Free Samples EAS437 Reliability Centred Maintenance
EAS437 Reliability Centred Maintenance
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Course Code: EAS437 University: Singapore University Of Social Sciences
Country: Singapore
Question:
Failure Data Analysis using Weibull Probability Plot
The air conditioning and pressurisation (ATA 21) system on aircraft is an essential system that regulates the pressure and the temperature in the cabin as well as supplies continuous flow of fresh air to ensure the comfort of the passengers throughout a flight.
The system schematic diagram of a passenger transport aircraft is shown, while the failure data for the aircraft components collected from the aircraft fleet is compiled in
Table 1.
(a)Using the MS Excel application, construct the Weibull Probability Plot for each of the ten (10) components in the air conditioning and pressurisation system. The cumulative distribution function may be estimated using the median rank method.
(b)For each of the Weibull Probability Plots drawn in (a), construct the best-fit straight line(s) connecting failure data points, and estimate the shape parameter and scale parameter for each best-fit straight line(s).
(c)Comment on the failure patterns exhibited by each of the ten (10) components in the air conditioning and pressurisation system and estimate the MTBF values, where applicable. Appraise one usage of the MTBF data.
(d)Recommend the appropriate maintenance option for each of the ten (10) components in the air conditioning and pressurisation system. Note that some of these components may exhibit more than one failure pattern over its lifetime.
Answer:
Using the MS excel application; construct the Weibull Probability Plot for each of the ten components in the air conditioning and pressurization system. The cumulative distribution function may be estimated using the median rank method
Weibull Plots And Analysis
In the first column of the tables 1 to 10, i represents the rank of the observed failure, i is the failure hours, N is the rank of the last item observed, MR is Median Rank, Yi is the dependent variable derived by determining the natural logarithm of the natural logarithm of 1/1-MR and ln t is the natural logarithm of t (failure hours). Now, the median rank parameter is derived from the formula: MR= i-0.3/N+0.4 (standardized). Once this parameter is obtained by substituting i, and N appropriately, the other remaining columns can be filled using the formulae given. For example, to determine 1/1-MR in an excel program, we invoke the sum formula ensuring a negative is embedded, then raising this to the power of negative one (-1) as this is also mathematically equivalent to finding reciprocal of a number. It should be noted that once the first cell in each column is calculated, we simply replicate the rest by copy and pasting the 1st cell entry (the excel programme automatically generates the rest). The weibull probability plot entails a plot of Yi (vertical axis) against ln t.
Table 1: Cockpit temperature controller
Temperature
Controller
WEIBULL PLOT
Failure Hours
i
ln t
i-0.3
N+0.4
MR
1-MR
1/1-MR
ln(1/1-MR)
Yi, (ln2)
lnt
Yi
250
1
5.521461
0.7
31.4
0.022293
0.977707
1.022801
0.022545
-3.79224
5.521461
-3.79224
300
2
5.703782
1.7
31.4
0.05414
0.94586
1.057239
0.055661
-2.88848
5.703782
-2.88848
320
3
5.768321
2.7
31.4
0.085987
0.914013
1.094076
0.08991
-2.40894
5.768321
-2.40894
760
5
6.633318
4.7
31.4
0.149681
0.850319
1.176029
0.162144
-1.81927
6.633318
-1.81927
860
6
6.756932
5.7
31.4
0.181528
0.818472
1.221789
0.200316
-1.60786
6.756932
-1.60786
920
7
6.824374
6.7
31.4
0.213375
0.786625
1.271254
0.240004
-1.4271
6.824374
-1.4271
1,000
8
6.907755
7.7
31.4
0.245222
0.754778
1.324893
0.281331
-1.26822
6.907755
-1.26822
1,090
9
6.993933
8.7
31.4
0.277069
0.722931
1.383258
0.324441
-1.12565
6.993933
-1.12565
1,990
10
7.59589
9.7
31.4
0.308916
0.691084
1.447002
0.369494
-0.99562
7.59589
-0.99562
1,990
11
7.59589
10.7
31.4
