Is a tuned damper effective in forced vibration? 5

[GMarsh]

Hi,

I have a structure on which a tuned damper is mounted, tuned for one of the resonant modes of structure. Now this structure is also being subjected to a forced vibration.

Now I am interested to know if the tuned damper will have effect on minimising forced vibration (because of added mass or some other effect)? If yes, how does the effect vary with amplitude of forced vibration?

I am also interested in simulating this through FE ? Any idea how to apply forced vibration with known frequency and amplitude in FE?

Thank you

Regards
Geoff

 

[GMarsh]

Hi all, it's a quite a good learning here.

Greg, thanks for your simple and short explanation.

Denial - Thanks for referring me to the formula and sharing your experience. I could see the 3 pages (169, 170 and 172) of the book in Google books. So I saw the formula you wrote. Having known the ms, do we iterate on mt till we approach desired damping factor ? And then the damping factor so reached is called 'optimum'. That means below that corresponding mt, mass of tuned damper would be insufficient and above that the excessive inertia force the mt causes is of no use. That is incremental benefit is less. Am I right in my understanding?

And this mass based damping will be over and above the loss factor of the damping material we use, such as viscoelastic tape. Right?

While verifying the formula with your spreadsheet calculation, what output did you consider for comparing? Is it frequency response amplitude (receptance) ?

I have few more doubts on what you said regarding pedestrian bridge, but I reserve them considering the length of this post.

Many thanks for all your support.

Kind regards

 

[Denial]

You're beginning to ask me to ask too much of my memory, since I no longer have access to any of the work I did on that pedestrian bridge. And as your questions get more specific we begin to hit up against differing terminologies between structural and mechanical engineers. (Like, for example, "mass based damping".)

The model from which the formula comes is a relatively simple 2-dof model, a diagram of which is in the reference. The "primary structure" will not be a simple lumped mass sitting on a spring and a dashpot, that has to be capable of being reduced to the simpler form. If you have several different forms of damping in your actual structure (general structural damping plus "viscoelastic tape"), then maybe your structure cannot easily be reduced to the required simple form.

My "verification" of the formula consisted of applying a unit harmonic force to the spreadsheet 2-dof model, for a range of frequencies for the force (the range well covering the resonance values). That gave me the classic two-peaked response-versus-frequency plot. I repeated this exercise for several values of the assumed TMD damping. The damping value that gave me the lowest upper-peak-height corresponded with what the formula predicted. QED.

 

[electricpete]

Den Hartog's Mechanical Vibrations 4th ed also has a derivation of the same "optimum" damping (equation 3.36)

His "optimal" approach resulting in his equation 3.36 is a two step process:
step 1 - select an optimum frequency ratio based on the mass ratio: Fabsorber/Fsystem = 1/[1+(Mabs/Msystem)]
step 2 - select an optimum damping as given by Denial

Note that part 2 is predictated on part 1. i.e. it is not optimal damping for any absorber you throw in there, it is only optimal for the particular absorber that is tuned per step 1.

The sense in which the damping is optimum is that it minimizes the highest peak for any frequency of the damped 2DOF composite system. It may not be necessary to worry about any frequency if you know where your frequency of excitation will be and can tune accurately and use large enough absorber to provide sufficient excitation.

When compared with the undamped model, the damped absorber is much better at the frequencies of the undamped system poles (of course) and worse at the frequencies of the undamped system zero (of course). Whether that is the desired target I guess depends in part on your situation. For fixed speed machinery I think it may not be. The undamped absorber will do the best possible job of any absorber at the tuned frequency. It just does really bad at those adjacent frequencies of the poles. The undamped is the only kind I have ever installed (I work on fixed speed machines). Damped might possibly be better for some purposes, but it's a little foreign to me.

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(2B)+(2B)' ?

 

[electricpete]

Correction:

and use large enough absorber to provide sufficient excitationfrequency separation

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(2B)+(2B)' ?

 

[electricpete]

Some more thoughts. It’s not soley for your benefit....it’s helpful for me to summarize/repeat as I read through the information.

