Making the Case for Harmonics, Part 1
Alan Friedman, Cat IV, CRL, CMRP, Author; Founder/CEO of Zenco Vibration Experts
Posted 5/22/2025
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 gets 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 drumstick 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?
Food for thought…
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.
To recap, 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? – Asking LinkedIn
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.
A Single Impact
Previously in 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 drumstick.
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 drumstick 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 nonlinear system you get things in the output that were not present in the input.
Keep an eye out for Part 2 of this article coming soon! It will cover cases of natural frequencies that are harmonics, as well as AI’s take on the reality of harmonics.
Alan Friedman
Alan, aka the Vibe Guru, has over 30 years of vibration analysis experience, He has trained 1000’s of students around the world up to Category IV. One of the things that makes Alan a great teacher is his ability to teach people where they are at. Whether you are a math challenged millwright, an engineer or a PhD, Alan will challenge you without overwhelming you. If you are interested in condition monitoring you can also check out his book: Audit It. Improve It! Getting the Most from your Vibration Monitoring Program or hire him for an on-site program audit.
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