6.4.5.3.3.3 Modification for
NonExponential Failure Densities (General Case)
Although the exponential technique indicated in the previous
sections can be used in most applications with little error, it must be
modified (1) if the system contains parts for which the density function of
failure times cannot be approximated by an exponential distribution over the
time period of interest; or (2) if the parts which are the dominant factor in
overall system unreliability do not follow an exponential density
function of times to failure. Mechanical parts such as gears, motors, and
bearings usually fall into this category.
In these cases, one cannot add the failure rates of all parts
because there are some parts whose failure rates vary significantly with time.
The method used is to consider separately within each block diagram the
portion of the block containing parts with constant failure rates, and the
portion containing parts with time varying failure rates. If the former
portion contains n parts, then the reliability of this portion is
The reliability of the
second portion at time t is formed by using the appropriate failure density
function for each part whose parameters have been determined through field
experience or testing. If this portion contains B parts, then
where:
and f_{i}(t) is the
probability density function, general expression, of each of the B
parts.
As discussed in 5.3.6, the Weibull
distribution can be used to describe the distribution of times to failure in a
wide variety of cases. If we use the Weibull to describe the portion of the
block diagram containing parts with varying failure rate, equation 6.47
becomes:
where:
B 
= 
numbered parts 
t 
= 
time 
q_{i} 
= 
Weibull scale parameter for part i 
b_{i} 
= 
Weibull shape parameter for part
i 
The reliability for the
block diagram, under the assumption of independence between the two portions,
is

R(t)
= R_{1}(t) R_{2}(t) 
(6.49) 
For example, consider the failure rates of two
elements, x and y, that make up a system. Let x be a microprocessor controller
with a constant failure of 2 failures per million hours. Let y be a roller
bearing operating at 1000 revolutions per minute for which 90% of the
population will operate without failure for 3.6 x 10^{9} revolutions. Bearing life test results have been fitted
to the Weibull distribution with a shape parameter, b, of 1.5.
STEP 1: The
microcircuit reliability is found by using equation
6.38.
R_{1}(t) 
= 
exp(l_{t})


= 
exp [  (2 x 10^{6})(50,000)
]

R_{1}(t) 
= 
0.905 
STEP 2: The bearing
reliability is determined by converting the revolutions into hours given that
the speed is 60,000 revolutions per hour. This is 3.6 x
10^{9} revolutions divided by 60,000 revolutions per hours
which equals 60,000 hours.
Then scale parameter q, is determined from the
standard Weibull equation shown as 6.48.
"1">
where:
R(t) 
= 
0.9 at 60,000 hours (given) 
t 
= 
60,000 hours 
b 
= 
Weibull shape of 1.5 for product
characteristic of early wearout 
q 
= 
meantimetofailure 
R(t) 
= 

q 
= 
60,000/(ln 0.9)^{1/1.5} 
q 
= 
268,967
hours 
This scale parameter is used to determine the
reliability at the 50,000hour point using equation 6.48.
STEP 3:
The system
reliability is found using equation 6.49 where
R_{1}(t) 
= 
R1(t) R2(t)


= 
(0.905) (0.923)


= 
0.835 
STEP 4: Calculate the system MTBF as follows:
"1">
where T is the time period of interest (T = 50,000
hours in this case).