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Curran919

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Reply with quote  #1 

I wonder how many people am I gonna scare off with that title?

I was performing some bump tests on a vertical OH pump and looking at detuning that first bending mode. With the baseplate fully bolted to the foundation, the 1st bending modes were at 6.5 (X) and 9.0 (Y) Hz. I untightened two adjacent corners of the baseplate and eigenmodes shifted to 5.9 and 6.3, except they were no longer parallel to my global coordinates and had shifted 45 degrees to yield a X' and Y' mode. This made it quite difficult to track the modes from one configuration to the other, and was overall a bit confusing. It got me thinking about what eigenmodes actually represent when it comes to beams.

I have namely three questions:

  • Is there always a mode at the axis of absolute minimal stiffness?
  • Considering a 2D point mass system with many non-orthogonal springs, will there only ever be 2 modes?
  • Will those two modes always be exactly orthogonal?

This is how I understand it all conceptually, but I may be overthinking this. So lets imagine a cantilever beam of constant cross section and look at the first bending modes. Ideally, I'll be talking about a 2D system with a single mass and a large number of springs around it that 'approximate' the cross section shape. I found it easier to imagine the former, though:

  • If the x-section is rectangular, it will have two orthogonal modes... easy. This makes sense, as the bending in the x and y direction each represent local minima in the bending stiffness as a function of angle, k(θ).
  • If it is a square, there will still be two orthogonal modes, and these will still be orthogonal and parallel to x/y axes as these still represent local stiffness minima, but they will be equal to eachother. I understand this to still be two fully uncoupled modes. If we excite the beam at 45 degrees from X, then the beam will vibrate in that direction, at the equal eigenfrequencies, as if it had an eigenmode in that direction.
  • If it is a perfect circle, there is no longer a direction of lowest stiffness. So in what angles will the eigenmodes be in? I imagine small imperfections in the symmetry would determine the angle of the first mode at some angle of minimum stiffness. Would the second eigenmode then always be exactly orthogonal to the first, at some infinitesimally higher frequency?
  • If it is an ellipse, things start to get a lot more screwy in my head. There is a minimum stiffness along the minor axis, but orthogonal to this would be the major axis, which is the maximum stiffness. Is this still how the 'eigenmodes' are 'determined'?
  • A step further, if we have a 5-pointed star, with 5 equal local stiffness minima and 5 equal local stiffness maxima, each 36 degrees apart. We can combine assumptions from the circle and ellipse examples to say that there will still be 2 eigenmodes, with the first at the absolute minimum stiffness and the second at a local maximum, orthogonal to the first. Or do we really have 5 modes?
  • Finally, we have an irregular geometry, where there are three local minima and maxima (none equal). If the absolute minimum has no maximum or minimum orthogonal to it, will the second mode be orthogonal regardless? If there is a maximum (say 25 N/m) orthogonal to the absolute minimum (say 10 N/m), but the second and third minima are orthogonal to eachother, and only minorly more stiff than the absolute minimum (say 11 N/m each), would these become the axes of the modes?

 

MarkL

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Reply with quote  #2 
I knew it was either you or Pete before I looked at the author...I'm now going to hidE in the corner and rock uncontrollably

:-)
Danny Harvey

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Reply with quote  #3 
Curran,

I just sent in a report requesting that a pump manufacturer make changes to move two natural frequencies in some newly installed pumps like you describe. Altering the stiffness was the most practical option but that's up to the manufacturer.

I'm glad I don't have to do the actual work to predict the natural frequencies of a fabrication. 


Curran919

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Quote:
Originally Posted by MarkL
I knew it was either you or Pete before I looked at the author...I'm now going to hidE in the corner and rock uncontrollably :-)


But there are all of those shapes! I bolded them so its like kindergarten again. Maybe I shouldn't have bolded irregular. I know my least favourite lectures in university were the ones that started with "so we have a rigid potato". Oh great... theory time.

