47 lines
2.5 KiB
TeX
47 lines
2.5 KiB
TeX
\appendix
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\section{Calculations}
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\label{appendix:calculations}
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\paragraph{Moment of inertia of rectangle}
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\begin{equation}
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\label{eq:rect_moment_long}
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\begin{split}
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I_{\text{rect}} & =
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\rho \int_{-\frac{h}{2}}^{\frac{h}{2}} \int_{-\frac{w}{2}}^{\frac{w}{2}} x^2 + y^2 \diff x \diff y \\
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& = 4 \rho \int_{0}^{\frac{h}{2}} \int_{0}^{\frac{w}{2}} x^2 + y^2 \diff x \diff y \\
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& = 4 \rho \int_{0}^{\frac{h}{2}} \Biggl[\frac{1}{3} x^3 + x y^2 \Biggr]_{0}^{\frac{w}{2}} dy \\
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& = 4 \rho \int_{0}^{\frac{h}{2}} \frac{1}{3} \frac{w^3}{8} + \frac{w}{2} y^2 dy \\
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& = 2 \rho \int_{0}^{\frac{h}{2}} \frac{w^3}{12} + w y^2 dy \\
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& = 2 \rho \Biggl[ \frac{w^3}{12} y + \frac{w}{3} y^3 \Biggr]_{0}^{\frac{h}{2}} \\
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& = 2 \rho \frac{w}{3} \Biggl[ \frac{w^2}{4}y + y^3 \Biggr]_{0}^{\frac{h}{2}} \\
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& = 2 \rho \frac{w}{3} \left( \frac{w^2}{4}\frac{h}{2} + \frac{h^3}{8} \right) \\
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& = \rho \frac{w}{3} \left( \frac{w^2}{4}h + \frac{h^3}{4} \right) \\
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& = \frac{\rho wh}{12} \left( w^2 + h^3 \right) \\
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\end{split}
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\end{equation}
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\newpage
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\paragraph{Moment of inertia of sub-triangle of regular polygon}
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Before starting the calculations, it is to be noted that according to Figure
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\ref{fig:subtriangle}, we have that
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$$ \tan\left(\frac{\theta}{2}\right) = \frac{\frac{l}{2}}{h} = \frac{l}{2h} $$
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it will be useful to simplify the result of the integral.
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\begin{equation}
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\label{eq:subtriangle_moment_long}
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\begin{split}
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I_{T} &= \rho \int_0^h\int_{-\frac{lx}{2h}}^{\frac{lx}{2h}}x^2 + y^2 \diff y\diff x\\
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&= 2\rho \int_0^h\int_0^{\frac{lx}{2h}} x^2 + y^2 \diff y\diff x\\
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&= 2\rho \int_0^h \Biggl[x^2y + \frac{1}{3} y^3\Biggr]_0^{\frac{lx}{2h}} \diff x\\
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&= 2\rho \int_0^h x^2 \frac{lx}{2h} + \frac{1}{3} \frac{l^3x^3}{8h^3} \diff x\\
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&= 2\rho \left(\frac{l}{2h} + \frac{l^3}{24h^3}\right) \int_0^h x^3 \diff x\\
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&= 2\rho \left(\frac{l}{2h} + \frac{l^3}{24h^3}\right) \Biggl[ \frac{1}{4} x^4\Biggr]_0^h \\
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&= \frac{h^4\rho}{2} \left(\frac{l}{2h} + \frac{l^3}{24h^3}\right) \\
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&= \frac{\rho lh^3}{4} \left(1 + \frac{l^2}{12h^2}\right) \\
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&= \frac{m_T h^2}{2} \left(1 + \frac{l^2}{12h^2}\right) \\
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&= \frac{m_T}{2} \frac{l^2}{4 \tan^2\left(\frac{\theta}{2}\right)}\left(1 + \frac{4 \tan^2\left(\frac{\theta}{2}\right)}{12}\right) \\
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&= \frac{m_Tl^2}{24} \left(1 + 3\cot^2\left(\frac{\theta}{2}\right)\right)
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\end{split}
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\end{equation}
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