I just discovered the reason for the 2pi and I was searching to be sure. This is my answer to why the 2pi is there.
A wave frequency f is the reciprocal of the wave period of oscillation T, where...
f = 1/T or T = 1/f
Frequencies f are often confused with angular frequencies w, where...
w=2*pi*f or f=w/(2*pi) or w=2*pi/T or T=2*pi/w
A system natural frequency is commonly written w_n "omega sub-n", where...
w_n = 2*pi*f_n where f_n is the frequency of the natural undamped oscillation of the system.
The time constant tau makes the negative exponential decay part of a system response take the form...
exp(-t/tau)
A stable underdamped system response to an excitation may take a form containing a negative exponential decay function within, such as...
y(t) = A*exp(zeta*w_n*t)*sin(Bt+C)
where zeta is the damping ratio,
zeta = (critically damped time constant)/(underdamped time constant)
critically damped time constant, tau_cd = 1/(w_n)
underdamped time constant, tau_ud = 1/(zeta*w_n)
overdamped --> zeta>1 (no oscillation)
underdamped --> zeta<1
critically damped --> zeta = 1
The time constant and the period of oscilation are two different things.
Underdamped systems have a natural frequency and a damped natural frequency w_d.
w_d = (w_n)*(sqrt(1-zeta^2)), where 0 <= zeta <= 1
The frequency of oscillation is effected by the damping ratio.
So, the point I am trying to make is that the "natural frequency" of a system is not a frequency. It is 2*pi*f.
The frequency of oscillation of an underdamped signal is something like
f_d = w_d/(2*pi)
And the period of oscillation of an underdamped signal is something like
T_d = (2*pi)/w_d
I hope this helps.