= Overdamping
= Overdamped
{synonym}
For $\zeta>1$, let $r_+>r_-$ be the real <characteristic roots>. The response with the specified <initial conditions> is
$$
x(t)=C_+e^{r_+t}+C_-e^{r_-t},\qquad
C_+=\frac{v_0-r_-x_0}{r_+-r_-},\quad C_-=\frac{r_+x_0-v_0}{r_+-r_-}.
$$
The generic late response is governed by $r_+$, the root nearer zero. For very large <damping ratio>, $r_+\sim-\omega_0/(2\zeta)=-k/b$: an increasingly strong brake delays the return to <equilibrium>. This approximation comes from balancing the <spring> and <viscous damping> terms after the fast transient. Choosing $v_0=r_-x_0$ makes $C_+=0$ and leaves only the faster root; that exception requires different initial <velocities> as $b$ varies.
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