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# Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

## Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi
I am designing a TMD for a microscope application. The microscope is mounted on a standard computer monitor stand. In effect, this is a weight on the end of a cantilever. The lower arm acts as a torsion spring. Unsurprisingly, this has flex and undamped harmonic motion that makes the microscope difficult to use. Commercial stands from reputable manufacturers are no better.

In order to design a TMD, I need to characterize the motion. I have done this in 2 ways.
I have precisely measured the force and deflection to find the spring constant k. k is surprisingly linear and shows almost no hysteresis.

I measured the natural harmonic frequency of oscillation and weighed the mass of the scope. From this I have calculated k.

The problem is that the values of k measured by the two methods vary by a factor of about 7. Too much to be explained by measurement uncertainty. I think my simple cantilever spring model is the source of the gross error.

I am thinking the simplest way to adjust the model is to use an effective length of the cantilever. If I doubled the effective length, that would reduce the calculated harmonic frequency by 1/4x. It would increase the mass inertia by 4x.

Am I on the right track??

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

Is the weight of the arms insignificant compared to the weight of the microscope? I'm curious what you used for mass and where you put it.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi
The microscope and adapter are the main mass. The stand arms do contribute, and I have included an estimate of that contribution to the modeled mass.
I am thinking that simply shifting the mass away from the spring would be a reasonable model, as shown in the attached sketch.

With this model, the spring rate and static mass would be "correct". The added leverage would have the effect of increasing the dynamic mass to match the measurements.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
On further thought, I could identify the pivot point by measuring k at a different location along the arm. The ratio of the difference between the measured k values would provide an estimate of the leverage and therefor the effective pivot point.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

"I measured the natural harmonic frequency of oscillation."

https://www.nytimes.com/1997/01/30/nyregion/belate...

How did you measure it? How heavy was the accelerometer/sensor? Was the measurement direction vertical, horizontal left, horizontal right or triaxial ?
Do you have full spectrums from 1 Hz to 500 Hz or so? Did you happen to also measure the vibration of the floor, at all the monitor stand legs, in great detail ?

I suspect the cantilevered OSB and Formica top of the Monitor stand ain't very stiff, at all.
Do you need to slide test specimens under the Monitor stand ?
Installing A tightly fitting (wedged) strut between the floor and the monitor stand underside, right at the microscope bracket, might be interesting.

https://m.media-amazon.com/images/I/81jxjgcFi2S._S...

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi
I measured the oscillation with a Gill Blade25 contactless position transducer. You can see it here
The photograph of the go-kart throttle position sensor on this page is mine.

The Gill Blade 25 was fixed. An activator, in the form of a light metal U, was mounted on the microscope.

The microscope is mounted to a 10mm steel plate base, and this sat on the bench top, on a non-slip mat. It is rigid enough for this application.
The natural frequency is only 5Hz and the disturbing force was less than 0.5N.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

Do you have any idea what is exciting this motion?

If it's what I'm imagining (random jostling of the support structures including floor and adjacent walls etc), you have a broadband excitation causing motion at the system natural frequency.
I don't think tuned mass damper would typically help much in that scenario (undamped TMD is best suited to avoid resonant coincidence between a fixed discrete sinusoidal excitation frequency and the system natural frequency).
The tuned mass damper creates peaks in the transfer function on either side of the original peak in the transfer function. If your excitation is broadband, that isn't going to reduce your vibration much if at all (unless you are including some damping)

On the other hand if there is a fan nearby shaking the building at a fixed frequency then undamped TMD could help.

There may be some worthwhile easy trial and error things. Maybe use a C-clamp or two on the overhang part of the counter to clamp onto that plate. I realize someone else was concerned about a flexible counter, but it looks to me like the plate/scope assembly could very easily rock.... I'm used to seeing things clamped down to help stop them from vibrating.

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

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

Hi E-Pete,

The OP has said "The microscope is mounted to a 10mm steel plate base, and this sat on the bench top, on a non-slip mat. It is rigid enough for this application."

