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

The correct option is:
𝜔_p < 𝜔_d < 𝜔_n

To understand why this relation holds true, let us analyze each of the three frequencies involved in the system: the undamped natural frequency (${\omega }_{\mathrm{n}}$), the damped natural frequency (${\omega }_{\mathrm{d}}$), and the frequency at which the peak steady-state amplitude occurs (${\omega }_{\mathrm{p}}$).

1. Damped Natural Frequency (${\omega }_{\mathrm{d}}$)
For a single degree-of-freedom system with a damping ratio $\zeta$ (where $0<\zeta <1$ for an underdamped system), the damped natural frequency is defined by the relation:
$${\omega }_{\mathrm{d}}={\omega }_{\mathrm{n}}\sqrt{1-{\zeta }^2}$$
Since $\sqrt{1-{\zeta }^2}<1$ for any non-zero damping ratio ($\zeta >0$), it is clear that:
$${\omega }_{\mathrm{d}}<{\omega }_{\mathrm{n}}$$

2. Peak Amplitude Frequency (${\omega }_{\mathrm{p}}$)
The steady-state response amplitude is given by:
$$X(r)={[{(1-r^2)}^2+{(2\zeta r)}^2]}^{-1/2}$$
To find the value of $r$ that maximizes the amplitude $X(r)$, we minimize the denominator expression inside the bracket:
$$g(r)={(1-r^2)}^2+4{\zeta }^2{r}^2$$
Taking the derivative of $g(r)$ with respect to $r$ and setting it to zero:
$$\frac{dg}{dr}=2(1-r^2)(-2r)+8{\zeta }^{2}r=0$$
Dividing by $-4r$ (assuming $r\ne 0$):
$$(1-r^2)-2{\zeta }^2=0$$
$$r^2=1-2{\zeta }^2$$
Thus, the resonance ratio $r_{\mathrm{p}}={\omega }_{\mathrm{p}}/{\omega }_{\mathrm{n}}$ at which the peak response occurs is:
$$r_{\mathrm{p}}=\sqrt{1-2{\zeta }^2}$$
Multiplying by ${\omega }_{\mathrm{n}}$ gives the peak frequency:
$${\omega }_{\mathrm{p}}={\omega }_{\mathrm{n}}\sqrt{1-2{\zeta }^2}$$

3. Comparison
Now we compare the three frequency expressions:
$${\omega }_{\mathrm{p}}={\omega }_{\mathrm{n}}\sqrt{1-2{\zeta }^2}$$
$${\omega }_{\mathrm{d}}={\omega }_{\mathrm{n}}\sqrt{1-{\zeta }^2}$$
$${\omega }_{\mathrm{n}}={\omega }_{\mathrm{n}}\sqrt{1}$$
For any non-zero, stable underdamped system ($0<\zeta <\frac{1}{\sqrt{2}}$), we have:
$$(1-2{\zeta }^2)<(1-{\zeta }^2)<1$$
Taking the square root of these terms preserves the inequality order:
$$\sqrt{1-2{\zeta }^2}<\sqrt{1-{\zeta }^2}<1$$
Therefore, scaling by the positive constant ${\omega }_{\mathrm{n}}$ yields:
$${\omega }_{\mathrm{p}}<{\omega }_{\mathrm{d}}<{\omega }_{\mathrm{n}}$$