ME 6105 Modeling and Simulation

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ME 6105 Modeling and Simulation Powered By Docstoc
					       ME 6105 Modeling and Simulation
                   Spring 2007
          Instructor: Dr. Chris Paredis




             Homework #4
               01 May 2007



                  Project 13


Preference Modeling and Optimization
 Of a Landing Gear Shock Absorber




                Prepared by:                Thomas T. Mowery
                               Georgia Institute of Technology
                                             Distance Learning
ME 6105: Modeling and Simulation                                                             HW4: Preference Modeling and Optimization




Table of Contents
   Table of Contents ...................................................................................................................................... 1
   Table of Figures ......................................................................................................................................... 1
1 Introduction ............................................................................................................................................ 2
2 .................................................................................................................................................................... 2
   2.1 The Decision Situation ..................................................................................................................... 2
      2.1.1          Design Variables ..................................................................................................................... 3
      2.1.2          Uncertainty .............................................................................................................................. 3
      2.1.3          Attributes ................................................................................................................................. 4
   2.2 Utility Function Development ........................................................................................................... 4
      2.2.1          Overall Utility Structure ............................................................................................................ 4
      2.2.2          Individual Utility Functions ....................................................................................................... 4
      2.2.3          Multi-attribute Utility Functions ................................................................................................ 7
      2.2.4          Utility Verification ..................................................................................................................... 8
   2.3 Exploration of the Design Space ...................................................................................................... 9
   2.4 The Deterministic Design Solution ................................................................................................. 11
   2.5 The Design Solution under Uncertainty ......................................................................................... 12
   2.6 Sensitivity Analysis ......................................................................................................................... 13
      2.6.1          Utility Function ....................................................................................................................... 13
      2.6.2          Landing versus Taxi (Bump) conditions ................................................................................ 13
   2.7 Lessons Learned ............................................................................................................................ 14
   2.8 Project web-page ........................................................................................................................... 15




Table of Figures
Figure 1 The Landing Gear Design Influence Diagram ............................................................................. 2
Figure 2 Model Overview ........................................................................................................................... 3
Figure 3 Utility Hierarchy .............................................................................................................................. 4
Figure 4 Utility of Ground Reaction Load ................................................................................................... 5
Figure 5 Utility of Shock Absorber Volume ................................................................................................. 5
Figure 6 Utility of Air Precharge Pressure ................................................................................................... 6
Figure 7 Utility of Peak Operating Pressures .............................................................................................. 6
Figure 8 Deriviation Schematic for Simultaneous Equations ...................................................................... 7
Figure 9 Total Utility Cube, Version 1 ......................................................................................................... 8
Figure 10 Relative contribution to Total Utility, Version 1 (“Not 3,5,9”) ...................................................... 9
Figure 11 Relative contribution to Total Utility, Version 2 (“Not 2,4,6’) ....................................................... 9
Figure 12 Sink Rate and Bump Height Effect on Utility ............................................................................. 10
Figure 13 Utility as function of Piston Diameter and Air Precharge Pressure ........................................... 11
Figure 14 Solutions Under Uncertainty .................................................................................................... 12
Figure 15 Sensitivity cases for sink rate and bump height ........................................................................ 13




T.T.Mowery                                                                                                                                     Page 1
ME 6105: Modeling and Simulation                                        HW4: Preference Modeling and Optimization




 1 Introduction
The objective of this homework assignment was to solve a complete design problem using the energy-
based model of a landing gear developed in Homework #2, and under uncertainty as explored in
Homework #3. This fourth and final homework introduces the decision maker’s preferences to the
problem, allowing tradeoffs between several design choices. The balance of the design variables and
their resulting attributes should result, hopefully, in the design solution of maximum utility.

 2
  2.1 The Decision Situation
In Homework 1 I presented fundamental design objectives of minimizing weight and space requirements
while maximizing load attenuation and reliability/maintainability of the landing gear shock absorber.
Figure 1 presents the influence diagram that corresponds to this design problem. I believe the basic
structure of the problem is still sound and propose no significant changes. The scope however, has to be
bounded to fit this course. Although I explored uncertainty due to servicing pressure variation, tire
characteristics, and friction, in addition to others, in HW #3, I set those aside for this homework to focus
on the boxes marked in Figure 1. I did this for two reasons: a) they were found to be lesser influences in
HW #3, and b) I have added other model capability that makes the problem more complete and
interesting and needed to reduce the scope elsewhere to keep run times reasonable.