Your observation and concern about rocking looks to me to be well justified.
Perhaps it is the camera angle, but the OP's posted picture suggests to me the steel plate has compressed the non slip mat to 2 mm or so on the loaded edge, The further, lightly loaded edge of the 10 mm steel plate looks to be about 10 mm in the air.

Rigid is not a word I would use to describe this design.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

Seems odd that the plate would be behind the bottom of the arm, rather than in front; I thin you've essentially turned the leading edge of the base a pivot point. Consider where the CG is, compared to the case where the bottom of the arm is mounted on the back edge of the base plate.

TTFN (ta ta for now)
I can do absolutely anything. I'm an expert! https://www.youtube.com/watch?v=BKorP55Aqvg
FAQ731-376: Eng-Tips.com Forum Policies forum1529: Translation Assistance for Engineers Entire Forum list http://www.eng-tips.com/forumlist.cfm

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi
I setup 2x dial indicator gauges to measure the deflection each side of an applied force. The DTIs were spaced at a measured distance. The difference between the measured deflections allowed me to identify the virtual point of rotation with some basic trig. The maximum deflection was less than 0.4mm. The maximum force was less than 0.4N, applied in 0.05N steps.

The measured results are shown in an X,Y plot in mm. They were surprisingly linear. I ran the test twice and plotted the results on the same graph. The value 0.0156 in the best fit equation represents an offset error of less than 0.02mm.

The attached image shows the setup. The step force was applied by adding 50ml to an empty milk container hanging on a string, run through a small pulley to apply a torque to the monitor arm.

Knowing the virtual pivot point will allow me to calculate the effects of moving the spring and damping away from the C of G of the microscope hanging off the end of the monitor arm.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi
I see there is some discussion about the source and nature of the input noise.

This is a microscope so in use, it is going to be knocked by the user, or the table may be bumped. At present the microscope wobbles for ages because the damping is close to zero.

A bump is best represented by an impulse, which is equivalent to a broad band noise input. I don't have a broad band movement generator so I use an impulse (a bump). The really nice thing about an impulse is that it is equivalent to a unity function.

So a TMD filters the input noise. It changes the frequency response of the monitor stand ( the output) and adds damping to rapidly stabilize the image. That's why I am using a TMD.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

> The attached image shows the setup. The step force was applied by adding 50ml to an empty milk container hanging on a string, run through a small pulley to apply a torque to the monitor arm.

> A bump is best represented by an impulse, which is equivalent to a broad band noise input. I don't have a broad band movement generator so I use an impulse (a bump). The really nice thing about an impulse is that it is equivalent to a unity function.

A bump test or step test would both be good ways to characterize a linear system AS LONG AS you record and properly analyse the dynamic response, but I'm not convinced that what you did. First imagine that you were reading the position of the dial indicator in steady state after the system comes to rest, then you are seeing the static response to a force at zero frequency (dc) rather than the dynamic response to the component of the ac excitation that is at the natural frequency, and it would not be a good represntation of the movement at the natural frequency. I don't think that's what you did, but I bring it up because I think it's intuitive to see that wouldn't work and it sets the stage for what comes next. I think what you actually did was to try to capture the peak movement during repeated drops from the same height... even in that case the peak response is likely still heavily influenced by the dc component of your applied dynamic force. It's unlikely (unless you're just lucky) that peak response would represent the modeshape of the first resonant frequency of interest. If I misunderstood your test or the way in which you think it will represent the modeshape of interest, please clarify.

> This is a microscope so in use, it is going to be knocked by the user, or the table may be bumped....;So a TMD filters the input noise. It changes the frequency response of the monitor stand ( the output) and adds damping to rapidly stabilize the image. That's why I am using a TMD.

Again undamped TMD is generally not a good solution for broadband excitation. All it does is shift your resonant frequency slightly (in both directions, replaces SDOF single resonant frequency with 2DOF two resonant frequencies on either side of original resonant requency), so broadband will likely still excite the new frequencies.