0.340763
0.659237
1.516905
0.416672
-0.87546
7.59589
-0.87546
2,040
12
7.620705
11.7
31.4
0.37261
0.62739
1.593905
0.466187
-0.76317
7.620705
-0.76317
2,810
13
7.94094
12.7
31.4
0.404457
0.595543
1.67914
0.518282
-0.65724
7.94094
-0.65724
2,980
14
7.999679
13.7
31.4
0.436304
0.563696
1.774006
0.57324
-0.55645
7.999679
-0.55645
3,540
15
8.171882
14.7
31.4
0.468151
0.531849
1.880233
0.631395
-0.45982
8.171882
-0.45982
3,680
16
8.210668
15.7
31.4
0.499998
0.500002
1.999992
0.693143
-0.36652
8.210668
-0.36652
4,990
17
8.515191
16.7
31.4
0.531845
0.468155
2.136044
0.758956
-0.27581
8.515191
-0.27581
5,320
18
8.579229
17.7
31.4
0.563692
0.436308
2.291958
0.829407
-0.18704
8.579229
-0.18704
8,010
19
8.988446
18.7
31.4
0.595539
0.404461
2.472426
0.9052
-0.0996
8.988446
-0.0996
8,600
20
9.059517
19.7
31.4
0.627386
0.372614
2.683742
0.987212
-0.01287
9.059517
-0.01287
8,930
21
9.097172
20.7
31.4
0.659233
0.340767
2.934556
1.076556
0.073767
9.097172
0.073767
10,280
22
9.237956
21.7
31.4
0.69108
0.30892
3.237083
1.174673
0.160989
9.237956
0.160989
10,570
23
9.265775
22.7
31.4
0.722927
0.277073
3.609156
1.283474
0.24957
9.265775
0.24957
10,690
24
9.277064
23.7
31.4
0.754774
0.245226
4.077869
1.405575
0.340446
9.277064
0.340446
12,150
25
9.405084
24.7
31.4
0.786621
0.213379
4.686495
1.544685
0.43482
9.405084
0.43482
12,690
26
9.44857
25.7
31.4
0.818468
0.181532
5.508668
1.706323
0.534341
9.44857
0.534341
15,060
27
9.619798
26.7
31.4
0.850315
0.149685
6.680692
1.899222
0.641444
9.619798
0.641444
19,670
28
9.88685
27.7
31.4
0.882162
0.117838
8.48622
2.138444
0.760078
9.88685
0.760078
19,830
29
9.894951
28.7
31.4
0.914009
0.085991
11.62911
2.453511
0.89752
9.894951
0.89752
20,880
30
9.946547
29.7
31.4
0.945856
0.054144
18.46923
2.916106
1.070249
9.946547
1.070249
22,850
31
10.03671
30.7
31.4
0.977703
0.022297
44.84888
3.803299
1.335869
10.03671
1.335869
N=31
Table 2: Overtemp switch
Over-Temp Switch
WEIBULL PLOT
Failure Hours
i
ln t
i-0.3
N+0.4
MR
1-MR
1/1-MR
ln(1/1-MR)
Yi, (ln2)
lnt
Yi
280
1
5.63479
0.7
30.4
0.02303
0.97697
1.02357
0.0232955
-3.75949
5.63479
-3.75949
930
2
6.83518
1.7
30.4
0.05592
0.94408
1.05923
0.0575454
-2.85518
6.835185
-2.85518
3,030
3
8.01632
2.7
30.4
0.08882
0.91118
1.09747
0.0930101
-2.37505
8.016318
-2.37505
3,340
4
8.11373
3.7
30.4
0.12171
0.87829
1.13858
0.1297789
-2.04192
8.113726
-2.04192
3,960
5
8.28400
4.7
30.4
0.15461
0.84539
1.18288
0.1679514
-1.78408
8.283999
-1.78408
4,810
6
8.47845
5.7
30.4
0.18750
0.81250
1.23077
0.2076391
-1.57195
8.478452
-1.57195
5,100
7
8.53700
6.7
30.4
0.22039
0.77961
1.28270
0.2489672
-1.39043
8.536996
-1.39043
5,780
8
8.66216
7.7
30.4
0.25329
0.74671
1.33921
0.2920773
-1.23074
8.662159
-1.23074
6,030
9
8.70450
8.7
30.4
0.28618
0.71382
1.40092
0.3371299
-1.08729
8.704502
-1.08729
7,070
10
8.86362
9.7
30.4
0.31908
0.68092
1.46860
0.3843084
-0.95631
8.863616
-0.95631
7,350
11
8.90246
10.7
30.4
0.35197
0.64803
1.54315
0.4338234
-0.83512
8.902456
-0.83512
7,780
12
8.95931
11.7
30.4
0.38487
0.61513
1.62567
0.4859184
-0.72171
8.959312
-0.72171
8,860
13
9.08930
12.7
30.4
0.41776
0.58224
1.71751
0.5408772
-0.61456
9.089302
-0.61456
8,960
14
9.10053
13.7
30.4
0.45066
0.54934
1.82036
0.599033
-0.51244
9.100526
-0.51244
9,570
15
9.16639
14.7
30.4
0.48355
0.51645
1.93630
0.6607808
-0.41433
9.166388
-0.41433
10,040
16
9.21433
15.7
30.4
0.51645
0.48355
2.06802
0.7265939
-0.31939
9.214332
-0.31939
12,290
17
9.41654
16.7
30.4
0.54934
0.45066
2.21898
0.7970454
-0.22684
9.416541
-0.22684
14,510
18
9.58259
17.7
30.4
0.58224
0.41776
2.39370
0.8728391
-0.136
9.582593
-0.136
15,430
19
9.64407
18.7
30.4
0.61513
0.38487
2.59829
0.954852
-0.0462
9.644069
-0.0462
17,910
20
9.79311
19.7
30.4
0.64803
0.35197
2.84112
1.0441968
0.043248
9.793114
0.043248
19,560
21
9.88124
20.7
30.4
0.68092
0.31908
3.13401
1.1423143
0.133056
9.881242
0.133056
20,090
22
9.90798
21.7
30.4
0.71381
0.28619
3.49424
1.2511168
0.224037
9.907977
0.224037
21,670
23
9.98368
22.7
30.4
0.74671
0.25329
3.94804
1.373219
0.317158