           Elements of HW#4
           Legend D Decision.
                           Uncertainty  .
                           Attributes   .                                                Strength
                                                                                                      Not part of
                                                                                                         this
                                                                                        Requireme
                                                                                                       Design
                                                                                            nts
                                 Size (volume)                                                          Study
     Landing Gear                 of Landing                                                          Cost of
                                     Gear                        Sliding             Material
       Stowage                                                                                        Landing
                                                                 Friction            Choice
       Volume                                       F                                                  Gear
                                     Reliability                                    Decisions
                                                                                                 C
                                                          Fluid
                                                        Properties
                                    C

  D
                     Air Chamber D
       Stroke                             Pressure-      D  Orifice             Ground                            Overall
     Length and                            Volume                                                Weight of
                     Arrangement                          Parameter              Loads                            Aircraft
       Piston                            Relationship                                            Landing
                        Decision                           Decision            (reaction                        Performance
      Diameter                            Decision                                                Gear
                                                                                forces)
            “The
      Decision LG Strut Design Decision”

                                                                          Tire
                                                    Servicing        Characteristics                     F
                                  Ease of           Pressure
                Aircraft
                                Maintenance         Variation
               Operating                                                                                 Weight of
                                                        s                                   Runway
                Weight                                                   Aircraft                        Fuselage
                                                                                           Roughnes
                                                                        Operating              s
                                                                         Speeds


                             Figure 1       The Landing Gear Design Influence Diagram




T.T.Mowery                                                                                                   Page 2
ME 6105: Modeling and Simulation                            HW4: Preference Modeling and Optimization


 2.1.1 Design Variables
I added the ability to select shock absorber piston diameter as an input design variable. Previous models
had this “hardwired” into several locations in the Modelica models. I now use a ModelCenter script model
to access and modify the various features that depend on piston diameter. This allows me to investigate
the first decision box in Figure 1 (marked with a yellow square “D”) in addition to the other two decisions
regarding shock strut servicing pressure and orifice diameters. These three (actually four since I have
two separate orifices) are the design variables for this problem.

I also added the capability to evaluate loads from both landing and taxiing across bumps in the same
model run. Previously these were separate models. This allows me to model the utility from loads in
general, as opposed to running separate optimizations of shock absorbers for landing and taxi.

The four design variables are shown in the upper left box in Figure 2. The fifth variable, strutVolume, is
not a free variable, but a simple calculation based on pistonDiameter.




                                       Figure 2    Model Overview


 2.1.2 Uncertainty
I have chosen aircraft mass, sink rate at landing, and runway bump amplitude as the uncertain variables
for this exercise. These are marked with red stars in Figure 1. All three are “usage” variables and have a
large influence on the “best” design. These are shown in the lower left box in Figure 2.




T.T.Mowery                                                                                   Page 3
ME 6105: Modeling and Simulation                           HW4: Preference Modeling and Optimization


 2.1.3 Attributes
The attributes used are:
Peak ground load, N. This it directly aligned with the ground load reaction outcome in the influence
diagram (blue triangle).
Strut volume, m^3: This is a partial measure of the “size of landing gear” outcome from figure 1. I have
simply calculated the volume of the shock absorber portion of the gear. Loads which dictate structural
thickness and extent of the gear would also factor into size; I am not addressing that aspect here.
Pressures, Pa. I have used the initial servicing pressure (air chamber precharge) and the peak hydraulic
chamber and rebound chamber pressures as a measure of both the reliability and maintainability of the
gear. Again, these are partial measures, but suffice for the learning purposes of this homework. I explain
how and why pressures relate to R&M in the utility elicitation section.

  2.2 Utility Function Development
 2.2.1 Overall Utility Structure
Figure 3 summarizes the utility structure used in this assignment. As suggested in class lecture, I
combined three lower level attributes into a composite utility before combining with the two remaining
attributes to get the total utility. This kept the cross terms to a manageable number.


                                   Total Utility


               U(Peak                U(Shock               U(Hydraulic
               Ground                Absorber              Reliability &
                Load)                Volume)              Maintainability
               Figure 4               Figure 5




                                      U(Servicing            U(Peak                 U(Peak
                                      (Precharge)           Hydraulic              Rebound
                                       Pressure)            Chamber                Chamber
                                                            Pressure)              Pressure)
                                         Figure 6

                                                                        Figure 7

                                       Figure 3 Utility Hierarchy

 2.2.2 Individual Utility Functions
For elicitation of each individual utility relation, I followed the same basic steps:
         a) defined reasonable max and minimum values for the attribute and assign u=0 and u=1 to these
points.
         b) split the range between the min and max and ask the question: “If I could have a shock
absorber with attribute value “m”, or take a 50/50 gamble on getting one with the max or min of the range,
which would I prefer?” and then adjusted “m” up or down until I had trouble deciding between the two
(indifference point).
         c) split the range again, and repeated until I had 5 or 6 points.
         d) entered the points into ZunZun.com’s equation finder and found an approximate fit to my
elicited points.


