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 #2
              09 March 2007




        Energy-Based Modeling
                 of an
Aircraft Landing Gear Shock Absorber




                Prepared by:                Thomas T. Mowery
                               Georgia Institute of Technology
                                             Distance Learning
ME 6105: Modeling and Simulation                                                                                                    HW2: The Model




Table of Contents
    Table of Contents ...................................................................................................................................... 1
    Table of Figures ......................................................................................................................................... 1
1      Goals and Problem Domain ................................................................................................................. 2
2      System and Simulation Specification ................................................................................................... 2
    2.1 System Decomposition .................................................................................................................... 2
    2.2 Connections and Interactions .......................................................................................................... 3
    2.3 Assumptions ..................................................................................................................................... 4
3      Model Description ................................................................................................................................. 4
    3.1 Overall Arrangement ........................................................................................................................ 4
    3.2 External Models ............................................................................................................................... 6
      3.2.1       Aircraft Model ...................................................................................................................... 6
      3.2.2       Tire Model ........................................................................................................................... 6
      3.2.3       Bump Model ........................................................................................................................ 7
    3.3 Internal Models ................................................................................................................................. 7
      3.3.1       Shock Structure (Piston-Cylinder) Model ............................................................................ 7
      3.3.2       Air Chamber Model ............................................................................................................. 9
      3.3.3       Hydraulic Chamber Model .................................................................................................. 9
      3.3.4       Rebound Chamber Model ................................................................................................... 9
4     Model Verification ................................................................................................................................ 10
5     Experimentation and Interpretation ...................................................................................................... 12
    5.1 Landing, Fixed Orifice Damping ..................................................................................................... 12
    5.2 Landing, Variable Orifice Damping ................................................................................................ 13
    5.3 Taxi, Rough Runway ...................................................................................................................... 16
6     Lessons Learned ................................................................................................................................. 16




Table of Figures
Figure 1        Landing Gear Schematic Arrangement ....................................................................................... 3
Figure 2        Landing Gear Model Arrangement .............................................................................................. 5
Figure 3        Aircraft Model ............................................................................................................................... 6
Figure 4        Tire Model .................................................................................................................................... 6
Figure 5        Rough Runway "Bump" Model .................................................................................................... 7
Figure 6        Shock Strut Structure Model ........................................................................................................ 8
Figure 7        Air Chamber (or Air Spring) Model .............................................................................................. 9
Figure 8        Hydraulic Chamber Model ........................................................................................................... 9
Figure 9        Rebound Chamber Model .......................................................................................................... 10
Figure 10        Slow Load-Stroke Curve .......................................................................................................... 11
Figure 11        Actual Load-Stroke Curve, Test Data ...................................................................................... 11
Figure 12        Landing Simulation, 12,382 kg, 3.148 m/s, Fixed Orifice ........................................................ 12
Figure 13        Ground Load Reaction, Fixed Orifice ...................................................................................... 13
Figure 14        Actual Ground Load Reaction, Test Data ................................................................................ 13
Figure 15        Variable Orifice Profile, time based ......................................................................................... 14
Figure 16        Ground Load Reaction, Variable Orifice .................................................................................. 14
Figure 17        Ground Load Reaction versus Shock Stroke, Fixed Orifice .................................................... 15
Figure 18        Actual Ground Load Reaction versus Shock Stroke, Test Data, (Variable Orifice) ................ 15
Figure 19        Ground Load Reaction versus Shock Stroke, Variable Orifice ................................................ 15
Figure 20        Rough Runway Simulation, Double Bump, Fixed Orifice ........................................................ 16



T.T.Mowery                                                                                                                               Page 1
ME 6105: Modeling and Simulation                                                          HW2: The Model




 1 Goals and Problem Domain
In Homework #1 I described the overall landing gear shock absorber design problem to be a series of four
sub-decision problems as follows:
         a. Stroke length and piston diameter decision
         b. Air chamber arrangement decision
         c. Pressure-Volume relationship decision
         d. Orifice parameter(s) decision
The decisions would be executed in the order shown, and with the primary objective of minimizing ground
load reactions. Through the execution of HW #2 I have found that basic problem structure to be sound
and not in need of revision. Therefore, the overall goal of the HW#2 modeling effort is to create a model
with sufficient flexibility to allow the designer to exercise each of these area in turn, with measurable
output of loads and related system responses such as internal pressures and dynamic instability.

