= Solution
Take $m,k,b>0$, and define $\omega_0=\sqrt{k/m}$ and $\gamma=b/(2m)$. Substitution of $e^{rt}$ into the unforced equation gives $r^2+2\gamma r+\omega_0^2=0$. The roots determine whether the return is oscillatory.
For return after a <displacement> in this ideal model, \b[<critical damping>, case (b), is the usual comfort choice]: it removes repeated bouncing while avoiding the slow relaxation mode of strong <overdamping>. For release from rest it gives a monotone return with the fastest asymptotic decay available without oscillation. This criterion does not prove optimal comfort for every road profile or initial <velocity>; actual comfort also depends on <acceleration>, road forcing and suspension travel.
For the forced response, set $\Omega=2\pi u/d$ and use $x_{\rm ss}=\operatorname{Re}(\widehat x e^{i\Omega t})$. Then
$$
(k-m\Omega^2+ib\Omega)\widehat x=kA,\qquad
\boxed{x_{\rm ss}(t)=X(\Omega)\cos(\Omega t-\delta),}
$$
$$
X(\Omega)=\frac{kA}{\sqrt{(k-m\Omega^2)^2+b^2\Omega^2}},\qquad
\delta=\operatorname{atan2}(b\Omega,k-m\Omega^2).
$$
The general forced solution adds the appropriate homogeneous transient from (a), (b) or (c). Initial data were not specified, so its constants cannot be fixed. Positive <damping> removes this transient at long times. This solution uses the particular driving <force> in the supplied equation; it does not silently replace it by a different moving-support model.
Maximising $X$ is minimising its squared denominator $D$. Differentiation gives
$$
\frac{dD}{d\Omega}=2\Omega[-2mk+2m^2\Omega^2+b^2],\qquad
\Omega_r^2=\frac{k}{m}-\frac{b^2}{2m^2}.
$$
A nonzero <displacement> <resonance> exists only if $b^2<2mk$. Here $\Omega_r^2=0.175\,\mathrm{s^{-2}}$, and therefore
$$
\boxed{u_r=\frac{d}{2\pi}\sqrt{0.175}=0.666\,\mathrm{m\,s^{-1}}\simeq2.40\,\mathrm{km\,h^{-1}},}
$$
$$
\boxed{X_{\max}=\frac{kA}{b\sqrt{k/m-b^2/(4m^2)}}=0.0616\,\mathrm m.}
$$
The low-frequency limit is $X(0)=A$, whereas $X\to0$ at large $\Omega$. The parameters give <damping ratio> $\zeta=b/(2\sqrt{mk})=0.456$, consistent with an underdamped system and the existence of this amplitude maximum.
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