T.T.Mowery                                                                                 Page 4
ME 6105: Modeling and Simulation                                      HW4: Preference Modeling and Optimization


My elicited points and the curve fit function are shown in Figures 4, 5, 6 and 7.

Elicited Utility Points
         0           1                                 Utility of (Minimized) Peak Ground Load
        72           1              1.20
       170        0.75
       200         0.5
       230        0.25              1.00
       255        0.08
       350           0
                                    0.80
a          -1.01482
b           200.164
c           26.4367
                                    0.60
d             1.004
                          Utility




Sigmoid w/ Offset
      60     0.9990
      80     0.9933
                                    0.40
     100     0.9816
     120     0.9573
     140     0.9095                 0.20
     160     0.8218
     180     0.6812
     200     0.4982                 0.00
     220     0.3147                         0     50       100      150      200      250         300     350      400
     240     0.1733
     260     0.0848                 -0.20
     280     0.0364
     300     0.0119                                                        Load, kN
     320     0.0000
                                       Figure 4     Utility of Ground Reaction Load



Elicited Utility Points
     0.003           1                                      Utility of Shock Absorber Volume
                                    1.10
     0.004           1
    0.0065        0.75              1.00
    0.0075         0.5
     0.009        0.25              0.90
       0.01      0.125
     0.014           0              0.80
       0.02          0
                                    0.70
     a         -1.0373
     b       0.007537               0.60
                          Utility




     c       0.001199
     d         1.03845              0.50
Sigmoid w/ Offset
          0     1.0365              0.40
     0.002      1.0283
                                    0.30
     0.004      0.9869
     0.006      0.8132              0.20
     0.008      0.4210
       0.01     0.1191              0.10
    0.0107      0.0704
     0.011      0.0559              0.00
     0.012      0.0256                      0     0.002     0.004    0.006     0.008           0.01     0.012     0.014
     0.014      0.0059                                      Shock Absorber Volume, m^3
       0.02     0.0012
                                       Figure 5 Utility of Shock Absorber Volume



T.T.Mowery                                                                                               Page 5
ME 6105: Modeling and Simulation                                             HW4: Preference Modeling and Optimization



Elicited Utility Points
 2.00E+06             1                                          Utility of Air Precharge Pressure
 3.00E+06             1                1.00
 5.25E+06          0.75
 5.75E+06           0.5
                                       0.90
 6.00E+06          0.25
 6.25E+06         0.125
 7.00E+06         0.063                0.80
 8.00E+06             0
     a         -6.85814                0.70
     b       5.25E+06
                                       0.60
                             Utility




Weibull CDF                            0.50
2.00E+06      1.00000
3.00E+06      1.00000
                                       0.40
3.25E+06      1.00000
3.80E+06      0.99989
4.20E+06      0.98991                  0.30
4.50E+06      0.94294
5.00E+06      0.75100                  0.20
5.20E+06      0.65438
5.50E+06      0.51478                  0.10
5.75E+06      0.41323
6.00E+06      0.32845                  0.00
6.25E+06      0.25988
                                        1.00E+06 2.00E+06 3.00E+06 4.00E+06 5.00E+06 6.00E+06 7.00E+06 8.00E+06
6.50E+06      0.20544
7.00E+06      0.12920                                     Air Precharge Pressure (servicing Pressure), Pa
8.00E+06      0.05386
                                              Figure 6 Utility of Air Precharge Pressure


Elicited Utility Points
1.00E+07              1                                         Utility of Peak Operational Pressures
2.20E+07           0.75
3.00E+07            0.5             1.00
3.60E+07           0.25
4.00E+07              0
                                    0.80


Inverse Exponential
with offset                         0.60
    a          -6.4905
                          Utility




    b       -7.65E+07
    c             1.00              0.40


 5.00E+06      1.00000              0.20
 1.00E+07      0.99692
 1.70E+07      0.92810
 2.20E+07      0.79993
                                    0.00
 3.00E+07      0.49400
 3.60E+07      0.22583               0.00E+00        1.00E+07       2.00E+07      3.00E+07      4.00E+07    5.00E+07
 4.00E+07      0.04241
 5.00E+07     -0.40410         -0.20
 5.50E+07     -0.61377                                               Peak Internal Pressures, Pa

                                           Figure 7 Utility of Peak Operating Pressures



T.T.Mowery                                                                                                   Page 6
ME 6105: Modeling and Simulation                                HW4: Preference Modeling and Optimization



The utility of air precharge pressure was based on two considerations. First, I believe the lower the
pressure, the lower the tendency to leak, although I do not have information to quantify that. Second,
shock absorbers must be serviced with ground carts equipped with pressurized nitrogen bottles. Knowing
that those carts are often limited to approximately 8e6Pa, and would provide useful volume down to about
6e6 Pa, I formulated my preference to stay below these pressures with a rather steep drop off if they are
exceeded (indicating new support equipment would be required). While I could imagine such a utility
function, ZunZun had a hard time matching it. The Weibull CDF came the closest.