Stiffening can sometimes be a good solution even for broadband force excitation because
1 - all real-world broadband has an upper frequency limit. If you can stiffen to move the first resonant frequency above that limit, then you will reduce vibration.
2- The excitation may be quasi broadband... maybe it starts out broadband elsewhere in the room, but what gets through to your tabletop depends on resonant transmission frequency of an intermediate component. Flexible systems have higher density of modes at low frequencies than stiff systems. Even if you are not successful in moving the first resonant frequency above all excitation, the odds of having a resonant frequency land near an strong exciting frequency are still lower in a stiff system than in a flexible system due to the difference in density of resonant frequencies (how closely they clump together in frequency).
3 - Let's say the broadband is effectively ideal to infinite frequency (constant force as a function of frequency). Assuming your system is a SDOF with constant damping, then increasing stiffness will still reduce peak displacement because the new resonant frequency is higher and peak displacement magnitude... is |Force|/(c*w) which is inversely proportional to frequency. Aslo the entire X(w)/F(w) transfer function curve everywhere to the left of the resonant frequency decreases in magnitude as the stiffness increases.

I'm still thinking a c-clamp is well worth your time to try (btw even if we believe you have properly captured the modeshape of interest, then it still suggests that clamping of the plate will reduce or at least alter that particular mode). I don't know your situation... I'm curious how much time/effort does it take to see if you have fixed the problem after a trial fix?

EDIT - I see now that you've got a C-clamp in there now in the test photo (sorry, that's a duhhh moment on my part). So maybe my last several paragraphs are irrelevant (I changed them to grey). Did the c-clamp it change the vibration experienced by the microscope?

I'm not a vib expert but undamped dynamic absorbers (TMD) is something I have worked on. Attached is a presentation I gave at an industry group regarding successful use of a dynamic absorber solution at our plant (for a fixed frequency excitation). Slide 14 gives an example qualtative view of the before/after transfer function. Slide 17 tells you how far the two new natural frequencies are from the original natural frequency. The higher the attached mass in relation to the original SDOF model mass, the more the separation from the original frequency. If you don't get much separation the likelihood of solving the problem is low. Even if you do get separation by attaching a relatively large mass, for broadband excitation it's still questionable.

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

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

>A bump test or step test would both be good ways to characterize a linear system AS LONG AS you record and properly analyse the dynamic response, but I'm not convinced that what you did. First imagine that you were reading the position of the dial indicator in steady state after the system comes to rest, then you are seeing the static response to a force at zero frequency (dc) rather than the dynamic response to the component of the ac excitation that is at the natural frequency, and it would not be a good represntation of the movement at the natural frequency.

My reading is that this was one of two ways dazz made a measurement. Which amounts to applying a fixed force and measuring spring deflection to find the spring constant. He then calculated an effective mass and used the spring constant to find the natural frequency... This is apart from a dynamic measurement of the oscillation using a position transducer. It's not entirely clear to me what provided the input for the dynamic testing.

I do agree that stiffening/damping might be the more straight-forward and effective approach, especially if the excitation is broadband.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

Pete, the C-clamp only clamps the steep plate holding the magnetic base. It does not appear to clamp the primary steel plate base to the table top.

The structure is a very flimsy support for a microscope! It would be amazing if a TMD would consistantly reduce vibrations for multiple uses of the scope on the same tabletop location or when moved to other locations.

Many alternatives to the flexible computer monitor stand:
Bausch & Lomb StereoZoom 5 Microscope on Boom Stand
From <https://microscopecentral.com/collections/used-bau...;

Walt

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

I'm surprised Bausch and Lamb would sell it with that problem. Is the arm support also from them? Is there an adjustment of the pivots for the holding torque? Maybe a little tighter would give more damping.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi
Just for some clarification.
This type of articulated lightweight arm is not an original idea. Here is a link to a commercial version. Link This looks as flexible as a soggy noddle with no obvious damping. Being sold under the Leica brand will add \$ to the price.

