Vibration modes for fixed-fixed beam with added mass

[nathanb99]

I have been stumped on this for several days now, wondering if someone might be able to point out some obvious error in my thought process.

So, the issue seems simple enough. I have a fixed-fixed beam, with known E, I, A, density. However, there is an added mass at a certain distance (not necessarily centered). The ultimate goal would be to play around with the size of the added mass, or its position, in order to avoid certain known input frequencies from other components on the vehicle.

I have combed the internet, and there are plenty of good solutions for finding the fundamental natural frequency of this. (Jump to page 81 here (p.599 in text), for a very useful table:

http://www.media.rmutt.ac.th/media/... Strain, and Structural Matrices/32212_11.pdf

) < note that this does not include mass of the beam itself... but there are examples out there with that included.

However, the issue arises when I try to find the 2nd / 3rd modes and higher. Every reference simply gives you the fundamental natural frequency and nothing else.

I have tried a couple of things, such as the separation of variables solution (

http://www.dtic.mil/dtic/tr/fulltext/u2/a399357.pdf

see p.5, or 16th page in pdf), and tried including the added mass as a boundary condition - which doesn't seem to work.

Thoughts? Should I give up on continuous models and just work with an FEA model?

Thanks for any input / help!

 

[GregLocock]

Did you look in Blevins? My copy is elsewhere so I can't check. To be honest i think FEA is probably the best approach for the higher order modes, for instance you might want to mess with the rotational inertia of your masss, which would start to make the hand calc cumbersome.

I doubt you'll get modes>1 if you don't include the self mass of the beam, otherwise all you've got is a mass on a spring.




Cheers

Greg Locock


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[electricpete]

Transfer matrix method (Mykelstahd method) is a numerical method well suited to beams.
It has the advantage imo that it is simpler than FE.
The disadvantage is that geometries which can be studied are limited mostly to beam-like structures with attached masses... not a disadvantage if that's the problem you want to study.

It includes moment of inertia effects, for example large disks attached to the beam.

I have a spreadsheet built for this method. Works very well to give frequencies and modeshapes for higher orders.

Instructions for use of the spreadsheet are not entirely self-evident.

Attached is the spreadsheet with a problem similar to yours already set up and run. Use of the spreadsheet is a little tricky. In your case:
1 - Delete Geometry and Modeshape tabs first before you change anything… those are static results tabs and will not be updated for changes.
2 – Adjust geometry in rotorsections tab.
3 – if desired press the “generate a profile” tab for graph of your geometry
4 – adjust E and rho in Main tab
5 – press run in Main tab
6 – Go to outsheet, and read instructions on the button labeled “to plot a modeshape”

Or you can give me the info and I will update it and re-run it for you.


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[electricpete]

4 – adjust E and rho in Main tab

Those values need to be transferred to the rotorsections tab. Either by typing them there or by pressing "fill in defaults" in that tab. Right now E and rho are set up for steel.

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[btrueblood]

What Greg said. We did this as an excersize in university, using Rayleigh-Ritz methods. Adding in the rotational intertia terms gets messy quickly, and you find that they start to become important as you move towards higher modes.

 

[electricpete]

Was there some disagreement with what pete said?
Or just bypassed it (which is also understandable, not everyone reads all responses).
Just curious.


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[btrueblood]

Sorry, no offense e-pete, but was not familiar enough with your method to comment.

 

[electricpete]

I must have had an extra dose of touchy-pills this morning. Thanks for putting my mind at ease.

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[nathanb99]

Thanks for the input everyone!

I went out and borrowed a copy of Blevins - quite a helpful resource, actually. However, the closest it comes to helping is a list of different scenarios for slender beams with concentrated mass - and again, only provide the fundamental natural frequency. Most of these do include the mass of the beam, so there should be more than one mode.... If only they would provide the derivation, maybe I could find the first 3 or 4 natural frequencies.

I played around a bit in ABAQUS, as it is what I have access too - turns out you cannot add loads in the frequency analysis bit.. I'll have to keep looking... surely you can have added masses, etc.

And Pete, I had a look over your spreadsheet - is this for torsional natural frequencies? Or is bearing stiffness / gyro stuff simply for boundary condition related stuff? Since it is all cylindrical, I guess finding equivalent mass / moment of inertia might give me similar results?

 

[GregLocock]

I suggets you learn to use your FEA program properly.

Cheers

Greg Locock


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[electricpete]

is this for torsional natural frequencies?

No, for lateral vibration (perpendicular to the beam axis)

Or is bearing stiffness / gyro stuff simply for boundary condition related stuff?

