What is the cause and meaning of harmonics?

Question

Let's suppose that a cantilever is vibrating at a frequency of 2Hz. That means that it goes up and it comes down twice per second. I don't understand what the term harmonic means. I can't grasp how the vibrating rod moves twice and, at the same time, four times or eight times per second.

Descriptions of harmonics in engineering texts that I study explain nothing. These explanations suggest " an event that happens twice, four times, etc at the period of a vibration".

Any comment, suggested text or answer will be useful.

Answer 1

The fundamental mode of a cantilever looks like this:

And the displacement of the end of the cantilever will be given by:

$$ y = A_0 \sin (\omega_0 t) $$

for some frequency $\omega_0$. But the cantilever can also oscillate as well as bending i.e. something like:

In this case the position of the end of the rod will be given by:

$$ y = A_0 \sin (\omega_0 t) + A_1 \sin (\omega_1 t + \phi_1) $$

where $\omega_1$ is a higher frequency than $\omega_0$ and is related to the way the cantilever oscillates along its length. So in this case we have more than one frequency describing the motion. Although I won't attempt to draw them, there can be higher frequency oscillations as well. In general the position of the end of the rod will be given by:

$$ y = \sum_i A_i \sin (\omega_i t + \phi_i) $$

The $\omega_i$ for $i \gt 0$ are the harmonic frequencies or overtones.

Answer 2

Try to plot the function $\sin x$. It goes up, back to zero, down, and back to zero once in $2 \pi$. Now plot the function $\sin 2x$. It goes up and down twice that of $\sin x$ in the same period $2\pi$.

Now what does it mean by going up and down once, and also twice, in the same period? Just plot $\sin x + \sin 2x$. That's what it means.