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.m^{2}. 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).

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GATE ME 2021 Official Paper: Shift 1

Option 4 : 4739

CT 1: Ratio and Proportion

3742

10 Questions
16 Marks
30 Mins

**Explanation:**

**Given:**

(I_{rad})_{A} = 10^{-4} kg/m^{2}, S_{1} = 15 N/mm, S_{2} = 10 N/mm, a = 3.5 cm, b = 2.5 cm

**Now,**

Applying Torque method

I_{A} θ̈ + (S_{1} a^{2} + S_{2} b^{2}) θ = 0

\(\mathop \theta \limits^ . \frac{{\left( {{S_1}{a^2} + {S_2}{b^2}} \right)\theta }}{{{I_A}}} = 0\)

**Now,**

By comparing the above equation with the equation of motion,

\(\mathop \theta \limits^ . \omega _n^2\;\theta = 0\)

\({\omega _n} = \sqrt {\frac{{{S_1}{a^2} + {S_2}{b^2}}}{{{I_A}}}} \)

\({\omega _n} = \sqrt {\frac{{1.5 \times 1000 \times {{\left( {0.035} \right)}^2} + 10 \times 1000 \times {{\left( {0.025} \right)}^2}}}{{{{10}^{ - 4}}}}} \)

**ω _{n} = 496.236 rad/s**

**Now,**

\(N = \frac{{60 \times {\omega _n}}}{{2\pi }} = \frac{{60 \times 496.236}}{{2\pi }} = 4738.7\;rpm\)

**∴**** N ****≈**** 4739 rpm**