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Consider a forced single degree-of-freedom system governed by x¨(t) + 2ζωₙ x˙(t) + ωₙ² x(t) = ωₙ² cos(ωt), where ζ and ωₙ are the damping ratio and undamped natural frequency of the system, respectively, while ω is the forcing frequency. The amplitude of the forced steady state response of this system is given by [(1 − r²)² + (2ζr)²]−1/2, where r = ω/ωₙ. The peak amplitude of this response occurs at a frequency ω = ωₚ. If ω_d denotes the damped natural frequency of this system, which one of the following options is true?

1. ωₚ < ω_d < ωₙ
2. ωₚ = ω_d < ωₙ
3. ω_d < ωₙ = ωₚ
4. ω_d < ωₙ < ωₚ

Correct answer: ωₚ < ω_d < ωₙ

Solution

The peak amplitude of the forced response occurs at a frequency lower than the damped natural frequency, which is itself less than the undamped natural frequency. This relationship indicates that as the system is damped, the frequency at which resonance occurs shifts, confirming that ωₚ is less than ω_d, which in turn is less than ωₙ.

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