# ME 6105 Modeling and Simulation

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```					         ME 6105 Modeling and Simulation
Spring 2007
Instructor: Dr. Chris Paredis

Homework #3
02 April 2007

Uncertainty Analysis

Part 1: of an Arbitrary Function
Part 2: of a Landing Gear Shock Absorber

Prepared by:                Thomas T. Mowery
Georgia Institute of Technology
Distance Learning
ME 6105: Modeling and Simulation                                                                                     HW3: Uncertainty Analysis

Table of Figures ......................................................................................................................................... 1
1     Introduction ............................................................................................................................................ 2
2     Part I: Uncertainty Analysis of an Arbitrary Function ............................................................................ 2
2.1 Task 1: Become Familiar with ModelCenter .................................................................................... 2
2.2 Task 2: Explore Monte Carlo Simulation.......................................................................................... 2
2.3 Task 3: Determine Main Effects ....................................................................................................... 4
2.4 Task 4: Compare Convergence of Monte Carlo and Latin Hypercube ............................................ 6
3     Part II: Uncertainty Analysis of a Landing Gear Shock Absorber ......................................................... 8
3.1 Task 5: Identify and Model Uncertainty............................................................................................ 8
3.1.1     Uncertain variables .................................................................................................................. 8
3.1.2     Determine Main Effects ........................................................................................................... 9
3.1.3     Distribution Development ...................................................................................................... 11
3.2 Task 6: Determine the Expectation and Standard Deviation ......................................................... 12
3.2.1     Monte Carlo Experiment ........................................................................................................ 12
3.2.2     Latin Hyper Cube Experiment ............................................................................................... 13
3.3 Task 7: Perform Sensitivity Analysis .............................................................................................. 15
4     Task 8: Lessons Learned .................................................................................................................... 16
5     Task 9: Project web-page .................................................................................................................... 16

Table of Figures
Figure 1 Monte Carlo Simulation, N = 10,000 samples ............................................................................. 2
Figure 2 Variable “a4” Distribution, N = 10,000 ......................................................................................... 3
Figure 3 Full Factorial Design with 10% and 90% Distribution Limits ....................................................... 4
Figure 4 Main Effects from Level 3 Full Factorial Experiment .................................................................... 5
Figure 5 Main Effects from Central Composite Experiment ........................................................................ 5
Figure 6 Main Effects from Monte Carlo Simulation, N = 10,000 (Task 2) .................................................. 6
Figure 7 Convergence of Monte Carlo versus Latin Hypercube sampling ................................................. 7
Figure 8 Influence Diagram for Landing Gear Design ................................................................................ 8
Figure 9 Main Effects on Peak Landing Load ........................................................................................... 10
Figure 10 Main Effects (non-usage) on Peak Landing Load .................................................................... 10
Figure 11 Monte Carlo Landing Simulation Variables and Outputs ......................................................... 12
Figure 12 Monte Carlo Landing Experiment, N = 1,000 landings ............................................................. 13
Figure 13 Latin Hypercube Landing Experiment, N = 50 ......................................................................... 14
Figure 14 Expectation and Standard Deviation with Differing Friction Distribution ................................. 15

T.T.Mowery                                                                                                                               Page 1
ME 6105: Modeling and Simulation                                                HW3: Uncertainty Analysis

1 Introduction
The objective of this homework assignment was to increase our understanding of uncertainty in
engineering design problems, and practice methods for accommodating uncertainty in design. Several
ModelCenter models were developed to explore the computation methods and to exercise one’s
interpretation of the results. Part I applies these skills to a simple, six-variable mathematical algorithm.
Part II then applies the same skills to the model developed in Homework #2, which is an aircraft landing
gear in my case.

2 Part I: Uncertainty Analysis of an Arbitrary Function
2.1 Task 1: Become Familiar with ModelCenter
Downloading the necessary files went smoothly. The introductory materials, both within ModelCenter and
on the ME6105 course website were straight forward and helpful. It took me longer to complete this task
than the 1 hour estimate mentioned in class, (took me about 3 hours, but still not too bad for an “old guy”).
Nothing else was noteworthy from task 1.

