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GATE 2024 CE (CE2) – Question 37

Structural Analysis · Deflections of determinate beams, frames and trusses · 2 marks · Multiple choice

A linearly elastic beam of length $2l$ with flexural rigidity $EI$ has negligible mass. A massless spring with a spring constant $k$ and a rigid block of mass $m$ are attached to the beam as shown in the figure.

[Figure: a simply supported beam of span 2l. At mid-span a spring of stiffness k attached to the ceiling pulls on the beam and the mass m hangs from the same point.]

The natural frequency of this system is

Diagram for GATE 2024 CE (CE2) question 37
  1. $\sqrt{\frac{kl^3 + 6EI}{ml^3}}$
  2. $\sqrt{\frac{kl^3 + 48EI}{ml^3}}$
  3. $\sqrt{\frac{6EIk}{(kl^3 + 6EI)m}}$
  4. $\sqrt{\frac{48EIk}{(kl^3 + 48EI)m}}$

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Show answer and explanation

Correct answer: (A) $\sqrt{\frac{kl^3 + 6EI}{ml^3}}$

Explanation

The mid-span stiffness of a simply supported beam of span $2l$ is $\frac{48EI}{(2l)^3} = \frac{6EI}{l^3}$. The spring acts at the same point and in parallel with the beam (both resist the displacement of the mass), so the total stiffness is $k + \frac{6EI}{l^3}$. The natural frequency is $\omega = \sqrt{\frac{k + 6EI/l^3}{m}} = \sqrt{\frac{kl^3 + 6EI}{ml^3}}$.