components of the OH-58A gearbox obtained from the SBCN

are included in Table 27.8. The results indicate that the diagnostics associated with individual

components are not as accurate as those obtained for the subsystems. Briefly, for test 1, the SBP fault in

subsystem 1 and the sun gear (SG) fault in subsystem 3 were correctly identified only on days 5 and 8,

respectively, while other components were assigned higher fault possibility values on other days. For test

3, the three bearing faults in subsystems 2 and 3 (BRG2 and BRG3, respectively) were correctly identified

on days 3, 4, and 12, but other components were also given high fault possibility values. In test 4, the

bearing fault in subsystem 3 (BRG3) was correctly identified only on day 10, while the SG fault remained

misdiagnosed. In test 5, the SG fault was correctly identified on day 9, while the planet gear (PG) fault was

misdiagnosed.

In view of the promising results obtained at the subsystem level, which confirm the validity of the

structural influences, the cause of diagnostic inaccuracies at the component level should be attributed

mainly to the deficiency of gear and bearing specific features used in this study. The strong cross-coupling

TABLE 27.8 Faulty Component Isolation by SBCN for the OH-58A Gearbox

Days Faulty Component Isolation for OH-58A

SS1 SS2 SS3

SBP SBG BRG1 BRG2 SG PG RG BRG3

Test 1

1 – 4 — — — — — — — —

5 0.90p 0.62 0.89 — 0.52p 0.73 0.12 0.86

6 —p — — — —p — — —

7 0.68p 0.43 0.79 — 0.67p 1.00 0.23 0.72

8 0.65p 0.74 0.18 — 0.98p 0.70 0.70 0.33

9 — — — — — — — —

Test 2

1 – 9 — — — — — — — —

Test 3

1 and2 — — — — — — — —

3 0.43 0.77 0.80 — 0.65 0.56 0.71 0.72p

4 0.38 0.60 0.78 — 0.56 0.47 0.04 0.79p

5 – 10 — — — — — — — —

11 — — — — 0.67 0.79 0.52 —p

12 — — — 0.74p 0.67 0.71 0.55 1.00p

13 — — — —p — — — —p

Test 4

1 – 9 — — — — — — — —

10 — — — — 0.34 0.41 0.75 0.79p

11 — — — — 0.54 0.53 0.79 —p

12 — — — — 0.59 0.50 0.91 0.64p

13 — — — — 0.72 0.85 0.83 1.00

14 — — — — 0.81p 0.90 0.88 0.68

15 — — — — 0.79p 0.90 0.93 0.48

Test 5

1 – 8 — — — — — — — —

9 0.58 0.24 0.68 — 0.60p 0.54p 0.50 0.58

10 and 11 — — — — —p —p — —

The components listed are SBP: spiral bevel pinion; SBG: spiral bevel gear; BRG1: bearings in subsystem (SS) 1; BRG2:

bearings in SS2; SG: sun gear; PG: planet gear; RG: ring gear; and BRG3: bearings in SS3. As before, “ p ” indicates observation

of the faulty component.

Fault Diagnosis of Helicopter Gearboxes 27-19

between these features is illustrated by the maximum values of the correlation coefficients between the gear

and bearing features and gear and bearing faults in Table 27.9. The results indicate reasonable correlation

values of 0.49 between the gear features and gear faults, and 0.44 between the bearing features and bearing

faults. These numbers, however, are not as impressive when they are compared with the cross-correlation

values of 0.57 between gear features and bearing faults, and 0.38 between bearing features and gear faults.

The high cross-correlation values in Table 27.9 indicate that the gear and bearing features do not provide

the resolution necessary for faulty component isolation. The manifestation of the resolution problem

caused by the coupling between gear and bearing features is observed in the similar fault possibility values

of 0.9 and 0.89 on day 5 of test 1, in Table 27.8, for the SBP and bearings in subsystem 1 (BRG1). Based on

the results in Table 27.8, one can conclude that the unsupervised pattern classification scheme

incorporated in this research cannot be a substitute for well-defined features, and that a more effective set of

features with smaller cross-correlation values are needed for diagnosis at the component level.