Likewise, I found the level of abstraction outlined in my HW#1 to be about right. The aircraft will continue
to be modeled as a lumped mass, although in HW#1 I stated I would not include aerodynamic effects --- it
turns out I need to at least crudely simulate lift to “fly” the mass to the ground instead of dropping it. For
the shock absorber itself, what I envisioned in HW#1 was appropriate, but had to be further broken down
into discrete chambers to allow modeling. I describe this below.

The successful execution of HW#2 should allow the designer to answer these types of questions:
Q1: How does changing the pre-charge air pressure in the shock change the resulting ground loads?
Q2: How do different types of orifice schemes change the resulting ground loads?
Q3: Given a certain shock absorber configuration, are landing loads or taxi (rough runway) loads going to
be more severe?

To demonstrate my model can investigation these types of questions, and others, I will run specific
landing scenarios, where a specified mass is “landed” at a specified decent rate (usually about 3 m/s for
limit load case, and 3.8 m/s for an ultimate load case), and observe peak vertical load reaction at the
ground (or at the aircraft-landing gear interface, which will be similar). Secondly, I will run the landing
gear over specified bump profiles and also observe peak vertical loads.

 2 System and Simulation Specification
  2.1 System Decomposition
The first division of the system is into those elements “internal” to the design subject, that is, the object
about which one is trying to make decisions, and those “external” elements which are required to
complete the system, but are themselves not the design. In this project the landing gear shock absorber
is my internal subject, and the aircraft mass and the ground the primary external interfaces. I will also
consider the tire an external component in this project, although it could also be the subject of design
studies from this same model.

Figure 1, re-used from HW#1, shows how the shock absorber is the connecting bridge between the
aircraft mass above and the tire/ground below. The figure also depicts the next level of decomposition
into components within the shock absorber. The piston and cylinder form the structural bones of the
device, with relative motion between them, and pneumatic and hydraulic chambers within them. Those
chambers are divided into three separate chambers: the air chamber, which provides the spring force and
the vast majority of the volume change for compression, the hydraulic chamber which resists
compression by metering fluid through the main orifice, and the rebound chamber which retards
extension be metering fluid out of the rebound chamber.




T.T.Mowery                                                                                    Page 2
ME 6105: Modeling and Simulation                                                                      HW2: The Model



                                                                                   Aircraft
                                                                                   Mass
                                                                                 Initial vertical
                                                                                 velocity (landing)




                      Cylinder                                           Air Chamber

                 Rebound Orifice
                 (controls extension stroke)
                                                                       Rebound Hydraulic
                                                                       Chamber
                Main Orifice
                (controls compression stroke)
                                                                   Hydraulic
                                                                   Chamber

                          Piston



                 Aircraft forward velocity                               Tire (flexible)



                Bump
                                                               Horizontal Load Reaction

                  Runway pavement
                                                        Vertical Load Reaction

                               Figure 1        Landing Gear Schematic Arrangement


  2.2 Connections and Interactions
Many of the connections are structural and fairly straight forward (tire to piston, cylinder to aircraft, etc)
and I will not discuss the here. The most interesting series of connections that required some thought to
model are those for the three hydraulic chambers. The behavior of the chambers can be described by
the pressure of the three chambers, P_air, P_hyd, and P_rebnd acting on three areas, A1 = piston outer
diameter, A2 = piston inner diameter, and A3 = cylinder inner diameter. Writing a force balance for either
the cylinder or piston and simplifying, one finds:

           P_air*(A3 – A2) + P_hyd*(A2) – P_rebnd*(A3 – A1) = Σ Fy (y = vertical in this case)

This pointed me to the necessary abstraction of the overall shock absorber into three separate piston
chambers (I used ChamberB from HyLib.Cylinders) with net areas as shown above.

The second essential relation is that while the hydraulic chamber and air chamber contract as the strut
compresses, the rebound chamber expands, and vice versa.

The placement of the orifices between the chambers is obvious from Figure 1. It is somewhat arbitrary as
to whether the orifices should be modeled with “upstream” or “downstream” chamber, or as separate
models themselves. I chose to include the main orifice with the hydraulic chamber, the rebound orifice
with the rebound chamber, and leave the air chamber unrestricted.




