変位を$w(x) \cdot \sin\omega t$とおくと,変位の最大値は$w(x) \cdot \omega$となることから,
\[
\begin{align}
T_{max}
&=\frac{1}{2} \int_{0}^{\frac{l}{2}} \rho A \left[ W \left\{3 \frac{x}{l}-4 \left( \frac{x}{l} \right)^3 \right\} ・\omega\right]^2 dx \times 2 \\
&=\frac{1}{2} \rho A (W \omega)^2 \times 2 \int_{0}^{\frac{l}{2}} \left\{ 9 \frac{x^2}{l^2} -24 \frac{x^4}{l^4} -16 \frac{x^6}{l^6} \right\} dx \\
\end{align}
\]
\[
\begin{align}
\int_{0}^{\frac{l}{2}} \left\{ 9 \frac{x^2}{l^2} -24 \frac{x^4}{l^4} -16 \frac{x^6}{l^6} \right\} dx
&=\left[ \frac{9^3}{l^2} \cdot \frac{x^3}{3}-\frac{24}{l^4}\cdot \frac{x^5}{5}+ \frac{16}{l^6}\cdot \frac{x^7}{7} \right]_0^{\frac{l}{2}} \\
&=\frac{3}{l^2} \left( \frac{l}{2} \right)^3- \frac{24}{5 l^4} \left( \frac{l}{2} \right)^5 + \frac{16}{7l^6} \left( \frac{l}{2} \right)^7 \\
&=\frac{3 l^3}{8l^2}- \frac{24l^5}{5\cdot 32l^4}+ \frac{16 l^7}{7\cdot 128l^6} \\
&= \left( \frac{3}{8}-\frac{3}{20}+\frac{1}{56} \right)l \\
&= \left( \frac{21}{56}+\frac{1}{56}-\frac{3}{20} \right)l \\
&= \left( \frac{22}{58}-\frac{3}{20} \right)l \\
&= \left( \frac{55}{140}-\frac{21}{140} \right)l \\
&=\frac{34}{140}l \\
&=\frac{17}{70}l
\end{align}
\]
\[
\begin{align}
T_{max}
&=\frac{1}{2} \rho A (W \omega)^2 \times 2 \times \frac{17}{70}l \\
&=\frac{1}{2} \rho A (W \omega)^2 \cdot \frac{17}{35}l \\
\end{align}\]
\[
\begin{align}\frac{d^2 w}{dx^2}
&=\frac{d^2}{dx^2} \left[W\left\{ 3\cdot \frac{x}{l}-4 \left( \frac{x}{l} \right)^3 \right\}\right] \\
&=\frac{d}{dx} \left[ W \left\{ \frac{3}{l}- \frac{4}{l^3} \cdot 3x^2 \right\}\right] \\
&=W \left\{ -\frac{12}{l^3}\cdot 2x \right\} \\
&=-W\cdot \frac{24}{l^3}x
\end{align}
\]
\[
\begin{align}
U_{max}
&=\frac{1}{2} \int_{0}^{\frac{l}{2}} EI \left(-W \cdot \frac{24}{l^3}x\right)^2 dx \cdot 2 \\
&=\frac{1}{2} EI W^2\left( \frac{24}{l^3} \right)^2\times 2\int_{0}^{\frac{l}{2}} x^2dx \\
&=\frac{1}{2}EI W^2\left( \frac{24}{l^3} \right)^2 \times 2\left[ \frac{x^3}{3} \right]^{\frac{l}{2}}_0 \\
&=\frac{1}{2} EI W^2 \left( \frac{24}{l^3} \right)\times 2 \left\{ \frac{1}{3} \cdot \left( \frac{l}{2} \right)^3 \right\} \\
&=\frac{1}{2}EI W^2 \frac{24^2}{l^6}\times 2 \times \frac{l^3}{3\times 8} \\
&=\frac{1}{2} EI W^2 \cdot \frac{48}{l^3}
\end{align}
\]
$T_{max}=U_{max}$より
\[
\frac{1}{2} \rho A l W^2 \omega^2 \cdot \frac{17}{35}
=\frac{1}{2}EI W^2 \cdot \frac{48}{l^3}
\]
\[
\begin{align}
\omega^2
&= \frac{48EI \frac{1}{l^3}}{\frac{17}{35} \cdot \rho A l} \\
\therefore m_b &= \rho A l\\
\omega
&= \sqrt{\frac{48 \times 35 \times EI}{17m_b l^3}} \\
&=9.941 \sqrt{\frac{EI}{m_b l^3}}
\end{align} \\
\]
\[
\therefore
\omega = 9.94 \sqrt{\frac{EI}{m_b l^3}}
\]