2.2 Task 2: Explore Monte Carlo Simulation
I modeled the given “y” equation in a ModelCenter script model. Figure 1 shows the resulting histogram
for my 10,000 sample Monte Carlo experiment, run with the given distributions for x1,x2,a1,a2,a3, and a4.

Figure 1    Monte Carlo Simulation, N = 10,000 samples

T.T.Mowery                                                                                    Page 2
ME 6105: Modeling and Simulation                                                  HW3: Uncertainty Analysis

It is clearly a very skewed distribution. The maximum value recorded (487.8) is over 14 standard
deviations away from the mean.

I also programmed the given equation into a Dymola model by modifying a feedback block model with six
inputs. The primary reason for doing this was to get more practice with Dymola and wrapping Dymola
models for ModelCenter. Fortunately, I got very similar results with the two methods:

For 10,000 samples of each:
ModelCenter Script equation:      Mean = 26.5648            Std Deviation = 32.1538
Dymola equation:                  Mean = 26.6621            Std Deviation = 32.0176

The most interesting thing was the time required to run 10,000 samples with the ModelCenter “script”
version took about 3 minutes; running 10,000 samples for the wrapped Dymola model took about 30
minutes, an order of magnitude longer.

Table 1 summarizes the resulting input variable statistics. The results for x1, x2, a1, and a2 are verifiable
“by inspection” that they represent the intended distributions. The Weibull distributions for a3 and a4 are
less intuitive (for me at least). I also notice that different applications use different order for the shape and
scale parameters (e.g, Arena uses (scale, shape), Excel uses (shape, scale).
10%              Mean              90%            StdDev
x1         3.71051          5.00345           6.31327          1.01115
x2         -3.63491         -2.99901         -2.36605          0.49884

a1      11.6259         12.6613      13.8912      0.85178
a2      4.19000         4.98989      5.79294      0.57666
a3      0.31202         0.88193      1.51047      0.46378
a4      0.05043         9.72525      25.3930      21.8908
Table 1 Variable Distribution Summary, from Monte Carlo N = 10,000

For the N = 10,000 experiment, ModelCenter calculated a 95% confidence interval on the mean as +/-
0.63. I confirmed this with a hand calculation. Using the same hand calculation of L = 2*t*s/Sqrt(N), I
found that only 3972 samples would be required to obtain a +/- 1 confidence interval. In this relation, L =
total interval length, t = the critical point from Student distribution, and s = sample standard deviation.

Comparing Figure 1 to Figure 2 one may conclude that variable a4 has a large influence on the resulting
distribution of “y”. This is explored more in the next section, but clearly a4 has the largest standard
deviation, and by nature of the equation model, a4 is directly additive to the result “y” (a one-to-one
relationship.)

Figure 2    Variable “a4” Distribution, N = 10,000

T.T.Mowery                                                                                      Page 3
ME 6105: Modeling and Simulation                                             HW3: Uncertainty Analysis

2.3 Task 3: Determine Main Effects
A full factorial, level 3 DOE simulation was run on the model using the 10%/90% limits from Table 1.

Figure 3    Full Factorial Design with 10% and 90% Distribution Limits

The main effects results shown in Figure 4 confirm the observation above that variable “a4” dominates
the output. Running the same experiment as a Central Composite design produced essentially the same
Main Effects results, for far fewer runs (77 versus 729 in Full Fatorial, Level 3). See Figure 5.

T.T.Mowery                                                                                Page 4
ME 6105: Modeling and Simulation                                      HW3: Uncertainty Analysis

Figure 4 Main Effects from Level 3 Full Factorial Experiment

Figure 5 Main Effects from Central Composite Experiment

T.T.Mowery                                                                        Page 5
ME 6105: Modeling and Simulation                                               HW3: Uncertainty Analysis

I also examined the Main Effects results from the Monte Carlo experiment (Figure 6). Variable “a4” again
dominate, but this time the other variables are in a different rank from the DOE experiments. I am not
sure I exactly understand this, but I think it is from the way ModelCenter computes effects as the average
change in response for a variable change. In the DOE versions, the changes are essentially the “corners”
of the boxes, while the Monte Carlo version, by random combination, uses “non-corners” which result in
less extreme differences in the output. If I am off track, please correct me.

