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Do resonant frequencies produce harmonics
2

Do resonant frequencies produce harmonics

Do resonant frequencies produce harmonics

(OP)
Hi,

I have a plate vibrating due to some forced excitation. When I take FFT of measured acceleration signal on the plate, I am getting a dominant mode and its harmonics. I see no link to this dominant frequency and exciting frequency (they are very wide and far). But the dominant frequency in FFT is very near to one of the resonant frequencies of the structure.

Before I conclude that the external excitation is exciting one of the resonant modes, I want to know if a resonant mode ever displays harmonics in FFT of acceleration signal.

Thank you.

Regards
Geoff

RE: Do resonant frequencies produce harmonics

Yes, sort of. In a non linear system you often get 3rd, 5th etc multiples of a strong frequency, due to contact/clearance/rattles.

What you shouldn't get is much response at frequencies other than harmonics of the drive signal. It is possible, but you need a rather elaborate non linearity to get that.

Oh, and stick slip can cause some pretty odd responses.

If you do a sine sweep and plot a waterfall of the response then things may become clearer.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?

RE: Do resonant frequencies produce harmonics

(OP)
Hi Greg,

Many thanks for such a prompt and useful response. Happy to see the answer! But as I said I am noticing dominant response and its harmonics which are no way connected to exciting frequency. In numbers, I am exciting at 26.6 Hz but I am finding dominant response around 4870Hz and then 9740Hz, etc.

There is a resonant frequency for the structure at 4874Hz. So I am wondering even if resonant mode gets excited, will it show harmonics like this ? What could be physical meaning of these harmonics ? Structure is vibrating at multiples of its mode shape ?? Not fully convinced.

Thank you.

Kind regards
Geoff

RE: Do resonant frequencies produce harmonics

(OP)
Greg, Thank you again.

Unfortunately I have only impact hammers and few accelerometers.

I have the above finding by doing FFT of acceleration signal with one of the accelerometers mounted on structure.

Any other ideas ?

Kind regards
Geoff

RE: Do resonant frequencies produce harmonics

Geoff
Some questions for carification.

Is 4870 Hz the lowest mode of the plate (is it the 1st natural frequency) or are there lower modes that you have missed in the measurements?
Is the plate only a square flat plate?
What are the boundary conditions?

To me it seems like something is omitted in the test (input or output)I cannot see where you should get much non-linearity in the sytem you are describing.

RE: Do resonant frequencies produce harmonics

(OP)
Hi izax1,

Thank you.

I am sorry. I should have written shell in the first place. It is a 2mm thin shell of 100mm height, clamped at bottom. By mistake I mentioned as plate as on a theoretical perspective I am looking at it as a curved plate. This shell is being excited at the top. By mounting the accelerometers on shell I am acquiring the vibration signal and doing FFT of it.

4870 Hz is not the 1st mode. Fundamental frequency is around 1300 Hz. When I did modal testing on structure, I got all the modes till 8000 Hz. Compared with FE also.

In fact there is some non linearity in structure. I won't get identical drive point FRF even if I hit at same point.

Puzzled with this.

Kind regards
Geoff

RE: Do resonant frequencies produce harmonics

(OP)
Greg, Thank you.

A milling tool with 2 cutting edges is rotating at 800 rpm and is machining it at top of shell - so basically the cutting edges are impacting at 800 * 2 = 1600 rev/min which translates to 1600 / 60 = 26.6 Hz.

Being a circular shell the tool moves along the periphery of the shell at feed of 0.2mm / rev of tool while rotating at 800 rpm all the time.

When the acceleration signal acquired on the shell near to machining zone was converted through FFT, I get dominant response peak at 4870 Hz and its multiples.

Geoff

RE: Do resonant frequencies produce harmonics

this only gets better...

have you given any thought to the frequency content of a sharp impact generally?

RE: Do resonant frequencies produce harmonics

(OP)
hacksaw, Thanks for your response.

In fact whatever frequencies I am saying as dominating are observed for one impact only. That is, I took FFT of time signal corresponding to one impact - start to end of impact.