The arm I am using is a standard computer monitor arm. I made an adapter to attach to the microscope. Monitor arms are readily available and relatively low cost.

Any test I do with a Dial Test Indicator is a time invariant static test.
The only dynamic test done was using the Gill contactless transducer. This was done to find the natural resonant frequency.
I applied an impulse by tapping the microscope. A step input would have required sudden shifting of the steel base (like an earthquake). That would have been difficult to achieve.
The kitchen cabinetry and bench are rigid and highly damped.
The anti slip mat is thin and highly damping. More importantly it stops the steel plate wobbling on the bench top, because neither are a perfectly flat surface.
All measurements are relative to the steel plate, so any flex in the anti-slip mat or bench top have not added errors.

In this application, the TMD role is to stabilize the viewed image as quickly as possible after the microscope is disturbed (by a knock). It does that by converting motion to heat. The TDM damping action is focused on the natural frequency of the microscope hanging on the end of a stand.

The reason I am not using/buying a standard Bausch & Lomb rigid stand is because the weight would make shipping cost prohibitive. I looked at making one but the lack of reach is not ideal for my application (electronics). The major advantage of the long reach stand I have is that the microscope can be swung clear of the workspace without have to lift anything. That is why the mounting points for my base are on the corners of the steel plate.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
For electricpete
I had a look at your presentation. We are applying different methods. You are applying a spring and mass (tuned resonator) but no damping.

I will be applying a spring, mass and damping explained here.
I intend using magnetic eddy current damping demonstrated here

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

> Pete, the C-clamp only clamps the steep plate holding the magnetic base. It does not appear to clamp the primary steel plate base to the table top.

Thanks Walt. So op can read what I greyed out to explain why trying stiffening makes a a lot more sense to me than trying TMD. If you are using damping then maybe you can accomplish something but it's still not clear why your first solution is such a complicated one (perhaps you can let us know if you tried something else before you got to this).

> Any test I do with a Dial Test Indicator is a time invariant static test.

Ok, so it appears this test will not reflect your resonant modeshape, as I discussed above.

> The anti slip mat is thin and highly damping. More importantly it stops the steel plate wobbling on the bench top, because neither are a perfectly flat surface

ok, so why not leave the mat there and move your C-clamp to clamp the plate to the table (with mat still between)? Or better yet two clamps at different distances from the scope. If that doesn't work then try without the matt. If you believe your dial indicator modeshape analysis, then the plate is moving as part of your modeshape and restraining it will reduce or alter your modeshape (personally, I don't believe your modeshape, but I still want you to try the clamps).

If clamps are completely out of the question for some reason that you haven't explained, then maybe try adjusting that articulating arm so that the horizontal distance between the plate and the miscroscope is less or maybe just put a heavy weight on to of that plate.

I'll repeat a question I asked before (honest question, not a criticism), how much time and effort is involved in checking whether you have fixed the problem after a trial fix like clamping? That is an input that will help guide the type of suggestions people can offer.

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

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi electricpete

I am going to be using the microscope on a working bench that is everything except rigid. Even if the microscope and stand were the most rigid a thing could be, I would still have a problem.

Not sure what you mean by "resonant modeshape". Not a term I am familiar with. Frequency response curve maybe (as in x= frequency Hz, y= amplitude dB)?

The C clamp is only there so I could locate a magnetic stand holding a pulley out over the edge of the plate. This allowed me to hang the weight used to apply torque to the stand. With a resonance frequency of 5Hz, the properties of the non-slip mat and bench are a non-issue. It would be different if this was a motor. Gravity is doing a perfectly adequate job of clamping the plate to the bench. Even if I did clamp the plate, it would have zero effect on the dynamics of the stand.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

The magnification on these isn't even that high, is it? We've used B&L microscopes with similar stands as below without any problems

TTFN (ta ta for now)
I can do absolutely anything. I'm an expert! https://www.youtube.com/watch?v=BKorP55Aqvg
FAQ731-376: Eng-Tips.com Forum Policies forum1529: Translation Assistance for Engineers Entire Forum list http://www.eng-tips.com/forumlist.cfm

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

#### Quote:

Gravity is doing a perfectly adequate job of clamping the plate to the bench. Even if I did clamp the plate, it would have zero effect on the dynamics of the stand.
You may be right. But personally, I would check rather than assume. It doesn't take much flexibility/movement at the bottom of the structure to have a big effect further up. As a wise man Tmoose told me long ago, watch those boundary condition assumptions.

