# Interpretation of Free Damped Vibrations

I'm studying vibrations; so I'm using Beer-Johnston-Cornwell Dynamics book. I am worry about the equation for Underdamped Vibration, which in the book it is: $$x_{(t)}=x_0e^{-\lambda t}\sin(\omega_d+\phi)$$; where $$\omega_d=\sqrt{\omega_n^2+c^2/4m^2}$$.

I think that $x_0$ would be replaced by the result vector of constants $c_1$ and $c_2$ affected by the factor $e^{-\lambda t}$. It means, an $x_m$ or an amplitud, but not the initial position, because it could be 0, with an inicial velocity.

Also the graphic, it depicts the boundary equation with $x_0$. I attached a picture.

Can you help me and comment? How I should interpret $x_0$? Maybe I misunderstood this topic.

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$x_0$ is proportional to $\sqrt{E_{initial}}$ square root of initial energy in the system.
For example suppose we have dumped motion of a mass on a spring and $x$ is distance from equilibrium position. Initial energy of the system can be fully in the tension of the spring (zero initial velocity).
Initial energy is $E_{initial} = \frac12k\,x_{initial}^2 = \frac12k\,x_0^2$
Generaly for system in the example: $x_0=\sqrt{2\frac{E_{initial}}{k}}$
We can change distribution of initial energy between potential and kinetic energy (changing $\phi$) but if sum of initial energy is the same then $x_0$ is also the same.
I agree with you. But, it's clear that $x_0$ is not the position in $t=0$, isn't? – Isai Oct 27 '13 at 21:38