= Added-load step response of a supported table
Adding a load $M$ to a spring-supported table shifts its static downward position by $\Delta=Mg/k$. If the load is deposited with negligible impulse and moves with the table, the moving <mass> is $m+M$. Starting at the former equilibrium with zero <velocity>, an <underdamped> response is
$$
x(t)=\Delta\left[1-e^{-\beta t}\left(\cos\omega_dt+\frac\beta{\omega_d}\sin\omega_dt\right)\right],\qquad
\beta=\frac{b}{2(m+M)},\quad \omega_d^2=\frac{k}{m+M}-\beta^2.
$$
Its transient envelope decays over $1/\beta$. <Critical damping> $b=2\sqrt{k(m+M)}$ gives $x=\Delta[1-(1+\omega_0t)e^{-\omega_0t}]$. A nonzero impact impulse changes the initial <velocity> and therefore the transient.
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