GATE ME 2020 Set 1 — Question 41
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Theory of Machines & Vibrations → Free Vibration → Undamped Free Vibration
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Question
A rigid mass-less rod of length is connected to a disc (pulley) of mass and radius through a friction-less revolute joint. The other end of that rod is attached to a wall through a friction-less hinge. A spring of stiffness is attached to the rod at its mid-span. An inextensible rope passes over half the disc periphery and is securely tied to a spring of stiffness at point C as shown in the figure. There is no slip between the rope and the pulley. The system is in static equilibrium in the configuration shown in the figure and the rope is always taut.
Neglecting the influence of gravity, the natural frequency of the system for small amplitude vibration is
Correct answer
(C) √(3) √((k)/(m))
Solution
Using the energy method for small angular displacement :
.
1.Potential Energy ():
- Stretch in spring at mid-span: . .
- Stretch in spring at point C: If the rod rotates by , the center of the pulley moves by . Due to the rope being fixed at the wall and passing over the pulley, the end at C moves by . .
- Total .
- Velocity of pulley center: .
- Angular velocity of pulley : Since there is no slip and the rope is fixed at the wall, .
- .
.
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