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GATE 2016 ME (ME1) – Question 47

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

A solid disc of radius a and mass M is connected to a spring at a point d above its centre; the other spring end is fixed to a wall. The disc rolls without slipping. The spring constant is K and the disc polar moment is $J=Ma^2/2$. The natural frequency is ______ rad/s.

Diagram for GATE 2016 ME (ME1) question 47
  1. $\sqrt{2K(a+d)^2/(3Ma^2)}$
  2. $\sqrt{2K/(3M)}$
  3. $\sqrt{2K(a+d)^2/(Ma^2)}$
  4. $\sqrt{K(a+d)^2/(Ma^2)}$

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Show answer and explanation

Correct answer: (A) $\sqrt{2K(a+d)^2/(3Ma^2)}$

Explanation

For a small disc rotation θ, centre translation is aθ and the spring attachment moves(a+d)θ. Effective rotational inertia about the ground contact is J+Ma²=3Ma²/2. Spring rotational stiffness is K(a+d)². Thus ωn=sqrt[2K(a+d)²/(3Ma²)], **A**, assuming small oscillations and no slip.