Figure 6 Main Effects from Monte Carlo Simulation, N = 10,000 (Task 2)

2.4 Task 4: Compare Convergence of Monte Carlo and Latin
Hypercube
Figure 7 shows three separate trials comparing the convergence of the expectation for Monte Carlo
sampling and Latin Hyper Cube sampling. These were generated by incrementally incorporating each
sample to the cumulative mean.

LHC was not as remarkably superior as shown in the class lecture “Computing with Uncertain
Information” example. Overall, it appears that the LHC mean does converge sooner, making it useful if
one wishes to limit the number of samples. It also seems that the Monte Carlo approach suffered larger
swings. Note that the y axis scale for Trial 2 is twice that of trials 1 and 3. In Trial 2, the Monte Carlo
mean suffered a large excursion, and took nearly 30 samples to correct. However, once it did correct, it
was well behaved.

T.T.Mowery                                                                                   Page 6
ME 6105: Modeling and Simulation                                                     HW3: Uncertainty Analysis

Trial 1: Convergence of Monte Carlo v. Latin HyperCube
50.0
45.0
40.0
35.0
Mean, ybar

30.0
25.0
20.0
15.0                                                                      Monte Carlo
10.0                                                                      Latin HyperCube

5.0
0.0
1     10     19     28     37       46   55        64     73          82      91    100
Run Number

Trial 2: Convergence of Monte Carlo v. Latin HyperCube
100.0
90.0
80.0
Monte Carlo
70.0
Latin HyperCube
Mean, ybar

60.0
50.0
40.0
30.0
20.0
10.0
0.0
1        10     19     28     37       46        55   64     73          82      91    100
Run Numbe r

Trial 3: Convergence of Monte Carlo v. Latin HyperCube
50.0
45.0
40.0

35.0
Mean, ybar

30.0

25.0
20.0
15.0
Monte Carlo
10.0                                                                     Latin HyperCube
5.0
0.0
1        10     19     28     37        46        55   64     73      82          91    100
Run Numbe r

Figure 7 Convergence of Monte Carlo versus Latin Hypercube sampling

T.T.Mowery                                                                                            Page 7
ME 6105: Modeling and Simulation                                                                   HW3: Uncertainty Analysis

3 Part II: Uncertainty Analysis of a Landing Gear Shock
Absorber
3.1 Task 5: Identify and Model Uncertainty
3.1.1 Uncertain variables
Returning to the Influence Diagram from Homework #1, I have indicated by a red star the variables on
which I have concentrated for this HW #3. I have also concentrated the core of this design problem, as
indicated by the red dashed line box. That is, my model picks up from the point where the earlier
decisions regarding stroke length, piston diameter, air chamber arrangement and the basic
pressure/volume relationships have been made. The primary attribute of interest is the ground load,
which the designer would like to minimize to keep the weight of the landing gear and fuselage structure to
a minimum, and thereby maximize aircraft performance.

Not part of
Strength
this
Requireme
Design
Landing Gear                                                                               nts
Study
Stowage
Volume                    Size (volume)                                                             Cost of
of Landing                          Sliding
Material          Landing
Gear                             Friction
F                                  Choice             Gear
Reliability                                        Decisions      C
Fluid
Properti
C
es
Stroke                                Pressure-
Air Chamber                             Orifice             Ground
Length and                               Volume                                                   Weight of        Overall
Piston                               Relationship                                               Landing          Aircraft
t Decision                           Decision            (reaction
Diameter                               Decision                                                   Gear          Performance
“The LG Strut Design Decision”                                              forces)
Decision

Tire
Servicing        Characterist                           F

Aircraft          Ease of             Pressure             ics
Operating         Maintenanc           Variation
s                                   Runway          Weight of
Weight                e                                      Aircraft        Roughnes         Fuselage
Operatin            s
g
Speeds

