Transcript failure rates

+
Failure rates
+
outline

Failure rates as continuous distributions

Weibull distribution – an example of an exponential
distribution with the failure rate proportional to a power of
time. The shape parameter (a) can be interpreted with
respect to decreasing or increasing failure rates.

Example problem. MWNT average lengths as a function of
time. Construct the experimental failure rate curve, model
with a Weibull distribution, related to reliability, hazard rate
function.
+
Failure rates as
continuous
distributions
Relevance to quality control,
reliability testiing
Mean Time Between Failures
+
+
+
+
+
+
+
Example data
+
What functions do we need?
Failure rate (density function for time to 1st failure), Reliability
function
+
+
How do we model f(t), R(t)?
+
Weibull distribution
One example of an exponentialdecaying failure rate distribution
+
+
The shape
factor, a relates
to change in
failure rates
with t
+
Weibull probability density
function, f(t)
+
Weibull cumulative distribution
F(t)
+
Example problem
Use the MWNT fracture data in
sonication experiment to generate a
failure rate density function.
Model this function with a Weibull
distribution
Interpret the coefficients with respect
to rates/mechanisms
+
Interpretation of Weibull
distribution for failure rate

We have length vs. time data for fracture of MWNTs under
sonication

Failure rate for the fracture/comminution process would be
the derivative of this curve.

Approach:

Fit empirical eqn. to L vs. t data

Take the derivative of this function to generate the
fracture/comminuation rate (failure rate)

Model this using the Weibull distribution
+
Model for MWNT average length
failure example.xlsx. Empirical fit
-
Review trendline
fits of power,
exponential, and
polynomical
functions to L vs. t
data.
-
Power law has the
best R2 value, and
its differential is
well-behaved over
the time interval
+
Next step.
Using the best empirical fit, take the derivative, which is the
fracture/failure rate for the average length particle
L = 48.336 × t -0.578
dL
=
dt
d (a × t n )
dt
= n×a×t
n-1
27.94
= 1.578
t
The equation for the rate, dL/dt, can now be inserted
into a moment equation – it represents f(t).
When we try to fit a Weibull distribution to this model,
the attempt fails. A likely problem is that our
experimental distribution is not normalized.
+
Fit Weibull
distribution to data

Note: most commercial software for
fitting probability density functions use
normalized equations.

A pdf from raw data is not necessarily
normalized.

We can use moment analysis to do this.

Sidebar: if we are after molecular weight
distributions, the moment generating
functions can be particularly handy for
doing this
+
Check for normalization
Maple program. The moment is normalized if its integral over
the independent variable space equals 1.0.
+
Normalize the distribution
Maple program. Dividing the rate function by its integral values
will normalize the probability density function.
+
Oth moment
plot(moment0(t),t=1..tinf/20.);
+
1st moment
plot(moment1(t),t=1..tinf/20);
+
2nd moment
plot(moment2(t),t=1..tinf/20);
+
Comparison to data
+
Weibull fit to data points
Note: the Weibull
distribution and this
fitted function diverge
significantly for t < 5
min.
+
Weibull cumulative distribution.
Failure rate falls significantly with time; a < 1.
+
Failure rate, h(t)
+
Alternative fitting
methods for
distributions
We have been fitting directly to the
cumulative frequency method, which
does well for estimating the average.
We could also minimize total error, or
compare data and model across
quartiles, which may be useful for
regulatory actions.
+
Linearized distributions.
+
Linearized Weibull model
+
Linearized lognormal.
25 min data. nanocomposites design.xlsx