GATE ME 2022 Set 1 — Question 39

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MCQ+2 / -0.67MediumHarmonic Excitation ResponseForced VibrationTheory of Machines & Vibrations

Theory of Machines & Vibrations → Forced Vibration → Harmonic Excitation Response

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Question

Consider a forced single degree-of-freedom system governed by x¨(t)+2ζωnx˙(t)+ωn2x(t)=ωn2cos(ωt)\ddot{x}(t) + 2\zeta\omega_n\dot{x}(t) + \omega_n^2 x(t) = \omega_n^2 \cos(\omega t), where ζ\zeta and ωn\omega_n are the damping ratio and undamped natural frequency of the system, respectively, while ω\omega is the forcing frequency. The amplitude of the forced steady state response of this system is given by [(1r2)2+(2ζr)2]1/2[(1 - r^2)^2 + (2\zeta r)^2]^{-1/2}, where r=ω/ωnr = \omega/\omega_n. The peak amplitude of this response occurs at a frequency ω=ωp\omega = \omega_p. If ωd\omega_d denotes the damped natural frequency of this system, which one of the following options is true?
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