In Part 1 of this article I noted that a sine wave, perhaps created by a mass on a spring bouncing up and down, results in a single peak in the spectrum. If the spring gets more rigid as it gets compressed, the mass will move less in one direction than the other. We can describe the form of the wave this makes as distorted or clipped. When we pass this clipped wave through the FFT, we get harmonics or multiples of the fundamental frequency.
For example, if the mass bounced up and down 30 times per second, we would see a peak in the spectrum at 30 Hz. If it was clipped, we would also see peaks at 60, 90, 120 and 150 Hz etc. These are called harmonics. The question I posed is: Are the harmonics real? In other words, although the mass is still only bouncing up and down 30 times per second, is there actually real vibration occurring at 60, 90 and 120 Hz Etc? OR are these just artifacts of the FFT?
Are Harmonics Real?
I posted this question to LinkedIN. Here are the responses:
Apparently, 65% of the respondents believe that harmonics are NOT real, but just a product of the FFT. Only 21% think harmonics are real. Perhaps popular opinion is not a good gauge of truth?
Natural Frequencies and Resonance
I am going to go off on a little tangent to talk about natural frequencies and resonance. In this tangent, I will propose an experiment that people can try themselves to prove if harmonics are real or not.
A single mass spring system has a single natural frequency. If you pull the mass down and let it go, it will vibrate at this frequency. A tuning fork is the same. It has one primary frequency it likes to vibrate at. A bell has numerous natural frequencies, but they also won’t ring if they are not excited by the same frequency.
A natural frequency is a property of the structure, related to its mass, stiffness and damping. Resonance is a condition where the natural frequency is excited by a forcing frequency. In the case of a tuning fork, it has a natural frequency, BUT if it is just sitting on a table, that natural frequency is not excited (it is not in resonance) and the tuning fork does not make a sound. If you shake it back and forth in your hand – Eg. vibrate it at a frequency other than its natural frequency, it also doesn’t make a sound. It only makes a sound when the natural frequency is excited.
This concept is very straight forward in vibration. I am not sure why the LinkedIN community got this wrong. If the natural frequency is NOT excited, the bell WILL NOT ring. On a positive note, I know where people can get some great vibration training! Y’all need it! 😉
A Single Impact
In Part 1 of the article I noted that a single impact in time creates broadband noise in the spectrum. The FFT basically says: “Give me a wave and I’ll give you a bunch of sine waves. If you add these sine waves together, they will look like the original wave.” People who don’t think harmonics are “real” might say that they just come from the FFT trying to define the shape of the wave by adding sine waves together.
If you have a single impact in time, it looks nothing like a sine wave at all and the FFT spits out sine waves at lots of frequencies to try to define it. We call this broadband noise. People who believe that harmonics are not real will also believe the broadband noise is not real for the same reasons.
Looking at the animation above. I am only tapping the structure once, but you see vibration all the way across the spectrum. There’s a hump just above 100 Hz, but I am not tapping anything 100 times per second. Therefore, this vibration must not be real. Right? I am also not tapping it 300 times per second and yet the spectrum shows vibration at that frequency (and all the frequencies between) as well.
I am asking the same question here. Is the broadband noise “real” or just an artifact of the FFT?
An Easy Experiment
I am using the example of broadband noise to make the case for harmonics being real because this one is more intuitive and easier to test. If you think harmonics are NOT real, you will also think broadband noise is NOT real for the same reasons. I am clearly only tapping the object once, not 100 times per second or 300 times per second or anywhere in between, so these frequencies must not be real… That’s how the thinking would go.
So, let’s say a bell or a tuning fork has a natural frequency of 1000 Hz. If you hit it once, does it ring at that frequency? Yes it does! That means that there really is vibration at 1000 Hz being input into the bell. That single tap does include real vibration at 1000 Hz. It is NOT just the FFT trying to come up with a bunch of sine waves that when added together will look like our input.
You could also prove that different “taps” include different frequencies, even if the tap itself only happens once. A drum head has a lot of natural frequencies – none of which will make a sound if it is not excited by the same frequency. If you hit a drum with a soft mallet, it makes a different sound than if you hit is with a wooden drum stick. Why is this? It is because the soft mallet only injects (or contains) lower frequencies and therefore only excites the lower modes or natural frequencies of the drum head. It makes a lower sound. A wooden drum stick injects lower and higher frequencies, so in addition to the low tones, you also get higher ones.
An Easy Experiment for Harmonics
With the last experiment in mind, we can prove the existence of harmonics in the same way. Let’s say we have the mass and spring bouncing up and down 30 times per second (30 Hz) but the wave is clipped due to the spring getting more rigid as it compresses. The FFT will contain 30 Hz and harmonics at 60, 90, 120 Hz etc.
To prove that the harmonics are “real,” we could attach another spring with a natural frequency of 60 Hz to the first spring. If the spring tuned to 60 Hz vibrates, it means there is in fact really vibration at 60 Hz, it is not an artifact or output of the FFT.
In the case of a machine, if it rotates at 1800 RPM (30 Hz) you could mount a spring on it with natural frequency of 60 Hz. If the spring vibrates, then there really is vibration occurring at this frequency.
Back in the old days (before my time) they had mechanical vibration sensors that were just a series of little masses on springs tuned to different frequencies. You placed it on a machine and looked to see which springs vibrated. This is how you knew which frequencies were present.
Non Linearity
Why are there frequencies in the output that do not exist in the input? If I bounce the mass on the spring up and down at 30 Hz, why do I also get frequencies at 60, 90, 120 Hz Etc? The answer is non-linearity. In a non linear system you get things in the output that were not present in the input. You can read more about that here: https://zencovibrations.com/linear-and-non-linear-vibrations/
Another Example of Harmonics that Real in a Different Way
In next week’s article, I’ll offer up a couple of cases where natural frequencies are also harmonics. Keep an eye out for it!
In the meantime…