It was intended to analyse rotating equipment, but can certainly be used for stationary.

Set gyroscopic actions to 2 and don't ever think about gyroscopic effects again.

Bearings provide a way to connect any point on the beam to ground through a stiffness (can transfer force but not moment... like a simple support on a beam). If you have not supports along the span, you needn't concern yourself with bearings (leave them out). The main page provides a way to introduce an ideal boundary condition (fixed, free or clamped) on each end.

Since it is all cylindrical, I guess finding equivalent mass / moment of inertia might give me similar results?

Yes, that should work.

If you want to map a beam element element, don’t change the length, just match the following properties of the cross section
I*rho
A*rho
I*E
where I is the area moment of inertia
You have four variables to work with (ri, ro, E, rho) and three equations to match, so shouldn’t be a challenge.

An attached disk is similar, except it carries no stiffness, you just have to I*rho and A*rho by manipulating ri, ro, rho. (2 equations, 3 unknowns).

The program assumes the radial properties are the same in all directions. If you have differeng properties in x and y directions, then build one model for the x direction and another one for the y direction.

Should work fine to implement the model that is stated (read the instructions and understand the limitations). It is based on Euler Bernoulli beam model. Does not include shear deformation. Does not include curved flexing of disks (disks move with the beam, and tilt with the slope of the beam, but otherwise retain their geometry). Perhaps in the example I provided, those disks may have flexed at the higher frequencies so may not have been valid for the highest of the modes I showed.

I have done many test cases to validate the program (within assumptions). Once upon a time I had a website which showed many of those test cases. I have the nested subdirectory in html folder. In the old days I could drag/drop that onto a website, but I can’t figure out how to do that anymore. (anyone have suggestions?). Guess that must mean I'm getting old

The guts of the web page is 3 files. I'll try to attach them
1 – discussion3.doc = How does the program work (the theory)
2 - xfmatdemos.doc = Demonstrations which show
*How to use the program to solve example problems
* Example problems compared against analytical results for program
3 xfmatdifficulties.doc = Difficulties - some small questions about the program I am still trying to figure out

Any sub-links probably won’t work - (ask and I’ll post those also).

If Greg or anyone has objections to this method, I’d be interested to hear them.

http://files.engineering.com/getfil...e-b4bc-32514bf7098d&file=xfmatdiscussion3.doc

http://files.engineering.com/getfil...22-4d52-91e1-9b7d579b281f&file=xfmatdemos.doc

http://files.engineering.com/getfil...-bb2f-717a90662ebc&file=xfmatdifficulties.doc

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[electricpete]

Regarding the website. I have directory of html linked files created using ms frontpage that used to work (copied by FTP onto the free webserver of my ISP... which doesn't exist anymore).
I read somewhere it could be done by copying the directory to dropbox and sharing the home page index file. I tried it and doesn't seem to work. Here's the link, which doesn't work for me... but if you figure out a way to use the link to navigate the website, let me know:

https://www.dropbox.com/s/s4u7cns8wllm6ay/index.htm

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[GregLocock]

Another method would be rayleigh ritz.

Cheers

Greg Locock


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[electricpete]

In simple textbook form, the simple Raleigh Ritz works for first resonant frequency only.

I'm vaguely remembering that there are procedures available to extend some of the approximate methods to higher modes beyond the first. Does it apply to Raleigh or R/R?



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[GregLocock]

...although quite how you stop that searching for the first mode rather than higher modes is not at all obvious to me but a google search suggests it is possible

Cheers

Greg Locock


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[electricpete]

Google leads me here:

http://ilanko.org/vib_chap5.pdf

5.4 Lagrangian Multiplier Method
The difficulty in choosing admissible functions has been the most significant drawback of the Rayleigh-Ritz method. One could relax the admissibility requirement by using the Lagrangian multiplier method (see Ref 5.4), where individual functions need not satisfy the geometric boundary conditions but the series as a whole is forced to satisfy these by including additional constraint equations….

Table 5.4.1 on page 19 gives results for geometry of Figure 5.4.1. As could be anticipated, it takes more terms to approximate the higher order shapes, so results for better higher order frequency estimation require more terms.


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[GregLocock]

Yes but I suspect it is one of those things where if you know the right answer you can select appropriate eigenvectors, but a blind search for the second mode will degenerate to the first mode. Perhaps you could caharacterise the second mode as in point L/4 will move in the opposite direction to point 3L/4, but tat seems dangerous in the general case.

Cheers

Greg Locock


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