#### Quote:

I am going to be using the microscope on a working bench that is everything except rigid.
You are going to move your tuned mass damper to a low stiffness table.
I think the system may well have a different resonant frequency when you transplant it to a different table (may affect your TMD tuning).

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

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
electricpete
I am trying to figure out why you went for the harmonic resonator in your presentation. I have never seen that type of device fitted to any rotating electrical machine. Why not just balance the motor and generator rotors?
Just curious.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

>I am trying to figure out why you went for the harmonic resonator in your presentation. I have never seen that type of device fitted to any rotating electrical machine. Why not just balance the motor and generator rotors?

There are excitations in electric motors that can't be balanced away. If these coincide with structural resonances at running speed, it could drastically reduce lifetime in a variety of ways; in pete's case, the foundation was cracking. The TMD moves the resonances away from the running speed. As mentioned above several times, this works well in his case since the running speed produces steady harmonic excitations (well several harmonic excitations...) and its quite easy to design the tmd so that the new system resonances don't coincide with these running speed excitations.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

Thanks onatirec, I agree with those points.

I’ll add some things and respond in generalities first and then talk about our case.

Generalities:
• For fixed speed machines, it is generally a design objective to keep natural frequencies separated from running speeds by some margin, often 10 or 15% (The separation on this machine was about 7%). This is because there are a variety of possible excitations at 1x, and there may be limits on how low you can get the exciting forces with reasonable effort (you have to get them lower than normal to achieve acceptable vibration when resonant amplification is present).
• When faced with, let’s say, a fan with obvious resonantly amplified unbalance at running speed, I think op is correct that many vib analysts would be inclined to balance as a first option. Correcting the resonance might be considered a prudent followup action (if it can be easily accomplished) in order to restore design objective frequency separation and to lessen likelihood of having to come back and balance it later after only a small amount of dirt accumulates.
Our case:
• In our case we did not attempt mechanical balance. It is not an easy solution and not a guarantee of success. We have four of these machines (30 years) and have never balanced any of them (we have never refurbished them yet, but first one will be done next year)
• The balance planes inboard (generator side) of both generator bearings are not easy to access. One end is obstructed by the voltage regulator and the other end is obstructed by the overhung flywheel. I would have to think very carefully before drilling on that flywheel itself (catastrophic burst hazard) although it may have some balance holes or other balance provisions provided by the manufacturer (I’m not sure).
• In addition to simple mechanical sources of 1x (unbalance, misalignment), this machine is also subject to other load related influences either electrical or mechanical. I say that because we observe that the 1X vibration changes slightly when the excitation is applied and even more when the load is applied to the generator. Repeated balance runs in the loaded condition is not a practical option based on the plantwide impact of establishing and securing the particular load fed by the generator
• Another way to look at it is that we saw an increase in vibration on a machine that previously ran smooth. We asked "what changed" since the machine was new. The answer was not likely a change in mechanical balance. The motor and generator are both air cooled, but they sit in very controlled atmospheric environment which makes rotor dirt buildup unlikely (and an identical machine in the same room 10 it 15' away had no vibration increase). On the other hand we could see the cracks in the foundation and observe the characteristics of resonance that did not exist on the other machines. It suggests that maybe what changed was the cracked foundation, which in turn changed our resonance characteristics. The resonance characteristics are what we fixed.
• The dynamic absorber was an easy-to-install relatively non-intrusive fix.