Likewise, the utility of the peak operating pressures in the hydraulic and rebound chambers was also a
“less is better” function. Reliable sealing systems exist for aerospace applications operating at 2.8e7 Pa.
Some go to 3.4e7 Pa, but not much higher. I used these as the approximate range where utility drops off.
Other measures of reliability such as part count or complexity are not incorporated.

    2.2.3 Multi-attribute Utility Functions
I had the most trouble with solving the system of utility tradeoff equations for 7 coefficients to combine the
three individual utilities. First I attempted this via tradeoffs on the surface of the “attribute cube” or “utility
cube”. I elicited tradeoff between a point of known corner utility (like 1,1,0) and an equivalent point. This
produced degenerate solutions. On suggestion of the instructor I moved to a system of equations based
on internal tradeoffs, comparing points to the central point. This also produced degenerate solutions. I
finally created an over-prescribed system of equations using both surface and internal tradeoffs, and
                                                                                              1
through trial and error, found a combination that would produce a meaningful solution . The tradeoffs I
used are schematically shown in Figure 8 for the Load – Volume – Pressure Reliability/Maintainability
combination.




                      Figure 8 Derivation Schematic for Simultaneous Equations



1
  I solved the equation set using Engineering Equation Solver (EES) software, which is well suited to
simultaneous equation solutions. It was easy to comment in/comment out various equations to explore
the solutions. I am not including the EES code in this report, but it is available upon request.


T.T.Mowery                                                                                        Page 7
ME 6105: Modeling and Simulation                                                                     HW4: Preference Modeling and Optimization


 2.2.4 Utility Verification
Omitting tradeoffs 3, 5, and 9 produced a reasonable utility function. So did omitting relations 2, 4 and 6.
How did I know which to use? How did I know any were correct? I performed a level three DOE on the
utility function and verified the results matched my preferences and did not produce abnormal trends.
(Some combinations did produce abnormalities, like decreasing utility in a direction expected to increase
utility, or negative utility over portions of the cube). The comparison of relative contribution of each
parameter to total utility was a good sanity check. The two versions are shown below. I used version 1,
which put more emphasis on minimizing loads. I will return to version 2 in the sensitivity analysis.




                                                    Figure 9 Total Utility Cube, Version 1




                    peakGroundLoad                                                                                   66%




                shockAbsorberVolume                                  21%




                                                                                          Load    66%
                                                                                          Volume  21%
                                                                                          Hyd R&M 13%

                        HydRMUtility                      13%




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




T.T.Mowery                                                                                                                                                Page 8
ME 6105: Modeling and Simulation                                                                HW4: Preference Modeling and Optimization


               Figure 10 Relative contribution to Total Utility, Version 1 (“Not 3,5,9”)




                    peakGroundLoad                                                        45%




                shockAbsorberVolume                                           34%




                                                                                            Load    45%
                                                                                            Volume 34%
                                                                                            Hyd R&M 21%
                        HydRMUtility                          21%




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


                                                                                                                                                   ’
               Figure 11 Relative contribution to Total Utility, Version 2 (“Not 2,4,6’)

One final comment on my utility function: because it uses peak load reaction as an attribute in calculating
utility, and loads are related to usage (a harder landing will generate higher loads, all else being equal),
one cannot compare utility across different usage. One cannot say a design with U = .85 is better than
one with U = .73 if the one of U = .73 was used at more severe conditions.


  2.3 Exploration of the Design Space
I used three primary tools in exploring the design space:
a) Dymola runs looking at system behavior in a “what if” fashion. I did many ad hoc studies to familiarize
myself with the limits of my model landing gear. I used Dymola because it provides better insight into
behavior (time histories). Through this I was able to see, for example, that at certain combinations of
(small) piston diameters and (low) servicing pressures, the piston stroke would exceed that available and
the model would crash.

b) Carpet plots in ModelCenter. I evaluated each of the paired design variable combinations (e.g., piston
diameter and orifice size) to evaluate system behavior. Some I did looking at utility; some I did looking at
an attribute behind the utility, such as peak ground load.

c) Full Factorial DOE. I ran full factorial experiments using the four design variables. I used piston
diameter and servicing pressure as the “primary” design variables. Orifice sizes are farther removed from
the utility (Do I really care what the orifice sizes are? No.), but I learned they need to be independent
variables. That is, I thought for awhile that I could optimize the orifices to reasonable values, and then
consider them fixed for the rest of the study. While physically possible, to do so would have unfairly
skewed the results because each piston/pressure combination needed slightly different orifice
combinations to maximize utility.