Attached is an Excerpt from Den Hartog’s Mechanical Vibrations 3rd Ed.
It is the same as the 4th ed discussed above except the equation numbers changed.
Equation 63 is the step 1 equation for “optimum” frequency of the absorber.
Equation 69 is the step 2 equation for “optimum” damping of the absorber.

By the way, I downloaded the entire Den Hartog’s 3rd Ed at one time from the Open Library Archive. It’s no longer available there for some reason, but Timoshenko vibration book is still there available for free download

http://archive.org/details/vibrationproblem031611mbp

Timoshenko section 41 is dynamic absorber. He uses different symbols. He uses a lot of the same analysis as Den Hartog. I saw he came up with the step 1 equation in his equation 81. I didn’t see where he got to the step 2 equation (I got a little lost along the way).

The big picture contrast between undamped and damped dynamic absorber I mentioned above:
* undamped absorber has ideally zero response at the tuned frequency and infinite at the nearby poles. It is good if we have fixed exciting frequency and are confident through design of our absorber that frequencies of excitation and poles won’t drift too close together.

*damped absorber would make more sense for broadband or variable frequency excitation.

Some new ideas:
* The optimum design of the damped absorber relies on assumptions about the system you attach it to. How will it change if attached to system other than SDOF is a little unknown to me.

* In contrast, it is easy to predict ideal performance of undamped absorber regardless of the system you attach it to, in the following sense: The undamped dynamic absorber ideally puts the vibration at point of attachment to zero regardless of the system you attach it to. That’s a pretty simple result, made more valuable imo by the fact that the proof is pretty simple:
1 – you can prove it using electrical analogy to vibration of the flavor described in the quote at end of this post. In that case, the absorber is a resonant series L/C circuit connected to ground... which represents a short circuit to ground (zero electrical impedance [infinite mechanical impedance]. It will drag the voltage [velocity] to ground at that point, but possible at the expense of very high current [forces]. We also know true short circuit rarely exists, but still a useful tool for understanding the ideal limiting behavior.
or
2 – Look at transfer function Dm(w) / Db(w) for undamped SDOF system where Dm is mass position and Db is base position (input). It becomes infinite at w=wn. ASSUME (proof by contradiction) that we have motion at the base. That means we need infinite motion at the mass. We know we cannot have infinite motion at the mass, so our assumption of motion of the base must have been wrong. For any finite motion of the mass, the motion of the base must be zero. This is an ideal case but again still a useful tool for understanding the ideal limiting behavior and the ideal behavior is not affected by the system to which the absorber is attached.

One final thought to add – tuning of the undamped absorber for fixed frequency excitation can be fairly critical and faces some unknowns, but it doesn’t need be finalized on paper. The absorber should have tuning adjustment (move the mass in and out on the bar). Then we can install it and do a bump test near its point of attachment (looking for minimum of response at proper frequency). We may also be able to tune it to minimum response after the machine is started, although movement of absorber makes that difficult.

1 - mechanical force f plays the role of electrical current I

2 - mechanical velocity difference v plays the role of electr voltage difference V

3 - mechanical mobility v/f plays the role of electrical impedance Z = V/I
(use the symbol Z throughout. We use the term impedance to represent either electrical impedance or mechanical mobility .. avoid the term mechanical impedance which is the reciprocal of mobility).

4 - A spring (k) acts like an electrical inductor of inductance L=1/k
The impedance is Zk = j*w/*k

5 - A damper (c) acts like an electrical resistor of resistance R = 1/c
The impedance is Zc = 1/c

6 - A mass (m) acts like an electrical capacitor of capacitance C =m **
The impedance is Zm = 1/(j*w*m)
** one terminal of this mass/capacitance is connected to ground.

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(2B)+(2B)' ?

 

[GMarsh]

Hi electricpete, Denial,

Many thanks for your time and detailed response. I am really inspired and motivated by your interest in explaining the things.

