Proving System Decay Time: A Detailed Explanation

[SteveDB]

hi all.
Some clarification on this would be helpful to get me going in the correct direction.
For a specified system, I'm trying to prove that the time it takes for the system to decay to 1/e of its original value (which works out to ~36.8%), takes a certain amount of time. The actual values are unimportant but the process is.
I have gone through my classical mech book-- 4th ed of Marion Thornton, as well as my diff/eq book-- 5th ed of Nagle, Saff, Snider, and of course my waves and oscillation text-- A.P.French, and cannot decipher what seems-- or I thought would be-- a fairly straightforward problem.
I'm not schooled/skilled in latex, so please bear with my "hand version."
I've taken the time derivative of the energy, and get a m/s^3 function for my acceleration value. With the values for b, k, and m, I do not get the time I'm looking to prove.
m*x_dbldot + b*x_dot + k*x = 0
Where x(t) = (A*exp(omega*t) +B*exp(-omega*t)
The rate given for decay to 1/e is:
E/delta_E
Where delta_E is given by -b*E/(m*nu)
where nu is given by omega_o/2pi.
I've also tried the quality value Q for this. I know I'm missing something, but can't quite identify it.
A detailed explanation of this would be deeply appreciated.

Thank you.

 

[lightgrav]

I'm not clear what you want ...

You know that for this viscous-damped oscillator
the envelope function is exponential, and you
even have the right exponent!

You already have t in the envelope function.

OK, did you forget that the E envelope
and the x-envelope are related by E = ½k x^2 ?

for reasonably small values of "b", the solution is
x approx. A exp(-omega*t)*sin(w_o*t) ,
because the natural frequency isn't changed much.

You don't want exponential growth curve, do you?
(I mean, set your A=0 and rename B=Amplitude)

 

[SteveDB]

that's part of what I meant when I said that I'd taken the time derivative of the energy equation.
E= m/2 (x_dot)^2 + k/2 *x^2
Based on my energy of the system, I need to then proof that the time only takes a certain amount of time to decay to 1/e.
E(t)= E(0)/e
Sounds like I'm not the only one that's struggling with the decay function.
I don't know how to explain it any better. that's part of what's confusing me, and why I posted.
Thanks.