Vibration and exploring the Fast Fourier Transform with CAD

[SillyOldDufferModerator @sillyoldduffer]

This won’t be to everyone’s taste, but it is Engineering and I am going Model it!  Hopefully others will join in.  Plenty on here whose maths is much better than mine, plus a few electronics/computer gurus who might understand FFT.

This isn’t a digital/electronics/computer/maths/CAD project – it’s intended to help isolate whatever fault(s) cause a machine tool or engine to vibrate.

In Nigel’s Graham’s Fine Chatter topic , I mentioned a failed project where I attempted to identify the individual frequencies my lathe vibrates at by sticking a microphone on the bed and recording an audio file whilst cutting. The file is analysed with a computer to extract the frequencies with a Fast Fourier Transform. I almost certainly failed because I don’t understand FFT and digitilisation well enough yet.

Fulmen expressed an interest:  Fast Fourier Transform: That’s very clever. I’d like to see a sensible guide on that some day. I did a quick search for digital motor stethoscopes but couldn’t find any. That would make for a sweet product , a bluetooth stethoscope with a spectrum analyzer.  

I agree and as the hardware isn’t difficult, basically a microphone and  laptop,  I thought I’d have another go, this time modelling the FFT with a CAD tool that we can all download and play with.

The tool is the GNU Radio Companion.  Free, and no programming required – models are created by dragging and dropping function blocks, and then linking them.  Download from here.

When the tool is installed and started, try this:

Should open a screen like this.  On the right, is a list of available function blocks.

Go to the Waveforms section (bottom of list), expand it, and click on Signal Source, hold the button, drag left and drop.  The box can be moved about later.  Then go to the Audio Section and drag and drop an ‘Audio Sink’.  Should get:

 

Note Signal Source has a blue out box and Audio Sink has an orange in.   Blue means ‘complex number’ and orange means ‘float’.  A booby trap because they aren’t compatible, so double click on Signal Source and change its Output Type from complex to float, making both out and in orange:

Then move the mouse onto Signal Source out, click hold, and drag the resulting arrow ended line onto Audio Sink in:

This models a 1kHz audio oscillator connected to the computer’s sound card.  Pressing the black triangular Play button top right on the tool-bar should activate it, making an annoying whistle.   No programming, soldering transistors, or plugging in cables!

Tomorrow I’ll model 3 or 4 Signal Sources mixed together and try and retrieve the input frequencies from the muddle.   Watch this space.  More function blocks are needed.   Didn’t work last time I tried.

Dave

 

[Michael GilliganParticipant @michaelgilligan61133]

For “the rest of us” … this is a very good introduction:

 

https://youtu.be/QmgJmh2I3Fw?feature=shared

MichaelG.

[Robert Atkinson 2Participant @robertatkinson2]

Hi Dave,
Looks interesting. I appreciate that this is a learning exercise but if you want someting a bit more ready to go have a look at Spectrum Lab.
https://www.qsl.net/dl4yhf/spectra1.html
For a DIY pickup rather than a microphone the “speaker” from a pair of cheap ear-buds works well and even better if you add a bit of mass to the center of the diaphram. A small nut fixed with something flexible like contact adhesive works well.

A lot of digital oscilloscopes will do FFT but the limited amplitude resolution of most ‘scopes of 8 bits limits their usefulness. Even though a sound card has poor amplitude accuracy their high resolution and dynamic range make them a good choice for FFT. I have a couple of 12bit Pico technology USB ‘scopes that work well. One of them is an automotive model and their automotive software has an FFT function specifically intended for noise and vibration analysis.

Robert.

[John HaineParticipant @johnhaine32865]

Worth looking at the Physics Toolbox Suite, and app for Android, possibly iPhone too.  Can do loads of stuff including audio fft using the phone sensors.

[Michael GilliganParticipant @michaelgilligan61133]

Yes, the excellent Physics Toolbox Suite is available for iOS

I would also recommend this:

https://apps.apple.com/gb/app/sonic-tools-svm/id1245046029

[cleverer people than I should read the developer’s comments]

MichaelG.

