Editing Help:Determining the Moment of Inertia Tensor (section)

=== Koordinatenursprung im Massenschwerpunkt, Koordinatenachsen entlang der Symmetrieachsen ===

[[Datei:Sketch Moment of Inertia Tensor Cuboid - Coordinate System.svg|miniatur|300px|rechts|Skizze zur Problemstellung]]

$$
\begin{split}
I_{xx} &= \int r_x^2 \dd m \xlongequal{r_x^2 = y^2 + z^2} \int (y^2 + z^2) \dd m \xlongequal{\mathrm{d} m = \varrho \dd V} \int_V \varrho (y^2 + z^2) \dd V \\
&\xlongequal{\varrho \stackrel{!}{=} \text{const.}} \varrho \int_V (y^2 + z^2) \dd V \xlongequal{\mathrm{d} V = \dd x \dd y \dd z} \varrho \iiint_V (y^2 + z^2) \dd z \dd y \dd x \\
&= \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \int\limits_{-\frac{t}{2}}^{\frac{t}{2}} (y^2 + z^2) \dd z \dd y \dd x = \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \left. \left[ y^2 z + \frac{1}{3} z^3 \right] \right|_{z = -\frac{t}{2}}^{z = \frac{t}{2}} \dd y \dd x \\
&= \varrho t \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \left( y^2 + \frac{1}{12} t^2 \right) \dd y \dd x = \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \left. \left[ \frac{1}{3} y^3 + \frac{1}{12} t^2 y \right] \right|_{y = -\frac{h}{2}}^{y = \frac{h}{2}} \dd x \\
&= \frac{1}{12} \varrho h t \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} (h^2 + t^2) \dd x = \frac{1}{12} \varrho h t \left. \left[h^2 x + t^2 x \right] \right|_{x = -\frac{b}{2}}^{x = \frac{b}{2}} = \frac{1}{12} \underbrace{\varrho b h t}_{= m} (h^2 + t^2) = {\color{red} \frac{1}{12} m (h^2 + t^2)}
\end{split}
$$


$$
\begin{split}
I_{yy} &= \int r_y^2 \dd m \xlongequal{r_y^2 = x^2 + z^2} \int (x^2 + z^2) \dd m \xlongequal{\mathrm{d} m = \varrho \dd V} \int_V \varrho (x^2 + z^2) \dd V \\
&\xlongequal{\varrho \stackrel{!}{=} \text{const.}} \varrho \int_V (x^2 + z^2) \dd V \xlongequal{\mathrm{d} V = \dd x \dd y \dd z} \varrho \iiint_V (x^2 + z^2) \dd z \dd y \dd x \\
&= \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \int\limits_{-\frac{t}{2}}^{\frac{t}{2}} (x^2 + z^2) \dd z \dd y \dd x = \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \left. \left[ x^2 z + \frac{1}{3} z^3 \right] \right|_{z = -\frac{t}{2}}^{z = \frac{t}{2}} \dd y \dd x \\
&= \varrho t \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \left( x^2 + \frac{1}{12} t^2 \right) \dd y \dd x = \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \left. \left[ x^2 y + \frac{1}{12} t^2 y \right] \right|_{y = -\frac{h}{2}}^{y = \frac{h}{2}} \dd x \\
&= \varrho h t \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \left( x^2 + \frac{1}{12} t^2 \right) \dd x = \varrho h t \left. \left[ \frac{1}{3} x^3 + \frac{1}{12} t^2 x \right] \right|_{x = -\frac{b}{2}}^{x = \frac{b}{2}} = \frac{1}{12} \underbrace{\varrho b h t}_{= m} (b^2 + t^2) = {\color{red} \frac{1}{12} m (b^2 + t^2)}
\end{split}
$$


