Question

Consider a cantilever beam, having negligible mass and uniform flexural rigidity, with length 0.01 m. The frequency of vibration of the beam with a 0.5 kg mass attached at the free tip, is 100 Hz. The flexural rigidity (in Nm${}^2$) of the beam is

Answer 1

$P-mg=0.5\times 9.81=4.905$ N
$L=0.01m,f_n=100$ Hz
${\omega }_n=2\pi f_n=628$ rad/s
${\omega }_n=\sqrt{\frac{k}{m}}$
or $k=m{\omega }_n^2=0.5\times (628{)}^2=197192$N/m
$\delta =\frac{P{L}^2}{3EI}$
$\Rightarrow \frac{P}{\delta }=\frac{3EI}{L^3}$
or $EI=\frac{P}{\delta }\times \frac{L^3}{3}$
$=197192\times \frac{(0.01{)}^3}{3}=0.06573N{m}^2$