T.T.Mowery                                                                                               Page 3
ME 6105: Modeling and Simulation                                                               HW2: The Model


  2.3 Assumptions
The most significant assumption made in this model is that the behavior is primarily a 1-dimensional
phenomenon. The horizontal load depicted in Figure 1 is reacted by a force couple between the piston
and cylinder. That force couple changes with stroke of the piston, and that in turn would change the
friction force between the piston and cylinder. I attempted to include some of that behavior through
inclusion of a Stribeck friction model, but as I will discuss below, it is not yet a good representation. The
1-D assumption also means I have lost the ability to investigate behavior with respect to differing aircraft
pitch angles.

I have made no modeling provisions for changes in temperature, assuming that near-standard
temperatures would exist. This assumption may breakdown for very cold weather operations (-20 to 40C)
when viscosity changes in the fluid may be significant.

My model structure divided the masses into three lumps, but I think this abstraction caused little loss of
fidelity. Certainly the aircraft mass can be lumped in this case, since we have already abstracted away 3-
D motion like roll, pitch and yaw. The second lump includes the piston wheel and tire. Again this seems
very reasonable assumption since they move as one unit, except for minor deflections. The third lump is
more arbitrary. I assigned the mass of the cylinder to the Stribeck friction sliding mass. This mass has
little basis in “reality”, but also has little effect on the overall behavior provided I do not let the sliding mass
oscillate wildly.

The tire is modeled as a point contact with the ground. This is fine for landing cases, but could cause
some concern for rough runway simulation, where the tire “traverses” the bump. In this 1-D model, the
bump is actually a forced displacement of the tire contact point. For gradual bumps this works well, but
for step inputs (the edge of mis-matched sections of taxiway concrete perhaps) this abstraction will taint
the model results. A real tire with fore-aft extent would roll up onto the step with some gradual increase in
force.

Finally, I originally thought I would have to make assumptions (simplifications) regarding ideal gas
behavior and fluid compressibility characteristics. However, by making use of several Dymola library
models for hydraulic components which already include real gas and real fluid behavior, those
simplifications were not required. The library models already included effects such as cavitation and
laminar versus turbulent flow characteristics. I could never have modeled such behavior in the weeks
allowed for this assignment. This demonstrates the power of object-oriented programming and
knowledge re-use.

 3 Model Description
  3.1 Overall Arrangement
Figure 2 shows the overall model arrangement of the primary components discussed above, that is a
shock absorber sandwiched between and aircraft above and the tire and ground below. The shock
absorber is composed of the piston/cylinder structure and the three hydraulic chambers. I could have
grouped the chambers and the piston/cylinder into a higher level model “shock absorber” but elected to
keep them as separate components at this level. I’m not sure it matters much, but as those are the
component subjects of the design study, it seemed keeping them “exposed” would be more convenient.

I should say a few words about the rod on the LH side of Figure 2. I used this rod to ground the rebound
chamber in opposition to the grounding of the other two chambers, so that as those two contract the
rebound chamber expands. There may be a more elegant attachment method, or alternately I could have
submerged the rod into the rebound chamber model or the piston/cylinder structure, but it helped me to
keep it at this level to make obvious the reverse nature of the rebound chamber.




T.T.Mowery                                                                                         Page 4
ME 6105: Modeling and Simulation                                                              HW2: The Model


                            12,382 kg mass, 3.148 m/s landing, constant orif ice damping




                                                                                  Air...




                                           Hyd...                              Pisto...
                                                             AirC...
                      rod




                                 Reb...




                                                                                     Tir...




                                                                                 fixed

                             Figure 2     Landing Gear Model Arrangement




The next sections contain a brief, high-level specification of the components, and an outline of how each
was verified.




T.T.Mowery                                                                                       Page 5
ME 6105: Modeling and Simulation                                                                                HW2: The Model




  3.2 External Models
 3.2.1 Aircraft Model
The aircraft model needed to be more than a simple mass in order to represent the transition from flying
(where lift = force due to gravity) to being on the ground (force due to gravity only). This was
implemented by prescribing a wing lift decay function that gradually ramps the “weight” of the mass from 0
up to 9.81*Mass. Presently this is manually calculated and entered as the magnitude of the decay
function. It would be nice if the mass was an input into a function that automatically did this. In most of
the simulations I used 3.5 seconds to decay the lift. In reality, it can take much longer, but that keeps the
simulation short without loss of fidelity.

