Wolfram Language™

A Crankshaft's Eigenvalues and Eigenfunctions

Compute the eigenvalues and eigenfunctions of a clamped crankshaft.

Specify a Graphics3D.

In[1]:=
```gr = \!\(\* Graphics3DBox[ {EdgeForm[None], FaceForm[GrayLevel[0.5]], GraphicsComplex3DBox[CompressedData[" 1:eJx0fXVc1973/0CRsMEuJiiCiqKIIgpMVMAGDFBRJ3Z310AxsLDFnIHYHaio swPsVtRZ2IEFGPCT87x7vTe/n9/rD1+P19PDdnd37+lzbsWIISG9zDmOsxzC cXk47VNSyP3XjCsi4HcLJfff6u/sjHhss1xcdGY4x3321eFyZnrT3G8lO8cl 9//58z99c3/zS140yf2WcnIcCS+oeuvphWyGa/Q7+KDc32Lixxm53/LsMU1y v5Uqh0Lo94r7hEsja/vROIZ4EC553iCc95jvT78Pj/Cj+3o3H0Z/33UEXVde ubGxAe/7qAXhfUYIelzd1a8lXe/0ClyvWFgE4VM6exB9WlgA4YecCRdPqLVo PPMeE718cHN3+v9TtevQt8ThuUreHUl0xfoGE75sehv6fdduFP1/5SGEq+en BelxYZdAOLe9tBGfa0G4uGdILZrXoWcnEr56dyjN75hBBlzydwvL/a1GjnDV 41xJK8KFGx6N6Lf2udrejf4uYIpgwHcXcKf5c6/vp8fFt5b0vEI/Ez2tE0ll 9MLLmvR8G+6NoHEHJ9HzSMfKVqbf+wTMT4G3NC9S2D66P98fuOhwj+jVuw0c 9PdVgyV630J0RWc9LhctTjjn7FzJgB8MoXWpfGqoXQfjLDiF6NXssKb6cUr5 3/vQ+EMqNNHj6sh59Jy8wwgDvXLgEs2L1Eim6/FPF0s0L0H1+dzfYgcPrM+T wLmtC5yIzn4O4eLt44TLvisr0ngOtalK9Iu6zqb7xXvXpuftWdyB/v/8dMKl hGV16LmOZLrQczvMBv3hcKLnf0fVo/F9dw6l+fT+RPMrhifW0ePK8iIV6XvU 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In[2]:=
```Needs["NDSolve`FEM`"] mesh = ToElementMesh[DiscretizeGraphics[gr], "MeshOrder" -> 1]```
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Specify the three-dimensional stress operator.

show complete Wolfram Language input
In[3]:=
```stressOperator[ Y_, \[Nu]_] := {Inactive[ Div][({{0, 0, -((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu])))}, {0, 0, 0}, {-(Y/(2 (1 + \[Nu]))), 0, 0}}.Inactive[Grad][ w[x, y, z], {x, y, z}]), {x, y, z}] + Inactive[ Div][({{0, -((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu]))), 0}, {-(Y/(2 (1 + \[Nu]))), 0, 0}, {0, 0, 0}}.Inactive[Grad][ v[x, y, z], {x, y, z}]), {x, y, z}] + Inactive[ Div][({{-((Y (1 - \[Nu]))/((1 - 2 \[Nu]) (1 + \[Nu]))), 0, 0}, {0, -(Y/(2 (1 + \[Nu]))), 0}, {0, 0, -(Y/(2 (1 + \[Nu])))}}.Inactive[Grad][ u[x, y, z], {x, y, z}]), {x, y, z}], Inactive[Div][({{0, 0, 0}, {0, 0, -((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu])))}, {0, -(Y/( 2 (1 + \[Nu]))), 0}}.Inactive[Grad][ w[x, y, z], {x, y, z}]), {x, y, z}] + Inactive[ Div][({{0, -(Y/(2 (1 + \[Nu]))), 0}, {-((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu]))), 0, 0}, {0, 0, 0}}.Inactive[Grad][u[x, y, z], {x, y, z}]), {x, y, z}] + Inactive[ Div][({{-(Y/(2 (1 + \[Nu]))), 0, 0}, {0, -((Y (1 - \[Nu]))/((1 - 2 \[Nu]) (1 + \[Nu]))), 0}, {0, 0, -(Y/(2 (1 + \[Nu])))}}.Inactive[Grad][ v[x, y, z], {x, y, z}]), {x, y, z}], Inactive[Div][({{0, 0, 0}, {0, 0, -(Y/(2 (1 + \[Nu])))}, {0, -(( Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu]))), 0}}.Inactive[Grad][ v[x, y, z], {x, y, z}]), {x, y, z}] + Inactive[ Div][({{0, 0, -(Y/(2 (1 + \[Nu])))}, {0, 0, 0}, {-((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu]))), 0, 0}}.Inactive[Grad][u[x, y, z], {x, y, z}]), {x, y, z}] + Inactive[ Div][({{-(Y/(2 (1 + \[Nu]))), 0, 0}, {0, -(Y/(2 (1 + \[Nu]))), 0}, {0, 0, -(( Y (1 - \[Nu]))/((1 - 2 \[Nu]) (1 + \[Nu])))}}.Inactive[Grad][ w[x, y, z], {x, y, z}]), {x, y, z}]}```

Specify constraints such that the boundary of the crankshaft is held fixed at these positions.

In[4]:=
```Subscript[\[CapitalGamma], D] = DirichletCondition[#, {y <= 0.1, y >= 890}] & /@ {u[x, y, z] == 0., v[x, y, z] == 0., w[x, y, z] == 0.};```

Find five eigenvalues and eigenfunctions.

In[5]:=
```{vals, funs} = NDEigensystem[ Join[stressOperator[10^9, 33/100], Subscript[\[CapitalGamma], D]], {u, v, w}, {x, y, z} \[Element] mesh, 5];```

Inspect the eigenvalues.

In[6]:=
`vals`
Out[6]=

Visualize the fifth eigenfunction of the crankshaft compared to its position at rest using the finite element package.

show complete Wolfram Language input
In[7]:=
```Show[ ToBoundaryMesh[mesh][ "Wireframe"[ "MeshElementStyle" -> {Directive[Opacity[0.15], EdgeForm[], FaceForm[Gray]]}]], ElementMeshDeformation[mesh, funs[[5]], "ScalingFactor" -> 100000][ "Wireframe"[AspectRatio -> Automatic, "MeshElementStyle" -> Directive[EdgeForm[], FaceForm[Gray]]]], ImageSize -> 600, ViewPoint -> Left ]```
Out[7]=