Figures 12 and 13 present two of the more interesting, and useful, explorations I performed. Figure 12 is
an assessment of peak ground loads for a particular shock absorber design (previously chosen through
some ModelCenter optimization runs) based on various combinations of sink rate and bump heights. The



T.T.Mowery                                                                                                                                         Page 9
                 ME 6105: Modeling and Simulation                                                                                              HW4: Preference Modeling and Optimization


                 plot shows that the lower/left section of the plot is driven by landing sink rate, and the upper/right section
                 of the plot is driven by bump height. My mean usage point is near the center of the ranges, and with
                 uncertainty, will produce some cases driven by sink rate, and some cases driven by bumps.

                               Plot Variable: design variable 2 (LandingTaxiHW4.LandTaxiHW4.bumpHeight)
                        0.12

                                                                                                                                                                                                                               249437
                       0.115

                        0.11
                                        Max Usage (2σ)                                                                                                                                                                         235350
                                                                                                                                                                                                                               221262
                                      Sink Rate -3.16 m/s                                                                                                                                                                      207175
                       0.105                                                                                                                                                                                                   193088

                  B                  Bump Height = 0.10 m                                            Deterministic Usage (1σ)                                                                                                  179001
                         0.1
                                                                                                                                                                                                                               164914

                  u    0.095
                                                                                                       Sink Rate -2.53 m/s                                                                                                     150827
                                                                                                                                                                                                                               136740
                                                                                                      Bump Height = 0.08 m
                  m     0.09                                                                                                                                                                                                   122652

                       0.085
                  p     0.08

                  H    0.075
                                                                                                                                   Mean Design Usage
                  ei    0.07
                                                                                                                                    Sink Rate -1.9 m/s
                                                                                                                                  Bump Height = 0.06 m
bumpHeight (m)




                  gh   0.065

                        0.06
                  t,   0.055

                  m     0.05

                       0.045

                        0.04

                       0.035

                        0.03

                       0.025

                        0.02

                       0.015

                        0.01

                       0.005

                          0
                                -3    -2.9   -2.8   -2.7   -2.6   -2.5   -2.4   -2.3   -2.2   -2.1   -2   -1.9   -1.8   -1.7   -1.6   -1.5   -1.4   -1.3   -1.2   -1.1   -1   -0.9   -0.8   -0.7   -0.6   -0.5   -0.4   -0.3
                                                                                                          sinkRate (m/sec)
                                                                                       Landing Sink Rate, m/sec
                                                             Figure 12 Sink Rate and Bump Height Effect on Utility

                 I should briefly note that the mean usage here is not the same as the distribution explored in HW 3. In
                 that exercise, I developed a mean sink rate of -0.85 m/sec. While I still believe that is reasonable
                 expected usage, it seems to me that for safety critical equipment it is still appropriate to use artificially
                 more severe expected usage to assure a robust design. That is what I have done here, developing a
                 more severe sink rate distribution with a mean of -1.9 m/sec. (Chris, if you view this as inappropriate or
                 ineffective design philosophy, I would like to discuss it with you at your convenience. TTM).

                 Figure 13 is a carpet plot of utility as a function of piston diameter and air precharge pressure. The Utility
                 has a “ridge” shape, with a strong roll off of utility with piston diameter, and a weaker relationship with
                 pressure. The peak of the ridge is along piston diameter of about 10.5 cm.

                 Figure 13 also shows where I applied several DOE experiments. The first was a general exploration, but
                 in it I noticed the optimizer took all of the solutions in the lower right quadrant to the maximum orifice
                 ranges I had set in the optimizer. There was no need to be so constraining (they were not unreasonably
                 large). I reran a portion of the DOE with wider limits and as one would expect, utility increased. The point
                 is one cannot just look at end results without examining the details.

                 DOE #3 explored the lower left corner in a finer grid. As mentioned above, combinations of small pistons
                 with low pressure can exceed the allowable stroke. Dymola aborts the run; in real life, the landing gear
                 would “bottom out” internally and cease to be an air spring. Neither is good. However, my preferences
                 favor a small shock absorber and low pressures, so utility is high in that lower left corner. This is an
                 example of potentially pushing the design too close to the infeasible range, as discussed in the Multi-
                 Attribute Utility Theory lecture.




