GATE ME 2024 Set 1 — Question 55
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Theory of Machines & Vibrations → Forced Vibration → Resonance
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Question
A vibratory system consists of mass , a vertical spring of stiffness and a horizontal spring of stiffness . The end A of the horizontal spring is given a horizontal motion . The other end of the spring is connected to an inextensible rope that passes over two massless pulleys as shown. Assume kg, kN/m, and neglect friction. The magnitude of critical driving frequency for which the oscillations of mass tend to become excessively large is _________ rad/s (answer in integer).
Correct answer
29 to 31
Solution
The critical driving frequency occurs at resonance, where the driving frequency equals the natural frequency of the system.Let be the downward displacement of the mass . The pulley attached to the mass moves down by , which requires the rope to move by . Thus, the extension of the horizontal spring is . The vertical spring is stretched by .The total potential energy of the system is:The equivalent stiffness of the system is .The natural frequency is given by:Given kN/m N/m and kg:Thus, the critical driving frequency is 30 rad/s.
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