From DYNAMICS OF STRUCTURES, Third edition, by Ray W. Clough and Joseph Penzien
Damping has much less importance in controlling the maximum response of a structure to impulsive loads than for periodic or harmonic loads because the maximum response to a particular impulsive load will be reached in a very short time, before the damping forces can absorb much energy from the structure.
And about the impulsive load:
Such a load consists of a single principal impulse of arbitrary form and generally is of relatively short duration.
The differential equation of motion of a single degree of freedom system is
$$m\ddot{x}+c\dot{x}+kx=p$$
$m$: mass
$c$: damping coefficient
$k$: elastic coefficient
$x$: displacement
$p$: excitation force
If damping has much less importance in controlling the maximum response of a structure to impulsive loads, why doesn’t elasticity have?
In other words, what is the difference between damping and elasticity forces that we can remove the damping term $c\dot{x}$ from the equation but cannot do the same about the elastic force term $kx$ for impulsive loading?
If time interval is very short for damping force, isn’t it very short for elastic force too?
Best Answer
This is simply because for (most) structures the damping is weak, in the sense that $c^2\ll4km$, when the damping time $2m/c$ (the time over which the damping term takes energy out of the system) is much longer than the period $2\pi\sqrt{m/k}$ of oscillations. The first maximum amplitude after an impulsive perturbation occurs after $\approx\tfrac{1}{4}$ of a period, when damping has hardly reduced the energy.
If you like it mathematical, consider an impulsive perturbation as one that acts instantly and changes the velocity $\dot{x}$ (but not the amplitude $x$), pumping the energy $E=\tfrac{1}{2}m\dot{x}^2$ into the system. After the impulse, the system follows the evolution for $p=0$ (assuming $c^2<4km$) $$ x(t) = \sqrt{2E/k}\; \mathrm{e}^{-\lambda t}\,\sin\sqrt{\omega_0^2-\lambda^2}t $$ where $\lambda=c/2m$ and $\omega_0=\sqrt{k/m}$. For weak damping $\lambda\ll\omega_0$ and the first maximum amplitude is almost $\sqrt{2E/k}$.
Of course, the situation is different for strong damping (but that does not apply to most structures).