6.2 Data Collection and Evaluation
6.2.2.1 Least Squares Curve Fitting Method
Exponential, Weibull, and normal are the most used distributions considered in this approach.
Least-Squares Method: Exponential Distribution Case The cumulative distribution of the exponential dis- tribution is well known:
F .t /D1et: (6.23) Applying the natural logarithm on both sides gives
lnŒ1F .t /Dln 1
1F .t /
Dt: (6.24) The slope of the line produced by consideringyi D ln 1
1 OF .ti/
andxiDtirepresents an estimation of. Performing the least-squares method in the form of yDbx, one obtains
bD OD Pn
iD1xiyi
Pn
iD1xi2 : (6.25) Least-Squares Method: Weibull Distribution Case
The Weibull cumulative distribution (Eq. 5.68) pro- vides
F .t /D1e.t =˛/ˇ: (6.26) Taking two natural logarithms in sequence, one ob- tains
ln ln 1
1F .t /
Dˇlntˇln˛: (6.27) The linear regression form is obtained by considering yiDln ln 1
1F .ti/
andxi Dlnti, and especially yiDaCbxi;
where
bD OˇD Pn
iD1.xi Nx/.yi Ny/
Pn
iD1.xi Nx/2 (6.28) and
aD Oˇln˛O D Nybx:N (6.29) ˇOis derived from Eq. 6.28 and then˛Ois estimated by Eq. 6.29.
Least-Squares Method: Normal Distribution Case Assuming the cumulative functionF .t /is a normal distribution, the normalized variablezcan be used. In particular,
F .t /D .z/D
t
D
Zz 1
p1
2ey2=2dy;
(6.30) whereis the standard deviation andis the average value of the normal distribution (int).
The link betweenzand .z/can be obtained quite quickly using the inverse function of the standardized
normal distribution, which is usually tabulated (Ap- pendix A.1).
Using the inverse function,
1ŒF .ti/D 1Œ .zi/DziD ti D ti
: (6.31) This function is linear in t, so the least-squares fit- ting process is applied to the following variables:yi D ziD 1ŒF .ti/andxi Dti.
From application of the least-squares fit, O
D 1
b (6.32)
and
O
D aO D a
b: (6.33)
Table 6.9 presents the fundamental information col- lected using the least-squares approach according to the main distributions mentioned above.
The index of fit in the least-squares method is cal- culated by
rD 1n Pn
iD1.xi Nx/.yi Ny/
qPn
iD1.xi Nx/2 n
qPn
iD1.yi Ny/2 n
; (6.34)
where yN and xN are, respectively, the average values ofyi andxi, andnis the number of couples.xi; yi/ available.
Application
The same complete data set used as in the EFDD approach (Table 6.2) was used in the research into a theoretical distribution of cumulative functionF .t / using the least-squares method:
1. Exponential distribution (Table 6.10).
Solving Eq. 6.25 forb, one obtains bD
Pn
iD1xiyi
Pn
iD1xi2 D OD0:000501:
The linear regression is represented byyiDaCbxi D 0:000501xiand the index of fitris 0.6601.
In terms of a cumulative distribution, the equation of the exponential distribution fitting the real-world data isF .t /D1et D1e0:000501t. The dashed line in Fig. 6.10 represents the linear regression: the approximation is not satisfactory, as the index of fit is very poor. In conclusion, the exponential distribution is not very appropriate.
Table 6.9 Least squares curve fitting method
Distribution Cumulative function Linear regression functionyiDaCbxi
xi yi Parameters (a; b)
Exponential F .t /D1et ti ln
1 1 OF .ti/
aD0 bD
Pn
iD1xiyi Pn
iD1xi2 D O Weibull F .t /D1e.t=˛/ˇ lnti ln ln
1 1F .t /
aD NybxND Oˇln˛O bD
Pn
iD1.xi Nx/.yi Ny/
Pn
iD1.xi Nx/2 D Oˇ
Normal F .t /D.z/D
t
D
Zz 1
p1
2 ey2=2dy
ti ziD1F .ti/ aD NybxND Ob bD
Pn
iD1.xi Nx/.yi Ny/
Pn
iD1.xi Nx/2 D 1 O
Function1ŒF .t /in Appendix A.1 Table 6.10 Exponential distribution
ti(h) F .tO i/ yiDln 1
1 OF .ti/
ti(h) F .tO i/ yiDln 1
1 OF .ti/
667 0.032 0.033 2,056 0.516 0.726
980 0.065 0.067 2,128 0.548 0.795
1,124 0.097 0.102 2,461 0.581 0.869
1,246 0.129 0.138 2,489 0.613 0.949
1,348 0.161 0.176 2,497 0.645 1.036
1,478 0.194 0.215 2,674 0.677 1.131
1,642 0.226 0.256 2,687 0.710 1.237
1,684 0.258 0.298 2,745 0.742 1.355
1,689 0.290 0.343 2,756 0.774 1.488
1,695 0.323 0.389 2,785 0.806 1.642
1,745 0.355 0.438 2,894 0.839 1.825
1,879 0.387 0.490 2,976 0.871 2.048
1,945 0.419 0.544 3,097 0.903 2.335
1,974 0.452 0.601 3,467 0.935 2.741
1,998 0.484 0.661 4,562 0.968 3.434
Estimated using the improved direct method
2. Weibull distribution (Table 6.11).
In solving Eqs. 6.28 and 6.29 fora, one can derive the estimates for the following directly:
bD OˇD Pn
iD1.xi Nx/.yi Ny/
Pn
iD1.xi Nx/2 D2:766 and
aD Oˇln˛O D NybxN D 21:593:
The linear regression is represented byyi DaCbxiD 21:593C2:766xiand the index of fitris 0.9801.