Geoff

RE: Do resonant frequencies produce harmonics

a sharp impact can excite more than one mode

RE: Do resonant frequencies produce harmonics

I think there are two possibilities:

1 (unlikely imo) - the system may have two resonant frequencies which happen to differ by a factor of 2 (would be quite a coincidence for anything but very simple system).

2 - the high vibration at 4870 is interacting with some non-linearity of the system to create non-sinusoidal motion with fundamental frequency of 4870 and (since non-sinusoidal), harmonics of that fundamental frequency so 2*4870 = 9740.

To tell the difference, try taking a time waveform sampled at much higher than twice the highest frequency of interest. Zoom in on the time waveform and see what you've got. The difference between 1 and 2 would be obvious in time waveform.

=====================================
(2B)+(2B)' ?

RE: Do resonant frequencies produce harmonics

(OP)
Thanks for all your replies.

electricpete - By 'fundamental frequency' do you mean this is first mode? It's not. As said earlier, the fundamental mode is around 1300Hz.

Reg non-linearity - the structure is held with bolts : 8 no.s M6 and 16 no.s M12

Do you mean we see harmonics whenever non-sinusoidal vibration exists ?

I am attaching the zoomed in time waveform and its FFT. I have marked the frequencies also. This is not constant always (i.e. depending on which cutting tooth impact I analyse for FFT). It moves anywhere from 4800Hz to 4880Hz. But mostly occurring around 4870Hz.

When I did experimental modal analysis in that same set up I found two similar mode shapes at 4790Hz and 4874Hz. In fact they are symmetric modes of same mode shape. I think due to nonlinearity these symmetric modes are spaced that apart. But I don't know out of all the modes available what makes this 4874 mode favourable.

Many thanks & kind regards
Geoff

RE: Do resonant frequencies produce harmonics

Your waveform shows an impact-like event at 17.0065 followed by another at 17.0457
Spacing is 0.0392 or roughly 25.51 hz which is roughly the 26hz frequency you mentioned.

I’d like to see zoom in much more so you can see a waveform oscillating at 4870 and see to what extent is is sinusoidal. For example, zoom in the high-amplitude time region from 17.0125 17.015 and you’d get about 10 cycles of 4870 so it’s spread out enough to judge it.

By "fundamental frequency", I meant not the first resonant frequency, but the frequency at which [assumed] periodic vibration repeats.

Quote:

Do you mean we see harmonics whenever non-sinusoidal vibration exists ?
Yes (IF the vibration is periodic... a characteristic which is a traditional assumption of spectrum analysis...without the assumption everything is murkier).

=====================================
(2B)+(2B)' ?

RE: Do resonant frequencies produce harmonics

(OP)
Thanks a lot for your support electricpete.

Please find attached the two zoomed signals from 17.0125 to 17.015 and 17.0225 to 17.0250. First one is when the impacting tool is in touch with structure. Second one is when tool has left structure and workpiece is vibrating freely (of course due to previous forced vibration). They don't look purely sine.

Sampling rate: 1e6 points/sec

Also I took FFT of the acceleration in these bands. In the first band you can see the peak shows 4804Hz and its multiples. The next point on frequency spectrum is again 5204 only. In second band, 5196 is dominant, but 4796 and its harmonic 9592 are also clearly seen. As said earlier the two symmetric modes are at 4791 and 4874 Hz. So not quite sure if structure is vibrating at one of these.

Now if this peak varies like this from 4869Hz to 4804Hz to 4796Hz, etc. can I consider that it is a resonant vibration of structure - this is my main focus.

Kind regards
Geoff

RE: Do resonant frequencies produce harmonics

I think you are overanalysing this situation. I think you can see where the 4800 +/-200 Hz is coming from the question is where is the 9600+/-200 Hz coming from.

Have you tried a modal with the cutter in contact with the workpiece?

If we ignore health and safety, can you try a modal on the running system?

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?

RE: Do resonant frequencies produce harmonics

if you are dealing with a 100 mm dia s.s. cylinder having a 2 mm wall, the circumerential modes fall in a sequence like

i fo(Hz)
2 524.2
3 1482.4
4 2842.3
5 4596.5
6 6742.8
7 9280.6

what are the dimensions of your work piece


RE: Do resonant frequencies produce harmonics

How is your accelerometer mounted?

