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LABORE
THE UNIVERSITY
OF QUEENSLAND
AUSTRALIA
SCIENTIA
AC
ENGG 4103 Engineering Asset Maintenance and Management
Practical No. 1 2024
Attach this to your submission.
Case Study: Weibull distributions and Optimal Replacement
Policies
The largest optical telescope array in the world is the Very Large Telescope (VLT)
array owned and operated by the European Southern Observatory and located on the
2,500 m peak of Cerro Paranal in the Atacama Desert in Chile. The array consists of
four telescopes with 8 m diameter mirrors (see Fig 1). These telescopes produced the
first direct images of planets outside of our solar system, weighed distant stars and have
made important observations concerning black holes.
The surface of each mirror has a curvature that has to be carefully controlled as the
telescope track stars that are moving relative to the earth. To control this curvature, 64
hydraulic cylinders are mounted around the perimeter of each mirror, and 150 axial
cylinders are mounted beneath.
Fig. 1 ESO Telescope, Cerro Paranal
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Fig.2 Base of 8 m mirror PART A
In the attached file "Paranal_cylinder_replacements” you will find a list of the axial
cylinders that were replaced during 2006. The data is classified as either a failure (F) or
suspension (S) according to the nature of the replacement (suspensions refers to those
components that are still in service at the time of analysis). It is required to fit a Weibull
function to the failure data in order to determine the dominant failure mode of the
cylinders.
To do this you will have to:
(i)
(ii)
(iii)
(iv)
An event is the name given to a failed or suspended item. Beginning by
ordering all of the events by days of service in ascending order Determine
the number of events, si (suspended items or failures) preceding each failure.
(for the first event, so = 0)
Create a new page by copying and pasting a copy of the current work page.
On the new page, filter out all suspended items, leaving just failed items.
List the failed components, i, ranked according to the hours of operation
N+1J
Determine the mean rank ji, for each failure where: j₁ = ji1 +
N+1S₁
where jo = 0. This is a way of taking into account the rank of each failed
item given where it is located in the ordered list of all events (failed plus
suspended items)
(v)
Calculate the median rank using: F(t) =
J0.3
N+0.4
This is the best
(vi)
(vii)
estimate for the cumulative probability of failure F(t).
Plot ln(ln(1/(1F(t)) against ln(tto), where to is the failure free time. Adjust a
linear regression fit to the graph and vary to to obtain the best fit.
Determine the shape factor, ẞ, from the gradient of the graph.
(viii) Determine the scale factor, n = (b/³) where b is the y intersect of the graph.
Marking criteria:
e
1 Mark  Discussion of use of Weibull curves for failure rate data
1 Mark – Inclusion of diagram showing Weibull failure rate curves for different ranges

of ẞ values
1 Mark  Inclusion of spreadsheet summarising results
2 Marks Correct estimations for Weibull parameters.
2 PART B
(i) If the cost of performing a preventive replacement is C₁ = US$ 5,000 versus the cost
of performing an unscheduled replacement Cf= US$ 20,000 calculate the interval at
which the cylinders should be preventively replaced in order to optimise hourly
operating costs. (Adapt the associated Excel file "Prac 1 optimal replacement”).
Marking criteria
1 Mark  Brief inclusion of theory, including pdf graph
1 Mark  Inclusion of spreadsheet
1 Mark  inclusion of graph
1 Mark  Correct interpretation of replacement window.
1 Marks  neat, summarised report <4 pages in length plus spreadsheet appendices.
Attach a copy of your working calculations
Submit via Turnitin by 16:00 hrs on Friday 22nd March.
Late penalty: zero marks in the absence of preapproval from the course
coordinator.
P. F. Knights
15032024.
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