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

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

OP's image 1060 posted on Oct 21 looked to me like a rig to excite the system with a horizontal force input.
I was thinking the troublesome vibration was vertical, but now I am sure I am unsure.

In image 1060 The plate is shown rotated 90 on the table, and the jointed arm is folded up so short the microscope is now over the steel plate.
Also the non-slip pad material is now quite different. Thin Black rubber/plastic mesh vs white foam blanket.
For my money this is a vastly different system than the one shown in IMG 1004 in the OP.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

OK, this was (a small but important part of...) my job for 20 long years. Assuming you don't want to change the system itself, that is, the damper is a band aid:

Forget k. You've measured the frequency. Now identify the point and direction of maximum amplitude for that frequency. The tip of a small screwdriver is a very good tool at this point.

Now you need to mount a TMD at that location/orientation. It needs to have some damping, but not too much, and it needs to be of sufficient mass. Generally less damping is easier to work with than too much. The more mass the better. Usually you tune to the exact frequency that is the problem, but it can be helpful to undertune or overtune in a given application.

This all sounds like a lot of hassle, because you haven't got all the instrumentation to develop the tuned damper. An untuned damper may be sufficiently effective, and obviously doesn't need tuning. This could be as simple as a bob weight dangling in a cup of water, or a vial of loosely packed sand.

Having said that changing the system is usually more cost effective in production. that base unit looks awfully rocky, and the wrong way round. is that a compliant mat under the steel base?

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: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi
The early images showed the plate laid on a towel so I didn't scratch the bench.
The articulate arm was folded to allow me to keep the DTIs on the plate to provide a fixed reference plane unaffected by any movement of the plate on the towel (later anti-slip matting). The frequencies and forces are so low that any movement of the plate on the towel/mat is of no consequence. You might speculate but I did the measurements.

Experiments showed that folding the arm made no significant differences to anything except the direction of harmonic motion of the microscope head. I need a 2-axis TMD.

The differential displacement with force results plotted above identified the actual point of rotation. In this application, the most practical position for the TMD is not through the CoG of the microscope head. The location of the TMD away from the CoG of the microscope head will affect all of the major characteristics of the TMD to get it to work properly.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi
The CoG is important only because the TMD is not acting through the CoG.
I need to scale the TMD to account for that.

The dominant source of resonance is from the torsion spring formed by the lower arm of the stand. Changing the relative direction of the upper arm is found to have no significant effect on spring rate or resonance freq. It does affect the preferred direction of oscillation of the upper arm and scope.
For that reason, I need a 2 axis TMD.

Flex of the monitor stand arms acting as cantilevers (no torsion)is less and the spring rates are higher and less problematic in this application.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

"I need to scale the TMD to account for that."
You are trying to extract energy from the system. The easiest place to do that is where it is moving most.

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: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi

"You are trying to extract energy from the system. The easiest place to do that is where it is moving most. "
Agreed but the scope can be rotated on the end of the arm about 270 degrees. That would require the TMD to be much more complex to design/make if it was mounted further out closer to the scope.
To keep things simple, it will be easier to mount the TMD on the end of the arm to keep it correctly orientated with the motion to be damped. The penalty is that the TMD is less effective, unless scaled to do the job.

The microscope on this stand has a large working envelope. It can work close over the base plate, or be swung half way across the room (almost).

No pendulums will be harmed in the construction of this TMD. I plan to use a lead mass suspended by 3 radially mounted springs to give a 2 axis TMD. The suspended mass will include aluminium sheet with magnets mounted in close proximity to provide damping.
At present, air fills the place where this will be mounted, but I have a 3D printer. The TMD will be fixed to the excessively long bolt on the end of the stand.

The motion on the end of the monitor stand is less than 0.5mm. Anymore than that and the position of the stand changes. If the TMD has an effective mass of about 10% of the stand and microscope, the TMD will have a movement range of about 5mm. Scaling will increase the range.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi
By my calculations, a 300g weight suspended by 3 radially spaced springs, will require springs with a rate k of 0.005N/mm.
I think I may need to make the springs I need.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

Is it my imagination or did a bunch of posts disappear from this thread? (why?)

