The existence of a spring and damper in the system introduces a phase-shift between the excitation at one end that the response at the other, so the term "initial force" seems somewhat ambiguous.
An "ideal damper" has no effect on the displacement. The displacement is controlled by the spring. The damper affects the velocity. In reality ,"shock absorbers" have a non-zero spring constant.
A stiffer damper provides a smaller phase shift.
An effective automotive suspension also tends to "spread out" the response over a longer time then the excitation. I think that you are not considering the time-variant nature of the event.