# 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.

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?

• I don't agree. Damping forces can be dominant in the initial response to an impulse. – ja72 Mar 2 '17 at 17:56
• The initial quote reads as though it ought to be embedded in a context where relatively weak damping is assumed. As @ja72 says, the completely general case includes situation where the quote is too strong or even completely wrong. But context is everything. – dmckee Mar 2 '17 at 18:56
• Try diving from 50ft into water and see if you feel any damping forces when you hit the water. – ja72 Mar 2 '17 at 19:55
• What @ja72 said. This is total nonsense, unless there is some context that the OP didn't include. Typically the maximum response to an impulse load is after half the period of the lowest vibration mode, and that time has nothing to do with the fact that the loading is impulsive. It doesn't have anything to do with the damping level either - and higher levels of damping will increase the time, not decrease it. – alephzero Mar 3 '17 at 6:32
• The quote is at the start of Chapter 5 of the book, and looks like a badly thought out explanation why the authors don't bother to include damping at all in their discussion of impulse response in the rest of the chapter! Don't take it too seriously. – alephzero Mar 3 '17 at 6:50

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).

• To be honest, my professor gave me the same answer; but I cannot understand, if this is the case why hasn't the book mentioned it? It could easily say "because in most structures we have $c\ll k$". Anyway, thank you! – lucas Mar 3 '17 at 11:12

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.