[FulmenParticipant @fulmen]

Thank you, Dave. I’ll take a dive into the subject tonight.

[SillyOldDuffer]

On 5 March 2025 at 20:31 SillyOldDuffer Said:

…who might understand FFT.

I might be in that category; part of my Ph.D. was to do with extending the mathematics of FFTs and creating hardware architectures to compute FFTs.

It’s a common misconception that the FFT is an algorithm developed by Cooley and Tukey. The algorithm by Cooley and Tukey is simply one of many fast Fourier transform algorithms, some of the others being the prime factor algorithm, the Rader algorithm and that by Winograd.

Another misconception is that the Cooley and Tukey FFT is based on powers of 2. The original Cooley and Tukey paper simply states that the length of the input vector needs to be a highly composite number. It turns out that using powers of 3 is slightly more efficient than powers 2.

Of course from a practical point of view using powers of 2 is helpful when calculating via a digital computer. An FFT using prime, or mutually prime, factors may be computational efficient but indexing the data can be a pain.

The video posted by Michael is wrong, the FFT doesn’t exploit redundancy, it simply transforms one complex matrix to a series of sparse matrices that leads to fewer overall calculations.

Julie

[Michael GilliganParticipant @michaelgilligan61133]

Thanks for the correction, Julie

… I bow to your wisdom.

My personal experience was limited to using random vibration [and swept sine, of course] as a test-tool … I never really understood the mathematics.

MichaelG.

[SillyOldDufferModerator @sillyoldduffer]

I’m going to stick with GNU Radio Companion for the moment.  Whatever the merits of the alternatives it’s too early for me to change tool.  But if anyone wishes to follow along with a tool they’re familiar with feel free!

My goal is to better understand the maths and configuration options needed to get FFT to work.  I have an inkling, and will post some examples.  With luck Julie will put me right!  She has the experience and skills.

Back to scene setting before I tackle FFT though.  Yesterday’s example showed how to output a sine wave to the sound-card, but that’s not useful here.  Just a demo.   What’s needed is a graphical oscilloscope display, and this is found in the Instrumentation list.

Drag and Drop ‘QT GUI Time Sink’ onto the main page.  On my machine it defaults to Complex (Blue) input, so double click and change it to float.  Then connect the Signal Source to it, giving:

Playing the model displays the waveform:

I’ve added an oscilloscope!  Though the Audio Sink could be left in parallel, it’s annoying, so delete it, unless listening to the output helps.

Then drag and drop ‘QT GUI Frequency Sink’ from the Instrumentation list, configure it for float input, and connect it:

GUI Frequency Sink applies FFT and displays the result.  FFT generates a positive and negative result, so two spikes expected.  (GUI Frequency Sink can be configured to show only the positive result.)

Now the instrumentation shows a waveform in the time domain as an oscilloscope does, and the frequency domain extracted by FFT.   The frequency domain identifies the frequency and amplitude of the signal.  Roughly hovering the mouse on the positive spike says 0.93kHz, which is close to the 1kHz input.  Hurrah, this works!   But it’s only on one signal, and I need to identify several frequencies.  More work needed.

Instrument displays can be zoomed, doing so allows me to see FFT correctly puts the peak at 1.0kHz.

There’s a practical problem.  Running the model eats 100% of a CPU, so the cooling fans come on.   Reason, I think, is that the Audio Source is a short loop of code generating the numbers representing a sine wave running flat out continuously.   Real audio input is much slower, constrained by the microphone.   This being a simulation, it’s necessary to throttle the racing Audio Sink, for which purpose there’s a Throttle block in Misc. Change the model to:

 

Now, the fans don’t come on!   Why is the throttle set to 32kb/s?  Seems to be a compromise.  Too slow and the model becomes sluggish, too fast and the CPU gets hot.  16kB/s is fine, but 8kB/s causes a noticeable delay.

