Dear Artur,

The answer to your question is yes, for nonlinear systems.

A nonlinear system can exhibit a hardening or softening behavior depending on the kind of nonlinearity, i.e. the natural frequency of the system can go up or down, as the oscillations' amplitude increase. The relationship between natural frequency and amplitude is commonly known as Backbone curve.

As a classic example on this topic I would suggest you look into the Duffing oscillator, for which the backbone curve can be obtained analytically.

Hope this helps.

Best,

G.

Thanks for quick responses.

Is there, according to your knowledge, any fundamental difference between non-linearity caused by nonlinear stiffness (by stress-stiffening or significant change in geometry) and caused by presence of contact?

Specifically, in complex system, is there a method to make a distinction between those two kinds of nonlinear normal modes? Maybe by a shape of backbone?

Hi

As indicated by the above answer, Yes, and I would add - of course.

Amplitude varying stiffness affects, frequency dependent properties matters and makes systems nonlinear. However, contacts tend to be nastier.

It is more the rule than the exception for systems involving gaps, e.g. mechanisms, motors, gears etc.

Take a look here for simulation

https://www.code-aster.org/V2/spip.php?article750&debut_articles=80

and here for experimental evidence of how vibration amplitude and gaps affect mode shapes

Article FE Model Correlation & Mode Shape Updating using Qualificati...

Sincerely

Claes

Claudio,

I'm well aware of material non-linearity, I have made some work for crash in automotive, but in case of vibrations usually it is only the nonlinear material elasticity. In my opinion its not a point.

Your beam with contact in the middle is just was I mean in question, but your interpretation is not correct for me. The beam after contact is more or less L/2 in length, but only in first half of cycle, and this is a point. In the second half of cycle its length is again L.

It should be checked, but idea of my question was if can be any correlation between "sharp" change of direction of backbone curve for such "sharp" kind of nonlinearity as a hard contact (like in your beam).

Artur,

In some of my previous work I used impacts as a source of nonlinearity to implement a nonlinear vibration absorber (Article Experimental investigation and analytical description of a v...

The FRFs I obtained experimentally presented a very complex behavior. In general, impacts introduce a non-smooth behavior in your system that you should be able to find in your FRF. Impacts/contacts do not necessarily make the natural frequency vary with the amplitude (like a cubic stiffness), so I would say that in this case it is much less trivial to find a well-established relationship frequency-amplitude.

Best,

G.

Dear Claes,

I agree with your "off course". My question was a little provocative :)

The picture from your first link (attached) with "sharp" backbone curve is exactly what I mean in my answer to Claudio.

The question is, if can we interpret "sharp" backbone curve as an evidence of dominant role of contact in nonlinear response. Even then we have only backbone curve.

Hi Artur

For the Olympus Satellite Model, the only things that were changed was first a sine sweep on a shaker with one vibration amplitude, then with another. Presumably from low to high amplitude.

Repeatability was tested simply by lifting the structure off the shaker and placing it in the floor next to the shaker, unloading it and lifting it back on.

As you can see, some modes retain their shape while other modes change significantly - also from repeatability.

The global system modes behaved reasonably linear while panel modes changed with amplitude, which is not strange, as the panels were hinged and have gaps at their ends.

I find when doing Experimental Modal Analysis on things like large machinery that the rigid body modes tend to stay reasonably linear, i.e. I am able to identify the same modes when impacting on different parts of the system, while higher order modes very much depend on how/where they are excited, which ties in with the feature of non-linear systems - the order in which you do things may affect the result, not only the energy or amplitude.

Another straightforward example on amplitude dependence is so called bean bag dampers where balls, e.g. are thrown upward by vibration at one or more frequencies, to fall down and impact and thereby redistribute energy across a wide frequency range. The amplitude dependence then, of course, is that acceleration must exceed 1 g for the lift off to occur.

However, you need not go very far before things become indeterminate. Take a look at the FE/FE correlation made here where the same FE model was chewed by two difference (major vendor) codes. Rigid body data matched with all decimals presented in the output data deck & the first ~10 modes match quite well before things start to become more dissimilar than alike - and this is for a 100% linear situation. Presentation Analysis Technology:; A discussion on some of the fundamentals

There is a numerical noise level also for computers. I have been told by people doing multi body simulation that if you execute the same system model N times, you end up with N results if you run the simulation long enough for noise to matter.

It is a big world out there & we are only scratching on its surface.

/C

Claes,

The word is big, but I'm trying to focus on a small part of that. Complete satellite is too complex to see single phenomenon. Interpreting the test results in such case can be more similar to black magic than science :)

For me, the shock/impact, similar to case mentioned by Giuseppe, is something slightly different than hard contact of vibrating structure with support. Impact always will generate wide spectrum excitation. Literally, moment of contact of structure with hard support is an impact too, but it is a little more gentle than just hit with the hammer. For that reason I hope to find some correlation with the backbone curve shape.