Figure 8 Influence Diagram for Landing Gear Design

T.T.Mowery                                                                                                       Page 8
ME 6105: Modeling and Simulation                                              HW3: Uncertainty Analysis

Table 2 present in brief summary the model variables and my first pass distribution:

Source of
Uncertainty
Aircraft Mass              9000 – 12,400 kg     Reasonable range for empty to full
Usage                                                              mass of aircraft (the portion for one
main landing gear).
Landing Sink Rate          -3.81 to -1 m/s      -3.81 is typical ultimate design
condition. -1 is arbitrary.
Air Pre-charge             4e6 to 4.5e6 Pa      Judgment of +/- 10 to represent off-
Maintenance        Pressure                                        nominal maintenance and/or leaks,
temperature changes, etc., then
rounded outward.
Tire Characteristics       1.35e6 to 1.63e6     Judgment of +/- 5% to represent off-
(spring stiffness)         N/m                  nominal maintenance (pressure in tire)
and/or leaks, temperature changes,
etc.
Various            Coulomb Friction Force     2500 to 5500 N       4000 N from original correlation of
model to test results (HW#2).
Estimated that to be coefficient of
friction of 0.08. Reasonable range
might be 0.05 to .11, therefore scaled
force accordingly.
Main Orifice Diameter      0.03765 to
0.03848 m            Ran ModelCenter Optimizer to
Design/Manufact    Rebound Chamber “In”       0.01605 to           minimize loads via orifice sizing. Then
ure                Orifice Diameter           0.01641 m            added and subtracted small dimension
Rebound Chamber            0.01390 to           from those values to develop range.
“Out” Orifice Diameter     0.01421 m            First pass at manufacturing tolerances.

Table 2 Uncertain Variable Summary

3.1.2 Determine Main Effects
Figure 9 was produced by a Central Composite experiment with the variables in Table 2, looking at
resulting peak landing load as the system output. Clearly the landing sink rate (vertical decent speed at
landing) dominates the resulting ground load. Aircraft mass is a distant second. These two comprise the
“usage” variables in the system. Since the designer may not have control over these, or they may be
customer specified, I removed them and re-ran the Central Composite study on the remaining variables.
See figure 10. It is interesting that the variables changed rank. In either case, it seems that the rebound
chamber orifices have little impact on the peak landing load. I think this is reasonable since the rebound
chamber is pressurized by the extension stroke, after the end of the compression stroke, and therefore
after the peak landing load has been developed.

T.T.Mowery                                                                                  Page 9
ME 6105: Modeling and Simulation                                                                                                                               HW3: Uncertainty Analysis

sinkRate                                                                                                                                            80%

aircraf tMass                                        13%

f rictionCoulomb                      3%

mainOrif iceDia                 2%

airPressurePrecharge                     1%

tireInitialStif f ness              1%

reboundChamberValveInDia                       0%

reboundChamberValveOutDia                       0%

0        5        10        15        20        25        30        35        40        45    50    55     60    65    70   75   80   85    90   95   100

Figure 9 Main Effects on Peak Landing Load

f rictionCoulomb                                                                                                               52%

airPressurePrecharge                                                         23%

tireInitialStif f ness                                        16%

mainOrif iceDia                             8%

reboundChamberValveInDia                  1%

reboundChamberValveOutDia              0%

0       5        10        15        20        25        30        35        40        45        50    55     60    65    70    75   80   85    90   95   100

Figure 10 Main Effects (non-usage) on Peak Landing Load

T.T.Mowery                                                                                                                                                                      Page 10
ME 6105: Modeling and Simulation                                                HW3: Uncertainty Analysis

3.1.3 Distribution Development
From the above results, I decided to elicit more detailed distributions for landing sink rate (largest
influence in Figure 9), air pre-charge pressure, and Coulomb friction force (largest influences in Figure
10). For these I used Arena’s Input Analyzer. I also did additional distribution definition of the orifices
and tire characteristics based on engineering analysis, not elicited distributions.

