Measuring the natural frequency of a spring-mass system driving force

[haruspex]

$\frac{Ce^{-\Gamma t_0/2}e^{-\Gamma T/4}}{-Ce^{-\Gamma t_0/2}} = 4/-8$

$\Gamma = 0.69$

Like that?

Yes.

 

[happyparticle]

So basically, $\Gamma$ is how the system decrease "speed"?

 

[haruspex]

So basically, $\Gamma$ is how the system decrease "speed"?

It's how the transient dies away, allowing the period to be controlled by the driving force. It continues to affect the amplitude.

 

[happyparticle]

I love you! Seriously big big thanks

one more thing if you want. $A(\omega_d) = 2.8$ is that correct?

 

[haruspex]

I love you! Seriously big big thanks

one more thing if you want. $A(\omega_d) = 2.8$ is that correct?

That's how it looks on the graph you posted.
I encourage you to try to repeat the path I took: observe that the general equation is the sum of a decaying oscillation and a steady oscillation; deduce the steady oscillation parameters from the right hand part of the graph; subtract the equation for the steady part from the graph as a whole to reveal the transient oscillation; from the period and attenuation of the transient deduce its parameters; recreate the curve in a spreadsheet using the parameters calculated.