I verified the aircraft model by testing that it started and ended with the proper force function.
                                                                                                         flange_b




w ingLif tDec ay
                                     f


                                                    force



 duration=3.5



                                                                 tireInitialStroke




                                                                                         tireMidStroke




                                                                                                                             tireBottom
                                     aircraftMass




                                flange_a                                               flange_a



        Figure 3    Aircraft Model                                                   Figure 4              Tire Model


 3.2.2 Tire Model
The tire can be adequately modeled as spring-damper arrangement, but real tires are generally not linear
springs. The initial deflection is the least stiff, and then the bulk of the stroke is stiffer, then after the tire
bottoms (flat) it is the stiffest, essentially compressing solid rubber. I approximated the tire as three
parallel spring-dampers. (Figure 4) The springs have different free lengths so they become active at
different deflections.

I calculated the necessary spring constants from actual aircraft tire load-deflection curve data. I verified
the tire model by applying an increasing load and confirming deflection and three distinct slopes. Tire
damping is light, and in the absence of data, I estimated it by “bouncing” the tire and adjusting to get
reasonable decay of the motion.



T.T.Mowery                                                                                                          Page 6
                   ME 6105: Modeling and Simulation                                                                        HW2: The Model




                       3.2.3 Bump Model
                   The landing simulations simply use the “Fixed” model from the library as the ground, but to enforce travel
                   across a bump, an active bump was required. Many different profiles could be modeled; Figure 5 shows
                   one for a double sine shaped bump. Sine functions are superimposed on a constant ground (flat) profile.
                   Adjusting the phase of the sine functions allows the selection of multiple bumps.


                   .
ss, Double 10 cm bumps, 1Hz [25m bump, 50 m/s (97 kts)], constant orifice damping               flange_b




                                          Air...




    Hyd...                             Pisto...
                                                                    DoubleSineBump
                       AirC...                                            const




                                                                          k=1.05




                                                                                                       position
                                                                                     sine3




                                                                                                                  s _...
                                             Tir...
                                                                       sine1

                                                                                    f reqHz=1
                                                                                       sine2


                                                                     f reqHz=1
                                                                        sine0                   sum1

                                        Double...                                   f reqHz=1



                                                                     f reqHz=1
                                                        Figure 5      Rough Runway "Bump" Model




                       3.3 Internal Models
                       3.3.1 Shock Structure (Piston-Cylinder) Model
                   Upper and lower stop gap elements limit the travel of the piston inside the cylinder. The difference
                   between the length of the cylinder minus the stops and the length of the upper piston mass set the stroke
                   of the shock absorber.

                   Also attached to the cylinder is the Stribeck friction sliding mass. This is one of the areas where the
                   model is not very representative of reality. In an actual shock absorber, the primary static and dynamic
                   friction is between the cylinder and piston. In the Stribeck friction model available in the Dymola library
                   the friction is between the sliding mass and the “housing” which seems to be the unmoving reference
                   frame (Earth). It is an approximation to attach the sliding friction mass to the cylinder. While the cylinder
                   is moving down it generates a resistive force. This is appropriate if the cylinder is moving relative to the



                   T.T.Mowery                                                                                                 Page 7
ME 6105: Modeling and Simulation                                                                                      HW2: The Model


piston, as when compressing the shock, but not appropriate when the cylinder is simply moving relative to
the earth, as in the time shortly before landing. More work could be done to make the Stribeck friction
between the cylinder and piston instead of cylinder and earth.

The other requirement for this model is a third flange (flange_b1) to transmit the stroke of the piston. It is
used to actuate the hydraulic chambers.
                                                                                           flange_b




                                                                     upperStop




                                                                                                      spring 2
                   flange_b1




                                                       pistonUpper




                                                                                                       frictionMass
                                                                                 housing




                                                                                                      spring 1
                                                                     lowerStop
                                                       pistonFork




                                                    flange_a



                                 Figure 6    Shock Strut Structure Model




T.T.Mowery                                                                                                               Page 8
ME 6105: Modeling and Simulation                                                              HW2: The Model




 3.3.2 Air Chamber Model
The air chamber is a simple combination of a hydraulic chamber (to incorporate the correct net piston
area) and a pneumatic chamber set to the air pre-charge pressure and volume. There is no restriction
between them as they are in direct fluid contact with each other. See Figure 7.

This model was verified by compressing the model and confirming the correct pressure increase per
volume decrease.