Landing Sink Rate Distribution. I began by defining the upper and lower bounds on sink rate. From
experience, I know that -10 ft/sec (~-3 m/s) is the upper limit on sink rate for most aircraft and rarely
experienced in operation. The other extreme is clearly 0, as any positive values would never touch down.
I started with CDF points at 100% of the landings exceed 0 ft/sec and 0% exceed 10 ft/sec (excuse the
non-SI units….I am used to sink rate in ft/sec). I then split that range as asked “what percentage would
exceed 5 ft/sec?” I estimated (believed) only 10% would exceed 5 ft/sec. I then split that range (0 to 5)
and estimated another percentage and continued until I completed a CDF. I converted the CDF into bins
and entered appropriate quantities into Arena Input Analyzer. While the best fit was a Beta distribution, I
settled on a Normal (-0.847, 0.517) in m/sec. I should also mention that I iterated this several times, as
once I drew the CDF, then saw the distribution, and worked with it a while, I tweaked my beliefs.

Air Pre-charge Distribution. I first experimented with ModelCenter optimizer to minimize landing load
by varying air pressure. It recommended a pressure of 4.24e6 Pa (615 psi). This is very reasonable
based on similar sized landing gear shock absorbers with which I am familiar. For this distribution I did
not draw a CDF, but a PDF. I estimated the upper and lower limits to be perhaps +10% and -20% (there
are more causes of a low pressure than there are causes of high pressure (leaks, cold temps at altitude,
etc). I divided the range into equally spaced bins and asked “If I could know the pressure in 100 landing
gears, what percent would I believe would be within +/- 1.5% of the nominal pressure servicing. I believe
that would be the majority, since leaks are rare and pressures are checked often by maintenance crews,
and serviced to strict instructions. I said about 60%. I then asked “of the remaining 40%, what percent
would be higher and lower? I said 10% and 30% respectively. I then distributed those percentages into
the bins, with a larger tail to the low side.

Again, I iterated this several times, and ended with a Normal (4.2e6, 0.143e6) distribution.

Coulomb friction force distribution. This development required some static force calculations, some
product research, and some experience. The Stribeck friction model from the Dymola library, which built
into my landing gear shock absorber, treats the friction force as a constant. I correlated that model in
HW#2 to real test data, and found a force o about 4,000 N produced reasonable results.

However, in an actual landing gear, the friction force between the piston and cylinder depend on three
things: the strut geometry (the force couple between upper and lower bearings changes with stroke),
applied force (higher ground loads cause higher bearing reactions), and bearing coefficient of friction.
Friction force is not a constant, so how should one represent it as a distribution of uncertainty?

I chose to average the bearing normal force over the expected load range (calculated about 55 kN). This
I then treated as a “constant” to which I could apply a variable coefficient of friction. I knew from
experience and little internet reseach that Teflon lined bearings on chrome pistons would have a
coefficient of friction around 0.08. As a sanity check, 55,000 N x 0.08 = 4,400 N which compares well
with my correlated value of 4,000 N. I also found the best (lowest coefficient would be about 0.03, and
new that badly worn bearings (meta on metal was on the order of 0.20. I used these a the bound of the
CDF and went through the typical elicitation process. I ten applied the coefficients to my “constant” force
to get a friction force distribution, Maximum Likelihood estimation recommended a Triangular distribution
TRIA[1650, 4120, 11600]

A higher fidelity method would be to go back to the Dymola model and develop the relationship between
stroke and bearing forces, and use those to calculate friction force.

T.T.Mowery                                                                                     Page 11
ME 6105: Modeling and Simulation                                              HW3: Uncertainty Analysis

Orifices. I used standard (recommended) drilled hole machining tolerances to calculate a min and max
orifice diameter. I then assumed these manufacturing limits would equate to +/-3 standard deviation
limits. I could then define Normal distributions for the three orifice diameters, which I did.

