Two-Dimensional Finite-Volume Solution Method for Material Loss and Heat Transfer in an Aircraft Skid Landing Wheels Up.

Using Mathematica as a low-dimensional multi-physics design tool

George W. Carroll

Hawker Beechcraft Corporation

The Problem: Gear-up Landing

SetDirectory["~/DeskTop/MathematicaConference"];

Run["open /Applications/QuickTimePlayer.app","/Users/Joe/Desktop/MathematicaConference/gulCine.avi"];

Tire blowout - > No hydraulics, no flaps, gear up,
 Little fuel, tanks ruptured, only a small fire
Everyone safe

Design for Gear-Up Landing

Land aircraft without puncturing fuel tanks

Absorb impact

Dissipate kinetic energy

Skid for 17000 kg business jet, side view;
Heavy, more accurate design could reduce weight.

Highly Coupled, Multiphysics

Coupling Key

Yield Strength is function of temperature: σ = f(T)
Gouging & friction are function of yield strength

Skid – Pavement Interface
( not the classical smooth wear surface )

Other areas that deal with non - smooth abrasion

Skidding car tire

Farm tractor on soil

Physics of interface between skid and runway

•	FrictionForce = μ (Pressure) (Area)
•	Energy = FrictionForce ( Distance)

•	μ = (Subscript[C, μ]) (Yield Strength)

•        Yield Strength = f [ T ]
•	 T = f[ Energy ]
•        WearVolume = K (Pressure) (Distance) / (Hardness)
•        Hardness = f [Yield Strength]

•        Wear is in a straight line ( it fits the runway )

•        Fraction of energy entering skid = f [wear] 
	     Examples
	     -- Hard disk brake = 1
	     -- Soft, color crayon = 0

Aircraft Dynamics

Solution Method

Energy Equation

ρ (Subscript[C, p ]) ( ((∂ T)/(∂ t))+ v . ∇T) = k(∇^2)T + q

Momentum Equation

m (dv/dt) =(Subscript[F, x])

Finite-Volume Method with Moving Grid to Account for Wear

Manipulate[Graphics[Table[{Line[{{0,4-(1-a) j},{5,4-j}}],Line[{{i,4a(5-i)/5},{i,4}}]},{j,0,4},{i,0,5}]],{a,0.,1.}]

Regular grid for energy equation on the skid
Time stepping is implicit for the energy equation and explicit for aircraft dynamics
Central differencing is used in space and Crank-Nicolson in time

Mathematica built-in functions were used for array manipuation and linear equation solutions

SparseArray function makes neat code

m2d = SparseArray[
   {Band[{2, 1}] -> bndYm,
    Band[{1, 2}] -> bndYp,
    Band[{1, nj + 1}] -> bndX,
    Band[{nj + 1, 1}] -> bndX}];
coefMat = SparseArray[ Band[{1, 1}] -> (Total[-m2d] + Flatten[sp])] + m2d

LinearSolve is fast

Partition[LinearSolve[coefMat, Flatten[su]], nj]

Functional paradigm makes for good code organization

oneStep[tT_List] := Module[{ a, b}, ... ....

sol = Nest[oneStep, tT, 200];

Calibration of Model

Test data is required to calibrate empirical models

Test Data, temperature on a sample skid a few centimeters above the bottom

Plot[x^2,{x,.1,.5},Ticks->None,Filling->Axis,ImageSize->Large,
AxesLabel->{"time","temperature"},BaseStyle->{FontWeight->"Plain",FontSize->14,FontFamily->"Helvetica"}]

Coefficients Calibrated by Skid Test

•	 K       Coefficient of Wear
•	 Cμ         Friction factor proportionality constant
•	 Cq         Fraction of energy conducted into skid

Example Aircraft Landing

•	10,000 kg , small business jet
•	 70 m/s (136 knots)
•	 30°C day
•	 Skid: 1.5 m x 0.2 m x 0.1 m

Temperature Distribution in Skid at End of Test

frames = Import["skidResults.avi",{"Frames",Range[28]}];
Manipulate[Show[frames[[i]],ImageSize->{950,150}],{i,1,28,1}]

Landing Dynamics

Friction

Manipulate[ListLinePlot[solPlt[[i,6]],Filling->Axis,Ticks->{None,Automatic},
PlotRange->{0,.6},AxesLabel->{"meters","(Subscript[C, f])"},PlotLabel->"Friction Coefficient along Skid",BaseStyle->{FontWeight->"Plain",FontSize->14,FontFamily->"Helvetica"},ImageSize->Large],{i,1,28,1}]

Pressure

Manipulate[ListLinePlot[solPlt[[i,7]],Filling->Axis,Ticks->{None,Automatic},
PlotRange->{0,800000},AxesLabel->{"meters"," Pascals"},PlotLabel->"Pressure along Skid",BaseStyle->{FontWeight->"Plain",FontSize->14,FontFamily->"Helvetica"},ImageSize->Large],{i,1,28,1}]

Closure

The two-dimensional skid model written in Mathematica is useful for:
  •            understanding skid-test-data physics
  •            making parameter studies for aircraft skids
  •            designing skid tests

Mathematica is a great tool for writing specialized multiphysics models