                 T.T.Mowery                                                                                                                                                                                             Page 10
ME 6105: Modeling and Simulation                                                                 HW4: Preference Modeling and Optimization




      Plot Variable: design variable 2 (LandingTaxiHW4.LandTaxiHW4.airPressurePrecharge)
       4.8e+006
      4.75e+006
       4.7e+006                                                                                                DOE #1                                        0.83039
                                                                                                                                                             0.78746
      4.65e+006                                                                                                                                              0.74452
       4.6e+006                                                                                                                                              0.70159
      4.55e+006                                                                                                                                              0.65865
       4.5e+006                                                                                                                                              0.61572


Air
      4.45e+006
       4.4e+006
                                                                Deterministic                                                                                0.57278
                                                                                                                                                             0.52985
      4.35e+006                                                 Solution “A”                                                  DOE #2                         0.48691
Pr     4.3e+006
      4.25e+006
                                                                                                                                                             0.44398


es     4.2e+006
      4.15e+006

sur    4.1e+006
      4.05e+006

e,      4e+006
      3.95e+006
       3.9e+006
      3.85e+006

Pa     3.8e+006
      3.75e+006                                                                Deterministic
       3.7e+006
      3.65e+006                                                                Solution “B”
       3.6e+006
      3.55e+006
       3.5e+006
      3.45e+006
       3.4e+006
      3.35e+006
       3.3e+006
      3.25e+006
       3.2e+006
      3.15e+006
       3.1e+006
      3.05e+006
        3e+006
      2.95e+006
       2.9e+006
      2.85e+006
                                                                        DOE #3
       2.8e+006
                  0.096   0.098   0.1   0.102   0.104   0.106   0.108   0.11   0.112   0.114   0.116   0.118   0.12   0.122   0.124   0.126   0.128   0.13
                                                                         pistonDiameter
                                  Max Stroke                            Piston Diameter, m
                                  Exceeded

                  Figure 13 Utility as function of Piston Diameter and Air Precharge Pressure


  2.4 The Deterministic Design Solution
Armed with the knowledge that there was a risk of infeasibility in the lower left corner, I increased my
usage input variables for sink rate and bump height to 1 sigma above my mean as my deterministic
design point. My intent was to assure the resulting design could handle heavy usage. This may have
been “cheating” with respect to the learning objective I suspect the instructor was trying to foster (to pick a
deterministic solution too close to the frontier, and then be forced back from it with uncertainty analysis).
However, I think I understand the learning point, and proceeded with this method. The model center
optimizer settled on the point labeled “deterministic solution ‘A’” in figure 13. The values of the design
variable at this point were:
Precharge Pressure = 4.30 e6 Pa, Piston Dia = 0.105 m, main orifice Dia = 0.0350 m and rebound
chamber orifice Dia = .0120 m. This became my starting point for most future runs.

I repeated this deterministic solution using even higher usage (2 sigma), which I consider “maximum
design usage. A larger piston, lower pressure design was selected (3.70e6 Pa, .1095 m, .0360 m, .0144
m). I believe these are reasonable solutions. They are similar to the size and pressure of the existing
fighter aircraft landing gear I used to develop and validate the model in HW #2.

I also did several optimization runs from different starting points. Most, (but not all), ended up in this
neighborhood. For those that did not, it seems the optimizer would head in that expected direction, but
stop short. I do not detect any local extrema in the utility function. Perhaps setting tighter tolerances
would drive the optimizer farther.


T.T.Mowery                                                                                                                                            Page 11
                            ME 6105: Modeling and Simulation                                                                           HW4: Preference Modeling and Optimization



                            My final comment on this portion of the assignment was that early in using my total utility function it
                            became clear that I had selected too high a range for load. That is, the actual loads coming out of the
                            simulation were lower than I expected, resulting in consistently high (>.90) utility. This seemed to be
                            “wasting” a large portion of my preference potential, so I returned to the load utility function and lowered it
                            by 50 kN for all elicitation points and obtained a new curve fit. The lowered function is the one shown in
                            Figure 4 and used for all the analysis described in this report.

                              2.5 The Design Solution under Uncertainty
                            I used the Latin HyperCube driver with the following uncertainty parameters:
                                     Aircraft mass:         Triangular, 9000, 10000, 12400 kg
                                     Sink Rate:             Normal, -1.90 m/sec mean, 0.63 std dev
                                     Bump Amplitude:        Normal, 0.06 m mean, 0.02 std dev

                            I ran cases with 5, 6, and 8 samples and noticed little difference. I used 6 or 8 samples for most runs. I
                            also set the reinitialization to “false”. I performed two trials and they produced two almost identical results
                            plotted on Figure 14. Optimizing on expected utility changed the deterministic solution to one of lower
                            servicing pressure: 4.3e6 was lowered to approx 3.85 e6 Pa. What does this mean? I think it means I
                            was too conservative in my selection of the deterministic solution. The variability-based design brought
                            me closer to the frontier.