Tire Characteristics. Similar to the air pre-charge discussion above, I estimated reasonable ranges of
off-nominal tire pressures. Using available tire load-deflection curves at various pressures, I was able to
estimate the change in tire stiffness per change in pressure. I used engineering judgment to guess that
+/- 10% would correspond to +/- 2 standard deviations, and could then define a Normal distribution.

3.2 Task 6: Determine the Expectation and Standard Deviation
3.2.1 Monte Carlo Experiment
For the Monte Carlo experiment I used 7 of the 8 variables in Table 2. I did not use the orifice size
variable for the restriction into the rebound chamber. From the main effects analysis it had the little
affect. I decided to keep the rebound orifice out of the chamber, as that controls one of the key
responses (rebound chamber pressure) I wanted to monitor. Peak landing load was the primary output of
interest, but I wanted to look at peak hydraulic chamber and rebound chamber pressures also. I was not
necessarily interested in the distribution of these, just in making sure the maximums did not venture into
unreasonably high values that could not be maintained in a real product. They did not.

Figure 11   Monte Carlo Landing Simulation Variables and Outputs

T.T.Mowery                                                                                  Page 12
ME 6105: Modeling and Simulation                                                 HW3: Uncertainty Analysis

Figure 12 Monte Carlo Landing Experiment, N = 1,000 landings

Does this result make sense? I think it does. One would expect The Central Limit theorem should drive
this towards a normal distribution, It does seem to be slightly heavier on the lower load side; this is
reasonable since sink rate is the driving effct, and sink rate is skewed to the lower side. The gap in the
histogram at about 45,000N worries me a little, but I think it is a statistical feature, not a system behavior.
The minimum should b near zero (it is) and the maximum should be less than the ultimate landing load
case I calculate in HW 2. In that HW I found a peak load of about 300 kN. After “optimization” via
ModelCenter, that ultimate case was about 245 kN. The maximum “real” load should be about 2/3 of
ultimate case, or about 163 kN. The experiment actually produced about 168 kN as the maximum. This
is very reasonable.

This was not the case in my first attempts at this experiment. The histogram in the initial experiment was
“bathtub” shaped, with a high occurrence of low load cases (this was believable), decreasing occurrences
as load increased (again believable), but then increasing occurrences at the highest loads (not
believable). It took me several hours of investigation to realize it was not a problem with my variable
distributions, but an unintended consequence of changing from a point-design model to a variation-based
model. My original model was “timed” so that the wing lift would begin to decay about when the aircraft
touched down, and that timing worked for high sink rates. My distribution of sink rates produced many
low sink rate cases, and in those the wing lift decay would occur too early, well before the aircraft landed.
The result was the aircraft would fall to earth, not fly to earth. The correlation was completely backwards:
low sink rates would fall to earth, resulting in the highest “landing” loads. I think this is an interesting,
semi-subtle anecdote regarding the dangers of adapting a model from one use to another.

3.2.2 Latin Hyper Cube Experiment
I repeated the experiment with the Latin Hypercube driver. It produced very similar results. The LHC
mean was within the 95% confidence band from the Monte Carlo experiment. With only 50 samples, the

T.T.Mowery                                                                                      Page 13
ME 6105: Modeling and Simulation                                                HW3: Uncertainty Analysis

LHC 95% confidence interval is much wider. For 5% of the computing time, I think the LHC version
produced excellent results.

I noticed one problem with the LHC version though. While the Probabilistic Analysis Tool allows one to
cap distribution with a high or low value, the LHC version did not. This mattered in my sink rate
distribution. With a mean of -.847 and std dev of 0.517, it would occasionally produce positive sink rates
(positive = aircraft gains altitude, not landing). I capped those at -0.1 m/s in Monte Carlo, but could not in
LHC. Therefore, there are two erroneous landings in the LHC that had positive sink rate.