Another issue, perhaps the mathematicians can help?   My maths limit is at the Complex Number / Calculus level.  I know what a complex number is, and why they are useful, but am hazy about applying them.  (Broadly, a complex number has a real part and an imaginary part, very helpful when calculating rotating problems, like vibrations, if you understand it.)  In this example I told GNU RC to work with floats, seems OK, but doing so must lose information, which might cause trouble later.  As unnecessary guessing isn’t engineering can anyone explain when it’s OK to model in floats and when complex numbers become vital?   Please be gentle, I’m out of my depth!

Dave

 

 

 

 

[peter1972Participant @peter1972]

But it’s only on one signal, and I need to identify several frequencies. More work needed.

Try simply setting the signal source to sawtooth wave. See the spectum of a sawtooth wave here:

https://en.wikipedia.org/wiki/Sawtooth_wave

[SillyOldDufferModerator @sillyoldduffer]

Moving on, here’s my first wild attempt at simulating a vibrating lathe.

I believe most lathe vibration due to worn bearings, set belts, bent lead-screw, worn rack / half-nut and loose gibs are all low frequency and hard to identify.  High-frequency squeal is likely more obvious: seized bearings, cutter vibrating due to being off centre height and such.

Does this analysis make sense?

Assuming a motor rotates properly at 3000rpm, it would vibrate at about 50Hz.     Conversely, a knackered ball-bearing in the motor, would vibrate at 50Hz x the number of faulty balls or less depending on how the race is brinelled.  Perhaps 50Hz x 20 = 1000Hz.

A flexible belt in good order would tend to stop motor vibration reaching the spindle. A set-belt (stiff at two points) would halve 50Hz vibration at the motor to 25Hz at the small pulley.

The spindle typically rotates at less than motor speed, so less than 50Hz, unless a bearing fault multiplies it.

The change gears and gearbox are a rich source of vibration.  All those crude teeth meshing and maybe with a maladjusted mesh. I might run the permutations later.  However, in general, these gears turn the lead-screw slowly, so rack, lead-screw, half-nut and gib problems should all be well below 50Hz, especially on fine feed.

For the moment I’ll model 1Hz, 5Hz and 7Hz.  Don’t take these values too literally – just examples for testing.  But if anyone knows the frequencies faulty lathes really vibrate at please say so!

Anyway here’s the Mk1 model.  Three signal sources at  1Hz, 5Hz and 7Hz are added together and then FFT applied to the mix.  The oscilloscope is disabled temporarily (greyed out) to get a big frequency display. It’s the FFT result that matters.

The model, using default sample rate and other settings:

Sort of works, but the scale looks wrong by a factor of 1024, and the display is noisy.  Big DC spike at 0Hz, the signals are low, and there’s an unexpected spike near 2Hz.

My guess is the default values in the blocks aren’t a good match to the input signal. The scaling error is a clue.   Possibly set for frequencies typical of radio and audio applications.  I shall have to try tuning them.

My belief is that digital signal processing works on streams of numbers arriving at a given sample rate.  This works in conjunction with other parameters to get the desired result, and I don’t understand the relationships.   I’m roughly equivalent to a learner driver attempting a hill start in 5th gear, without knowing why that’s wrong.

Reminds me, my late mother-in-law drove up hills in top gear, completely ignoring straining engine noises,  and then drop gear at the top, screaming along the flat at 50mph in second.   Horrible, and her being a lady of strong ignorant convictions meant no-one could persuade her to use gears properly.   Father-in-law, an otherwise intelligent man, drove with his foot riding the clutch; he couldn’t comprehend why he needed a new clutch every 6000 miles!

Dave

 

 

[Neil WyattModerator @neilwyatt]

I have an excellent, but fairly mathematical, article with me on ‘vibration’ although from the perspective of simple harmonic motion which would, of course, only give a single line of SOD’s graph.

It might be challenging for some readers, but this thread suggests there would be an audience for it. Any thoughts?