9.983684
0.317158
34,960
24
10.46196
23.7
30.4
0.77960
0.22040
4.53730
1.5123311
0.413652
10.46196
0.413652
36,420
25
10.50287
24.7
30.4
0.81250
0.18750
5.33331
1.6739716
0.515199
10.50287
0.515199
38,220
26
10.55111
25.7
30.4
0.84539
0.15461
6.46805
1.866874
0.624265
10.55111
0.624265
38,840
27
10.56721
26.7
30.4
0.87829
0.12171
8.21615
2.1061017
0.744839
10.56721
0.744839
44,410
28
10.70122
27.7
30.4
0.91118
0.08882
11.25913
2.4211793
0.884255
10.70122
0.884255
54,280
29
10.90191
28.7
30.4
0.94408
0.05592
17.88201
2.8837954
1.059107
10.90191
1.059107
70,570
30
11.16436
29.7
30.4
0.97697
0.02303
43.42651
3.77107
1.327359
11.16436
1.327359
TOTAL
Table3: Duct temperature sensor
Duct
Temperature
Sensor
WEIBULL PLOT
Failure Hours
i
ln t
i-0.3
N+0.4
MR
1-MR
1/1-MR
ln(1/1-MR)
Yi, (ln2)
lnt
Yi
600
1
6.39693
0.7
30.4
0.02303
0.97697
1.02357
0.023296
-3.75949
6.39693
-3.75949
1,000
2
6.907755
1.7
30.4
0.05592
0.94408
1.05923
0.057545
-2.85518
6.907755
-2.85518
1,600
3
7.377759
2.7
30.4
0.08882
0.91118
1.09747
0.09301
-2.37505
7.377759
-2.37505
1,600
4
7.377759
3.7
30.4
0.12171
0.87829
1.13858
0.129779
-2.04192
7.377759
-2.04192
1,800
5
7.495542
4.7
30.4
0.15461
0.84539
1.18288
0.167951
-1.78408
7.495542
-1.78408
2,140
6
7.668561
5.7
30.4
0.18750
0.81250
1.23077
0.207639
-1.57195
7.668561
-1.57195
2,200
7
7.696213
6.7
30.4
0.22039
0.77961
1.28270
0.248967
-1.39043
7.696213
-1.39043
2,460
8
7.807917
7.7
30.4
0.25329
0.74671
1.33921
0.292077
-1.23074
7.807917
-1.23074
3,320
9
8.10772
8.7
30.4
0.28618
0.71382
1.40092
0.33713
-1.08729
8.10772
-1.08729
3,330
10
8.110728
9.7
30.4
0.31908
0.68092
1.46860
0.384308
-0.95631
8.110728
-0.95631
3,590
11
8.185907
10.7
30.4
0.35197
0.64803
1.54315
0.433823
-0.83512
8.185907
-0.83512
3,860
12
8.258422
11.7
30.4
0.38487
0.61513
1.62567
0.485918
-0.72171
8.258422
-0.72171
4,250
13
8.354674
12.7
30.4
0.41776
0.58224
1.71751
0.540877
-0.61456
8.354674
-0.61456
5,240
14
8.564077
13.7
30.4
0.45066
0.54934
1.82036
0.599033
-0.51244
8.564077
-0.51244
5,480
15
8.60886
14.7
30.4
0.48355
0.51645
1.93630
0.660781
-0.41433
8.60886
-0.41433
5,590
16
8.628735
15.7
30.4
0.51645
0.48355
2.06802
0.726594
-0.31939
8.628735
-0.31939
9,090
17
9.11493
16.7
30.4
0.54934
0.45066
2.21898
0.797045
-0.22684
9.11493
-0.22684
9,570
18
9.166388
17.7
30.4
0.58224
0.41776
2.39370
0.872839
-0.136
9.166388
-0.136
11,110
19
9.315601
18.7
30.4
0.61513
0.38487
2.59829
0.954852
-0.0462
9.315601
-0.0462
14,110
20
9.554639
19.7
30.4
0.64803
0.35197
2.84112
1.044197
0.043248
9.554639
0.043248
14,840
21
9.605082
20.7
30.4
0.68092
0.31908
3.13401
1.142314
0.133056
9.605082
0.133056
17,310
22
9.75904
21.7
30.4
0.71381
0.28619
3.49424
1.251117
0.224037
9.75904
0.224037
17,980
23
9.797015
22.7
30.4
0.74671
0.25329
3.94804
1.373219
0.317158
9.797015
0.317158
23,150
24
10.04975
23.7
30.4
0.77960
0.22040
4.53730
1.512331
0.413652
10.04975
0.413652
23,180
25
10.05105
24.7
30.4
0.81250
0.18750
5.33331
1.673972
0.515199
10.05105
0.515199
28,690
26
10.2643
25.7
30.4
0.84539
0.15461
6.46805
1.866874
0.624265
10.2643
0.624265
30,670
27
10.33104
26.7
30.4
0.87829
0.12171
8.21615
2.106102
0.744839
10.33104
0.744839
36,780
28
10.51271
27.7
30.4
0.91118
0.08882
11.25913
2.421179
0.884255
10.51271
0.884255
36,790
29
10.51298
28.7
30.4
0.94408
0.05592
17.88201
2.883795
1.059107
10.51298
1.059107
98,830
30
11.50116
29.7
30.4
0.97697
0.02303
43.42651
3.77107
1.327359
11.50116
1.327359
TOTAL
Duct temperature sensor
Table 4: Control System shut off valve SOV
Environmental Pressure Regulating
Control System Shut-off Valve
(ECS) (SOV)
WEIBULL PLOT
Failure Hours
i
ln t
i-0.3
N+0.4
MR
1-MR
1/1-MR
ln(1/1-MR)
Yi, (ln2)
lnt
Yi
930
1
6.835185
0.7
30.4
0.02303
0.97697
1.02357
0.023296
-3.75949
6.835185
-3.75949
1,000
2
6.907755
1.7
30.4
0.05592
0.94408
1.05923
0.057545
-2.85518
6.907755
-2.85518
1,110
3
7.012115
2.7
30.4
0.08882
0.91118
1.09747
0.09301
-2.37505
7.012115
-2.37505
1,980
4
7.590852
3.7
30.4
0.12171
0.87829
1.13858
0.129779
-2.04192
7.590852
