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GATE 2021 ME (ME1) – Question 42

Vibrations · Forced vibration and resonance · 2 marks · Multiple choice

A tappet valve mechanism in an IC engine comprises a rocker arm ABC that is hinged at B as shown in the figure. The rocker is assumed rigid and it oscillates about the hinge B. The mass moment of inertia of the rocker about B is $10^{-4}$ kg.m². The rocker arm dimensions are $a=3.5$ cm and $b=2.5$ cm. A pushrod pushes the rocker at location A, when moved vertically by a cam that rotates at $N$ rpm. The pushrod is assumed massless and has a stiffness of 15 N/mm. At the other end C, the rocker pushes a valve against a spring of stiffness 10 N/mm. The valve is assumed massless and rigid. Resonance in the rocker system occurs when the cam shaft runs at a speed of _____ rpm (round off to the nearest integer).

Rocker ABC hinged at B, with pushrod and cam at A and valve spring at C; lever arms are a and b. See the attached source image.
  1. 496
  2. 4739
  3. 790
  4. 2369

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Show answer and explanation

Correct answer: (B) 4739

Explanation

For a small rotation θ, spring deflections are aθ,bθ, giving rotational stiffness $k_\theta=k_aa^2+k_bb^2=15000(0.035)^2+10000(0.025)^2=24.625$ Nm/rad.

$\omega_n=\sqrt{k_\theta/I}=496.236$ rad/s. Convert with $N=60\omega_n/(2\pi)=4738.71$ rpm. **Answer: 4739 rpm (B)**.