GATE Mechanical Engineering (ME) 2022 Shift 1 Solved Paper

Consider a forced single degree-of-freedom system governed by $\ddot{\mathit{x}}\left(\mathit{t}\right)+2\mathit{\zeta }{\mathit{\omega }}_{\mathit{n}}\dot{\mathit{x}}\left(\mathit{t}\right)+{\mathit{\omega }}_{\mathit{n}}^2\mathit{x}\left(\mathit{t}\right)={\mathit{\omega }}_{\mathit{n}}^2\cos \left(\mathit{\omega }\mathit{t}\right)$, 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/ωn. 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?