Topic: Theory of Machines / Vibrations
A cantilever beam of negligible weight is carrying a mass M at its free end, and is also resting on an elastic support of stiffness $k_{1}$ as shown in the figure below. If $k_{2}$ represents the bending stiffness of the beam, the natural frequency (rad/s) of the system is-
- A. $\sqrt{\frac{(k_{1}k_{2})}{M(k_{1}+k_{2})}}$
- B. $\sqrt{\frac{2(k_{1}+k_{2})}{M}}$
- C. $\sqrt{\frac{(k_{1}+k_{2})}{M}}$
- D. $\sqrt{\frac{(k_{1}-k_{2})}{M}}$
Answer: C. $\sqrt{\frac{(k_{1}+k_{2})}{M}}$
Explanation: From the equivalent system, the stiffness of the elastic support $k_{1}$ and the bending stiffness of the beam $k_{2}$ are in parallel. Therefore, equivalent stiffness $k_{eq} = k_{1} + k_{2}$. The natural frequency is given by $\omega_{n} = \sqrt{\frac{k_{eq}}{M}} = \sqrt{\frac{(k_{1}+k_{2})}{M}}$.