Interpretation of Free Damped Vibrations

Question

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 amplitude, but not the initial position, because it could be 0, with an initial velocity.
Also, the graphic 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.

user1086737 · Accepted Answer

$x_0$ 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:
$x_0=sqrt{2frac{E_{initial}}{k}}$

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.

Interpretation of Free Damped Vibrations

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