-2.04192
2,240
5
7.714231
4.7
30.4
0.15461
0.84539
1.18288
0.167951
-1.78408
7.714231
-1.78408
2,470
6
7.811973
5.7
30.4
0.18750
0.81250
1.23077
0.207639
-1.57195
7.811973
-1.57195
4,150
7
8.330864
6.7
30.4
0.22039
0.77961
1.28270
0.248967
-1.39043
8.330864
-1.39043
4,830
8
8.482602
7.7
30.4
0.25329
0.74671
1.33921
0.292077
-1.23074
8.482602
-1.23074
5,340
9
8.582981
8.7
30.4
0.28618
0.71382
1.40092
0.33713
-1.08729
8.582981
-1.08729
6,160
10
8.725832
9.7
30.4
0.31908
0.68092
1.46860
0.384308
-0.95631
8.725832
-0.95631
6,650
11
8.802372
10.7
30.4
0.35197
0.64803
1.54315
0.433823
-0.83512
8.802372
-0.83512
6,830
12
8.82908
11.7
30.4
0.38487
0.61513
1.62567
0.485918
-0.72171
8.82908
-0.72171
7,500
13
8.922658
12.7
30.4
0.41776
0.58224
1.71751
0.540877
-0.61456
8.922658
-0.61456
8,140
14
9.004545
13.7
30.4
0.45066
0.54934
1.82036
0.599033
-0.51244
9.004545
-0.51244
9,320
15
9.139918
14.7
30.4
0.48355
0.51645
1.93630
0.660781
-0.41433
9.139918
-0.41433
9,740
16
9.183996
15.7
30.4
0.51645
0.48355
2.06802
0.726594
-0.31939
9.183996
-0.31939
9,940
17
9.204322
16.7
30.4
0.54934
0.45066
2.21898
0.797045
-0.22684
9.204322
-0.22684
14,080
18
9.552511
17.7
30.4
0.58224
0.41776
2.39370
0.872839
-0.136
9.552511
-0.136
15,550
19
9.651816
18.7
30.4
0.61513
0.38487
2.59829
0.954852
-0.0462
9.651816
-0.0462
18,210
20
9.809726
19.7
30.4
0.64803
0.35197
2.84112
1.044197
0.043248
9.809726
0.043248
21,270
21
9.965053
20.7
30.4
0.68092
0.31908
3.13401
1.142314
0.133056
9.965053
0.133056
22,390
22
10.01637
21.7
30.4
0.71381
0.28619
3.49424
1.251117
0.224037
10.01637
0.224037
26,100
23
10.16969
22.7
30.4
0.74671
0.25329
3.94804
1.373219
0.317158
10.16969
0.317158
27,950
24
10.23817
23.7
30.4
0.77960
0.22040
4.53730
1.512331
0.413652
10.23817
0.413652
33,220
25
10.41091
24.7
30.4
0.81250
0.18750
5.33331
1.673972
0.515199
10.41091
0.515199
37,760
26
10.53901
25.7
30.4
0.84539
0.15461
6.46805
1.866874
0.624265
10.53901
0.624265
42,910
27
10.66686
26.7
30.4
0.87829
0.12171
8.21615
2.106102
0.744839
10.66686
0.744839
47,820
28
10.7752
27.7
30.4
0.91118
0.08882
11.25913
2.421179
0.884255
10.7752
0.884255
64,060
29
11.06758
28.7
30.4
0.94408
0.05592
17.88201
2.883795
1.059107
11.06758
1.059107
66,680
30
11.10766
29.7
30.4
0.97697
0.02303
43.42651
3.77107
1.327359
11.10766
1.327359
TOTAL
Control system shut off valve
Table 5
Environmental Pressure Regulating
Control System Shut-off Valve
(ECS) (SOV)
WEIBULL PLOT
Failure Hours
i
ln t
i-0.3
N+0.4
MR
1-MR
1/1-MR
ln(1/1-MR)
Yi, (ln2)
lnt
Yi
2,220
1
7.705262
0.7
10.4
0.067308
0.932692
1.072165
0.0696799
-2.66384
7.705262
-2.66384
2,320
2
7.749322
1.7
10.4
0.163461
0.836539
1.195402
0.1784827
-1.72326
7.749322
-1.72326
2,330
3
7.753624
2.7
10.4
0.259615
0.740385
1.350649
0.3005853
-1.20202
7.753624
-1.20202
2,940
4
7.986165
3.7
10.4
0.355769
0.644231
1.552238
0.439698
-0.82167
7.986165
-0.82167
3,140
5
8.051978
4.7
10.4
0.451923
0.548077
1.824561
0.6013392
-0.5086
8.051978
-0.5086
3,600
6
8.188689
5.7
10.4
0.548077
0.451923
2.212765
0.7942427
-0.23037
8.188689
-0.23037
4,500
7
8.411833
6.7
10.4
0.64423
0.35577
2.810808
1.0334721
0.032924
8.411833
0.032924
4,660
8
8.446771
7.7
10.4
0.740384
0.259616
3.851847
1.3485527
0.299032
8.446771
0.299032
4,700
9
8.455318
8.7
10.4
0.836538
0.163462
6.117632
1.8111751
0.593976
8.455318
0.593976
4,750
10
8.4659
9.7
10.4
0.932692
0.067308
14.85704
2.6984741
0.992686
8.4659
0.992686
TOTAL
10
Table 6: Temperature modulating valve
Temperature Modulating Valve
WEIBULL PLOT
Failure Hours
i
ln t
i-0.3
N+0.4
MR
1-MR
1/1-MR
ln(1/1-MR)
Yi, (ln2)
lnt
Yi
1,950
1
7.575585
0.7
15.4
0.045455
0.954546
1.047619
0.04652
-3.06787
7.575585
-3.06787
2,300
2
7.740664
1.7
15.4
0.11039
0.889611
1.124087
0.116972
-2.14582
7.740664
-2.14582
2,400
3
7.783224
2.7
15.4
0.175325
0.824676
1.212598
0.192765
-1.64628
7.783224
-1.64628
2,430
4
7.795647
3.7
15.4
0.24026
0.759741
1.316239
0.274778
-1.29179
7.795647
-1.29179
2,500
5
7.824046
4.7
15.4
0.305195
0.694806
1.439252
0.364123
-1.01026
