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

[dazz100]

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??

 

[BrianE22]

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.

 

[dazz100]

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.

 

[dazz100]

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.

 

[Tmoose]

"I measured the natural harmonic frequency of oscillation."

https://www.nytimes.com/1997/01/30/...-riddle-of-an-attack-on-rather-is-solved.html

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._SL1500_.jpg

 

[dazz100]

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.

 

[electricpete]

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)' ?

 

[Tmoose]

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.

 

[IRstuff]

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)
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[dazz100]

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.

 

[dazz100]

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.

 

[electricpete]

> 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)' ?

 

[onatirec]

>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.

 

[Strong]

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/colle...h-lomb-stereozoom-5-microscope-on-boom-stand>

https://davidwadesalon.com/pictures/bausch-and-lomb-stereozoom-4-manual-5.jpg

Walt

 

[BrianE22]

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.

 

[dazz100]

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.

 

[dazz100]

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

 

[electricpete]

> 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)' ?

 

[dazz100]

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.

 

[IRstuff]

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)
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