=====================================
(2B)+(2B)' ?

RE: Do resonant frequencies produce harmonics

(OP)
Greg, hacksaw, electricpete - I am thrilled and thankful to see your support.

Greg -

Are you saying a resonant mode can appear within +/-200Hz band during operational condition ?

By modal in running system, do you mean operational modal analysis ? I wanted to do this, but I have only two suitable accelerometers. So did modal by impact hammer. So at a given instance I may be able to acquire only a reference and one additional acceleration signal. So I think it is difficult to do operational modal.

hacksaw -

Dimensions of workpiece: 365mm OD, 2mm wall, 100mm wall height, material: nickel-based superalloy.
I did some analysis on circumferential modes of this casing, of course for FE modes. They fall in a pattern as shown in attached document. This is in-line with shell behaviour explained in textbooks.

electricpete -

What is your comment on sinusoidal behaviour of signal? I am inconclusive due to the nature of peaks. Do you think they are genuine harmonics?
Please see the actual structure and accel mounting in attached document. Please note that the accelerometers are in-fact mounted close to the top edge than shown in figure. Signal I have analysed is exactly behind that of accelerometer when tool is cutting at that point.

Many thanks to you all.

Kind regards
Geoff

RE: Do resonant frequencies produce harmonics

The +/-200 comes from the frequency resolution of your fft. You know absolutely nothing about that signal to a resolution better than +/-200 Hz, with confidence.

Incidentally, are you using a hanning window? and an AA filter?

Stick the accelerometer on the workpiece and hit the workpiece with your hammer. Do this with the tool touching the workpiece, without the tool touching the piece, and ideally, with the cutter running, ie 3 tests.

You are looking for frequencies, not amplitudes, so this last one can be rough and ready. You might look into a gated (synchronous) analysis if you want to get fancy.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?

RE: Do resonant frequencies produce harmonics

(OP)
Greg, I think +/-200 sounds too wide. I sampled the data at 1e6 points/sec. Then how come the resolution can be so low? Or do you mean something else?

Yes I am using hanning window. No filter.

I understood first two cases - tool touching and not touching. But what about 3rd one? If tool is touching and rotating then within no time it removes material and then it is equivalent to not touching (except for some rubbing at times).

I did synchronous analysis earlier with tool cutting force and acceleration signal and saw the evolution of 4870Hz as the tool excites the workpiece, and then gradual diminishing of the peak. But it didn't answer my question as to why that mode only is getting excited and mainly why harmonics are present for a resonant mode.

Thank you.
Geoff

RE: Do resonant frequencies produce harmonics

On the zoomed-in time waveform, you can see the binwidth of the spectrum is 400hz. For example 5204-4804 = 400hz.
This is 1/T where T = 17.015-17.0125 = 0.0025 sec (1/0.0025sec = 400hz).

Apparently when you zoomed in you limited the duration of the data input to the FFT, which made your bandwidth bigger.

You can get much higher resolution if you keep the time record longer. In fact you can get much higher resolution (theoretically unlimited except by numerical error) simply be zero padding.
thread384-208992: "Interpolating" to estimate exact frequency of peak from FFT results
Note resolution is not the same as accuracy. Accuracy in estimating frequency is complicated.

Quote (electricpete)

1 (unlikely imo) - the system may have two resonant frequencies which happen to differ by a factor of 2 (would be quite a coincidence for anything but very simple system).

2 - the high vibration at 4870 is interacting with some non-linearity of the system to create non-sinusoidal motion with fundamental frequency of 4870 and (since non-sinusoidal), harmonics of that fundamental frequency so 2*4870 = 9740.
Looking at the zoomed-in time waveform, it doesn't look like a non-sinusoidal waveform repeating at interval of 1/4870 (which would be #2). The phase of the 9740 and higher frequencies seems to be drifting with respect to the 4870, as if they are not exact multiples of 4870. That would seem to argue for #1, which doesn't make much sense sense to me. Still thinking.

The reason I asked about accelerometer mounting. There could be looseness introduced there. Ordinarily I'd think that would fall in the category of #1 and so doesn't really seem to be the case from your time wavform, but I don't to be too clever and rule it out by what I think it should look like. Sometimes looseness acts strangely.