Quote:
Originally Posted by Danny
I just sent in a report requesting that a pump manufacturer make changes to move two natural frequencies in some newly installed pumps like you describe. Altering the stiffness was the most practical option but that's up to the manufacturer. I'm glad I don't have to do the actual work to predict the natural frequencies of a fabrication. 


Yup, that's what I do partly. In practice, I think detuning a resonance on a vertical pump is reaaaally simple. With an hour of training, I could get people successfully detuning structures 80% of the time. The hardest part is knowing what to change without messing with something else on the pump, which is why I always have to run it by mechanics first. Its that extra 20% that's tricky. If I want someone to really understand all the steps so they are self sufficient, that's at least 25x as much work.
fburgos

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Reply with quote  #5 
I love this kind of questions, force me to read a lot of topics just to understand the question.
Danny Harvey

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Reply with quote  #6 
This fabrication is full of what I would call irregularities.  There are four vertical columns made of rectangular tubing that are fixed at 45% angles to one another. It has heavy plate on top and bottom and some thin plate intermediate braces. More lateral bracing is probably what's called for.

The bigger pumps from the same manufacturer are even more irregular. They must have been a nightmare to design and fabricate but you can tell at a glance that there will not be any resonances near operating speed on those because of their extremely high mass and stiffness.

If I had to design these things everything would be round or square and meet at 90 degree angles. I don't like doing things twice.
Curran919

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Reply with quote  #7 
Quote:
Originally Posted by Danny Harvey
This fabrication is full of what I would call irregularities.  There are four vertical columns made of rectangular tubing that are fixed at 45% angles to one another. It has heavy plate on top and bottom and some thin plate intermediate braces. More lateral bracing is probably what's called for.


What do you mean "this fabrication"? Did you mean to attach something?

Quote:
Originally Posted by Danny Harvey
The bigger pumps from the same manufacturer are even more irregular. They must have been a nightmare to design and fabricate but you can tell at a glance that there will not be any resonances near operating speed on those because of their extremely high mass and stiffness.

If I had to design these things everything would be round or square and meet at 90 degree angles. I don't like doing things twice.


High mass and high stiffness should be having the opposite effect on the natural frequencies. We have some very large vertical suspended pumps and they are actually more likely to be at resonance. I just had to install absorbers on a pair of 3MW cooling water pump at a mixed cycle power plant, where the top motor bearing is 3 floors above the deck.

I'm curious what installation you are referring to specifically. Right angles in castings are usually not the best option.
Curran919

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Reply with quote  #8 

I had a coffee with a phd, so just to answer my own questions:

  • Is there always a mode at the axis of absolute minimal stiffness?
  • Considering a 2D point mass system with many non-orthogonal springs, will there only ever be 2 modes?
  • Will those two modes always be exactly orthogonal?

There will only ever be two modes. In a 3D point mass system, there will be three. In each case, the three will always be orthogonal/perpendicular. These directions will be determined similarly as the primary axes in Mohr's circle (anyone remember that shit?). In other words, the direction of minimal stiffness will be the first primary axis. The 2nd must be perpendicular. In 3D space, the 2nd will be the resulting vector that is the minimum stiffness of all the perpendicular vectors. The 3rd is simply whats left, perpendicular to the first two.

A little deeper:
If we have as in the example, a massless beam with constant cross section and a point mass at the end, the cross section is what determines the stiffness. We have to find the centroidal axis in any direction that yields the lowest 2nd moment of area. If there is a line of geometric symmetry (assuming bulk isotropic young's modulus), the symmetrical line must also be a centroidal line giving either a maximum or minimum moment. For a perfect circle, there are inifinite lines of symmetry, so the system is technically indeterminate. In practice or in FEA (with a finite mesh and weak springs), there will always be a direction with the lowest stiffness. With a perfect 5-pointed star, it has 5 lines of symmetry. Does one of those directions give a maximum or minimum? I don't know and it doesn't matter, its still a primary axis, and the other is at 90 degrees.
Danny Harvey

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Reply with quote  #9 
Curran,

These are fabricated steel and they appear to be custom fabrications to me.