                                                                                                          flange_b

                                           flange_b




                                                                          const




                                                                          k=.01




                                                                                                                hydChamber
                                                 AirChamber




                          gasVolume
port_A
                                                                                     meteringOri

                                                              hydPort_A




                                      flange_a                                                       flange_a



Figure 7   Air Chamber (or Air Spring) Model                         Figure 8     Hydraulic Chamber Model


 3.3.3 Hydraulic Chamber Model
The hydraulic chamber model also utilizes a hydraulic chamber with the correct net piston area. Flow into
or out of this chamber must pass through a metering orifice. I used a variable orifice in the model which
allows it to be either a constant area (when used with a constant source as shown in Figure 8) or variable
as a function.

As currently configured the orifice is variable as a function of time. This is obviously not realistic or
generally useful. In most landing gears the main orifice varies as a function of shock absorber stroke.
Additional modeling could be done to use the relative motion between flange_a and _b as an input to a
function to control the metering orifice.


 3.3.4 Rebound Chamber Model
The rebound chamber model is similar to the hydraulic chamber except that instead of a variable orifice, it
uses two check valves. This allows different restriction into and out of the chamber. This is necessary to
avoid cavitation of the chamber on the strut compression stroke, but provide stiff restriction on the


T.T.Mowery                                                                                         Page 9
ME 6105: Modeling and Simulation                                                              HW2: The Model


extension stroke. This is an example of where some experience with the system being modeled is
helpful. I knew from my work experience that rebound chambers usually have two distinct resistances,
and could structure the model accordingly. Someone without that insight could eventually come to the
same conclusion with more work.


                                                                       flange_b




                                               checkValveOut




                                                                             reboundChamber
                                                checkValveIn
                             hydPort_A




                                                                  flange_a



                                  Figure 9   Rebound Chamber Model




 4 Model Verification
The individual models described above were tested to confirm they produced reasonable results. I then
used an existing aircraft landing gear as the baseline model to correlate mode performance. I used the
dimensions (piston and cylinder diameters, total stroke), pre-charge pressure and volume, tire data, and
aircraft mass (the appropriate portion of the total aircraft mass). This set the basic shock absorber
design. I adjusted the friction and damping variables to approximate the load reactions of the baseline
shock absorber.

The performance of the air spring and friction are best observed in a slow compression test (no delta
pressure across orifices). [ref: StrutLib.Tests.SlowLoadStrokeCurve] Figures 10 and 11 compare the
correlated model to an actual landing gear slow compression test (load plotted on vertical axis). The
overall performance looks similar and they begin and end at the same forces. One difference is that the
model uses constant friction values, while the real landing gear friction increases with load (wider
hysteresis as load increases). Also the steps due to static friction are much more numerous in the real
test. I believe I cannot get this exact behavior with the Stribeck model configured relative to earth as I
have it.




T.T.Mowery                                                                                       Page 10
ME 6105: Modeling and Simulation                                                                                                  HW2: The Model


              tire.flange_a.f [N]   reboundChamber.reboundChamber.s_rel [m]     force.flange_b.f [N]



     3.6E5




     3.2E5




     2.8E5




     2.4E5




     2.0E5




     1.6E5




     1.2E5




     8.0E4




     4.0E4




     0.0E0



             0.30                             0.35                            0.40                                0.45     0.50       0.55

                                                                                     reboundChamber.reboundChamber.s_rel

                                                           Figure 10                 Slow Load-Stroke Curve




                                              Figure 11                 Actual Load-Stroke Curve, Test Data


I used a similar correlation to actual data to adjust damping parameters. The results are used in section
5.




T.T.Mowery                                                                                                                           Page 11
ME 6105: Modeling and Simulation                                                                                                          HW2: The Model



 5 Experimentation and Interpretation
     5.1 Landing, Fixed Orifice Damping
Figure 12 shows landing on the fixed orifice gear [ref: LandingTaxiSimulations.NormalMassLimitLanding].
The upper curve shows the aircraft elevation with its initial decent. The second plot shows the load
reaction at the ground. The initial landing impact lasts about 0.5 second, then the aircraft bounces back
into the air for about 1 second, then lands a second time and settle out. The third plot shows the
pressures in the three chambers


                   aircraft.aircraftMass.s
         4.0



         3.5
 [m ]




         3.0



         2.5



         2.0
               0                               1                           2                          3                               4                    5




                   tire.flange_a.f



         2E5
 [N ]