Neil

[Neil Wyatt]

On Neil Wyatt Said:

[…] It might be challenging for some readers, but this thread suggests there would be an audience for it. Any thoughts?

Yes please … Go for it

MichaelG.

[peter1972]

On peter1972 Said:

But it’s only on one signal, and I need to identify several frequencies. More work needed.

Try simply setting the signal source to sawtooth wave. See the spectum of a sawtooth wave here:

https://en.wikipedia.org/wiki/Sawtooth_wave

A sawtooth?  A few seconds work:

But wow, that really upsets the FFT:

I’ll have to come back to it, square and triangular waveforms are a complication.   For the moment, best to keep to sine waves, because that’s the problem I have at the moment.  However I think a loose gib is likely to generate a triangular wave form, so they are part of the problem.  Why not download GNU Radio Companion and try it yourself?  If you can get FFT to work with triangular waves I shall be delighted!

Ta,

Dave

 

 

[SillyOldDuffer]

On SillyOldDuffer Said:

[…] For the moment I’ll model 1Hz, 5Hz and 7Hz.  Don’t take these values too literally – just examples for testing.  But if anyone knows the frequencies faulty lathes really vibrate at please say so!

[…]

This is a genuinely exciting project, Dave … and I wish you every success with it … but I must mention one point which may be significant:

For most vibration analysis, levels are expressed in ‘g’ and the relevant physical displacements vary enormously with frequency … I predict that you will have great difficulty in extracting useful low frequency data from a noisy beast like a lathe.

MichaelG.

.

Edit: __ we used some nifty little stickers in the test-lab which illustrated this very well, but I fear my last remaining sample has probably gone … Meanwhile, this is worth a look:

https://www.modalshop.com/rental/learn/vibration/why-shake-at-159-hz

[peter1972Participant @peter1972]

I was suggesting just one sawtooth wave.

It looks like you have sawtooth waves at 1Hz, 5Hz and 7Hz but your FFT goes up to over 16,000Hz.

Square, triangular and sawtooth waveforms can be regarded as being a number of sine waveforms at harmonic frequencies. So if you use a sawtooth wave you should see even and odd harmonics on your FFT display.

Just to throw a spanner into the works: when I was doing FFT, I used a Hanning Window to taper the edges of a series of signal samples. This can improve the FFT result.

 

[peter1972]

On peter1972 Said:

I was suggesting just one sawtooth wave.

It looks like you have sawtooth waves at 1Hz, 5Hz and 7Hz but your FFT goes up to over 16,000Hz.

Square, triangular and sawtooth waveforms can be regarded as being a number of sine waveforms at harmonic frequencies. So if you use a sawtooth wave you should see even and odd harmonics on your FFT display.

Just to throw a spanner into the works: when I was doing FFT, I used a Hanning Window to taper the edges of a series of signal samples. This can improve the FFT result.

 

Yes, I tried just one, no coconut!   I expected adding 3 sawtooth waves to result in a complex frequency domain, and it didn’t.  Surprising because as you say all (?) waveforms are sine wave combinations.  I’m doing something wrong!   Like running the FFT at 16000Hz, many thanks I’ll reduce it.

Also, for the Hanning suggestion.  GRC supports that, but in my ignorance I don’t know what the different windows are best at.

Interrupted after lunch by family, so haven’t been able to play.

Cheers,

Dave

[Neil Wyatt]

On Neil Wyatt Said:

…but this thread suggests there would be an audience for it. Any thoughts?

Fine with me – Julie

[peter1972]

On peter1972 Said:

…when I was doing FFT, I used a Hanning Window to taper the edges of a series of signal samples. This can improve the FFT result.

I’ll let you explain what advantages windowing has and the common window functions.

Julie

[Michael GilliganParticipant @michaelgilligan61133]

 

Different sticker, same concept:

https://youtu.be/9WmHjv5HGac?feature=shared

.

