Are Harmonics Real?
Are harmonics real or just an artifact or result of processing the waveform through the FFT? This is a question that comes up a lot among vibration analysts, or anyone working with signals and signal processing.
What Are Harmonics?
A sine wave has a single amplitude and frequency. Let’s say our sine wave has a frequency of 25 Hz and a peak amplitude of 10. If you use the FFT algorithm to convert a sine wave to a spectrum, you will see a single peak at 25 Hz with a peak amplitude of 10. You can see this at the beginning of the animation above. At the starting point there is a sine wave at the top and the spectrum below with a single peak at 25 Hz.
A wave that repeats itself, but is not a sine wave results in harmonics in the spectrum. You can also see this in the animation above. As I clip or flatten the top of the wave, you can see the harmonics appearing in the spectrum. Harmonics are multiples of the base frequency. In this case they are 25 Hz x 2, 25 x 3, 25 x 4 etc. giving us peaks at 25 Hz, 50 Hz, 75 Hz, 100 Hz etc.
The Fast Fourier Transform (FFT)
The FFT is a tool that reveals the frequency components present in a signal. It breaks the signal down into a series of sine waves.
Another way of phrasing this is: The FFT essentially says “give me a wave and I will give you a bunch of sine waves. If you add all these sine waves together, it will look like the initial wave.” This is my layman’s way of understanding of it anyway.
Each peak in the spectrum represents a sine wave. If I were to add the sine waves represented by the peaks at 25, 50, 75 and 100 Hz etc at the end of the animation above, the resulting wave would look just like the clipped wave.
But Are Harmonics Real?
One view of this is that the wave I used for this explanation is not “real” in that it is something I generated in software. I also clipped it in software, so the peaks at 50, 75 and 100 Hz don’t correspond to anything real in the physical world, so they must just be artifacts or things stemming from running the data through the FFT. Right?
A Physical Representation of Clipping
Let’s say I move a mass and spring back and forth 30 times per second (30 Hz) with a sinusoidal force (a sine wave) but as the spring compresses, it gets more rigid and eventually stops compressing. This will create a clipped wave as in the image below. When I pull the mass out, it looks like a sine wave, when I push it in, compressing the spring, it get’s flattened.
As we noted before, the clipped wave is repetitive but not a sine wave and therefore we will get harmonics. As you can see from the image above, we have peaks at 30, 60, 90 and 120 Hz.
One might look at this physical example of a clipped wave and they might say: “I see the mass moving back and forth 30 times / second. I do not see anything happening at 60 times per second or at 90 times per second, so harmonics can’t be real. They must just come out of the FFT algorithm.”
Repetitive Impacting
Another example of a wave that is repetitive but not a sine wave is repetitive impacts. If I take a drum stick and I hit a drum 5 times per second (as in the time waveform image below) I would see peaks in the spectrum at 5, 10, 15, 20 Hz etc. In other words, harmonics. Intuitively you know that you are only hitting the drum 5 times per second, so why would there be frequencies at 10, 15 and 20 Hz etc. in the spectrum? They must just come out of the FFT right?
The pattern above is common in machinery vibration analysis. This could be a bearing with a defect on the race and the impacts are the balls or rollers slamming into that defect one after the other. This could also a be a broken gear tooth that causes a big impact every time it comes into contact with the other gear. In both of these cases, the analyst would expect to see harmonics of the defect frequency in the spectrum.
A single Impulse or Impact
A common test in vibration analysis it to strike an object with a calibrated hammer. The calibrated hammer has a force sensor in it’s tip, so it measures the actual hit. If we look at this in the time waveform, we see a single sharp, short duration pulse. When we convert this pulse to a spectrum, we get vibration at many frequencies or what is referred to as “broadband” noise.
One might say: “The FFT has a really hard time describing that single impact by adding sine waves together. The only way to do it is to add sine waves together at every frequency as shown in the image above.” These frequencies don’t exist in the real world though. (Or do they?)
I will ask again, using this example, are the frequencies shown in the spectrum for this single impact “real?” Do they really exist? Or are they just a result of passing that data through the FFT and the FFT trying to describe it by adding sine waves together?
Conclusion to Part 1
I want to leave this as an open question for now. Take some time, think about it, and decide for yourself if you think harmonics (and broadband noise) are real or not. In upcoming articles, I will lay out a case for what I think. I will also admit that my thinking about this has changed over the years. I used to take one side of this issue, now I lean towards the other. But, you can decide for yourself.