         1E5




         0E0



               0                               1                           2                          3                               4                    5




                   airSpring.AirChamber.port_A.p   hydChamberConstArea.hydChamber.port_A.p   reboundChamber.reboundChamber.port_A.p
         3E7




         2E7
 [Pa ]




         1E7




         0E0
               0                               1                           2                          3                               4                    5

                                     Figure 12     Landing Simulation, 12,382 kg, 3.148 m/s, Fixed Orifice


The initial touchdown produces the highest ground load. Figure 13 zooms in on that segment and Figure
14 provides the same curve for the actual baseline landing gear test. The duration of the event is almost
identical. The peak loads are comparable: 230 kN from the model versus 240 kN from the test.




T.T.Mowery                                                                                                                                   Page 12
ME 6105: Modeling and Simulation                                                               HW2: The Model


                   tire.flange_a.f
          2.6E5



          2.4E5



          2.2E5



          2.0E5




          1.8E5



          1.6E5



          1.4E5
    [N]




          1.2E5



          1.0E5



          8.0E4



          6.0E4



          4.0E4



          2.0E4



          0.0E0



          -2.0E4
                          0.2            0.3          0.4        0.5        0.6          0.7        0.8


                                      Figure 13    Ground Load Reaction, Fixed Orifice




                                     Figure 14    Actual Ground Load Reaction, Test Data

  5.2 Landing, Variable Orifice Damping
A primary difference between Figure 13 and 14 gears is that the actual gear used a variable orifice. I
approximated this by replacing the constant source with an exponential source, and then setting the rise
times (lower graph in Figure 15) to approximately correspond to the stroke position (the upper graph).
The results are in Figure 16. [Ref: LandingTaxiSimulations.NormalMassLimitLandingVariableOrifice]. The
peak load is lower (202 kN), and the load profile is wider (should be about the same energy, or area
under the curve, but I didn’t confirm that). Lowering the peak loads is one of the key objectives for the
design problem. However, the profile of the load curve is important as well. That is, it would desirable to
keep the load low at first, while the piston is extended from the cylinder, and allow the load to increase as
the piston strokes into the cylinder. This minimizes bending stresses.



T.T.Mowery                                                                                        Page 13
ME 6105: Modeling and Simulation                                                                                                                             HW2: The Model


                                           reboundChamber.reboundChamber.s_rel
                              0.7




                              0.6




                              0.5




                       [m ]
                              0.4




                              0.3




                              0.2




                              0.1
                                 0.00                       0.25                      0.50                    0.75          1.00         1.25         1.50




                                        hydChamberVariableArea.variableAreaOrifice1.exponOrificeDiameter.y
                       0.10



                       0.08



                       0.06



                       0.04



                       0.02



                       0.00



                      -0.02



                      -0.04
                              0.00                       0.25                       0.50                     0.75          1.00          1.25         1.50


                                                          Figure 15                     Variable Orifice Profile, time based



                   tire.flange_a.f
          2.4E5



          2.2E5



          2.0E5



          1.8E5




          1.6E5



          1.4E5



          1.2E5
    [N]




          1.0E5



          8.0E4



          6.0E4



          4.0E4



          2.0E4



          0.0E0



          -2.0E4



          -4.0E4
                              0.2                               0.3                          0.4                     0.5           0.6          0.7               0.8


                                                    Figure 16                     Ground Load Reaction, Variable Orifice


Figures 17, 18 and 19 compare the same three load profiles but plotted versus shock stroke instead of
time. This format allows one to see that more tuning of the orifice in both the fixed and variable case
could be done to delay the load peak to later in the stroke.

Also, it is hard to discern from the figures, but whereas the actual shock was designed to use the full
available stroke (94% in the test), the model version used only about 75%. Peak loads could be lowered
more by spreading them across a longer distance.