Actually I got hold of Den Hartog book 4th Ed. and reading it today whole day. I thought it is necessary before commenting anything here. But I see you also attached digital version of that part of text here. Good for my future reference. A treasure !

Denial - Now I understood what is 'mass based damping' Earlier my concept was wrong. I was thinking that the optimum damping equation which you wrote refers to the structural damping of the added masses in tuned vibration absorber. So I was creating such phrases! In fact Den Hartog explains so nicely about 'Optimum tuning' and 'Optimum damping'. What electricpete wrote simply in two steps is summary of that. First we do optimum tuning and then optimum damping. And the formula was actually derived to be a non-dimensional. So it looks like that.

My understanding of damping in that optimum damping equation is: It is for tuned vibration damper only. Also called as damped vibration absorber. It is not for tuned vibration absorber i.e. the one without damper. The equation is to choose optimum 'c/cc' for added viscous dashpot. I believe if we keep any other damping material also, such as viscoelastic tape, etc. we can use that equation by calculating its equivalent viscous damping from their structural damping factor.

The question which I wanted to ask about your pedestrian bridge is: can we design a tuned vibration absorber or a tuned vibration damper for a big experimental structure using analytical models of 2-DOF? In such case are we assuming it as lumped mass ? OR utilising experimental drive point FRFs? I want to use this analytical simulation for my structure also. No problem about incorporating damping. I can incorporate it as a complex stiffness = k(1+ i*eta). Sorry if this is testing your memory. You can then leave this. I just asked out of curiosity.

electricpete - While reading the Den Hartog book I was again puzzled by the question with which this thread started. In 4th ed, page 88, last line, he says "It was seen however that addition of an absorber has not much reason unless the original system is in resonance or at least near it." Again on page 92 he gives example of electric hair clipper where he is trying to design a tuned vibration absorber for 120Hz forced frequency of electromagnet. That is : he is targeting forced vibration which has nothing to do with resonance of hair clipper body on which he mounted the absorber.

I think I need few more reads before firmly understanding the concept.

Many thanks.

 

[electricpete]

electricpete - While reading the Den Hartog book I was again puzzled by the question with which this thread started. In 4th ed, page 88, last line, he says "It was seen however that addition of an absorber has not much reason unless the original system is in resonance or at least near it." Again on page 92 he gives example of electric hair clipper where he is trying to design a tuned vibration absorber for 120Hz forced frequency of electromagnet. That is : he is targeting forced vibration which has nothing to do with resonance of hair clipper body on which he mounted the absorber

That's a $64,000 question.

A few thoughts to consider:

Based on the proofs above, the undamped dynamic absorber ideally works to remove vibration at the absorber-tuned frequency, regardless of the attached system (so it is not limited to a system where the forcing frequency that we are tuning for corresponds to resonance of the original system). So it's not so much a question of whether we can be successful at reducing vibration (as long as we're using big enough absorber), as it is a question of whether application of absorber to reduce vibration will accomplish anything useful.

Does it make sense to add an absorber to reduce forced single-frequency vibration in a non-resonant system?
I'd say it depends.

Let’s say I have a 1200rpm 50hp horizontal rolling bearing motor with 1.0 ips horizontal, 0.5ips vertical running speed vibration measured on the bearing housings. Let’s also say the machine is operating far below resonance both directions, although a little further below in vertical direction than horizontal direction since stiffer vertical direction has higher resonance (all of that seems typical for this description machine to me). Let’s say I know or suspect that the cause is unbalance of the rotor. Obviously the ideal solution is to balance it but that maybe the machine needs to continue running and that’s not practical. For this scneario: Should I use absorber to reduce the vibration at the bearing housing?

The system resembles single degree of freedom system far below resonance. The SDOF mass is roughly the rotor mass. The bearing and frame transmit the force to the base. Since far below resonance, the system is spring dominated, the amount of unbalance force that goes into mass acceleration is small, almost the entire unbalance foce is trnasmitted through the bearing and frame to the foundation.