" the rigid body modes tend to stay reasonably linear "

I'm a FEM man, for me rigid modes are translations and rotations of structure without any deformation. Do you mean global modes?

" the first ~10 modes match quite well". It is well known effect. It is just a result of numerical iterative algorithms used in linear modal calculations. Every next mode is less accurate, and from my practice, looking at modes higher than 10 very risky and require big experience, cannot be just taken as reasonable ones.

The numerical noise you mentioned is an issue in complete other kind of simulation - dynamic/transient response of structure, especially with explicit approach.

Hi Artur

Correct, global modes is a better phrase. I had gensets in mind when writing my sentence and the global modes of these tend to be rigid body modes.

I do both FE, Test, SEA and Theory. As a consultant, I do not get to choose what tasks to deal with - they get presented to me and I must address them in 'realtime' mode.

I doubt the 10 modes rule-of-thumb can be explained only by solver precision as you then should get different results from running the same data deck through the same solver on the same computer and, in my experience, you get the same results in such situations.

That said, this was not the case when solving a linear case for complex modes. When asking the supplier, I got the response that "Solving for complex modes is a hobby for gentlemen", which I accepted. The situation improved in later releases.

/C

Hi

One more example as it is a fun one - forgetful modes.

Modes are the result from standing waves, i.e.reinforcing wave interaction. Nonlinear wave steepening creates shock waves (aka freak waves, monster waves or rough waves) where the hill travels faster than the valley. The wave front steepens until the hill catches up with the valley, at which point the wave collapses.

http://qringtech.com/2010/09/15/wave-steepening-increase-peak-pressure-piping-pumps/

Now, the fun thing is that the wave forgets what it was doing before the collapse. When measuring FRFs and this occurs, signal coherence suddenly is gone.

This implies that you can add also a memory effect to modes. The mode is there until response grows large enough for it to forgets to exist and the build up starts anew.

If you want a clean and neat thing to run in the lab as well numerically, this is one such case.

Have fun, because this is fun.

C

Hi

The number of modes you use affects convergence for the modal summation.

You should have enough of them or you end up sacrificing also precision on weight (mass/inertia) and static stiffness. In this respect, it does not matter of the other modes are wrong and not including them otherwise makes other, correct, aspects of the model incorrect.

Nastran has convergence checks for such. The number of modes depends very much on the situation, for a suspended system, you need lots of modes to get convergence on strain while weight converges almost immediately and, vice versa, the reverse is the situation for a cantilevered case.

Using static modes (Craig Bampton) or residual correction greatly speeds up convergence at the cost of a slightly more complicated analysis sequence.

I am afraid that there are no free lunches on this.

/C

Claudio

I made my PhD work on the use of FE for SEA.

You skip a bunch of things to avoid the use of modes, which does not really save you.

The enclosed presentation presents the basis as to when and why it makes sense to give up on analysis of deterministic properties.

/Claes

Claes,

a lot of posts:)

" I doubt the 10 modes rule-of-thumb can be explained only by solver precision as you then should get different results from running the same data deck through the same solver on the same computer and, in my experience, you get the same results in such situations. "

I didn't said "solver precision" but "iterative algorithms". You will always get the same results on the same machine, but results will by more and more distant from real ones. On other solver you can get different ones, because of slightly/significant algorithms.

"... hill travels faster than the valley. The wave front steepens until the hill catches up with the valley, at which point the wave collapses."

I spend some time of military test range and blast/shock wave is what I like very much, but your description is a perfect definition of great wave for surfers :)

" The number of modes you use affects convergence for the modal summation "

Its true, if you are using FEM for modal analysis. Then you have of course worry about modal mass and get high enough number of modes. Exact frequencies and modal mass or higher modes are fortunately not so important. Anyway, modes above 10 are less exact.

The problem is significant in completely other area of FEM usage - in structural analysis and resonance avoiding. In complex structure one can be interested in exact resonance frequency of small part of big structure with normal mode number 35 - in this case it is not exact at all.

Hi Artur

A dark hole in my knowledge set - good. :)

I must admint that I never tried very many different solvers. Lanczos, SOL 103, it was.

/C

:)

Well, this what RG is all about as I see it.

Thx

C

upside down (inverted) pendulum?

The larger the amplitude, the more off center is the mass... If stiffness is low enough, at an amplitude large enough, it stops osculating all together.

Also - sand. Especially wet sand, fully saturated, under water. Sand stiffness drops as amplitude of deformation increases.

Smaller deformation - larger stiffness (and less damping).

Larger deformation - smaller stiffness (and more damping).

Stick a metal Ruler in sand, and push it left - right with large deformation, to create "disturbed soil state". Clamp pliers on for "mass", and allow to oscillate. Observe as stiffness will gradually recover, and the frequency of oscillations will go up. To drop stiffness - apply large deformation again.

Feel free to publish that paper in international geotechnical journals. It's a new thing. Offshore wind turbine foundation researchers need it. "deformation dependent sand stiffness properties" is the next big thing in that paradigm.