This is the standardized form and something significant has been achieved here. We now have a single angular frequency, \$\omega_{_0}\$, and a new dimensionless damping factor, \$\zeta\$. (There is something else called a damping value, which has units, so be cautions when reading and don't be confused. Older books may often use the damping value. Newer books have tended to standardize on using the damping factor.) If \$\zeta=1\$ then the system is critically damped. If \$\zeta \gt 1\$ then the system is over-damped. If \$\zeta\lt 1\$ then the system is under-damped.
[If you want to study the filter behavior without getting bogged down by the angular frequency itself, just set \$\omega_{_0}=1\:\frac{\text{rad}}{\text{s}}\$. The equation is a lot simpler now, \$H\left(s\right)=\frac{1}{s^2+2\,\zeta\,s+1}\$ and there's only one parameter to play with, \$\zeta\$. The amplitude is then: \$\frac{e_\text{out}}{e_\text{in}}=-20\operatorname{log}_{10}\left(\sqrt{\omega^4+\left(4\,\zeta^2-2\right)\omega^2+1}\right)\$. If you now plug in \$\omega=1\:\frac{\text{rad}}{\text{s}}\$ you'll find that the under-damped peak value is \$\frac{e_\text{out}}{e_\text{in}}=-20\operatorname{log}_{10}\left(2\,\zeta\right)\$. The phase response is also now just \$\phi=\operatorname{tan}^{-1}\left(\frac{2\,\zeta\,\omega}{1-\omega^2}\right)\$. So these two things make it very easy to study the amplitude and phase response in this special case. But once you know the case where \$\omega_{_0}=1\:\frac{\text{rad}}{\text{s}}\$, you have already studied every possible other value for \$\omega_{_0}\$. All you have to do is "re-normalize" things to different values of \$\omega_{_0}\$. What you learned from \$\omega_{_0}=1\:\frac{\text{rad}}{\text{s}}\$ covers every other case. So just keep it simple while studying.]
Note 1: There is something else called a damping value (or in Sallen
and Key's writings it is called a dissipation factor), which has
units. It is designated by the letter \$d\$ and not \$\zeta\$. So be
cautious when reading. Earlier writing will often use the damping value
or dissipation factor, \$d\$. This will be particularly true for earlier
authors who are familiar with Sallen and Key's, "A Practical Method of
Designing RC Active Filters", Technical Report No. 50 (now unclassified.)
Newer books have evolved to use the above approach and therefore have now
tended to standardize on using the unitless damping factor, \$\zeta\$, together with \$\omega_{_0}\$, instead.
Note 2: If you want to study the filter behavior without getting
bogged down by the angular frequency itself, just set
\$\omega_{_0}=1\:\frac{\text{rad}}{\text{s}}\$. The equation is a lot
simpler now, \$H\left(s\right)=\frac{1}{s^2+2\,\zeta\,s+1}\$ and
there's only one parameter to play with, \$\zeta\$. The amplitude is
then:
\$\frac{e_\text{out}}{e_\text{in}}=-20\operatorname{log}_{10}\left(\sqrt{\omega^4+\left(4\,\zeta^2-2\right)\omega^2+1}\right)\$
If you plug in \$\omega=1\:\frac{\text{rad}}{\text{s}}\$ you'll
find that the under-damped peak value is
\$\frac{e_\text{out}}{e_\text{in}}=-20\operatorname{log}_{10}\left(2\,\zeta\right)\$.
The phase response is also now just
\$\phi=\operatorname{tan}^{-1}\left(\frac{2\,\zeta\,\omega}{1-\omega^2}\right)\$.
So these two things make it very easy to study the amplitude and phase
response in this special case. But once you know the case where
\$\omega_{_0}=1\:\frac{\text{rad}}{\text{s}}\$, you have already
studied every possible other value for \$\omega_{_0}\$. All you have
to do is "re-normalize" things to different values of \$\omega_{_0}\$.
What you learned from \$\omega_{_0}=1\:\frac{\text{rad}}{\text{s}}\$
covers every other case. So just keep it simple while studying.
