Undamped natural frequency
Rather than the frequency domain, let's look at this in the time domain and particularly, the characteristic equation associated with a linear homogeneous 2nd order differential equation for some system:

\$r^2 + 2 \zeta \omega_n r + \omega^2_n = 0\$.

If the roots of the characteristic equation are real (which is the case if \$\zeta \ge 1\$), the general solution is the sum of real exponentials:

\$Ae^{\sigma_1 t} + Be^{\sigma_2t} \$

where

\$\sigma_1 = -\zeta \omega_n + \sqrt{(\zeta ^2 - 1)\omega^2_n} \$

\$\sigma_2 = -\zeta \omega_n - \sqrt{(\zeta ^2 - 1)\omega^2_n} \$

Since these are real exponentials, there is no oscillation in these solutions.



If the roots are complex conjugates (which is the case if \$\zeta &lt; 1\$), the general solution is the sum of complex exponentials:

\$e^{\sigma t}(Ae^{j\omega t} + Be^{-j\omega t})\$

where

\$\sigma = -\zeta \omega_n\$

\$\omega = \sqrt{(1 - \zeta ^2)\omega^2_n}\$ 

This solution is a sinusoid with angular frequency \$\omega\$ multiplied by a real exponential.  We say the system has a "natural frequency" of \$\omega\$ for a reason that I think is obvious.

Finally, setting \$\zeta = 0\$ (an undamped system) , this solution becomes:

\$Ae^{j\omega_n t} + Be^{-j\omega_n t}\$

which is just a sinusoid of angular frequency \$\omega_n\$.

In summary, a system may or may not have an associated natural frequency.  Only systems with \$\zeta &lt; 1\$ have a natural frequency \$\omega\$ and only in the case that \$\zeta = 0\$ will the natural frequency \$\omega = \omega_n\$, the undamped natural frequency.