Bottom line, I don't know what's going on with your data.

=====================================
(2B)+(2B)' ?

RE: Do resonant frequencies produce harmonics

Correction
which made your bandwidth bigger
should have been
which made your bin-width bigger

=====================================
(2B)+(2B)' ?

RE: Do resonant frequencies produce harmonics

(OP)
electricpete,

Many thanks for your detailed reply. It's very useful. Though I understood your bin width concept, I will look into the thread your forwarded.

I use to do zero padding to reaching to nearest 2^n when using FFT. But I never realised we can increase resolution. I am still wondering if it is ok to keep on adding zeros to increase resolution. Is there no limit to this? Anyway I always have a basic doubt about amplitude of FFT which I will post as a separate thread in this forum.

You commented
The phase of the 9740 and higher frequencies seems to be drifting with respect to the 4870, as if they are not exact multiples of 4870.

Can you briefly explain how you made this observation ? I see that there is some drift in double peaks and reduction in amplitude. But not clear and sure.

Accelerometer was mounted with superglue. I think the bond was very good and no looseness here.

Bit saddened to see your bottom line ! I was also so much puzzled with this that after all analyses failed I resorted here. Anyway learnt many good points here. Thanks a lot for that.

Kind regards
Geoff



RE: Do resonant frequencies produce harmonics

You don't get something for nothing. Zero padding does not affect the confidence of the estimate, although the cynic in me suggests it increases the likelihood of it increasing my confidence that the answer is wrong.

dF=1/T is an energy or information based statement. Adding zeroes adds neither energy nor information to the data.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?

RE: Do resonant frequencies produce harmonics

You're right, you don't get something for nothing. With the original short length FFT resuls (including phase), we can reconstruct a continuous function of magnitude and phase vs frequency. It is a sum of weighted, frequency-shifted continuos sync functions, one term for each FFT point). As a continuous function, the peaks are single values (not intervals). The individual FFT magnitudes represent samples of that continuos function (because all the other sync functions contribute zero at that particular point).

The resolution (not accuracy) is virtually unlimited because we can determine that single value. Accuracy is a different but related question. On average, the single value which represents the peak of the underlying continuous function is our most accurate estimate of the true frequency of the peak . If and engineer should just simplistically look for the highest peak and declares that bin center to be the peak frequency, he is using a less accurate estimate. It may be good enough, but it is not the most accurate estimate available from the data.

Do we gain something for nothing when we use the more exact estimate? No, we simply don't throw away information. The engineer who looks only at the single highest peak and ignores the rest of the spectrum is throwing information away.

A simple example.
Bin Center / Magnitude
100hz / 0.1
101hz / 0.2
102hz / 1.0
103hz / 1.1
104hz / 0.2
105hz / 0.1.

Where is the peak? The simplistic approach is pick the highest value. That would be 103hz.
I think a little intuition would tell us that 102.5 is a more accurate, and intuition would be correct.
We improved our estimate by not just looking at one peak but by looking at the neighboring peak. Although the peaks furhter away have less inflence on our estimate (the sync function disappears as we go away from the center), each additional point we consider from the FFT result improves our estimate. Best possible (most accurate) estimate comes from adding all the FFT contributions to determine the underlying continuos function and determine its peak.

=====================================
(2B)+(2B)' ?

RE: Do resonant frequencies produce harmonics

Just to step back, starting with a given time record, we can improve our estimate of peak frequency two roughly equivalent ways
1 - Time domain zero padding prior to FFT.
OR
2 - Take the original FFT results (a function over a discrete domain of frequencies) and convert it into a DTFT (a function over a continuous domain of frequencies). This can be accomplished using equation 6.17 and 6.18 here:
http://ens.ewi.tudelft.nl/education/courses/et2405...
Once we have the DTFT (a complex valued function over continuos frequencies), we can take it's magnitude (real valued) and apply the same peak-finding techniques we would use for any real-valued function on a continuos domain.

Further down on the page the expression P(w) is shown to be a sync function.
The DTFT will be sum of weighted frequency-shifted sync functions, one per FFT output point.
The FFT output magnitude and phase is the weight.
The sync function is centered on the bin center and decays away from there.
The first zero of the sync occurs one bin width away, then two bin-widths etc.
Maybe this supports my previous discussion better.

