1. MORE SOLVED EXAMPLES
Lecture 5
Prof. Dr. Ahmed Farouk Abdul Moneim

2. 1000 components were tested over an interval of time of 10 000 hours.
Hours of operation in
1000 hrs
Cumulative number of failures 4
729
0.5 10 5
874 1 25 6
909
1.5
122
7.5
953 2
212
984 3
541
Evaluate the parameters β, η
Evaluate MTTF
Evaluate the design life of a support manufactured from the polymer for
an engine seating having minimum allowable reliability of 0.9.
Could a burn-in test for 500 hrs extend the design life of the support
Prof. Dr. Ahmed Farouk Abdul Moneim

3. This is an equation of a Straight line with slope= β and intercept C
IF the Time to Failure is ASSUMED to be distributed by WEIBULL distribution, then we have:
Since η and β are constants, the variables R(t) and t have the above relation. If we put:
This is an equation of a Straight line with slope= β and intercept C
Prof. Dr. Ahmed Farouk Abdul Moneim

4. A Hint about LINEAR REGRESSION
If you have data about two variables Y and X. You want to construct a STRAIGHT LINE Relation between Y and X:
The Slope of the straight line β and the intercept C will be given by Y X y1 x1 y2 X2
….. ….
Y n
X n
The calculations is on an Excel sheet
Prof. Dr. Ahmed Farouk Abdul Moneim

5. Observation
Time in Hours
# of Failures F R
Y=ln(ln(1/R))
X=ln(t) XY
X^2 1
500 10
0.01
0.99
-4.600
6.215
38.621 2
1000 25
0.025
0.975
-3.676
6.908
47.717 3
1500
122
0.122
0.878
-2.039
7.313
53.483 4
2000
212
0.212
0.788
-1.434
7.601
57.774 5
3000
541
0.541
0.459
-0.250
8.006
-2.003
64.102 6
4000
729
0.729
0.271
0.267
8.294
2.212
68.791 7
5000
874
0.874
0.126
0.728
8.517
6.203
72.543 8
6000
909
0.909
0.091
8.700
7.605
75.682 9
7500
953
0.953
0.047
1.118
8.923
9.972
79.614
10000
984
0.984
0.016
1.420
9.210
13.074
84.830
-0.759
7.969
Matrix
Normal Algebra Bo
Beta B1
Eta

6. Find DESIGN LIFE with Reliability of 0.9DL
Could a burn-in test for 500 hrs extend
the design life of the support
R=0.9 DL
500
DL+500
Prof. Dr. Ahmed Farouk Abdul Moneim

7. Could a burn-in test for 500 hrs extend the design life of the support
But DL without Burn-in test =1457 hours
Therefore, the Burn-in test of 500 hours REDUCES the Design Life from 1457 hours down to hours
REMEMBER that β = 2.18 >> 1
Prof. Dr. Ahmed Farouk Abdul Moneim