resonance frequencies and harmonics

[Vegas125]

Hello,

if a system has got a basic resonance frequency, it's also supposed to swing in the harmonic frequencies, belonging to the basic resonance frequency, if excited, and these harmonic frequencies are supposed to be whole-number multiples of the first frequency(for example: 2..4..8...12...etc.)?
Should FEM simulations with nonlinear geometry also yield whole-number-multiples of eigenfrequencies? In my calculations the multiples don't seem to be whole-numbers.

 

[MAGNAFEAMAN]

In theory based on a single degree of freedom system, it is calculated to be as such. If you are calculating with ANSYS, do not use the full but instead use a reduced degrees of freedom you're interested in.
The moment you included the mass and shape factor in the analytical model, you have introduced part of the reality and you will never get the integer multiples you are looking for. Also part of the problem is that FEA is an approximation and not exact. You will always have calculation residuals in each of the matrix calculation resulting in variation between harmonics multiple.

Have fun

 

[electricpete]

Should FEM simulations with nonlinear geometry also yield whole-number-multiples of eigenfrequencies?

I agree roughly with the last post, except I would substitute "simple beam" for sdof (sdof only has one resonant frequency... no higher orders modes).

Here is my response, starting from scratch: For simple beam geometries, it sometimes works out that the higher order modes are exact integer order multiples of fundamental. For example simply supported beam. If fundamental mode is at frequency f1, the higher order modes occur at frequencies fk = k^2*f1 where k = 2, 3, 4 etc..

For typical real-world geometries, the higher modes do NOT occur as exact multiples of the fundamental frequency.

Separate, DIFFERENT subject, if you have non-sinusoidal periodic excitation (for example square wave) at fundamental frequency of f1, the signal will contain content at exact integer multiples of the fundamental.


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(2B)+(2B)' ?

 

[btrueblood]

Um, e-pete, I think you meant the frequencies of a simply-supported beam would be integer multiples, not square-of-integer multiples, i.e. the equation would be

f[sub]k[/sub] = k * f1

Eignevalues would step in squares-of-integers, but the freq's. are sqrt(eigenvalues).

That said, I can think of few real-world structures that show such simple behavior. Pipe organ oscillations, some simple torsion systesm, maybe. But even a cantilever beam has frequencies that scale by non-integer values, and real world objects that can vibrate in 3 axes and have combined modes involving all those axes just aren't gonna follow anybody's rules.

 

[electricpete]

I'm talking about lateral vibration of a beam simply supported on both ends. The natural frequencies follow the pattern fk = k^2*f1 where k = 2, 3, 4 etc exactly as I said. Some axial and torsional beam problems act the way you described fk = k*f1 k=2, 3, 4 etc, but not lateral vibration problems. To me "simply supported" beam implies we are discussing lateral vibration.


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(2B)+(2B)' ?

 

[btrueblood]

Sorry, you are correct.

 

[electricpete]

No problem. It probably would have been better for me to pick a simpler example like beam axial vibration that matched the pattern fk = k*f1 k=2, 3, 4 etc.

All responders agree these types of integer-multiple patterns only occur in extremely simple problems, which is not likely in a problem complicated enough to be solved by FE method.

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(2B)+(2B)' ?

 

[GregLocock]

"In theory based on a single degree of freedom system, it is calculated to be as such."

Good grief. SDOF has one resonance, by definition. I can't say i agree with much of the rest of that post except in the most general handwavy terms.

There are comapraitively few multi-dof systems that obey a strict integer (k*f1) harmonic series for their frequencies. Plucked taut strings, and organ pipes, and presumably some other 'transmission line' like systems do.




Cheers

Greg Locock


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[SomptingGuy]

Engine intake and exhaust systems are gratifyingly simple in this respect. If you ignore temperature gradients and bulk flow.

- Steve

 

[Vegas125]

Thanks a lot for the interesting and comprehensive answers!

 

[izax1]

Sorry, I'm taking up this thread again, but if we are talking about "FEM simulations with nonlinear geometry" I would think that we are not getting harmonics at all, no matter how "simple" the beam is. A basic resonant frequency is a linear motion, and cannot occur for a non-linear system (geometry) Please correct me if I am wrong!

 

[electricpete]

I agree with you based on the math definition of non-linear. Maybe op can clarify what he meant by nonlinear (a 2-d non-beam geometry could be considered very dissimilar to a line.... that does not equate to non-linear in math sense but one can imagine the term might be mis-used that way)

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(2B)+(2B)' ?

 

[GregLocock]

Non linear systems certainly exhibit resonant behaviour, that is, forces at quadrature wrt to accelerations, and large amplitudes at some frequencies (confusingly not necessarily the same frequencies).

So I don't get too hoity toity about that.

In the real world lab tests we often see much cleaner transfer functions from a swept sine modal test rather than random input, partly because the system is only being excited at the force level for each frequency.

Of course if you don't call them resonances, all you have to do is invent another word for them.



Cheers

Greg Locock


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[izax1]

I agree that you will get dynamic responses and amplificatiopns in a non-linear system, but if the system is truly non-linear, I would think that you will not get harmonics in the mathematical sense. I think it is important to understand what we are talking about. And I have not yet invented the new word.

 

[hacksaw]

one of the main uses of non-linear systems in a generalized sense is generating harmonics...and well described mathematically

 

[electricpete]

[quoute izax1]I agree that you will get dynamic responses and amplificatiopns in a non-linear system, but if the system is truly non-linear, I would think that you will not get harmonics [emphasis added] in the mathematical sense. I think it is important to understand what we are talking about. And I have not yet invented the new word. [/quote]
Did you mean to say "resonances" rather than "harmonics"

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(2B)+(2B)' ?

 

[SomptingGuy]

Signal processing.

If you are inputing a high amplitude single frequency and driving a system to its bump stops, you'll measure a squared-off response. That will naturally have harmonics of the driving frequency. People get obesessed trying to find the source of these additional frequencies. Somethimes you just need to go back to the time domain to understand a system.

- Steve

 

[electricpete]

SomptingGuy - to whom is your comment directed.

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(2B)+(2B)' ?

 

[IRstuff]

I don't get it, by definition, a linear system MUST have outputs proportional to inputs. Therefore, if you have a resonance, which is an output that is not proportional to the input, the system is, by definition nonlinear.

Likewise, a linear system cannot generate harmonics, therefore, only a nonlinear system can.

TTFN
faq731-376

 

[MikeyP]

A linear system has output proportional to input at a given frequency.

M

--
Dr Michael F Platten