SHM Math Explodes (Req: help finding Real parts)

[malweth]

Homework Statement

I'm not attempting a specific problem, I'm just trying to find a correct way of doing the math behind simple harmonic motion ODEs...

The example problem I've given myself (based on the books I'm using) is a dashpot with constant $$c$$, spring with constant $$k$$ and a mass $$m$$.

Specifically, I'm trying to understand the energy lost by the dashpot, but I'm in the "Calculus" subforum for a reason:

Homework Equations

(variables $$/dt$$)
$$m\ddot x + c\dot x + kx = 0$$ - (Unforced, damped SHM).
which is eqivalent to:
$$\ddot x + \gamma\dot x + \omega_0^2 x = 0$$

The solution is
$$x = \hat A e^{\alpha t}$$ and is differentiated & substituted back into get:

$$\alpha^2 + \gamma\alpha + \omega_0^2 = 0$$

which is easy to solve for alpha:

$$\alpha = -\gamma / 2 \pm j\sqrt{\omega_0^2 - \gamma^2/4}$$

The Attempt at a Solution

The problem I have is in substituting back in... at first it's not too bad, but I'm trying to get real values out... even with letting $$\omega_r = \sqrt{\omega_0^2 - \gamma^2/4}$$ it gets complicated quickly in my pen & paper scratches...

$$x = \hat A e^{-\gamma/2}(e^{j\omega_r t} + e^{-j\omega_r t})$$

gets differentiated to find velocity (making it more complicated, but manageable), and then "de-Eulered" to find a real part. Am I doing something wrong? I don't see anything half as complicated in the books I'm using (primary book is "Fundamentals of Acoustics," by Kinsler and my additional resource is from Feynman's first volume on physics).

The books make it seem as though I'm doing something wrong, but I can't find how. Any help would be appreciated!

 

[quantumdude]

$$x = \hat A e^{-\gamma/2}(e^{j\omega_r t} + e^{-j\omega_r t})$$

This is wrong. You've lost the decaying exponential part. Also, you only have one constant of integration. In a 2nd order ODE you must have 2 constants of integration in your general solution.

It should be:

$$x(t)=e^{-\gamma t/2}\left(Ae^{j\omega_rt}+Be^{-j\omega_rt}\right)$$

 

[malweth]

Thanks! I forgot about the other constant. The missing $$t$$ value in the exponent was a mistype.

Am I otherwise doing things correctly when trying to get back into real space (for Energy calculations)?

 

[quantumdude]

It looks to me like you are.

 

[malweth]

great! Thanks for the help.