If you saw the solid one, you'd likely know what I am talking about.
electricpete

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Reply with quote  #10 

Interesting stuff.

Let’s say I had a massless elliptical-shaped bar with fixed support at one end and mass at the other end.
Then I impact the mass in a direction 45 degrees from either axis of the bar.
What happens?..

My initial intuition says the mass would oscillate in the direction that I impacted it, and the resonant frequency of that oscillation  might be somewhere between the lower resonant frequency associated with impact along minor axis and higher resonant frequency associated with impact along major axis. 

I think you’re saying something completely different. You’re saying the mass will vibrate along the major axis at one frequency and the minor axis at another frequency.  And in fact it doesn’t oscillate sinusoidally along the axis that I impacted it…only during brief sporadic periods of time when the two different modes happen to drift roughly in-phase would it move along that 45-degree axis where I impacted it.  I believe you're right, I just have to think about that awhile. 

vogel

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Reply with quote  #11 
Nice topic. Orthogonality of beam modes was one of the first questions that came to my mind when I started doing modal tests. This topic has helped to answer some.

Quote:
Originally Posted by Curran919

In practice, I think detuning a resonance on a vertical pump is reaaaally simple. With an hour of training, I could get people successfully detuning structures 80% of the time. The hardest part is knowing what to change without messing with something else on the pump, which is why I always have to run it by mechanics first. Its that extra 20% that's tricky. If I want someone to really understand all the steps so they are self sufficient, that's at least 25x as much work.


Any book that you could recommend for those of us wanting to learn that 20%?
Curran919

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Reply with quote  #12 
Quote:
Originally Posted by electricpete

Interesting stuff.

Let’s say I had a massless elliptical-shaped bar with fixed support at one end and mass at the other end.
Then I impact the mass in a direction 45 degrees from either axis of the bar.
What happens?..

My intuition says the mass would oscillate in the direction that I impacted it, and the resonant frequency of that oscillation  might be somewhere between the lower resonant frequency associated with impact along minor axis and higher resonant frequency associated with impact along major axis.

I think you’re saying something completely different. You’re saying the mass will vibrate along the major axis at one frequency and the minor axis at another frequency.  And in fact it doesn’t oscillate sinusoidally along the axis that I impacted it…only during brief sporadic periods of time when the two different modes happen to drift roughly in-phase would it move along that 45-degree axis where I impacted it. Am I understanding that right?



Sounds like you are mostly right. If you have an ellipse where the ratio between the minor and major axes is 5, the ratio between the natural frequencies will be about 11 (I think 5^1.5?). In that case, as your minor axis experiences one period, your major axis will experience 11. It will have a lower displacement and probably dampen itself out faster, so you will probably just see mostly the lower frequency vibration.

Lets say the ratio is 1.1. You will wind up with a 15% beat. The frequencies aren't completely constructive, since they are technically independent. If you look down either of the axes, you will see no chance in the vibration. But if you look at 45 degrees, parallel to your strike angle, you will see what looks like a perfect beat. If we look at the beam head on, you will see the 'orbit' will beat between a line at 45 degrees, to a circle, to a line at -45 degrees, then back to a circle, etc.

Quote:
Any book that you could recommend for those of us wanting to learn that 20%?


I'd say that 20% is definitely not book stuff. Most of that would be the design background of the pump that may preclude certain modifications. Hence why I always pass modifications past a mechanic and/or design engineer. My favourite primer for modal analysis in general is this document by Agilent.

http://www.modalshop.com/techlibrary/Fundamentals%20of%20Modal%20Testing.pdf

Nothing exists satisfactory exists that really explores rotating equipment. I had to write my own for use in my organization, to try and close down that 20% gap, but I still have people put a gummy layer under a structural flange and then tighten the anchor bolts down to full torque.
electricpete

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Reply with quote  #13 

Thanks. Your answer to my question makes more sense now (I just had to think about it for awhile).

Danny Harvey

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Reply with quote  #14 
I just posted more about the pumps in my case in the topic about torsional analysis if anyone is interested.
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