27.5.5 Sensor Location Evaluation

The eight candidate accelerometer locations (see Figure 27.2 for their locations and orientations), which

were actually used for vibration measurement during accelerated fatigue tests of the gearbox, were

analyzed and ranked for their significance in monitoring. In order to obtain the overlap coefficients for

the OH-58A gearbox, the values of the orientation factors, Ejk, distance factors, Djk, and symmetry

factors, Sjk, for the eight accelerometer locations were defined as

Ejk ¼

1 0:5 0:3 1 0:4 0:8 1 0:8

0:5 1 0:3 0:5 0:4 0:7 0:5 0:7

0:3 0:3 1 0:4 1 0:3 0:3 0:3

1 0:5 0:4 1 0:4 0:5 1 0:8

0:4 0:4 1 0:4 1 0:4 0:4 0:4

0:8 0:7 0:3 0:5 0:4 1 0:8 0:6

1 0:5 0:3 1 0:4 0:8 1 0:8

0:8 0:7 0:3 0:8 0:4 0:6 0:8 1

66666666666666666664

77777777777777777775

; aDjk ¼

0 0 0 1:5 1:5 0:5 2:0 0:5

1:5 1:5 1:5 0 0:8 1:0 0:5 2:0

1:5 1:5 1:5 0:8 0 1:0 0:5 2:0

0:5 0:5 0:5 1:0 1:0 0 1:5 1:0

2:0 2:0 2:0 1:5 1:5 1:5 0 1:5

0:5 0:5 0:5 2:0 2:0 1:0 1:5 0

66666666666666666664

77777777777777777775

Sjk ¼

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1

1 1 1 1 0 0 0 0:5

0 0 0 0 0 1 0:5 0

66666666666666666664

77777777777777777775

TABLE 27.9 Maximum Values of Correlation Coefficients between Features and Faults

Fault-Feature Correlation Coefficients

Features/Faults Gear Faults Bearing Faults

Gear features 0.49 (1) 0.57 (0)

Bearing features 0.38 (0) 0.44 (1)

The values inside parentheses are the expected ideal values.

27-20 Vibration and Shock Handbook

Using the above values, the overlap coefficients for the eight locations were determined from

Equation 27.21 as

Ojk ¼

1 0:50 0:3 0:2 0:089 0:485 0:271 0:485

0:500 1 0:3 0:112 0:089 0:425 0:135 0:425

0:300 0:300 1 0:089 0:223 0:182 0:081 0:182

0:223 0:112 0:089 1 0:180 0:184 0:446 0:108

0:089 0:089 0:223 0:180 1 0:147 0:089 0:054

0:485 0:425 0:182 0:184 0:147 1 0:179 0:441

0:271 0:135 0:081 0:446 0:089 0:179 1 0:268

0:485 0:425 0:182 0:108 0:054 0:441 0:268 1

66666666666666666664

77777777777777777775

ð27:27Þ

Given the above values of coverage and overlap, the ME values were determined from Equation 27.24 for

various suites from the eight candidate accelerometer locations of the OH-58A main rotor gearbox. The

ME values can be used, for example, to select the best set of accelerometer locations given a number of

accelerometers to be used. A sample set of accelerometer locations with the highest and lowest rankings

(from the 254 possible suites of the eight OH-58A locations) are shown in Table 27.10. Note that, for some

suite sizes, several combinations of accelerometer locations are selected as the “best” and “worst” suites,

because in these cases the ME values were too close to render a suite better or worse than the others.

The ME values can also be used to assess the benefit of additional accelerometers, leading to larger suite

sizes. To demonstrate this utility of monitoring effectiveness, the maximum ME value for each suite size

was obtained (Figure 27.9). While additional accelerometers are expected to add to monitoring

effectiveness, the added coverage they provide will

diminish for larger suite sizes due to the increase in

overlap between the accelerometers. This is clearly

reflected in Figure 27.9 where the maximum ME

value of the eight-strong accelerometer suite is the

same as that of the seven-strong accelerometer

suite, and the suite with five accelerometers has a

maximum ME value of 0.90, only 10% less than

that of the suite with eight accelerometers.

27.5.6 Sensor Location Validation

The validity of the ME values was evaluated by

comparing the rankings they provided for various

accelerometer suites to the rankings obtained

TABLE 27.10 Best and Worst Accelerometer Suites within Each Suite Size

Suite Size Best Suites Worst Suites

1 accelerometer 5 3

2 accelerometers (5, 8); (2, 5) (4, 7); (1, 6)

3 accelerometers (2, 5, 7); (4, 5, 8); (2, 4, 5) (1, 6, 8); (1, 2, 6); (1, 2, 8)

4 accelerometers (3, 4, 5, 8); (2, 4, 5, 8); (2, 4, 5, 7) (1, 2, 7, 8); (1, 4, 7, 8); (1, 6, 7, 8)

