Simulating Aircraft Lifetime with ReliabilityDistribution

Use system reliability modeling and random sampling to estimate airplane lifetime and failure probabilities

Suppose that in order to fly, an airplane needs both wings, one engine out of two on each wing, and the air control system operating. Model the system.

Define component lifetime distributions for the engine system:

In[1]:=

{Motor,Fuel,Cooling,Propeller}={WeibullDistribution[1.58,4830],WeibullDistribution[1.44,5130],WeibullDistribution[1.60,4190],WeibullDistribution[1.63,3740]};

Model the reliability of a single engine:

In[2]:=

ℛSingleEngine=ReliabilityDistribution[motor∧fuel∧cooling∧propeller,{{motor,Motor},{fuel,Fuel},{cooling,Cooling},{propeller,Propeller}}];

Represent two engines per wing in parallel and both wings in series, so the aircraft needs one engine on each wing operational to fly:

In[3]:=

ℛEngines=ReliabilityDistribution[(engine1∨engine2)∧(engine3∨engine4),{{engine1,ℛSingleEngine},{engine2,ℛSingleEngine},{engine3,ℛSingleEngine},{engine4,ℛSingleEngine}}];

Provide Weibull shape and scale parameters for eight air-control elements in two lists (shape and scale):

In[4]:=

parameters={{1.85,1.57,2.25,0.95,0.73,1.14,0.92,1.65},{4729.02,4718.22,5843.58,3956,2672.1,3977.39,2895.62,3994.78}};

Define lifetime distributions for air-control components:

In[5]:=

{Directional,Longitudinal,Lateral,Flaps,Trim,Hydraulic,LandingGear,Steering}=Flatten[Transpose[{MapThread[WeibullDistribution,parameters]}]];

Build a series structure where the air-control system operates only if all eight elements are functional:

In[6]:=

ℛACS=ReliabilityDistribution[x1∧x2∧x3∧x4∧x5∧x6∧x7∧x8,{{x1,Directional},{x2,Longitudinal},{x3,Lateral},{x4,Flaps},{x5,Trim},{x6,Hydraulic},{x7,LandingGear},{x8,Steering}}];

Combine all subsystems into the complete airplane model:

In[7]:=

Wing=WeibullDistribution[1.79,4250];

In[8]:=

ℛAirplane=ReliabilityDistribution[enginesystem∧wing1∧wing2∧acs,{{enginesystem,ℛEngines},{wing1,Wing},{wing2,Wing},{acs,ℛACS}}];

Calculate the mean time to failure (MTTF) for the airplane:

In[9]:=

NExpectation[t,tℛAirplane]

Out[9]=

506.447

Calculate the probability that a six-hour flight will be successful:

In[10]:=

NProbability[t≥6,tℛAirplane]

Out[10]=

0.982331

Find the probability that an airplane working after 300 hours still works after an additional 200 hours:

In[11]:=

NProbability[t≥500t≥300,tℛAirplane]

Out[11]=

0.698215

Plot the survival function:

In[12]:=

Plot[SurvivalFunction[ℛAirplane,t]//Evaluate,{t,0,2000},FillingAxis]

Out[12]=

Simulate the lifetime of the airplane:

In[13]:=

Histogram[RandomVariate[ℛAirplane,100],AxesOrigin{0,0},ChartStyleDirective[Opacity[0.6],ColorData[97][3]],PlotRangeAll,FrameTrue,FrameLabel{"Lifetime","Frequency"},Epilog{{Thick,Red,Line[{{0,NExpectation[t,tℛAirplane]},{100,NExpectation[t,tℛAirplane]}}]}}]

Out[13]=

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Contributed by: Wolfram Staff

Simulating Aircraft Lifetime with ReliabilityDistribution

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