$$
\begin{split}
I_{zz} &= \int r_z^2 \dd m \xlongequal{r_z^2 = x^2 + y^2} \int (x^2 + y^2) \dd m \xlongequal{\mathrm{d} m = \varrho \dd V} \int_V \varrho (x^2 + y^2) \dd V \\
&\xlongequal{\varrho \stackrel{!}{=} \text{const.}} \varrho \int_V (x^2 + y^2) \dd V \xlongequal{\mathrm{d} V = \dd x \dd y \dd z} \varrho \iiint_V (x^2 + y^2) \dd z \dd y \dd x \\
&= \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \int\limits_{-\frac{t}{2}}^{\frac{t}{2}} (x^2 + y^2) \dd z \dd y \dd x = \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \left. \left[ x^2 z + y^2 z \right] \right|_{z = -\frac{t}{2}}^{z = \frac{t}{2}} \dd y \dd x \\
&= \varrho t \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \left( x^2 + y^2 \right) \dd y \dd x = \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \left. \left[ x^2 y + \frac{1}{3} y^3 \right] \right|_{y = -\frac{h}{2}}^{y = \frac{h}{2}} \dd x \\
&= \varrho h t \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \left( x^2 + \frac{1}{12} h^2 \right) \dd x = \varrho h t \left. \left[ \frac{1}{3} x^3 + \frac{1}{12} h^2 x \right] \right|_{x = -\frac{b}{2}}^{x = \frac{b}{2}} = \frac{1}{12} \underbrace{\varrho b h t}_{= m} (b^2 + h^2) = {\color{red} \frac{1}{12} m (b^2 + h^2)}
\end{split}
$$


$$
\begin{split}
I_{xy} = I_{yx} &= \int xy \dd m \xlongequal{\mathrm{d} m = \varrho \dd V} \int_V \varrho xy \dd V \xlongequal{\varrho \stackrel{!}{=} \text{const.}} \varrho \int_V xy \dd V \xlongequal{\mathrm{d} V = \dd x \dd y \dd z} \varrho \iiint_V xy \dd z \dd y \dd x \\
&= \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \int\limits_{-\frac{t}{2}}^{\frac{t}{2}} xy \dd z \dd y \dd x = \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \left. \left[ xyz \right] \right|_{z = -\frac{t}{2}}^{z = \frac{t}{2}} \dd y \dd x = \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} 0 \dd y \dd x = {\color{red} 0}
\end{split}
$$


$$
\begin{split}
I_{xz} = I_{zx} &= \int xz \dd m \xlongequal{\mathrm{d} m = \varrho \dd V} \int_V \varrho xz \dd V \xlongequal{\varrho \stackrel{!}{=} \text{const.}} \varrho \int_V xz \dd V \xlongequal{\mathrm{d} V = \dd x \dd y \dd z} \varrho \iiint_V xz \dd z \dd y \dd x \\
&= \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \int\limits_{-\frac{t}{2}}^{\frac{t}{2}} xz \dd z \dd y \dd x = \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \left. \left[ \frac{1}{2} x z^2 \right] \right|_{z = -\frac{t}{2}}^{z = \frac{t}{2}} \dd y \dd x = \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} 0 \dd y \dd x = {\color{red} 0}
\end{split}
$$


$$
\begin{split}
I_{yz} = I_{zy} &= \int yz \dd m \xlongequal{\mathrm{d} m = \varrho \dd V} \int_V \varrho yz \dd V \xlongequal{\varrho \stackrel{!}{=} \text{const.}} \varrho \int_V yz \dd V \xlongequal{\mathrm{d} V = \dd x \dd y \dd z} \varrho \iiint_V yz \dd z \dd y \dd x \\
&= \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \int\limits_{-\frac{t}{2}}^{\frac{t}{2}} yz \dd z \dd y \dd x = \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} \left. \left[ \frac{1}{2} y z^2 \right] \right|_{z = -\frac{t}{2}}^{z = \frac{t}{2}} \dd y \dd x = \varrho \int\limits_{-\frac{b}{2}}^{\frac{b}{2}} \int\limits_{-\frac{h}{2}}^{\frac{h}{2}} 0 \dd y \dd x = {\color{red} 0}
\end{split}
$$


$$
\color{red}
\mathbf{I} = \begin{pmatrix} \frac{1}{12} m (h^2 + t^2) & 0 & 0 \\ 0 & \frac{1}{12} m (b^2 + t^2) & 0 \\ 0 & 0 & \frac{1}{12} m (b^2 + h^2) \end{pmatrix}
$$