# How long does it take to enter the steady state response in forced vibration?

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

• 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). Sep 19, 2015 at 14:18