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GATE 2017 ME (ME1) – Question 56

Vibrations · Free vibration of single degree of freedom systems, damping · 2 marks · Multiple choice

A thin uniform rigid bar of length L and mass M is hinged at O, L/3 from its left end. Springs of stiffness k support both ends. A point mass $m=M/4$ is attached at the right end, as shown. For small angular oscillations about O, the natural angular frequency is

Diagram for GATE 2017 ME (ME1) question 56
  1. $\sqrt{5k/M}$
  2. $\sqrt{5k/(2M)}$
  3. $\sqrt{3k/(2M)}$
  4. $\sqrt{3k/M}$

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Correct answer: (B) $\sqrt{5k/(2M)}$

Explanation

The rotational spring stiffness is $K_\theta=k(L/3)^2+k(2L/3)^2=5kL^2/9$. Bar inertia about O is $ML^2/12+M(L/6)^2=ML^2/9$. The end mass adds $(M/4)(2L/3)^2=ML^2/9$. Thus total inertia is $2ML^2/9$ and $\omega_n=\sqrt{K_\theta/I_O}=\sqrt{5k/(2M)}$, B.

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