                            Of course, there is the chance, especially with the low number of samples I chose to run, that this design
                            is not robust enough to handle the most severe usage. As a check, I ran the solution from uncertainty
                            analysis at the maximum usage conditions. It passed. The model is built for maximum piston stroke of
                            28 cm. This maximum use run produced a stroke of 27.97 cm! Can’t get any closer to the frontier! Was
                            that good modeling and simulation, or an accident? I must confess I suspect it is an accident. My utility
                            function does not look at maximum piston stroke and try to stay within a limit. That would be a good
                            addition for future work.

                                        Plot Variable: design variable 2 (LandingTaxiHW4.LandTaxiHW4.airPressurePrecharge)
                                          4.8e+006
                                         4.75e+006
                                                                                                                                                                                                 0.83039
                                          4.7e+006                                                                                                                                               0.78746
                                         4.65e+006                                                                                                                                               0.74452
                                          4.6e+006                                                                                                                                               0.70159
                                         4.55e+006                                                                                                                                               0.65865
                                          4.5e+006                                                                                                                                               0.61572
                                         4.45e+006
                                          4.4e+006
                                                                                                   Deterministic                                                                                 0.57278
                                                                                                                                                                                                 0.52985
                                         4.35e+006                                                 Solution “A”                                                                                  0.48691
                                          4.3e+006                                                                                                                                               0.44398
                                         4.25e+006
                                          4.2e+006
                                         4.15e+006
                                          4.1e+006
                                         4.05e+006
airPressurePrecharge (Pa)




                                           4e+006
                                         3.95e+006                                                         Uncertainty
                                          3.9e+006
                                         3.85e+006
                                                                                                           Solutions
                                          3.8e+006                                                         Trial 1
                                         3.75e+006
                                          3.7e+006
                                                                                                           Trial 2
                                         3.65e+006
                                          3.6e+006
                                         3.55e+006
                                          3.5e+006
                                         3.45e+006
                                          3.4e+006
                                         3.35e+006
                                          3.3e+006
                                         3.25e+006
                                          3.2e+006
                                         3.15e+006
                                          3.1e+006
                                         3.05e+006
                                           3e+006
                                         2.95e+006
                                          2.9e+006
                                         2.85e+006
                                          2.8e+006
                                                     0.096   0.098   0.1   0.102   0.104   0.106   0.108    0.11   0.112   0.114   0.116   0.118   0.12   0.122   0.124   0.126   0.128   0.13
                                                                                                             pistonDiameter
                                                                     Max Stroke                            Piston Diameter, m
                                                                     Exceeded
                                                                              Figure 14                    Solutions Under Uncertainty



                            T.T.Mowery                                                                                                                                                           Page 12
                 ME 6105: Modeling and Simulation                                                                                     HW4: Preference Modeling and Optimization




                   2.6 Sensitivity Analysis
                 I looked at two areas for sensitivity analysis:

                  2.6.1 Utility Function
                 As mentioned in section 2.2.4, I found to similar solutions to the set of simultaneous multi-attribute utility
                 equations. While I used the one closest to my preferences for the exercises above, I also loaded the
                 other and reran the solution under uncertainty. I had trouble getting a complete solution to run, as the
                 model aborted on several occasions. It seems the optimizer was taking larger steps with this utility
                 function, and in doing so would occasionally cross into the unfeasible region. I do not know enough about
                 how the Model center optimizer chooses its points to troubleshoot this. I tightened some of the limits and
                 eventually got it to complete with this second utility function. When is did, it found a solution very similar
                 to the previous one. The piston size was the same, but the servicing pressure was slightly lower. This is
                 as I would expect since the second utility function puts more emphasis on pressure reliability (13% in the
                 first, 21% in the second, per figures 10 and 11).

                  2.6.2 Landing versus Taxi (Bump) conditions
                 As one enters the design process, one may not know what the ultimate usage of the product may be.
                 Perhaps the customers are unclear themselves how they will use the product. I examined the sensitivity
                 of the design solution to usage, first reducing mean sink rate to 33%, keeping all else the same, and
                 alternately, reducing mean bump height to 33%, keeping all else the same.

                               Plot Variable: design variable 2 (LandingTaxiHW4.LandTaxiHW4.bumpHeight)
                        0.12

                                                                                                                                                                                                                              249437
                       0.115
                                                                                                                                                                                                                              235350
                        0.11                                                                                                                                                                                                  221262
                                                                                                                                                                                                                              207175
                  B    0.105                                                                                                                                                                                                  193088

                  u      0.1
                                                                                                                                                                                                                              179001
                                                                                                                                                                                                                              164914

                  m    0.095
                                                                                                                                                                                                                              150827
                                                                                                                                                                                                                              136740

                  p     0.09                                                                                                                                                                                                  122652


                  H    0.085


                  ei    0.08
                                                              Mean Design Usage                                                                                            Sensitivity Trial B
                  gh   0.075                                   Sink Rate -1.9 m/s                                                                                          Reduce sink rate
                        0.07                                 Bump Height = 0.06 m                                                                                       sink rate = -.63 m/sec
                  t,
bumpHeight (m)




