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??

 

[electricpete]

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.

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).

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[dazz100]

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.

 

[onatirec]

>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.

 

[electricpete]

Thanks onatirec, I agree with those points.

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

Generalities:
[ul]
[li]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).[/li]
[li]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. [/li]
[/ul]

Our case:
[ul]
[li]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) [/li]
[li]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).[/li]
[li]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[/li]
[li]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. [/LI]
[li]The dynamic absorber was an easy-to-install relatively non-intrusive fix. [/li]
[/ul]


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[Tmoose]

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.

 

[GregLocock]

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


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[dazz100]

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.

 

[GregLocock]

" Now identify the point and direction of maximum amplitude for that frequency. "

Forget the centre of gravity.

Cheers

Greg Locock


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[dazz100]

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.

 

[GregLocock]

"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


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[dazz100]

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.

 

[dazz100]

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.

 

[electricpete]

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

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[dazz100]

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

 

[electricpete]

thanks, my bad. I was confusing different threads.

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[dazz100]

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.

 

[electricpete]

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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[dazz100]

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.

 

[electricpete]

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:
[ul]
[li]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.[/li]
[li]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). [/li]

[li]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.[/li]​

[/ul]
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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