I could could attach a dynamic absorber to the frame near the bearing in such a manner to drive the horizontal vibration towards zero.
If my concern is the force transmitted through bearing, I have done nothing to address it because the force transmitted through bearing is the same (the unbalance force). Adding the ideal dynamic absorber in this case is equivalent to adding an ideally infinitely-stiff brace to the frame near the bearing to drive vibration at that location to zero. We can easily see in the case of the brace that we have only further increased the resonant frequency of the system, it remains very far below resonance (Even moreso than before), so the force transmitted through the bearing is still the unbalance force in this far-below-resonance (spring controlled) system. The result for the dynamic absorber is the same as for the infiitely-stiff brace... I just thought it was a little easier to explain using the infinitely-stiff brace. I hope in this situation it is clear the dynamic absorber does nothing if my goal is to reduce load on the bearing.. . On the other hand if machine were operating near resonance, then the resonance could magnify the unbalance force such that the force seen at the bearing could be many times higher than the unbalance force. In this case, it is helpful (reduce bearing load) to add a brace or to add a dynamic absorber.

Consideration of the above scenarios may be part of the basic thought process by which Den Hartog implies dynamic absorber is generally not real helpful unless used near resonance.

So can we come up to an exception to the "rule" that absorber is only useful in resonant system?
Sure. Den Hartog gave one that you mentioned (hair clipper).

I’ll give another one: let’s go back to the machine above and put it back far below resonance. But let’s say there are cracks on the foundation below the machine running perpendicular to the shaft under the coupling. We suspect the vibration is aiding the crack growth. We also suspect the vibration crack is causing misalignment between motor and load which is causing our 1x vibration (new scenario... not unbalance anymore). Now should I add an absorber to the machine frame? Yes, it would reduce the force transmitted from frame to vibration and reduce the crack growth and help break the cycle. Maybe not a permanent solution but dynamic absorber should help. And it will help even if you are not near resonance.

I had to stretch to come up with scenarios that were simple enough to yield an obvious yes/no answer to the question: should we add a dynamic absorber” (although the cracked foundation is pretty close to a scenario at our plant where we have installed an absorber). In most cases I suspect the answer is not that clear.

Certainly the insight from people like Den Hartog and our own Greg Locock will be valuable as a starting point.

By the way, the particular sentence you quoted at bottom of page 88 occurs in the context of a seguee into a discussion of the particular behavior of dynamic absorber applied to a resonant system. It is Den Hartog’s way to start general and wander to many particular special cases. Maybe he is just justifying to small extent why it is worth to examine that special case (resonant) more closely than the others.

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(2B)+(2B)' ?

 

[Denial]

My pedestrian bridge work was different from what I understand you are trying to do. I had a significant vibration mode whose frequency corresponded horribly closely with the footfall frequency of pedestrians, so I was specifically worried about resonance effects. Bear that in mind as I attempt to answer your question.

I set up a detailed FE model of the bridge, and subjected that to a modal analysis. This showed (as I had expected) the potential resonance problem I mentioned in the preceding paragraph. If a structure has one predominant vibration mode of interest it is possible to produce an approximate 1-dof model, with an effective mass value, an effective stiffness value, and an effective damping value. This I did. I then converted this 1-dof model into a 2-dof model by adding a TMD. Using Bachmann's formulae and my spreadsheet-based 2-dof model, I configured my TMD. I then added the resulting TMD into the original detailed FE model to obtain final confirmation of the TMD's efficacy. (Actually, this wasn't the final confirmation at all. The final confirmation came once the bridge was constructed. The dynamics was tested with the TMD deactivated. Based on the results of this test the TMD was given a final tuning, and was then activated. The dynamics was then re-tested. The results were pretty much as predicted.)

I gave a talk on the dynamics of pedestrian bridges a couple of years ago, and this included some discussion of TMDs. The text of the talk is available on my web site (

http://rmniall.com).

