Discussion
Consider a forced single degree-of-freedom system governed by
\[ \begin{aligned} &\ddot{x}(t) + 2\zeta\omega_{n}\dot{x}(t) + \omega_{n}^{2}x(t) \\ &= \omega_{n}^{2}\cos(\omega t), \end{aligned} \]
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 \([(1 - r^{2})^{2} + (2\zeta r)^{2}]^{-1/2}\), where \(r = \omega/\omega_{n}\). 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?
Answer: A
The correct order is \(\omega_{p} \lt \omega_{d} \lt \omega_{n}\).
Minimising \((1 - r^{2})^{2} + (2\zeta r)^{2}\) with respect to \(r^{2}\) gives the peak of the amplitude curve:
\[ \begin{aligned} r_{p}^{2} &= 1 - 2\zeta^{2} \Rightarrow \omega_{p} = \omega_{n}\sqrt{1 - 2\zeta^{2}} \\ \omega_{d} &= \omega_{n}\sqrt{1 - \zeta^{2}} \end{aligned} \]
For any damping \(0 \lt \zeta \lt 1/\sqrt{2}\), \(1 - 2\zeta^{2} \lt 1 - \zeta^{2} \lt 1\), so the peak lies below the damped natural frequency, which lies below the undamped one.
\(\omega_{p} = \omega_{d}\) is a common confusion: the free vibration oscillates at \(\omega_{d}\), but the forced amplitude peaks lower. The peak equals \(\omega_{n}\) only with no damping and never exceeds it, which rules out the last two options.
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