[SillyOldDuffer]

On SillyOldDuffer Said:

Another issue, perhaps the mathematicians can help?   My maths limit is at the Complex Number / Calculus level.  I know what a complex number is, and why they are useful, but am hazy about applying them.  (Broadly, a complex number has a real part and an imaginary part, very helpful when calculating rotating problems, like vibrations, if you understand it.)  In this example I told GNU RC to work with floats, seems OK, but doing so must lose information, which might cause trouble later.  As unnecessary guessing isn’t engineering can anyone explain when it’s OK to model in floats and when complex numbers become vital?   Please be gentle, I’m out of my depth!

Complex numbers are fundamental to engineering, electrical in particular. The Fourier transform is defined in terms of complex numbers. So complex numbers are the default. Of course measurements are in real numbers, so if the input data to a FT is real some characteristics of the output can be specified. One interesting consequence is that if the input data is real and even then the output data of the FT is also real and even. A function is even if it is symmetric around zero.

I’m not sure I understand the question regarding floats? I would take a float to mean floating point, ie, mantissa and exponent, as opposed to an integer. It’s about arithmetic representation, not whether the number is real or complex. A complex number consists of two real numbers, however they are represented in the computer.

Using floating point allows a larger range of numbers to be used, but can lead to loss of precision in some operations. Traditionally an integer is represented by 32 bits. Single precision floating point is also represented by 32 bits. The exponent takes some bits, so the mantissa is less than 32 bits and hence can be less precise than an integer. Double precision floating point uses 64 bits and is the norm for calculations.

The question of precision versus range in engineering calculations is a huge one and is subject to ongoing debate. There is no clear cut answer to what number representation is best. To some extent it depends upon the nature of the calculations being made.

Julie

[garyashParticipant @garyash]

What an interesting thread. I’d like to know if the tools would be able to handle very low frequency input (0.1-1 Hz) as I am looking for a means of measuring structural performance of buildings excited by large complex low frequency forces.

On a more practical note, there are some very practical vibration measurement tools like the Ultraprobe

https://www.uesystems.com/en-gb/product/ultraprobe-9000/?srsltid=AfmBOoonjZdG4mB8qIjR52SO_emudNsA9ca7kNqOimfDjXlVVReLUgeF which present either a whole spectrum, or particular frequency as a level meter and as a sound frequency shifted into the audio for headphones. The aviation headphones supplied also act as ear defenders so it is possible to use the equipment to ‘listen’ to machinery even when the audio frequency levels are deafeningly high.

The microphone head can be replaced with a contact sensor and probes, and the gun touched onto the machine to find the loudest (ultrasonic) source. For condition monitoring purposes, repeated measurements are taken at the same points and analysed to find the point when the equipment is about to fail. For many types of equipment including rolling element bearings, the noise level follows a sort of bath tub curve – gradually rising with wear through it’s life, shifting to rapid increase as it nears failure. Of course it is also possible to do the detailed frequency analysis to determine if the damage is to a ball, impressions on inner race, the outer race, etc – but the outcome is much the same in all cases, time for a new bearing.

These devices aren’t cheap, but using the software described above one ought to make something comparable.

[Robert Atkinson 2Participant @robertatkinson2]

The software will handle low frequencies but a sound card won’t. However you can get a low frequency ADC with good resolution at fairly low cost.

[Nigel Graham 2Participant @nigelgraham2]

I read through that thread having seen I was cited as one of its inspirations!

I worked in acoustics, as a laboratory assistant, measuring things for them as knows about the analysis. I just gave the scientists the files of numbers, and lots of ‘Excel’ graphs. So although I could never have learnt the mathematics used by harmonic analysis, with all its advanced trigonometrical functions, matrices, complex numbers and Fourier Transforms, I do know the basics of signal measuring and have heard of those Very Hard Sums!

In later years we used PC-mounted virtual systems such as shown above; LabView the brand, I think similar to Matlab. It allowed building emulators of the various instruments graphically: a sort of 2D CAW. (Computer Assisted Wiring!)

.

