What's the definition of the undamped natural frequency? I've looked and I cannot find it.
A second order system has the following transfer function
$$
H(s) = \frac{A_o \omega_n^2}{s^2 + 2 \zeta \omega_n s + \omega_n^2}
$$
where \$\zeta\$ and \$\omega_n\$ are the damping coefficients and the undamped natural frequency respectively.
Let's suppose I have a transfer function as follows
$$
T(s) = \frac{200}{s^2 + 10s – 50}
$$
What would the undamped natural frequency be? I don't know if it'd be \$\sqrt{50}\$ or \$j\sqrt{50}\$ or something else… If \$\omega_n\$ were complex, this would also imply that \$\zeta\$ be complex. Is it even possible to have a complex valued damping coefficient? Also, since it's \$\omega_n^2\$, does this imply that there are always two undamped natural frequencies? A positive and a negative one?
Best Answer
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 < 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 < 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.