7.824046
-1.01026
2,890
6
7.969012
5.7
15.4
0.37013
0.629871
1.587628
0.462241
-0.77167
7.969012
-0.77167
3,000
7
8.006368
6.7
15.4
0.435065
0.564936
1.770114
0.571044
-0.56029
8.006368
-0.56029
3,010
8
8.009695
7.7
15.4
0.5
0.500001
1.999998
0.693146
-0.36651
8.009695
-0.36651
3,020
9
8.013012
8.7
15.4
0.564935
0.435066
2.298504
0.832259
-0.18361
8.013012
-0.18361
3,190
10
8.067776
9.7
15.4
0.62987
0.370131
2.70175
0.9939
-0.00612
8.067776
-0.00612
3,500
11
8.160518
10.7
15.4
0.694805
0.305196
3.276588
1.186803
0.171263
8.160518
0.171263
3,500
12
8.160518
11.7
15.4
0.75974
0.240261
4.162149
1.426032
0.354895
8.160518
0.354895
3,550
13
8.174703
12.7
15.4
0.824675
0.175326
5.703677
1.741111
0.554523
8.174703
0.554523
4,450
14
8.400659
13.7
15.4
0.88961
0.110391
9.058751
2.203731
0.790152
8.400659
0.790152
4,550
15
8.422883
14.7
15.4
0.954545
0.045456
21.99954
3.091021
1.128502
8.422883
1.128502
TOTAL
15
Table 7: Non-return valve
Non-Return Valve
WEIBULL PLOT
Failure Hours
i
ln t
i-0.3
N+0.4
MR
1-MR
1/1-MR
ln(1/1-MR)
Yi, (ln2)
lnt
Yi
20
1
2.995732
0.7
19.4
0.036082
0.963918
1.037433
0.036749
-3.30364
2.995732
-3.30364
70
2
4.248495
1.7
19.4
0.087628
0.912372
1.096044
0.091708
-2.38915
4.248495
-2.38915
250
3
5.521461
2.7
19.4
0.139174
0.860826
1.161675
0.149863
-1.89803
5.521461
-1.89803
320
4
5.768321
3.7
19.4
0.19072
0.80928
1.235667
0.211611
-1.55301
5.768321
-1.55301
350
5
5.857933
4.7
19.4
0.242266
0.757734
1.319725
0.277423
-1.28221
5.857933
-1.28221
360
6
5.886104
5.7
19.4
0.293812
0.706188
1.416054
0.347874
-1.05591
5.886104
-1.05591
440
7
6.086775
6.7
19.4
0.345358
0.654642
1.527553
0.423667
-0.85881
6.086775
-0.85881
480
8
6.173786
7.7
19.4
0.396904
0.603096
1.658111
0.505679
-0.68185
6.173786
-0.68185
730
9
6.593045
8.7
19.4
0.44845
0.55155
1.813073
0.595023
-0.51915
6.593045
-0.51915
800
10
6.684612
9.7
19.4
0.499996
0.500004
1.999985
0.69314
-0.36652
6.684612
-0.36652
950
11
6.856462
10.7
19.4
0.551542
0.448458
2.229864
0.801941
-0.22072
6.856462
-0.22072
1,050
12
6.956545
11.7
19.4
0.603088
0.396912
2.519451
0.924041
-0.079
6.956545
-0.079
1,100
13
7.003065
12.7
19.4
0.654634
0.345366
2.895481
1.063151
0.061237
7.003065
0.061237
1,520
14
7.326466
13.7
19.4
0.70618
0.29382
3.403447
1.224789
0.202768
7.326466
0.202768
1,580
15
7.36518
14.7
19.4
0.757726
0.242274
4.127561
1.417687
0.349027
7.36518
0.349027
1,640
16
7.402452
15.7
19.4
0.809272
0.190728
5.243074
1.656908
0.504953
7.402452
0.504953
1,830
17
7.512071
16.7
19.4
0.860818
0.139182
7.184847
1.971974
0.679035
7.512071
0.679035
1,850
18
7.522941
17.7
19.4
0.912364
0.087636
11.41086
2.434566
0.889768
7.522941
0.889768
1,900
19
7.549609
18.7
19.4
0.96391
0.03609
27.70866
3.321745
1.20049
7.549609
1.20049
TOTAL
N=19
Table 8: Shutoff valve
Shut-off Valve (SOV)
WEIBULL PLOT
Failure Hours
I
ln t
i-0.3
N+0.4
MR
1-MR
1/1-MR
ln(1/1-MR)
Yi, (ln2)
lnt
Yi
220
1
5.393628
0.7
20.4
0.034314
0.965686
1.035533
0.034916
-3.3548
5.393628
-3.3548
280
2
5.63479
1.7
20.4
0.083333
0.916667
1.090909
0.087011
-2.44172
5.63479
-2.44172
440
3
6.086775
2.7
20.4
0.132353
0.867647
1.152542
0.14197
-1.95214
6.086775
-1.95214
800
4
6.684612
3.7
20.4
0.181373
0.818627
1.221557
0.200126
-1.60881
6.684612
-1.60881
1,660
5
7.414573
4.7
20.4
0.230392
0.769608
1.299363
0.261874
-1.33989
7.414573
-1.33989
1,840
6
7.517521
5.7
20.4
0.279412
0.720588
1.387755
0.327687
-1.1157
7.517521
-1.1157
1,900
7
7.549609
6.7
20.4
0.328431
0.671569
1.489051
0.398139
-0.92095
7.549609
-0.92095
2,120
8
7.659171
7.7
20.4
0.377451
0.622549
1.606299
0.473933
-0.74669
7.659171
-0.74669
2,620
9
7.87093
8.7
20.4
0.426471
0.573529
1.74359
0.555946
-0.58708
7.87093
-0.58708
2,720
10
7.908387
9.7
20.4
0.47549
0.52451
1.906542
0.645291
-0.43805
7.908387
-0.43805
2,820
11
7.944492
10.7
20.4
0.52451
0.47549
2.103092
0.743409
-0.29651
7.944492
-0.29651
2,840
12
7.951559
11.7
20.4
0.573529
0.426471
2.344827
0.852212
-0.15992
7.951559
-0.15992
3,480
13
8.154788
12.7
20.4
0.622549
0.377451
2.64935
0.974314