T.T.Mowery                                                                                                                                                      Page 14
ME 6105: Modeling and Simulation                                                                                                                                HW2: The Model


                      tire.flange_a.f
          2.4E5




          2.2E5




          2.0E5




          1.8E5




          1.6E5




          1.4E5




          1.2E5
    [N]




          1.0E5




          8.0E4




          6.0E4




          4.0E4




          2.0E4




          0.0E0



                  0.300                   0.325                0.350                    0.375                 0.400               0.425          0.450          0.475

                                                                                    reboundChamber.reboundChamber.s_rel



                            Figure 17              Ground Load Reaction versus Shock Stroke, Fixed Orifice




   Figure 18                   Actual Ground Load Reaction versus Shock Stroke, Test Data, (Variable Orifice)

                     reboundChamber.reboundChamber.s_rel [m]           tire.flange_a.f [N]
     2.2E5




     2.0E5




     1.8E5




     1.6E5




     1.4E5




     1.2E5




     1.0E5




     8.0E4




     6.0E4




     4.0E4




     2.0E4




     0.0E0
                     0.30               0.32         0.34                 0.36                  0.38          0.40         0.42           0.44           0.46     0.48

                                                                                     reboundChamber.reboundChamber.s_rel



                          Figure 19               Ground Load Reaction versus Shock Stroke, Variable Orifice



T.T.Mowery                                                                                                                                                               Page 15
ME 6105: Modeling and Simulation                                                                                                                          HW2: The Model


  5.3 Taxi, Rough Runway
Figure 20 shows a simulation across a double bump. The top graph shows the elevation of the aircraft
mass, the second graph the bump profile, the third is the ground reaction load and the fourth is the plot of
chamber pressures. [Ref: LandingTaxiSimulations.RoughRunwayDouble10cmBump]. In this case the
rebound chamber pressures are the highest, but still within reasonable values. The peak load (170kN)
comes from the second bump.


                                 aircraft.aircraftMass.s



                   3.4
            [m ]




                   3.2



                         1.5                                 2.0                     2.5                         3.0                                3.5           4.0




                               doubleSineBump.sum1.y
            1.3



            1.2



            1.1



            1.0
                   1.5                                     2.0                     2.5                          3.0                             3.5               4.0




                                 tire.flange_a.f
                   2E5
            [N]




                   1E5




                   0E0
                         1.5                                 2.0                     2.5                         3.0                                3.5           4.0




                                 reboundChamber.reboundChamber.port_A.p   airSpring.AirChamber.port_A.p   hydChamberConstArea.hydChamber.port_A.p
              2.0E7



              1.5E7
    [Pa ]




              1.0E7



              5.0E6
                         1.5                                 2.0                     2.5                         3.0                                3.5           4.0


                                          Figure 20                Rough Runway Simulation, Double Bump, Fixed Orifice


I originally thought one would have to run many combinations of bump lengths and speeds to fully
investigate the load response over bumps. However, in defining the bump model I realized that a series of
different frequency bump forcing functions will do the same thing. That is, traversing a 25 m bump length
at 50 m/s is the same as traversing a 10 m bump length at 20 m/s --- both are 1 Hz wave functions. This
may be an interesting study for our ModelCenter work, treating the bump definition as uncertainty.

In summary, I believe these simulation cases show the model is able to investigate the behaviors of
interest to the designer, and have the correct inputs to explore design alternatives. More work could be
done on modeling the Stribeck friction, and the variable orifice function.



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

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

1a. I learned the difficulty in knowing when the model is “good enough”. I spent a lot of time tuning the
various parameters. Is that necessary? It is hard to judge when the performance is adequate for the
purpose. Generally, there is always something else that could be improved or tuned, and engineers want
it to be “correct”

1b. I also learned some lessons in configuration control. It is easy to propagate numerous models with
many slight differences. It is hard to capture that model definition in the title along for future retrieval, and



T.T.Mowery                                                                                                                                                   Page 16
ME 6105: Modeling and Simulation                                                       HW2: The Model


easy to get confused about which simulation used which inputs. I think I would put some pre-thought into
a numbering or naming convention if I was trying to do an extensive study with many options.

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

2a. I had the most trouble with initialization of the model. It was not always obvious where the initial
positions and speeds were coming from. I gradually learned which variables need to be initialized. Some
seem to matter to Modelica and others do not.

2b. I also learned that knowledge reuse is wonderful, but a dangerous double edge sword. Using library
models was a tremendous head start, and resulted in higher fidelity models than I could generate myself.
However, knowing what are the required and proper inputs, or even how the library class is structured is
still required. One cannot simply plug them in and go.

3. Increase my understanding of landing gear.

3a. I was a bit surprised to see the influence of the rebound chamber on the overall shock absorber
performance. I had previously considered the rebound chamber an “after thought” that was only there to
keep the strut from slamming back to full extension after a bounce. It actually has a more important role
in dissipating energy.




T.T.Mowery                                                                                 Page 17

				
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