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GATE 2025 ME – Question 42

Vibrations · Free vibration of single degree of freedom systems, damping · 2 marks · Multiple choice

The system shown in the figure below consists of a cantilever beam (with flexural rigidity $EI$ and negligible mass), a spring (with spring constant $K$ and negligible mass) and a block of mass $m$. Assuming a lumped parameter model for the system, the fundamental natural frequency ($\omega_n$) of the system is

A cantilever beam of length $L$ fixed at the wall O. At the free end the block of mass $m$ hangs from the beam, and a spring of stiffness $K$ connects the free end to the ceiling.
  1. $\sqrt{\dfrac{\frac{3EI}{L^3} + K}{m}}$
  2. $\sqrt{\dfrac{\frac{EI}{L^3} + K}{m}}$
  3. $\sqrt{\dfrac{\frac{3EI}{L^3} + K}{2m}}$
  4. $\sqrt{\dfrac{\frac{EI}{L^3} + K}{2m}}$

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Correct answer: (A) $\sqrt{\dfrac{\frac{3EI}{L^3} + K}{m}}$

Explanation

The beam acts as a spring with stiffness $\frac{3EI}{L^3}$ at its tip. The spring $K$ acts in parallel with the beam at the same point. So the equivalent stiffness is $\frac{3EI}{L^3} + K$, and the frequency is $\sqrt{\frac{k_{eq}}{m}}$.