
In this short lecture, we're going to determine a simple expression for the damped angular natural frequency, $\omega_d$, as a function of the damping ratio $\xi$. We'll see that for low levels of damping, $\omega_d \approx \omega_n$.

We saw in a previous lecture that the critical damping coefficient is given by,

$$
c_{cr}=2\sqrt{km}
\tag{1}
$$

If we now multiply the right side of this equation by $\sqrt{m/m}$, noting that this is equivalent to multiplying by $1$, we get,

$$
c_{cr} = 2\sqrt{\frac{k}{m}m^2} = 2m\sqrt{\frac{k}{m}}
\tag{2}
$$

Recall that we defined the angular natural frequency as,

$$
\omega_n=\sqrt{\frac{k}{m}}
\tag{3}
$$

Combining these two equations, we can state an expression for the critical damping coefficient as,

$$
c_{cr}=2m\omega_n
\tag{4}
$$

We also recall that we defined the damped angular natural frequency as,

$$
\omega_d=\sqrt{\frac{k}{m}-\left(\frac{c}{2m}\right)^2}
\tag{5}
$$

In order to simplify this expression, extract and consider the term,

$$
\frac{c}{2m}
\tag{6} \frac{c}{2m}
$$

We can manipulate this term as follows,

$$
\frac{c}{2m} =\frac{c}{2m} \frac{c_{cr}}{c_{cr}} =\frac{c}{2m}\frac{2m\omega_n}{c_{cr}}=\frac{c}{c_{cr}}\frac{2m\omega_n}{2m} = \xi\omega_n
\tag{7}
$$

Therefore the damped angular natural frequency is,

$$
\boxed{
\omega_d = \omega_n\sqrt{1-\xi^2}
}
\tag{8}
$$

From this expression we can clearly see that if the damping ratio is small, $\omega_d\approx\omega_n$. For example, recall that we said most civil engineering structures have damping ratios less than $5\%$; so, if $\xi=5\%$,

$$
\omega_d = \omega_n\sqrt{1-0.05^2}
\tag{9}
$$

$$
\omega_d = 0.999\omega_n
\tag{10}
$$

On this basis we often assume $\omega_d = \omega_n$ for low levels of damping.

In the next lecture we'll continue our discussion of damping and determine a method of evaluating the actual level of damping in a structure by examining its oscillation behaviour.


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