 

[GregLocock]

I think the Den Hartog problem is slightly different to why I'd use a TMD on a non resonant forced vibration issue. A typical steel structure has a q of the order of 1e2-1e3, and it is often impractical to add damping generally to the structure to get the forced response down. However, a small amount of damping can be exercised very effectively by a TMD, and without adding any damping in important load paths. This will help suppress the peak and hopefully lose it in the floor of all the other noise. Of course with pure tones you often need to pull 6-10 dB out of them to really kill them, and that is difficult.



Cheers

Greg Locock


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[GMarsh]

On the other hand if machine were operating near resonance, then the resonance could magnify the unbalance force such that the force seen at the bearing could be many times higher than the unbalance force. In this case, it is helpful (reduce bearing load) to add a brace or to add a dynamic absorber.

electricpete - I am sorry. I didn't understand why you said in a spring controlled system unbalance force gets transmitted to bearing, but not at resonance. When you mount dynamic absorber on frame, the load gets transmitted through bearing whether it is at resonance or not. Of course at resonance it gets amplified - so more force. It will get minimised only at the frame after passing through the bearing. I agree with your second example of crack growth. There mounting a dynamic absorber helps is minimising vibration transmitted from frame to foundation. But in 1st case (assuming you mounted it on frame), in any way the bearing sees the load - less or more.

Greg - your statement makes meaning for my application, of course in reverse. In fact as a conclusion to all this discussion I will post my problem where I seek all of your valuable opinions. I didn't post it in the beginning as the whole discussion goes biased and we would miss this wonderful discussion.

Denial - Though you targeted damping of resonant situation, I learned one very useful point from what you wrote - we can model a big system or structure with 1-DOF with its dominant mode of interest, of course with its effective mass, stiffness and damping. This will be useful in simulating analytically designing damper in resonant situations. Your website is useful for the downloads!

Many thanks to you all.

 

[electricpete]

On the other hand if machine were operating near resonance, then the resonance could magnify the unbalance force such that the force seen at the bearing could be many times higher than the unbalance force. In this case, it is helpful (reduce bearing load) to add a brace or to add a dynamic absorber.

electricpete - I am sorry. I didn't understand why you said in a spring controlled system unbalance force gets transmitted to bearing, but not at resonance. When you mount dynamic absorber on frame, the load gets transmitted through bearing whether it is at resonance or not. Of course at resonance it gets amplified - so more force. It will get minimised only at the frame after passing through the bearing. I agree with your second example of crack growth. There mounting a dynamic absorber helps is minimising vibration transmitted from frame to foundation. But in 1st case (assuming you mounted it on frame), in any way the bearing sees the load - less or more.

It sounds to me like you said roughly the same thing I said.
I'm not sure what the question or conflict is.
Sorry if I’m being verbose, but I’ll try to say it again in a (hopefully) more organized manner:

My assumed system:
Fubalance == Mrotor == Kbearing === Kframe===Ground

Far below resonance, the load on the bearing is approx Fub (because mass acceleration forces are negligible far below resonance).
At resonance, the load on the bearing is many times higher than Fub (resonant amplification)

Now what happens when we add a dynamic absorber or an infinitely stiff brace (both have same effect, which is to effectively stiffen the frame)
Far below resonance, the load on the bearing is still approx Fub. The absorber accomplished nothing.
At resonance (initially), the load on the bearing will decrease below what it was at resonance. The absorber helped.
This example illustrated a possible reason that Den Hartog suggests absorber is only useful at resonance.


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(2B)+(2B)' ?

 

[electricpete]

Clarication

Now what happens when we add a undamped dynamic absorber or an infinitely stiff brace (both have same effect, which is to effectively stiffen the frameat the tuned frequency)

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[GMarsh]

Ah, I got it. Bearing is connected to frame and absorber is mounted on frame, near to the bearing. So both the absorber and rotor will vibrate at same frequency. Net result is reduction of loading on bearing! Thank you, electricpete.

Now finally I am posting here my actual problem and the tuning dilemma!

