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GATE 2026 AE – Question 59

Structures · Structural dynamics: free and forced vibration of SDOF systems and free vibration of 2-DOF systems · 2 marks · Numerical answer

A system comprising a bar, spring and mass is shown in the figure below. The bar, having negligible mass, is made of a material having Young's modulus E = 200 GPa, cross-sectional area A = 100 mm$^2$, and length L = 100 mm. The spring stiffness k = 200 kN/mm and the mass M = 100 kg. The natural frequency of free vibration of the system is ________ rad/s (rounded off to the nearest integer).

A bar of length L, modulus E and area A is fixed at the top. A spring k hangs from its lower end, and the mass M hangs from the spring.

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Correct answer: 1000

Explanation

The bar acts as a spring of stiffness $k_b = \frac{EA}{L} = \frac{200 \times 10^9 \times 100 \times 10^{-6}}{0.1} = 2 \times 10^8$ N/m = 200 kN/mm. It is in series with the spring, so $k_{eq} = \frac{k_b k}{k_b + k} = \frac{200 \times 200}{400} = 100$ kN/mm $= 10^8$ N/m. Then $\omega_n = \sqrt{\frac{k_{eq}}{M}} = \sqrt{\frac{10^8}{100}} = 1000$ rad/s.