Topic: Theory of Machines
The fundamental natural frequency of the cantilever beam with point load P acting at the free end, in rad/sec is
- A. $\sqrt{\frac{3EIg}{PL^{3}}}$
- B. $\sqrt{\frac{PL^{3}}{3EIg}}$
- C. $\frac{1}{2\pi}\sqrt{\frac{3EIg}{PL^{3}}}$
- D. $\frac{1}{2\pi}\sqrt{\frac{PL^{3}}{3EIg}}$
Answer: A. $\sqrt{\frac{3EIg}{PL^{3}}}$
Explanation: The static deflection of a cantilever beam with a point load P at the free end is $\Delta = \frac{PL^{3}}{3EI}$. The natural circular frequency is given by $\omega_{n} = \sqrt{\frac{g}{\Delta}}$. Substituting the expression for deflection gives $\omega_{n} = \sqrt{\frac{g}{PL^{3}/3EI}} = \sqrt{\frac{3EIg}{PL^{3}}}$ rad/sec.