Wolfram言語

微分固有系

クランク軸の固有値と固有関数

固定されたクランク軸の固有値と固有関数を計算する.

Graphics3Dを指定する.

In[1]:=
Click for copyable input
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有限要素法パッケージをロードし,有限要素メッシュを生成する

In[2]:=
Click for copyable input
Needs["NDSolve`FEM`"] mesh = ToElementMesh[DiscretizeGraphics[gr], "MeshOrder" -> 1]
Out[2]=

三次元の応力演算子を指定する.

完全なWolfram言語入力を表示する
In[3]:=
Click for copyable input
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}]}

クランク軸の境界がこれらの位置で固定されたままであるような制約条件を指定する.

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

固有値と固有関数を5個求める.

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

固有値を調べる.

In[6]:=
Click for copyable input
vals
Out[6]=

有限要素法パッケージを使って,クランク軸の第5の固有関数を,動いていないときの位置と比較して可視化する.

完全なWolfram言語入力を表示する
In[7]:=
Click for copyable input
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]=

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