Designing a dynamic vibration absorber 3

[electricpete]

There is a free file available on-online entitled "Dynamic Absorbers for Solving Resonance Problems" about the 20th paper on the list at:

http://domino.automation.rockwell.c...Papers?OpenDocument&ExpandSection=1#_Section1

Page 7 gives a picture of the problem to be solved.

We have a cantilvered bar of length L with weight W2 at distance "a" from the cantilevered end. Bar is width b and thickness "h". The goal is to determine suitable parameters a, b, h, L to make the natural frequency match a specified Nf.

There is an equation posted in the paper. By the way the author clarified to me that the 2nd term should have L^4 (not L) in the numerator... as is seen by the example.

My question:
1 - Where do you come up with that answer.
2 - Putting aside his answer, how would you solve the problem?

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[ivymike]

The Hartog book he refers to calls this device a Frahm absorber, if I remember correctly. Also, if I remember correctly, a Frahm absorber has the following characteristics which I didn't notice mention of in the article:
* makes vibration worse at some off-tune points (he specifically says it has no effect off-tune)
* is adversely affected by internal damping (internal damping of the absorber, by hysteresis or other, will influence the Nf and vibratory amplitude, making it less effective)
* requires a relatively large reciprocating mass (in the neighborhood of that of the object to be damped) or else it will require a huge oscillatory amplitude to cancel the motion of the object

To solve a simplified version of the problem, represent the system as a massless spring (beam) with a lumped mass at the end (a=L). If you put the added mass and the beam mass at the end of the beam, you get:
Nf = sqrt(k/m)
k = F/y = 6*E*I/(2*a^3) (cantilever beam, see Shigley)
m = w2 + a*w
W2 = [3*E*I / (Nf^2 * a^3)] - [(w * a^4)/(a^3)]

trying to match the given result (paper) to this one, as closely as possible for the case where a = L,
subst (3*a^2*L - a^3) = 2*a^3
I get W2 = [6*E*I/(Fn^2 * (3*a^2*L - a^3))] - [2*L^4*w/(3*a^2*L - a^3)]

which doesn't quite match...

 

[ivymike]

The way I'd do it is to perform the "massless beam with lumped mass at the end" calculation, which will underestimate the mass of the system, and thus overestimate the natural frequency. Then I'd assemble a damper, and "tune" it to the right frequency by sliding the weight W2 inboard until the frequency had climbed sufficiently.

 

[ivymike]

I take back part of my earlier statement - he does touch on the need for a large enough absorber, on page 11.

 

[EnglishMuffin]

Just as a point of interest, there is a sort of damped version of this type of absorber, known as a Stockbridge damper, which you often see on high voltage transmission lines near the suspension points. It reduces aeolian vibration.

 

[GregLocock]

I hadn't been able to see that site before, for some reaon.

Undamped dynamic absorbers are not madly popular in the automotive world (doubtless counterexamples will follow), but the tuning principle is shared with the lightly damped harmonic absorbers (as we also call them) we often use.

In practice what we do is

(a) decide what frequency we are trying to kill.

(b) set up a simple, tunable system on an inertial block

(c) tune that system to the frequency identified in (a)

(d) fit the absorber to the original system, and remeasure the response.

At this point things get rather philosophical. We have to decide 1) is the modal mass sufficient? 2) is the damping sufficient? 3) is this the best place to fit the absorber?

FWIW a first iteration demonstration of this is one of the practical exercises our mechanics have to complete to get their top grade, at our noise and vibration lab. It is interesting to consider exactly what proportion of our /engineers/ could do this.




Cheers

Greg Locock

 

[GregLocock]

Perhaps I should also add that in the case of the three dampers I tuned most often: powertrain bending, crank torsional and crank bending, all 3 can be tuned by ear, no instrumentation is required.



Cheers

Greg Locock

 

[electricpete]

Thanks guys.

