GATE ME 2017 Set 1 — Question 46
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Theory of Machines & Vibrations → Free Vibration → SDOF Free Vibration
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Question
A thin uniform rigid bar of length and mass is hinged at point O, located at a distance of from one of its ends. The bar is further supported using springs, each of stiffness , located at the two ends. A particle of mass is fixed at one end of the bar, as shown in the figure. For small rotations of the bar about O, the natural frequency of the system is
Correct answer
(B) √((5k)/(2M))
Solution
To find the natural frequency, we use the torque method. Let be a small angular displacement about the hinge O.
By parallel axis theorem: .
The particle of mass is at a distance from O.
.
Total .
The restoring torque is .
.
1.Moment of Inertia about O ():
The bar's moment of inertia about its center of mass is . The distance from the center of mass to O is .By parallel axis theorem: .
The particle of mass is at a distance from O.
.
Total .
2.Restoring Torque ():
For a small rotation , the displacement of the left spring is and the right spring is .The restoring torque is .
3.Natural Frequency ():
The equation of motion is ..
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