
## Critically-damped systems

In the last lecture we saw that for a critically-damped system,

$$
c^2-4mk=0
\tag{1}
$$

Rearranging we have,

$$
c_{cr}=\sqrt{4mk}
\tag{2}
$$

Therefore there are two identical real roots to the characteristic equation,

$$
\lambda_1 = \lambda_2 = -\frac{c_{cr}}{2m}
\tag{3}
$$

**Critical damping represents the smallest amount of damping for which no oscillation occurs in the free response.** It also represents the boundary between under-damped and over-damped systems discussed below.

We typically express the damping in a system as a percentage of critical damping or as a damping ratio, $\xi$ where,

$$
\xi = \frac{c}{c_{cr}}
\tag{4}
$$

**Typical damping ratios for civil engineering structures are of the order of $1-5\%$** which is significantly less than critical damping. The inherent damping within a structure is heavily influenced by the materials and geometry of the structure. When the response of the structure to in-service loading is too noticeable or uncomfortable for occupants (e.g. London Millennium Bridge), the inherent structural damping ratio can be increased by adding additional dampers.

## Over-damped systems

For an over-damped system,

$$
c^2-4mk>0
\tag{5}
$$

Thus there are two real roots to the characteristic equation. Further solution of the differential equation yields a response that decays exponentially to zero without oscillation. Over-damped systems are very uncommon, particularly within civil engineering structures. We won't consider over-damped systems. Instead, we'll move on to discuss the more common under-damped system in the next lecture.


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