-0.02602
8.154788
-0.02602
4,240
14
8.352319
13.7
20.4
0.671569
0.328431
3.044775
1.113427
0.107443
8.352319
0.107443
4,540
15
8.420682
14.7
20.4
0.720588
0.279412
3.578946
1.275068
0.243
8.420682
0.243
5,460
16
8.605204
15.7
20.4
0.769608
0.230392
4.340423
1.467972
0.383882
8.605204
0.383882
5,840
17
8.672486
16.7
20.4
0.818627
0.181373
5.51351
1.707201
0.534855
8.672486
0.534855
5,940
18
8.689464
17.7
20.4
0.867647
0.132353
7.555548
2.022282
0.704227
8.689464
0.704227
6,760
19
8.818778
18.7
20.4
0.916667
0.083333
11.99998
2.484905
0.910234
8.818778
0.910234
7,640
20
8.941153
19.7
20.4
0.965686
0.034314
29.14273
3.372205
1.215567
8.941153
1.215567
TOTAL
20
Table 9: Cabin temperature controller
Cabin
Temperature
Controller
WEIBULL PLOT
Failure Hours
i
ln t
i-0.3
N+0.4
MR
1-MR
1/1-MR
ln(1/1-MR)
Yi, (ln2)
lnt
Yi
270
1
5.598422
0.7
30.4
0.02303
0.97697
1.02357
0.023296
-3.75949
5.598422
-3.75949
340
2
5.828946
1.7
30.4
0.05592
0.94408
1.05923
0.057545
-2.85518
5.828946
-2.85518
910
3
6.813445
2.7
30.4
0.08882
0.91118
1.09747
0.09301
-2.37505
6.813445
-2.37505
1,510
4
7.319865
3.7
30.4
0.12171
0.87829
1.13858
0.129779
-2.04192
7.319865
-2.04192
1,630
5
7.396335
4.7
30.4
0.15461
0.84539
1.18288
0.167951
-1.78408
7.396335
-1.78408
1,800
6
7.495542
5.7
30.4
0.18750
0.81250
1.23077
0.207639
-1.57195
7.495542
-1.57195
1,960
7
7.5807
6.7
30.4
0.22039
0.77961
1.28270
0.248967
-1.39043
7.5807
-1.39043
2,350
8
7.762171
7.7
30.4
0.25329
0.74671
1.33921
0.292077
-1.23074
7.762171
-1.23074
2,600
9
7.863267
8.7
30.4
0.28618
0.71382
1.40092
0.33713
-1.08729
7.863267
-1.08729
2,630
10
7.874739
9.7
30.4
0.31908
0.68092
1.46860
0.384308
-0.95631
7.874739
-0.95631
2,720
11
7.908387
10.7
30.4
0.35197
0.64803
1.54315
0.433823
-0.83512
7.908387
-0.83512
2,840
12
7.951559
11.7
30.4
0.38487
0.61513
1.62567
0.485918
-0.72171
7.951559
-0.72171
3,120
13
8.045588
12.7
30.4
0.41776
0.58224
1.71751
0.540877
-0.61456
8.045588
-0.61456
4,550
14
8.422883
13.7
30.4
0.45066
0.54934
1.82036
0.599033
-0.51244
8.422883
-0.51244
5,210
15
8.558335
14.7
30.4
0.48355
0.51645
1.93630
0.660781
-0.41433
8.558335
-0.41433
5,630
16
8.635865
15.7
30.4
0.51645
0.48355
2.06802
0.726594
-0.31939
8.635865
-0.31939
5,820
17
8.669056
16.7
30.4
0.54934
0.45066
2.21898
0.797045
-0.22684
8.669056
-0.22684
6,230
18
8.737132
17.7
30.4
0.58224
0.41776
2.39370
0.872839
-0.136
8.737132
-0.136
7,020
19
8.856518
18.7
30.4
0.61513
0.38487
2.59829
0.954852
-0.0462
8.856518
-0.0462
8,890
20
9.092682
19.7
30.4
0.64803
0.35197
2.84112
1.044197
0.043248
9.092682
0.043248
9,570
21
9.166388
20.7
30.4
0.68092
0.31908
3.13401
1.142314
0.133056
9.166388
0.133056
10,000
22
9.21034
21.7
30.4
0.71381
0.28619
3.49424
1.251117
0.224037
9.21034
0.224037
10,940
23
9.300181
22.7
30.4
0.74671
0.25329
3.94804
1.373219
0.317158
9.300181
0.317158
11,360
24
9.337854
23.7
30.4
0.77960
0.22040
4.53730
1.512331
0.413652
9.337854
0.413652
13,970
25
9.544667
24.7
30.4
0.81250
0.18750
5.33331
1.673972
0.515199
9.544667
0.515199
17,460
26
9.767668
25.7
30.4
0.84539
0.15461
6.46805
1.866874
0.624265
9.767668
0.624265
21,110
27
9.957502
26.7
30.4
0.87829
0.12171
8.21615
2.106102
0.744839
9.957502
0.744839
21,960
28
9.996978
27.7
30.4
0.91118
0.08882
11.25913
2.421179
0.884255
9.996978
0.884255
26,750
29
10.19429
28.7
30.4
0.94408
0.05592
17.88201
2.883795
1.059107
10.19429
1.059107
31,030
30
10.34271
29.7
30.4
0.97697
0.02303
43.42651
3.77107
1.327359
10.34271
1.327359
Table 10: Cockpit temperature controller
Cockpit
Temperature
Controller
WEIBULL PLOT
Failure Hours
i
ln t
i-0.3
N+0.4
MR
1-MR
1/1-MR
ln(1/1-MR)
Yi, (ln2)
lnt
Yi
250
1
5.521461
0.7
31.4
0.022293
0.977707
1.022801
0.022545
-3.79224
5.521461
-3.79224
300
2
5.703782
1.7
31.4
0.05414
0.94586
1.057239
0.055661
-2.88848
5.703782
-2.88848
320
3
5.768321
2.7
31.4
0.085987
0.914013
1.094076
0.08991
-2.40894
5.768321
-2.40894
760
5
6.633318
4.7
31.4
0.149681
0.850319
1.176029
0.162144
-1.81927
6.633318
-1.81927
860
6
6.756932
5.7
31.4
0.181528
0.818472
