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Consider a forced single degree-of-freedom system governed by $\ddot{\chi }\left ( t \right ) + 2 \zeta \omega _{n}\dot{\chi }\left ( t \right ) +\omega _{n}^{2}\chi(t)= \omega _{n}^{2} \cos \left ( \omega t \right )$, where $\zeta$ and $\omega_{n}$ are the damping ratio and undamped natural frequency of the system, respectively, while $\omega$ is the forcing frequency. The amplitude of the forced steady state response of this system is given by $\left [ \left ( 1 - r^{2} \right ) ^{2} + \left ( 2\zeta r \right )^{2}\right ]^{-\frac{1}{2}}$, where $r =\frac{\omega }{\omega _{n}}$. The peak amplitude of this response occurs at a frequency $\omega = \omega _{p}$. If  $\omega_{d}$ denotes the damped natural frequency of this system, which one of the following options is true?

  1. $\omega _{p} < \omega _{d} < \omega _{n}$
  2. $\omega _{p} = \omega _{d} < \omega _{n}$
  3. $\omega _{d} < \omega _{n} = \omega _{p}$
  4. $\omega _{d} < \omega _{n} < \omega _{p}$
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