Figure 6.11 shows the plots of real-world data and linear regression.
In terms of the failure cumulative distribution, the original equation is
F .t /D1e.t =˛/ˇ:
Parameters˛andˇare directly derived from parame- tersaandb, which characterize the linear regression.
In particular, ˇO D b D 2:766 and ˛O D eab D 2,463.66.
In conclusion, the equation of the cumulative fail- ure function is
F .t /D1e.t =2;463:66/2:766:
Table 6.11 Weibull distribution
ti(h) F .tO i/ xiDlnti yiDln ln 1
1 OF .ti/
ti(h) F .tO i/ xiDlnti yiDln ln 1
1 OF .ti/
667 0.032 6.503 3:418 2,056 0.516 7.629 0:320
980 0.065 6.888 2:708 2,128 0.548 7.663 0:230
1,124 0.097 7.025 2:285 2,461 0.581 7.808 0:140
1,246 0.129 7.128 1:979 2,489 0.613 7.820 0:052
1,348 0.161 7.206 1:738 2,497 0.645 7.823 0.035
1,478 0.194 7.298 1:537 2,674 0.677 7.891 0.123
1,642 0.226 7.404 1:363 2,687 0.710 7.896 0.212
1,684 0.258 7.429 1:209 2,745 0.742 7.918 0.303
1,689 0.290 7.432 1:070 2,756 0.774 7.922 0.397
1,695 0.323 7.435 0:943 2,785 0.806 7.932 0.496
1,745 0.355 7.465 0:825 2,894 0.839 7.970 0.601
1,879 0.387 7.538 0:714 2,976 0.871 7.998 0.717
1,945 0.419 7.573 0:610 3,097 0.903 8.038 0.848
1,974 0.452 7.588 0:510 3,467 0.935 8.151 1.008
1,998 0.484 7.600 0:413 4,562 0.968 8.426 1.234
Estimated using the improved direct method
0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000
0 1000 2000 3000 4000 5000
hours yi
Fig. 6.10 Exponential least-squares plot of failure data
3. Normal distribution. Using the well-known method (Table 6.9) and the data in Table 6.12,
bD Pn
iD1.xi Nx/.yi Ny/
Pn
iD1.xi Nx/2 D 1 O
D0:0011;
aD NybxN D ObD 2:3826:
The linear regression is represented byyi DaCbxiD 2:3826C0:0011xiand the index of fitris 0.9531.
Figure 6.12 shows the plots of real-world data and linear regression.
The resolution of Eqs. 6.32 and 6.33 makes it pos- sible to determine the cumulative failure distribution,
-4 -3 -2 -1 0 1 2
6 6.5 7 7.5 8 8.5 9
ln(hours) yi
Fig. 6.11 Weibull least-squares plot of failure data
-2.000 -1.000 0.000 1.000 2.000 3.000
0 1000 2000 3000 4000 5000
hours yi
Fig. 6.12 Normal least-squares plot of failure data and in particular
O D 1
b D914:528 and
O D aO D a
b D2;178:933:
Table 6.12 Normal distribution
ti(h)Dxi F .tO i/ yiDziD1ŒF .ti/ ti(h)Dxi F .tO i/ yiDziD1ŒF .ti/
667 0.032 1:849 2,056 0.516 0.040
980 0.065 1:518 2,128 0.548 0.122
1,124 0.097 1:300 2,461 0.581 0.204
1,246 0.129 1:131 2,489 0.613 0.287
1,348 0.161 0:989 2,497 0.645 0.372
1,478 0.194 0:865 2,674 0.677 0.460
1,642 0.226 0:753 2,687 0.710 0.552
1,684 0.258 0:649 2,745 0.742 0.649
1,689 0.290 0:552 2,756 0.774 0.753
1,695 0.323 0:460 2,785 0.806 0.865
1,745 0.355 0:372 2,894 0.839 0.989
1,879 0.387 0:287 2,976 0.871 1.131
1,945 0.419 0:204 3,097 0.903 1.300
1,974 0.452 0:122 3,467 0.935 1.518
1,998 0.484 0:040 4,562 0.968 1.849
Estimated using the improved direct method
In conclusion, the equation of the cumulative failure function is
F .t /D Zt 1
1 p
2e .y/2
22
dy
D Zt 1
1 914:528p
2e
.y2;178:933/2 2914:5282
dy:
Crossover Analysis and Final Observations
On comparing the three different equations repre- senting the cumulative failure function calculated us- ing the least-squares method, it is worth initially not- ing that as reported in Fig. 6.13 the exponential dis- tribution does not fit the real-world data well enough, whereas the two remaining distributions (i. e., Weibull and normal) are perfectly satisfactory. This observa- tion is confirmed by the respective index of fit results:
0.6601, 0.9801, and 0.9531.
The good fit of the Weibull and normal distributions is an indicator of the typical process of failure for the component analyzed. In fact, the failure rate according to the Weibull distribution is
.t /D ˇ
˛ t
˛ ˇ1
: (6.35)
Figure 6.13(d) presents the trend of the failure rate adopting the Weibull distribution. The increasing trend
demonstrates that the component tested is working in conditions of wear.