5 accelerometers (2, 3, 4, 5, 8); (2, 3, 5, 7, 8); (2, 3, 5, 6, 7) (1, 4, 6, 7, 8); (1, 2, 6, 7, 8); (1, 2, 4, 7, 8)

6 accelerometers (1, 2, 3, 4, 5, 7); (2, 3, 4, 5, 6, 8) (1, 2, 4, 6, 7, 8); (1, 3, 4, 6, 7, 8)

7 accelerometers (1, 2, 3, 4, 5, 6, 8) (1, 2, 3, 4, 6, 7, 8)

Number of accelerometers in the suite

Maximum monitoring effectiveness

0 1 3 4 7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2 5 6 8

FIGURE 27.9 Maximum values of monitoring effectiveness

for different accelerometer suite sizes.

Fault Diagnosis of Helicopter Gearboxes 27-21

empirically from a series of diagnostic tests. The suites were ranked into four evenly spaced categories.

The suites with the highest monitoring effectiveness were ranked as one, and those with the lowest values

were ranked into the fourth category. The empirical rankings were obtained from the SBCN according to

the fault possibility values, pi [ ð0; 1Þ; of the individual gearbox subsystems (Jammu et al., 1998). For

quantification purposes, a performance index, ps, was obtained for each suite of accelerometers to

represent the accuracy of the diagnostic results, as

ps ¼

faults

pi ð27:28Þ

where the summation was carried out over all the available faults that had occurred during the OH-58A

experiments. Under ideal conditions and with a perfect accelerometer suite, the values of pi should be all

equal to one (i.e., the value of ps should be equal to the number of faults). However, in practice, the fault

possibility values are smaller than one due to imperfect signal conditioning, presence of noise, and so on.

The understanding used here is that, everything else being the same, the value of ps is only affected by the

quality of the accelerometer suite when the vibration features from different suites are used as inputs to the

diagnostic system. The value of ps was computed for various accelerometer suites, and then normalized

against the largest value of ps obtained for the same suite size. These normalized ps values were then used as

the basis for ranking the suites. As with the monitoring effectiveness values, the suite with the highest

normalized ps value was assigned to the first category. It should also be noted that, in both cases, the

rankings provide only a relative measure of effectiveness among suites of the same size, and that they

should not be perceived as global measures. The normalized ps values and the associated rankings for

suites of seven accelerometers are included in Table 27.11, along with the ME values and their

corresponding rankings. The results indicate remarkably close agreement between the estimated and

empirical rankings and that, except for one mismatch, the rankings are identical. Similar analyses were

performed for suites of other sizes. A summary of matches and mismatches for all the suites is given in

Table 27.12. The results indicate that, out of the 254 possible suites, the estimated rankings of 174 suites

match exactly the empirical rankings, and 103 mismatch by only one rank.

The results summarized in Table 27.12 indicate that the proposed selection method is effective in

assessing the monitoring effectiveness of suites of accelerometers. The experimental data set, although

one of the most complete sets available in the industry, is not comprehensive enough to render a

complete evaluation of the method. The main limitation is the absence of faults in all of the components

of the gearbox. This could result in an overestimation of the significance of accelerometers that cover

faulty components during the experiments. Similarly, it could lead to devaluation of accelerometers

which cover healthy components during the experiments. For example, there was only a single fault in

subsystem 2 (mast bearing micropitting), therefore, accelerometer locations that covered this subsystem

were given a lower empirical ranking than they actually deserved.

TABLE 27.11 Rankings Obtained from the Monitoring Effectiveness Values and from

the Diagnostic Results for Suites of Seven Accelerometers

Accelerometers Included Monitoring

Effectiveness

Empirical

MEs Rank ps Rank

2, 3, 4, 5, 6, 7,8 0.983 1 0.888 1

1, 3, 4, 5, 6, 7,8 0.791 2 0.702 2

1, 2, 4, 5, 6, 7,8 0.705 2p 0.844 1

1, 2, 3, 5, 6, 7,8 0.908 1 0.926 1

1, 2, 3, 4, 6, 7,8 0.200 4 0.200 4

1, 2, 3, 4, 5, 7,8 0.990 1 1.000 1

1, 2, 3, 4, 5, 6,8 1.000 1 0.938 1

1, 2, 3, 4, 5, 6,7 0.940 1 0.843 1

“ p ” indicates a mismatch between the rankings.

27-22 Vibration and Shock Handbook

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