
The natural frequency, $f_n$, measured in cycles per second or hertz $(Hz)$ is the frequency at which a system oscillates if displaced from equilibrium and released. The angular natural frequency, $\omega_n$ is the natural frequency reported in radians per second $(rads/s)$. The relationship between natural frequency and angular natural frequency is,

$$
\omega_n = 2\pi f_n
\tag{1}
$$

We'll soon see that the natural frequency is a critically important property of any dynamic system. Related to the natural frequency is the period of oscillation, $T$. This is the time required to complete one complete oscillation cycle and is the inverse of the natural frequency,

$$
T=\frac{1}{f_n}
\tag{2}
$$

We can obtain the natural frequency directly from the properties of the system, specifically the mass $m$ and stiffness $k$,

$$
f_n = \frac{1}{2\pi}\sqrt{\frac{k}{m}}
\tag{3}
$$

Combining the three previous equations we can summarise the relationship between all quantities as,

$$
\boxed{\omega_n = 2\pi f_n = \frac{2\pi}{T}=\sqrt{\frac{k}{m}} }
\tag{4}
$$


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