But also we should note the sync function so defined is relatively smooth. Therefore simpler interpolation mechanisms that fit a smooth curve through a few neighboring FFT points (for example quadratic interpolation) work pretty darned well (a lot more accurate than just looking for the highest FFT ouptut magnitude, not quite the best estimate, but but a lot less work then the full calculation described previously)



=====================================
(2B)+(2B)' ?

RE: Do resonant frequencies produce harmonics

all well and good, but you are no better off unless you have more data,

RE: Do resonant frequencies produce harmonics

I agree 100%, longer time record data will of course give better ability to determine frequency peak accurately. The point I was trying to make is that for a given time record (and associated associated FFT output), the approach to obtain the most accurate estimate of the frequency of a peak is NOT to simply look for the highest FFT magnitude and select the center of the corresponding frequency bin. It is a common approach, good enough for many purposes, but it is not the most accurate approach. When reviewing FFT's from already-collected condition monitoring data in our E-monitor database, attempting to determine the source of peaks such as bearing defect-related peaks, having a "peak label" feature which interpolates rather than giving the bin center is simply invaluable in getting the most out of the data you have available. I'm sure that anyone who has used such a tool in such a task would agree.

=====================================
(2B)+(2B)' ?

RE: Do resonant frequencies produce harmonics

Quote:

The phase of the 9740 and higher frequencies seems to be drifting with respect to the 4870, as if they are not exact multiples of 4870.

Can you briefly explain how you made this observation ? I see that there is some drift in double peaks and reduction in amplitude. But not clear and sure.
Attached is an example of what I mean.
The tab labeled "calc" accepts inputs A1, A2, A3, F1, F2, F3 and plots the waveforms:
y1 =A1_*SIN(2*PI*F1_*t)
y2=A2_*SIN(2*PI*F2_*t)
y3==A3_*SIN(2*PI*F3_*t)
Ytotal=y1+y2+y3

In tab "HarmInPhase", the harmonics F2 and F3 are exact multiples of F1 and you can see they maintain a constant phase relationship to the fundamental and as a result the waveform repeats itself identically for each time period T = 1/F1.

In tab "HarmInPhase", the "harmonics" (not a great term) F2 and F3 are close but exact multiples of F1 and you can see they that their phase with respect to the fundamental drifts continuuously. As a result, the waveform does not repeat itself, but evolves slightly each period 1/T1.

Your waveform looks more like HarmDrift than HarmInPhase to me.

=====================================
(2B)+(2B)' ?

RE: Do resonant frequencies produce harmonics

Quote:

all well and good, but you are no better off unless you have more data,
More data is better.
For a given amount of data, you are in fact better off (in terms of accuracy in estimating frequency) to use one of the approaches I have suggested (for example quadratic interpolation based on the highest magnitude bin and one neighbor on each side) than to simply select the frequency of the center of the bin with the highest magnitude.

Out of respect to OP (unless he has interest to discuss this further), may I suggest any further discussion on the question of frequency resolution be in a new thread (I would be very happy to particpate).

=====================================
(2B)+(2B)' ?

RE: Do resonant frequencies produce harmonics

(OP)
electricpete,

I gave a try to zero padding concept with my data. Attached are the results in excel sheet.

As you can see, padding with zeros has effect on frequency and amplitude both. I used my data which we looked at earlier i.e. 17.0125 to 17.0150. Added zeros on left, right and equal on both sides. Looked at amplitude and frequency of first peak i.e 4804 in unpadded data.

While padding zeros on any side has no effect on frequency, amplitude is less effected (relatively) with equal padding on both sides. Still there is a significant reduction in amplitude.

With all these I feel zero padding may not be useful as neither the frequency nor the amplitude reduction converges.

I have another basic doubt on amplitude of FFT which I will post in a separate thread.

Thank you.

Kind regards
Geoff

RE: Do resonant frequencies produce harmonics

(OP)
Sorry electricpete, just now I saw that you advised posting in new thread on 'Frequency resolution and amplitude'. I will create one and please reply there.

Thank you.

Geoff

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