I have a thin walled casing - 2.5mm wall thickness and ~100mm height and ~400mm diameter. I mill this casing with a tool rotating around the periphery (schematic attached here). I observe tool's natural frequency strongly imposed in the FFT of the acquired acceleration signal. For lower depths of cut, I also see workpiece's first mode dominating along with the tool's frequency.

Now before this discussion I was of the view that a tuned damper is meant only for resonant vibrations and hence designed one for workpiece first mode. I used six of them as that first mode has 6 petals (circumferential waves) or antinodes. Now having damped the workpiece modes over wide range (with more tuned dampers and hence higher mass ratio), I am left with tool mode again. Of course the amplitude of vibration reduced significantly, nearly 3 times. But frequency spectrum of machining vibration signal still shows distinctly tool mode. I am not sure if I have to live with it or any further gains can be achieved by targeting the tool mode. As you can see this is forced vibration with forced natural frequency, and the source of vibration is moving around the casing.

Also considering the casing has wide range of frequencies 1K to 16K (that's my measuring range), I am not sure if I have to stick with traditional 5% mass ratio or go to some 20-25% to damp entire range. What do you say?

I see someone else recently asked question of tool mode appearance in milling in another post in this forum. Will be good if he can have a look at this.

Thanks to you all and look forward to your opinions.

 

[electricpete]

ok, now you’re way out of my area of knowledge so I have nothing useful to contribute.

fwiw, I had been thinking you had single frequency driving force which is why I talked about undamped absorber. Now it's sounds to me like you have a resonance of the tool maybe excited by broadband excitation from the cutting process. In that case, the undamped absorber would just give you two new resonances at slightly different frequencies which will also be excited by the same broadband excitation... doesn’t accomplish anything. Damped absorber attached to the tool now makes more sense to me. You're probably way ahead of me on that.

I’m sure you’ll get more useful advice from the others.


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(2B)+(2B)' ?

 

[GMarsh]

Thank you electricpete. Yes tool is excited by broadband excitation.

I wish to hear any other views from others as well - Greg, Denial, etc. In fact both Greg and electricpete had a look at this structure (photo) earlier as part of our discussion on FFT of signal acquired, modal testing, etc. where some of your views greatly helped me.

Many thanks.

 

[GregLocock]

Is it a resonant mode of the tool, or is it just the tooth cutting frequency of the tool?




Cheers

Greg Locock


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[GMarsh]

It is resonant mode of tool.

 

[MikeHalloran]

With a right handed cutter, you are 'climb-milling' (chip starts thick), which is known to excite pretty much every available vibratory response.

Reverse something, e.g. the orbit direction, so you are 'up-milling' (chip starts thin), and you should see a bit less response.



Mike Halloran
Pembroke Pines, FL, USA

 

[GMarsh]

Mike, Thank you. What you said correlates with my observation. I use down milling. So that's why tool mode is very strong in FFT.

I still have doubt if it is worth tuning a damper for this tool's frequency and mount the damper on workpiece? As tool is rotating, we cannot obviously put a damper on that.

electricpete - From this configuration - i.e. a thin walled ring - for a forced vibration situation, do you see this as stiffness (spring) controlled system or mass controlled system ? We can consider it as stiffness controlled as mass is very less. But the applied forced vibration amplitude is very high. So this leads me to doubt whether it can be called as stiffness or mass controlled? If it mass controlled, I would try to design a tuned damper with a high mass ratio. Do you think it is ok?

Many thanks.

 

[MikeHalloran]

Are you working near the cutter's optima?

By which I mean, maybe you can get away from the resonance by doubling the spindle speed, and maybe dropping the feed rate a little if that burns edges. Your cutting tool salesman should be working with you to find the sweet spots here. Trust him; it's in his interest to make you delighted with his brands of consumables.

If it's not possible to effect improvements with feed/speed adjustments, the next thing I'd try is building a dam around the worktable and doing the milling with the piece submerged in coolant. ... not for the cooling, but for the damping.



Mike Halloran
Pembroke Pines, FL, USA