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(2B)+(2B)' ?

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi
I don't think so. They seem to be all there.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

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

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi
I was planning on supporting a mass with 3 radially mounted springs with the mass in the centre. I assumed that the three springs would provide a constant k regardless of which 2D direction the mass moved.
When I did the maths, my assumption was wrong.

When pulling directly in line with a spring (any one of the 3):
k = 1/2 + 1/2 + 1 = 2

When pulling at 90 degress to a spring:
k = √3 = 1.73

A difference of 13.5%
Not massively different, but unexpected.

The problem I am also having is finding springs of the right form, size and k.
I will have to make my own springs.

Rather than using coil springs in compression or extension, I am thinking of using a coil in shear.
If I make the spring a spiral shape, it can be flat. This will avoid the problem with directional k.

It will take months to import the wire I need to make springs so this thread will go quiet for too long.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

That is interesting. I'm remembering that the bending stiffness of a rotor that has 120 degree symmetry (like 3 keyways in a shaft, or 3 arms in a spider structure) is axisymmetric (doesn't change when viewed from different angles). Either I'm remembering that wrong, or else there is some differences in looking at your linear springs arranged with 120 symmetry vs my bending stiffness of a shaft with 120 degree symmetry (although it's not immediately obvious to me what those differences would be)

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(2B)+(2B)' ?

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

(OP)
Hi
The difference is that when a mass is supported in series between two springs, it is as if the springs are mounted in parallel. So if k=1 for each spring, then mounting between two springs, k=2.
This was a surprise to me.

So, I think it is likely that a rotor with 120 degrees symmetry has stiffness (k) that is axis-asymetric, but the variation is likely to be a lot less than 13.5% because the the geometry.

### RE: Tuned Mass Damper (TMD) Design for cantilever with weight at the end.

I agree with you that in your 3-spring system the springs are in parallel, and the rotor stiffness system is a much different configuration from that. To combine the three 120-apart vectors associated with an element when describing their contribution about an axis in these two systems, two different approaches are used:
• In your 3-spring system, it is the sum of the absolute values of the projections of the vectors onto an axis (for example x axis). The absolute value arises because the direction of the spring force in response to a displacement perturbation is always restoring (regardless of whether the spring is in a location where the perturbation results in spring tension or spring compression). So as theta varies, the sum |cos(theta)| + |cos(theta-120)| + |cos(theta-240)| varies within the limits you mentioned.
• In the 3-element rotor bending stiffness system, it is the sum of the squares of the projections of the vectors onto an axis (for example x axis) because the definition of area moment of inertia involves the square of a coordinate. For example Iy=Integral{x^2} dA (assuming the centroid is at x=0). We can substitute x=r*cos(theta) and x^2=r^2*cos^2(theta). When we account for three 120-spaced symmetrical elements we end up with r^2 *[cos^2(theta) + cos^2(theta-120)+cos^2(theta-240)]. By trigonometric identity, that sum in the square brackets is always 1.5, regardless of the value of theta. So as theta varies, the sum is constant (and the bending stiffness is constant regardless of what axis angle we examine it from).

• Side note on the identity [cos^2(theta) + cos^2(theta-120)+cos^2(theta-240)=1.5... It is an unfamiliar identity vaguely reminiscent of the more familiar cos^2(theta)+sin^2(theta)=1 (both have theta on the left side but not the right). It is also recognizable from study of a balanced three phase power systems feeding a resistive load where the power transmitted is constant over time, even though it is sum of three I^2*R terms where all three currents are varying sinusoidally over time but shifted by 120 degrees from each other. I don't have a trig proof handy, but if anyone is skeptical of the identity it's easy to prove it by just plotting in excel.
I apologize for the detour. The lesson for me is don’t trust my first intuition (it's usually wrong)

I’ll be interested to hear more as your project progresses…

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(2B)+(2B)' ?

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