Figure 13    Latin Hypercube Landing Experiment, N = 50

T.T.Mowery                                                                                    Page 14
ME 6105: Modeling and Simulation                                                            HW3: Uncertainty Analysis

3.3 Task 7: Perform Sensitivity Analysis
I performed a sensitivity analysis on the Coulomb friction design variable in my landing gear model. The
baseline model uses a triangular distribution of [1650, 4120, 11600] Newtons. As explained above, I
derived this from some limited knowledge of Teflon lined bearing materials. As design team may be
wondering if this is adequate characterization of the friction characteristics, or if they should invest on
development testing to quantify their particular bearing selection? A sensitivity analysis could help

I arbitrarily assumed +/- 10% variation on the peak and high ends of the triangular distribution. I first ran
the base three times in the LHC simulation to get a feel for normal variation in the mean and standard
deviation of the ground load. Then, pairing these in a very short DOE I ran the 10% middle + 10% upper
combination through the LHC simulation, and separately ran the -10% / -10% combination (twice each).

The result was very little change in either the expectation or standard deviation. I would conclude from
this result that the design team probably did not need to invest in testing to better define the distribution; it
made little impact on the results.

100000

90000

80000
Peak Ground Load, N

70000

60000
Mean, N     StdDev
base         63405      26065                                50000

base         65409      25056                                40000
base         64027      27787
high-                                                        30000
high         63221      26242                                                                             Expectation
high-                                                        20000
high         62973      25329                                                                             -1 sigma
10000                                        +1 sigma
low-low      63836      27707
low-low      64233      25781                                    0
base   base   base   high-   high-   low-low low-low
high    high
Friction Characteristics

Figure 14   Expectation and Standard Deviation with Differing Friction Distribution

T.T.Mowery                                                                                                 Page 15
ME 6105: Modeling and Simulation                                                HW3: Uncertainty Analysis

4 Task 8: Lessons Learned
In HW #1 I outlined three learning objectives. I have organized my lessons under those three objectives:

1. Gain insight into how to efficiently structure and execute a model and simulation study.

I struggled for a awhile rebuilding Dymola models and trying to get them wrapped and running in
ModelCenter. I had two problems: first, ModelCenter analyzes only the ending values of variables, and
the peak values may occur at different times. That is, peak landing load may be earlier in a mild landing
than in a hard landing. Following the instructor’s suggestion, I added “max” recorders to the Dymola
model to solve this.

However, that brought to the surface another problem. The compiled variable list had two variables with
similar names (one a subset of the other) which confused ModelCenter (it took the first it came too, which
was the wrong one). Again, the instructor had an easy work around, but this points out the difficulties of
integrating different modeling applications. It can take time to get them to work together, and a fair
amount of familiarity,

I also struggled a bit with which variables I should include. Since this was primarily an exploration project,
not trying to answer any specific question, I mixed uncertainty from product usage with uncertainty from
product design, manufacturing and from product maintenance. While interesting, that is probably a
confusing way to run a study. If one was trying to answer a specific question, I think it would be better to
hold usage fixed, for example, to focus on pressure variation or orifice sizing alone.

2. Learn and practice current tools for executing simulation based studies.

I learned how powerful analysis of uncertainty can be. Doing this by hand would be impossible, but with
modern tools and reasonable computer power, it made the job fairly easy. The tools are very usable.
However, it is the interpretation that engineers (should) get paid for. I came across several example (a
couple described in this report) where the results were sort of reasonable (not obviously flawed) but not
correct on deeper thought. One cannot just trust the output. One must verify.

3. Increase my understanding of landing gear.

I was most surprised to see the difference between the ultimate landing loads (used for structural sizing of
the landing gear) I did in HW 2 and the more representative distribution of loads from a “real” usage
spectrum. I’ve known for years that day-in/day-out usage was much less severe than design cases.
Analysis with uncertainty made it clear.

While computer programs to produce a spectrum of loads have been used for many years in aerospace
design (to perform structural fatigue analysis), I am also thinking about the possible use of uncertainty
analysis in the development of diagnostic and predictive monitoring algorithms. Those are just now
starting to be employed in aerospace products. Thorough analysis with the right uncertainty could give
one a clear picture of the expected response (pressures perhaps) that could be monitored for health.

5 Task 9: Project web-page
The models used in completing this assignment have been loaded onto my project web page.

T.T.Mowery                                                                                    Page 16

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