                       0.065
                  m
                        0.06

                       0.055

                        0.05

                       0.045

                        0.04
                                                        Sensitivity Trial A
                                                       Reduce bump height
                       0.035
                                                      Bump Height = 0.02 m
                        0.03

                       0.025

                        0.02

                       0.015

                        0.01

                       0.005

                          0
                                -3   -2.9   -2.8   -2.7   -2.6   -2.5   -2.4   -2.3   -2.2   -2.1   -2   -1.9   -1.8   -1.7   -1.6   -1.5   -1.4   -1.3   -1.2   -1.1   -1   -0.9   -0.8   -0.7   -0.6   -0.5   -0.4   -0.3
                                                                                                         sinkRate (m/sec)
                                                                                      Landing Sink Rate, m/sec
                                                     Figure 15 Sensitivity cases for sink rate and bump height




                 T.T.Mowery                                                                                                                                                                                Page 13
ME 6105: Modeling and Simulation                            HW4: Preference Modeling and Optimization



Resulting design solutions are summarized here:
                           Baseline (Uncertainty       Sensitivity Trial A, lower   Sensitivity Trial B, lower
                           Trial 1)                    bump height                  sink rate
Piston diameter, m         10.5 cm                     10.5 cm                      10.5 cm
Precharge Pressure         3.90 e 6 Pa                 4.09 e6 Pa                   3.80 e6 Pa
Main Orifice Diameter      0.0357 m                    0.0365 m                     0.0280 m
Rebound Orifice Dia        0.0100 m                    0.0124 m                     0.0125 m

Are the results reasonable? It seems reasonable to expect a shock absorber designed by landing
conditions only (Trial A with small bumps) would need a stiffer air spring and larger main orifice (when
flow rates are high on landing impact). Conversely, a strut designed to traverse runway bumps, which are
generally a slower dynamic phenomenon, would need a softer strut and smaller main orifice. It is not so
clear why the rebound orifice was solved to be larger in both sensitivity trials. That may require some
additional investigation, although a good approach, at least early in the design, would be to make the
orifice design capable of handling the full range.

How could the results be used? If this were a real design situation, one could expand product offering by
designing one basic design, and by making relatively small adjustments to pressure and orifices, to offer
tailored performance. One could also gain some assurance that if initial predictions of usage turn out to
be wrong, small change could salvage the design.

  2.7 Lessons Learned
In HW #1 I outlined three learning objectives. This homework contributed to them as follows:

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

While I certainly learned the mechanics of how one can structure a mathematical model to complete a
design solution, I am still unsure on how to draw realistic boundaries on the model. That is, what to leave
in, what to leave out. To a large degree I am sure that judgment comes from experience and from model
experimentation. I spent time on details that didn’t really matter much, and other areas of my model are
crude at best (Stribeck friction representation for one).

I am leaving the course with a good foundation in how one could approach a design problem through
modeling and simulation, but the more I learn about what goes into good modeling and simulation, the
more questioning I become of “the answer”. I am leaving with belief that the real benefit of modeling and
simulation comes from the design space exploration and increased product knowledge, not from finding
“the answer”.

Related to this, I have a better understanding of the importance of uncertainty in design. I am almost
embarrassed by some of the single point designs I have been associated with in the past that turned out
to lack robustness when faced with variation. The tools from this course should help me recognize such
situations and deal with them when they arise.

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

I had the most trouble in this assignment with determining MAU constants. I understand the tradeoff of
equivalent utility and can generate many simultaneous equations. I am puzzled by the apparent
sensitivity and unpredictability of multi-linear solutions. Perhaps there are mathematical constraints or
guidelines that would help tell one where to look for non-degenerate solution. Trial and error worked
eventually, but is not a good method in general.

I gained good experience with ModelCenter and the more I work with it, the more I like it. It is a good
intuitive product and it seems robust even in the hands of a novice. I would like to know more about how




T.T.Mowery                                                                                    Page 14
ME 6105: Modeling and Simulation                              HW4: Preference Modeling and Optimization


it makes optimization decisions and about optimization routines in general. We touched on this in one
lecture; perhaps I will take another course on the subject.

3. Increase my understanding of landing gear.

It was very interesting (for me at least) so spend so much time considering the mutual optimization for
landing and taxi (bump) conditions. Usually I have seen that approached as design for one, then
evaluate the performance you get for the other. I think it is feasible, and wise, to try to bring them into a
simultaneous solution.


  2.8 Project web-page
The models used in completing this assignment have been loaded onto my project web page.




T.T.Mowery                                                                                     Page 15

				
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