Test Academy | Consider a forced single degree-of-freedom system governed by x ¨ ( t ) + 2 ζ ω n x ˙ ( t ) + ω n 2 x ( t ) = ω n 2 cos ⁡ ( ω t ) , where ζ and ω n 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 − r 2 ) 2 + (2ζr) 2 ] -1/2 , where 𝑟 = ω/ω n . The peak amplitude of this response occurs at a frequency ω = ω p . If ω d denotes the damped natural frequency of this system, which one of the following options is true? 𝜔p < 𝜔d < 𝜔n, 𝜔p = 𝜔d < 𝜔n, 𝜔d < 𝜔n = 𝜔p, 𝜔d < 𝜔n

Question

Consider a forced single degree-of-freedom system governed by $\ddot{x} (t) + 2 ζ ω_{n} \dot{x} (t) + ω_{n}^{2} x (t) = ω_{n}^{2} \cos (ω t)$ , where ζ and ω_n 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/ω_n. The peak amplitude of this response occurs at a frequency ω = ω_p. If ω_d denotes the damped natural frequency of this system, which one of the following options is true?

Options :

𝜔_p < 𝜔_d < 𝜔_n
𝜔_p = 𝜔_d < 𝜔_n
𝜔_d < 𝜔_n = 𝜔_p
𝜔_d < 𝜔_n < 𝜔_p

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