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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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