1.221789
0.200316
-1.60786
6.756932
-1.60786
920
7
6.824374
6.7
31.4
0.213375
0.786625
1.271254
0.240004
-1.4271
6.824374
-1.4271
1,000
8
6.907755
7.7
31.4
0.245222
0.754778
1.324893
0.281331
-1.26822
6.907755
-1.26822
1,090
9
6.993933
8.7
31.4
0.277069
0.722931
1.383258
0.324441
-1.12565
6.993933
-1.12565
1,990
10
7.59589
9.7
31.4
0.308916
0.691084
1.447002
0.369494
-0.99562
7.59589
-0.99562
1,990
11
7.59589
10.7
31.4
0.340763
0.659237
1.516905
0.416672
-0.87546
7.59589
-0.87546
2,040
12
7.620705
11.7
31.4
0.37261
0.62739
1.593905
0.466187
-0.76317
7.620705
-0.76317
2,810
13
7.94094
12.7
31.4
0.404457
0.595543
1.67914
0.518282
-0.65724
7.94094
-0.65724
2,980
14
7.999679
13.7
31.4
0.436304
0.563696
1.774006
0.57324
-0.55645
7.999679
-0.55645
3,540
15
8.171882
14.7
31.4
0.468151
0.531849
1.880233
0.631395
-0.45982
8.171882
-0.45982
3,680
16
8.210668
15.7
31.4
0.499998
0.500002
1.999992
0.693143
-0.36652
8.210668
-0.36652
4,990
17
8.515191
16.7
31.4
0.531845
0.468155
2.136044
0.758956
-0.27581
8.515191
-0.27581
5,320
18
8.579229
17.7
31.4
0.563692
0.436308
2.291958
0.829407
-0.18704
8.579229
-0.18704
8,010
19
8.988446
18.7
31.4
0.595539
0.404461
2.472426
0.9052
-0.0996
8.988446
-0.0996
8,600
20
9.059517
19.7
31.4
0.627386
0.372614
2.683742
0.987212
-0.01287
9.059517
-0.01287
8,930
21
9.097172
20.7
31.4
0.659233
0.340767
2.934556
1.076556
0.073767
9.097172
0.073767
10,280
22
9.237956
21.7
31.4
0.69108
0.30892
3.237083
1.174673
0.160989
9.237956
0.160989
10,570
23
9.265775
22.7
31.4
0.722927
0.277073
3.609156
1.283474
0.24957
9.265775
0.24957
10,690
24
9.277064
23.7
31.4
0.754774
0.245226
4.077869
1.405575
0.340446
9.277064
0.340446
12,150
25
9.405084
24.7
31.4
0.786621
0.213379
4.686495
1.544685
0.43482
9.405084
0.43482
12,690
26
9.44857
25.7
31.4
0.818468
0.181532
5.508668
1.706323
0.534341
9.44857
0.534341
15,060
27
9.619798
26.7
31.4
0.850315
0.149685
6.680692
1.899222
0.641444
9.619798
0.641444
19,670
28
9.88685
27.7
31.4
0.882162
0.117838
8.48622
2.138444
0.760078
9.88685
0.760078
19,830
29
9.894951
28.7
31.4
0.914009
0.085991
11.62911
2.453511
0.89752
9.894951
0.89752
20,880
30
9.946547
29.7
31.4
0.945856
0.054144
18.46923
2.916106
1.070249
9.946547
1.070249
22,850
31
10.03671
30.7
31.4
0.977703
0.022297
44.84888
3.803299
1.335869
10.03671
1.335869
N=31
Temperature controller
For each of the Weibull Probability Plots drawn in (a), construct the best-fit straight line(s) connecting failure data points, and estimate the shape parameter and scale parameter for each best-fit straight line(s).
ANSWERS
For the best-fit straight lines, check the figures (from fig 1 to 10).
The shape parameter is the slope of the best-fit straight line and is determined as follows:
Shape parameter= change in Yi/change in ln t
Hence : shape parameter 1: (0—2)/8.5-6= 0.76923 { Note that the points (8.5,0) and (6,-2) can be deduced from figure 1 in the line of best-fit. To get shape parameter, use the gradient formula, that is: change in Y/change in X and substitute the points above); repeat the same for all figures 2 to 10}
The scale parameter is derived from the intersection of the 63.2% of the y-axis range of data with the weibull distribution plot. {In other words, we determine the 63.2% of the total distance in the Y axis, then mark off the point on the Y axis, next draw a horizontal line that is parallel to the x axis and perpendicular to the Y axis crossing at the marked point, that is, the 63.2% of the total distance. Also note that, in order to get the 63.2% mark point, we must add it to the smallest data point. Note that the total distance is the difference between the largest data point and the smallest data point, and this is the range. In this case, from all figures, it can be seen that the smallest data point is negative hence the range will be just addition as shown below, therefore, the intersection of the horizontal line and the line of best-fit tracing it downwards to the X axis gives the value of scale parameter}.