It is clear there are a lot of ways to solve the problem.

I haven't looked in detail yet but it looks like the approach suggested by inhiding is the right one to get toward the solution in the paper. Understanding that particular formula is what I was really after and I think inhiding has cracked the code.

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[GregLocock]

Oh equations...

(Damn I'm being empirical again)

oh, there's also 4) Do we want to undertune or overtune the absorber? This is strongly tied up with the modal mass decision and the damping available, two times out of three we'd tune the resonant frequency of the absorber slightly under the original target frequency. This suppresses the lower frequency spike at the expense of raising the higher frequency spike.





Cheers

Greg Locock

 

[electricpete]

Greg - In our case (vertical pump with non-vfd induction motor) the excitation is usually single-frequency. I don't understand any reason why I would want to tune to slightly high or slightly low frequency. Allowance for possible future loosening?

What is the logic for overtuning/undertuning in the automative world? I suspect maybe it has something to do with the possibility that the excitation frequency varies and may excite either upper or lower sideband/hump thigamajig.

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[EnglishMuffin]

It's not quite clear to me what Greg is saying either. When you add a vibration absorber to a structure with a pronounced natural frequency, whether damped or undamped, you create two natural frequencies where before there was only one. And the textbook approach in the case of a damped absorber is to choose the natural frequency of the added absorber mass such that the two resonant peaks are the same height. This always requires that the natural frequency of the additional absorber mass is less than the natural frequency of the structure without the absorber, assuming of course that one has optimized the amount of damping. So perhaps Greg is referring to that. With an undamped absorber, you would tune the auxiliary mass exactly to the resonance that you are trying to eliminate. I have designed a number of damped vibration absorbers for use in the machine tool field over the years, and in that world, its always been my experience that the best results are obtained when the two displacement peaks are about the same height (ie classical solution of Den Hartog). Of course, in the machine tool world, undamped absorbers are almost useless since the vibration one is trying to damp out is usually self-excited.

 

[GregLocock]

In automotive world the general response increases with speed, and we have to cope with variations in rotational speed of the order of a factor of 10 (idle to redline), in which the general noise level will increase from say 50 to 75 dBA

So, to some extent the higher frequency spike will be masked by the rising noise from everything else.

There is also the practical aspect that typically the lower frequency spike will be encountered more often, and by more critical customers.

I agree if you have a single operating speed then you'll get the best result with equal tuning, however, even then durig run up you would get a better result with under-tuning. This is analagous to the case with a TV damper - typically our crank resonance is around 300 Hz, so if we manage a 50 Hz split then we can force the upper spike out of the running range of the engine, so, yet again, undertuning is better, since it makes the spike we never see worse, and the one we always run through better.

Does that make more sense?


Cheers

Greg Locock

 

[EnglishMuffin]

Yes - that makes sense - it just wasn't clear to me whether you were actually just talking about "undertuning" the frequency of the auxiliary mass in isolation, which you have to do regardless of anything else with a damped absorber even to get two equal amplitude peaks. In my experience you always have to end up tuning it by checking the total combined response experimentally anyway - knowing the natural frequency of the auxiliary mass/spring in isolation just allows you to get in the ball park. And in any case, Den Hartog's recommended tuning formulae are only exact for a system which is undamped without the absorber present.
"I agree if you have a single operating speed then you'll get the best result with equal tuning" - actually - I don't think I said that, although I don't disagree with it - but I do say that for self excited vibrations you will get the best results with equal tuning. If you have a single operating speed, I think you will actually get better results at that speed with a totally undamped absorber, since you operate in the "trough" between the two peaks - this trough is likely to be higher if you have a damped absorber. But admittedly you then have to contend with the fact that on run-up you must traverse a higher spike - I suppose it depends on how much "running up" you do and how much natural damping the system has to begin with. Of course, on the site which inspired electricpete's original question, the guy was talking essentially about undamped absorbers - as I recall.