Quality Factor in damped oscillation

[Mattofix]

Working through my lecture summaries, I have been given that Q (the quality factor) =\frac{2\pi}{(\Delta E/E)cycle}

and accepted this as a statement, taking \((\Delta E/E)cycle} to mean the 'energy loss per cycle'.

The notes carry on to say

'The frequency \widetilde{\omega} of under(damped) oscillator as function of the frequency \omega_{0} and the Q factor:

\widetilde{\omega} = \omega_{0}\sqrt{1 - (\frac{b}{2m\omega_{0})^{2}}} = \omega_{0}\sqrt{1 - \frac{1}{4Q^{2}}

My problem being that I cannot prove that \frac{b}{2m\omega_{0}} = \frac{1}{4Q^{2}}

Knowing that E = E_{0}exp^{-bt/m} i tried finding the energy loss per cycle by finding the difference between the energy at time t and the energy at time t + T (where T is the time period) but just ened up with an unhelpfull equation.

 

[Mattofix]

any help would be much appreciated so i can get rid of this irritating missing link.

 

[Mattofix]

ok - i appreciate that <br /> \((\Delta E/E)cycle}<br /> means energy loss per cycle divided by energy stored - where energy stored would be <br /> E = E_{0}exp^{-bt/m}<br />

but i still cannot prove it

 

[rbj]

Working through my lecture summaries, I have been given that Q (the quality factor) =\frac{2\pi}{(\Delta E/E)cycle}

and accepted this as a statement, taking \((\Delta E/E)cycle} to mean the 'energy loss per cycle'.

The notes carry on to say

'The frequency \widetilde{\omega} of under(damped) oscillator as function of the frequency \omega_{0} and the Q factor:

\widetilde{\omega} = \omega_{0}\sqrt{1 - (\frac{b}{2m\omega_{0})^{2}}} = \omega_{0}\sqrt{1 - \frac{1}{4Q^{2}}

My problem being that I cannot prove that \frac{b}{2m\omega_{0}} = \frac{1}{4Q^{2}}

Knowing that E = E_{0}exp^{-bt/m} i tried finding the energy loss per cycle by finding the difference between the energy at time t and the energy at time t + T (where T is the time period) but just ened up with an unhelpfull equation.

i can tell you why, if Q&gt;\sqrt{1/2} that the peak resonant frequency is

\omega_{0}\sqrt{1 - \frac{1}{4Q^{2}}

if \omega_0 the "natural" resonant frequency (i don't know what to call it) of the system. but i do not know what b and m are and can't tell from the context. is this a second order mechanical system or an electrical system?