I see in the example above, the waveform is the Cosine. Not the Sine wave, given the two are the same thing 90º apart. Is it to give a (0, 1) initial co-ordinate rather than (0, 0) for the calculations? (We measured everything by deciBels, as those signals shown are, and those need a mimimum 1-unit for the specific scale’s Reference Level to avoid the impossible division-by-nowt.)

.

Frequencies <1Hz: (Gary Nash’s question). This is in the realm of accelerometers for the probes themselves but the analytical tools will be the same. The technical difficulty may be in finding a suitable amplifier for the sensor but they do exist.  The lowest frequency I worked down to was about 10Hz at a push, but we were really only interested in well above that, and such a low frequency was a calibration minimum for piezo-electric reference sensors. Their amplifiers are charge types, using capacitance, but I don’t know the principle.

Accelerometers come in various flavours but those for higher frequencies typically consist of a piezo-electric sensor bonded to a mass so it is alternately stretched and compressed by the vibration. Such sensors can be used for diagnosing machinery vibrations, and indeed are.

Some are made with small tapped holes, e.g. 10-32UNF, so they can be screwed to the work, but where that was not possible we stuck them on with “super-glue”. For analysing lathe vibrations the sensor might be fitted to the tool-holder by an adaptor replacing one of the clamping-screws, so as near as possible to the tool tip and the work.

 

I don’t know what those for sub-Hertz sensing use, where a mass-reaction would likely be dormant. They might be capacitative or resistive strain-gauges. The latter at least will handle very low frequencies indeed.

……

Going back to peculiar patterns on turned surfaces, there is a similar thread presently live on this Forum.

I never really traced the cause in my case, on a fairly slender taper in mild(?) steel about four inches long; the stem of a connecting-rod. I have it in front of me now, and though the racoon-tail effect is not very visible the finish is not very good, so the vibration may well have been within the material from a slightly blunt or fractionally mis-set tool.

Another possible source of such trouble, I recall suggested to me, on long self-acting feeds, is the saddle yawing slightly or the feed varying cyclically, by machine wear. This, I now realise, might be assessed by running a DTI attached to the saddle, along the cleaned side of the bed. It will jump in any dents or scratches but a regularly-varying deflection will indicate yawing. One source of that might be eccentricity of the feed-shaft.

This was on an elderly Harrison L5 that has probably seen a lot of unwitting abuse as well as good use over the years.

If the finish varies cyclically though, might the bands be nodes and antinodes of vibrations within the steel? I cannot think how you might establish that, though.

[Michael Gilligan]

On Michael Gilligan Said:

On SillyOldDuffer Said:

[…] For the moment I’ll model 1Hz, 5Hz and 7Hz.  Don’t take these values too literally – just examples for testing.  But if anyone knows the frequencies faulty lathes really vibrate at please say so!

[…]

…

For most vibration analysis, levels are expressed in ‘g’ and the relevant physical displacements vary enormously with frequency … I predict that you will have great difficulty in extracting useful low frequency data from a noisy beast like a lathe.

…

Is g amplitude expressed relative to the acceleration due to gravity?   And if so, is there a practical reason for using g?  I suspect g facilitates the maths sense in a mechanical world, where metal objects are moving, how much and how quickly.   My experience of waveforms comes from electronics and amateur radio, where it’s convenient to express amplitude in decibels (like g but relative to a voltage or watts reference).

At the moment I’m concentrating on frequency and ignoring amplitude.   FFT extracts amplitude too, so I’m hoping the size of a frequency spike on the graph will decide if it needs investigation.  (Guess: only big spikes are bad,)   As the detector is electronic, I’ll stick with decibels for the moment.

Thanks for the hint about the difficulty of extracting low frequency data: you’re probably right!  Robert suggested: For a DIY pickup rather than a microphone the “speaker” from a pair of cheap ear-buds works well and even better if you add a bit of mass to the center of the diaphram. A small nut fixed with something flexible like contact adhesive works well.

I’m some distance from processing real data:  have to get FFT sorted first, and it’s easier to explore it with a mix of known frequencies.

Dave

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