Reliability explanation with graphical plot of failure density vs time, failure rate vs time and reliability vs time.

1. Reliability
G
r a p h i c a l
P
l o t s
Prepared By:-
Aziz Companiwala

2. Time (Hours) Failure
Density
Failure Rate Reliability
0 1
0.13 0.139
1 0.87
0.083 0.101
2 0.787
0.075 0.1
3 0.712
0.068 0.1
4 0.6644
0.062 0.101
5 0.582
0.056 0.101
6 0.526
0.051 0.101
7 0.475
0.046 0.101
8 0.429
0.041 0.1
9 0.388
0.037 0.1
Table
1
Time (Hours) Failure
Density
Failure Rate Reliability
10 0.351
0.034 0.101
11 0.317
0.031 0.103
12 0.286
0.028 0.103
13 0.258
0.064 0.283
14 0.194
0.076 0.486
15 0.118
0.062 0.714
16 0.056
0.04 1.11
17 0.016
0.012 1.2
18 0.004
0.004 2
19 0
Total = 1 Mean = 0.376
Continues…

3. 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Time interval V/S Failure Density
Time Value
Failure Density

4. Time interval V/S Failure Density
• Since the time interval chosen is one hour, the figure is drawn in from of bar chart.
• Height is equal to fd
• Area of each bar is equal to fdX1=fd
• n1,n2,n3…..nl are the number of units that fail respectivily, during 1st hour, 2nd hour, 3rd hour and so on.
• n1+n2+n3+….+nl=N
• l=period of last test
• N total population
• fd=n/N
• fd1+fd2…fdl=1
• This means sum of the areas of vertical bar in chart is 1

5. The time interval is made smaller, smooth curve as shown in fig is obtain
The advantage of smooth curve is to bring out certain generalized behavioral
characteristics of the component with regard to failure
Time interval V/S Failure Density
number of failures
total operating hours
Failure density =

6. Time interval V/S Failure Density
• Since the time interval chosen is one hour, the figure is drawn in from of bar chart.
• Height is equal to fd
• Area of each bar is equal to fdX1=fd
• n1,n2,n3…..nl are the number of units that fail respectivily, during 1st hour, 2nd hour, 3rd hour and so on.
• n1+n2+n3+….+nl=N
• l=period of last test
• N total population
• fd=n/N
• fd1+fd2…fdl=1
• This means sum of the areas of vertical bar in chart is 1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Failure Density V/S Time
Time Interval
Failure Density

7. 0
0.2
0.4
0.6
0.8
1
1.2
Failure Rate V/S Time Value
Time Value
Failure Rate

8. 0
0.05
0.1
0.15
0.2
0.25
0.3
Failure Rate V/S Time
Time Value
Failure Rate
Infant Failures Service
Failures
Wear-Out
Failures

9. Failure Rate V/S Time Value
• Since the time interval chosen is one hour, the figure is drawn in from of bar chart.
• Height is equal to fr
• The time interval is made smaller and smooth curve as shown in fig is obtain
• The advantage of smooth curve is to bring out certain generalized behavioral characteristics of the
component with regard to failure
• Failure rate is equal to number of unit fail in certain interval to average population in that interval.survival at (i-1)- survival at i
Average population during ith hour
Failure rate Z(i) =

10. Failure Rate V/S Time Value
• Dividing equation by N
• However,
• R(i-1) = [survival at (i-1)]/N
• R(i) = [survival at i]/N
• [average population during ith hour]/N = [survival at (i-1) + survival at (i)]/2N
= [R(i-1) + R(i)]/2
• Therefore,
[survival at (i-1)]/N- [survival at i]/N
[average population during ith hour]/N
Failure rate Z(i) =
R(i-1) - R(i)
R(i-1) + R(i)
Failure rate Z(i) = 2 X

11. Failure Rate V/S Time Value
• There are 3 clear zones in smooth curve obtain
●
Short Initial period during which failures are called INFANT OR EARLY FAILURES.
●
This period is break in period when component fail due to defects in manufacturing, being inherently weak
because of weak parts, bad assembly and so on.
●
Hazard rate is constant –this is termed as SERVICE FAILURES
●
The failure density curve for this period is exponential in character.
●
Relation between both failure density and failure rate is explain later.
●
WEAR OUT FAILURES
●
The incidence of failure in this zone is high since most of the components will have exceeded their service
life, and would have deteriorated.
●
Because of the shape it is also known as Bathtub Curve

12. 0
0.2
0.4
0.6
0.8
1
1.2
Reliability V/S Time Value
Time Value
Reliability

13. Reliability V/S Time Value
• Let total population be 1000
• n1 is the number of units that fail during 1st hour.
• n2 is the number of units that fail during 2nd hour and so on.
• Then total number of failed component till end of ith hour will be n1+n2+n3+….+ni
• Survival at end of ith hour will be [1000-(n1+n2+n3+….+ni)]
number of survivors till the ith hour
1000
Reliability =
[1000-(n1+n2+n3+….+ni)]
1000
= 1-(fd1+fd2…fdi )=
Sum of the areas of the bars from the end of the ith hour to the
end of the test
=

14. Important Points
1. fd1+fd2…fdi =
= 1
3. R(i) = 1 – ( fd1+fd2…fdi )
= 1 -
R(i-1) - R(i)
R(i-1) + R(i)
3. Z(i) =2 X
R(i-∆t) - R(i)
∆t[R(i- ∆t) + R(i)]
Z(i) =2 X

15. Important Points
• It is very important to note one thing about the time interval in these
equation.
• In the analysis of failure data we adopted one hour as the interval between
each observation.
• If the interval is ∆t hour instead of one hour then,
• This agrees our definition of failure rate.
• It is the rate at which failure take place at any given period, assuming that
no failure place prior to the period of observation.
• It is the base that we use average population in analysis during a period
instead of the original population at the beginning of the particular period.
R(i-∆t) - R(i)
∆t[R(i- ∆t) + R(i)]
Z(i) =2 X

16. MTTf in terms of Failure density
MTTF = 1/N
Where,
N is initial total population
n1 is the number of specimens that fail during the first ∆t interval
n2 is the number of specimens that fail during the second ∆t interval and
nk is the number of specimens that fail during the kth interval
Fd is the failure density then by definition
Fd = nk / (N ∆t)
Hence,
nk / N = Fd∆t
k∆t is the elasped time t.Hence
MTTF =
Where summation is for the period from the first interval to the ith interval

17. Thank You