What is the difference between damping and elasticity forces? 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?
 A: 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).
A: Elastic forces involve the storage and release of energy within the system. Damping forces are forces that lead to the dissipation of energy outside the system (energy loss).
Both damping and elastic forces are usually necessary to achieve suitable results in engineered systems. Example: your automobile suspension. Neither elastic (spring) or damping (hydraulic dashpot) are useful by themselves to achieve a smooth ride. You need just the right proportions of both.
Synonymous with elastic is stiffness and speed and restoring force. If you have no damping along with stiffness, your system will to bounce or ring. If you just have damping forces there is no force to return your system to a desired equilibrium position. You need both to get you there fast and smooth
Time intervals for system settling (time constants for example in linear systems) are usually a function of the elastic (spring), inertia (mass or moment of inertia), and damping parameters describing a system. None of these alone determine response, settling time.
