GATE ME 2022 Set 1 — Question 39
Go beyond PYQs with Success TrackerAI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply.MCQ+2 / -0.67MediumHarmonic Excitation ResponseForced VibrationTheory of Machines & Vibrations
Theory of Machines & Vibrations → Forced Vibration → Harmonic Excitation Response
Last updated
Question
Consider a forced single degree-of-freedom system governed by , where and are the damping ratio and undamped natural frequency of the system, respectively, while is the forcing frequency. The amplitude of the forced steady state response of this system is given by , where . The peak amplitude of this response occurs at a frequency . If denotes the damped natural frequency of this system, which one of the following options is true?
Correct answer
(A) ωₚ < ω_d < ωₙ
Solution
The equation of motion is given as . The steady-state amplitude is given by the magnification factor formula:where .To find the frequency at which the peak amplitude occurs, we minimize the term inside the square root with respect to :For , , so .
Thus, the resonant frequency is .The damped natural frequency is defined as .Comparing the three frequencies for a damped system ():
Therefore, .
Thus, the resonant frequency is .The damped natural frequency is defined as .Comparing the three frequencies for a damped system ():
1.
2.
3.
Since , it follows that .Therefore, .
Continue learning with Success Tracker
A step still unclear? Work through it with support
Use Success Tracker to ask about the reasoning, then try another GATE ME question to check your understanding.
AI-powered practice· Unlimited practice on eligible plans
- PYQs with solutions
- Attempt available previous-year questions, then compare your reasoning with the worked solution. Coverage varies by stream.
- Practice that adapts
- Choose a topic, work on weaker areas and bookmark questions to revisit. Your attempts feed your progress tracking.
- AI doubt support
- Ask follow-up questions about a step or concept while practising, instead of stopping at the final answer.
Unlimited practice is available on eligible plans. Free practice and AI usage have limits; check the current plan allowances before choosing.
This page stays readable without an account. AI responses can be wrong; check them against the solution and source material.
More questions on Forced Vibration
2026 Set 1 Q17A gear train with five gears is shown in the figure below. The number of teeth on each gear is…2026 Set 1 Q18In a single degree of freedom vibrating system with only viscous damping, the critical damping…2026 Set 1 Q20The rotor of an aeroplane engine has a mass moment of inertia . The engine…2026 Set 1 Q22Worm gearsets are used for transmitting rotary motion between2026 Set 1 Q42A 5 kg mass is suspended at the free end of an overhanging massless beam, having a pin support and…