GATE ME 2025 Set 1 — Question 35
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Theory of Machines & Vibrations → Free Vibration → Damped Free Vibration
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Question
A block of mass 1 kg connected to a spring of stiffness 10 N m⁻¹ is operating in a viscous medium such that the damping ratio (or damping factor) is equal to the ratio of the damped frequency to the natural frequency. The magnitude of the damping ratio for this system is __________ (rounded off to 2 decimal places).
Correct answer
0.7 to 0.72
Solution
Let be the damping ratio, be the damped frequency, and be the natural frequency.From the problem statement, we are given the condition:The standard formula for the damped frequency of a system is:Substitute this formula into the given condition:Cancel out from the numerator and denominator:To solve for , we square both sides of the equation:Rearrange the terms to solve for :Take the square root of both sides to find . Since the damping ratio is a positive quantity, we take the positive root:Now, we calculate the numerical value:The question asks to round the answer off to 2 decimal places.The values of mass ( kg) and stiffness () are extra information and not required for the calculation, as the relationship is defined in terms of frequency ratios.
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