This is done as follows:
In the first graph, the range is 6 , that is :(2—4)= 6 hence : 0.632 x 6 = 3.792
Then drawing the line (check figure 1), y= 3.792 + -4 = -0.208
Hence scale parameter is x= 8.5 (note that this is just but estimation as the graphs could not locate decimals)
Shape parameter 2: (0—2)/9.5-8= 1.3333
For this case, repeat the procedure in figure 1 above hence:
Range= 4+1.5 = 5.5
0.632x 5.5 = 3.476
Y= 3.476-4 = -0.524
Hence X= 9.5
Shape parameter 3: (0—3.5)/9.5-6 = 1.000
Scale parameter3: Range = 4+1 = 5
0.632 x5= 3.16
Y= 3.16+-4= -0.84
X= 9
Shape parameter 4: (0—3.5)/10-6= 0.875
Scale parameter 4: range= 4+1.2 = 5.2
0.632×5.2 = 3.286
Y= 3.286-4= -0.714
X= 9.0
Shape parameter 5: (6—1.5)/2-0= 3.75
Scale parameter 5: range= 1+2.5= 3.5
0.632×3.5= 2.212
Y= 2.212-1= 1.212
Hence x= 8.5
Shape parameter 6: 0.75—1.75/8-2=0.41667
Scale parameter 6: range = 3+1.5 = 4.5
0.632 x 4.5= 2.844
Y= 2.844+ -3= -0.156
X= 8.02 (refer to figure 7)
Shape parameter 7: (0—2.5)/7-5= 1.25
Scale parameter 7: range= 3.5+1.5= 5.0
0.632×5.0= 3.16
Y= 3.16-3.5 = -0.34
X= 6.5 (refer to figure 7)
Shape parameter 8: 0—2/8.2-7= 1.6667
Scale parameter 8: range= 1.2+3.5 = 4.7
0.632×4.7= 2.9704
Y= 2.9704-3.5 = -0.5296
X= 7.9 (refer to figure 8)
Shape parameter 9: 6.5-0/9—3=0.54167
Scale parameter 9: 1.5+4= 5.5
0.632×5.5= 3.476
Y= 3.476-4= -0.524
X= 8.5 (refer to figure 9)
Shape parameter 10: 1—3/10-5.5= 0.8889
Scale parameter 10: range = 1.2+4= 5.2
0.632×5.2= 3.2864
Y= 3.2864-4= -0.7136
Hence x= 8.1 (refer to figure 10)
Comment on the failure patterns exhibited by each of the ten (10) components in the air conditioning and pressurisation system and estimate the MTBF values, where applicable. Appraise one usage of the MTBF data.
ANSWERS
The graphs above exhibit a generally uniform failure pattern. Therefore, in selecting the maintenance option for each case will a bit simple as one strategy can fit all (of course with slight modifications) (SAE JA, 2002). The failure is almost linearly distributed as shown in the plots. Admittedly, the best approach would be preventive maintenance where failure is predicted before it occurs and necessary action is taken such as replacements, lubrication among others (NAVAIR, 2005). However in specific terms the following strategies shall be adopted:
MTBF is often used in predicting the subsequent failures by checking at the patterns of failure henc effective maintenance strategy can be drafted. The MTBF values are obtained via finding the antilog of the respective scale parameters:
The failure pattern exhibited by each is described as follows:
Graphs
PATTERN EXHIBITED
scale parameter
MTBF
1
Infant mortality since there is constant wear rate
8.5
4969.101
2
Similar pattern exhibited
9.5
13524.9
3
Similar pattern exhibited
9
8197.962
4
Similar pattern exhibited
9
8197.962
5
Similar pattern exhibited
8.5
4969.101
6
Similar pattern exhibited
8.02
3072.888
7
Similar pattern exhibited
6.5
670.7572
8
Similar pattern exhibited
7.9
2724.985
9
Similar pattern exhibited
8.5
4969.101
10
Similar pattern exhibited
8.1
3329.165
Recommend the appropriate maintenance option for each of the ten (10) components in the air conditioning and pressurization system. Note that some of these components may exhibit more than one failure pattern over its lifetime.
The air conditioning and pressurization system is one of the most essential subsystems in the aircraft. Failure of one component, either hidden or exposed, can have irreparable damage on the aircraft performance in the long run. It is therefore necessary that aircraft engineers stay abreast with the functioning state of each component. Reliability centred maintenance approach is one of the methods used to study the failure patterns of these components (SAE JA1011, 1999). Therefore, the report hereinafter provides an analysis of the failure patterns of these components using Weibull distribution method. Reliability centered maintenance analysis of the aircraft pressure control system was done via Weibull distribution for each part as illustrated in tables 1 to 10.
The table illustrates more specifically the maintenance strategies to adopted:
COMPONENT
MAINTENANCE STRATEGY
Cockpit temperature controller
Replacements since preventive maintenance would be cheaper to undertake and for safety reasons
Cabin temperature controller
Replacements is the best option due to cost effectiveness
SOV 2
Condition monitoring on a regular basis , it is a safety risk
Pressure SOV
Condition monitoring on a regular basis, it is a safety risk
Cockpit temperature modulating valve
Prior fault finding as a preventive inspection method
ECS
Replacements is most necessary
Cabin duct temp sensor
Condition monitoring is the best option since it is a safety risk
Cockpit overtemp S/W
Replacements is the best strategy in this case
Cabin temp controller
Regular inspection and servicing is necea
Cabin temp sensor
As illustrated above, failure of one component may lead to failure of the entire system. It is therefore crucial that the engineer develops an effective program to either prevent or minimize failure. Although preventive maintenance is costly to implement, however, in the long run it ensures operational stability and safety is maintained which then translates to better working conditions and increased profitability (SAE JA., 2000).
References
NAVAIR 00-25-403. (2005) “Guidelines for the Naval Aviation Reliability-Centered Maintenance Process,”
SAE JA1011. (1999). “Evaluation Criteria for Reliability-Centered Maintenance (RCM) Processes,”
SAE JA1012. (2002) “A Guide to the Reliability-Centered Maintenance (RCM) Standard,”
SAE JA. (2000).Defence Standard 02-45 (NES 45)
SAE JA. (2000). Requirements for the Application of Reliability-Centred Maintenance Techniques to HM Ships, Submarines, Royal Fleet Auxiliaries and other Naval Auxiliary Vessels
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