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According the basic theory of forced vibration[1], before entering the steady-state response, there is a short duration for so-called "transient" state vibration. I'm wondering how long is this "transient" state? How can I compute its duration?

[1]Introduction to Dynamics and Vibrations http://www.brown.edu/Departments/Engineering/Courses/En4/Notes/vibrations_forced/vibrations_forced.htm

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    $\begingroup$ That depends on the strength of the damping. For a toy-model simply consider a driven damped harmonic oscillator. $\ddot x + 2 \gamma \dot x + \omega^2 x = f(t)$ (one may choose $f(t) = \cos(\Omega t)$ for simplicity, the easiest way to solve the equation is by using the Green function). $\endgroup$ Sep 19, 2015 at 14:18

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Theoretically speaking, it takes infinite time. Because the transient state dies out exponentially and although after sometime its effect is negligible but its still there.

Although for all practical purposes we consider the transient state "dead" after its amplitude decreases 1/e times its original amplitude and putting this in the formula you should get a finite value.

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