GATE 2024 AE – Question 55
The equations of motion for a two degrees of freedom undamped spring-mass system are:
$m\ddot x_1 + 2kx_1 - kx_2 = 0$
$m\ddot x_2 - kx_1 + 2kx_2 = 0$
where m and k represent mass and stiffness respectively, in corresponding SI units, and $x_1$ and $x_2$ are the degrees of freedom. The larger of the two natural frequencies is given by: $\omega = \alpha\sqrt{\frac{k}{m}}$ rad/s. The value of $\alpha$ is ________ (rounded off to 2 decimal places).
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Show answer and explanation
Correct answer: 1.72 to 1.74
Explanation
Writing $x = X e^{i\omega t}$ gives $\begin{bmatrix}2k - m\omega^2 & -k\\-k & 2k - m\omega^2\end{bmatrix}X = 0$. The determinant gives $(2k - m\omega^2)^2 = k^2$, so $m\omega^2 = k$ or $3k$. The larger frequency is $\omega = \sqrt{3}\sqrt{\frac{k}{m}}$, so $\alpha = 1.73$.