Free vibration of a folded plate. 1

[Hussein Hussein]

Hi there everybody,
I'm calculating the natural frequencies of an ordinary cantilevered plate and a cantilevered folded plate using Galerkin-based finite elements analysis. I prepared a matlab code for the analysis. For the ordinary plate, I got excellent results comparing to Ansys, but for the folded plate, there is some difference.

Can anybody help me rectifying my results?.

 

[GregLocock]

Probably, but I doubt the information you have given is sufficient. Galerkin ain't no big clue. For a start you might like to define the dimensions of your folded plate, and how bad the results are. It's not top secret.


Cheers

Greg Locock


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[Hussein Hussein]

Mr GregLocock, Thank you very much for your reply.

A word file titled as (Hussein) related to my thread is attached. Thanks a lot.

Hussein

 

[GregLocock]

I'm actually more interested in the errors in the flat plate example. Assuming both are supposed to be exactly the same mathematical formulation then it seems something is wrong. Can you plot the mode shapes out? What's the theoretical result (Look in Blevins)?

a 10x10 mesh isn't really enough to capture the first 4 modes I suspect, try 100x100.

Aso you might find it easier to debug a long thin cantilever rather than s square plate with one edge encastre.

Cheers

Greg Locock


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[hacksaw]

sort out which modes your MatLab code is identifying

The first mode in both cases is that of a cantilevered plate, the first of 0.5 m the second of 1.0 m.

The corresponding natural frequencies of the first order cantilever modes are 16.3 Hz and 4.1 Hz, these agree with your matlab calcs. The ANSYS result is considering a true plate moding response. In the folded plate case, the folded tip stiffens the structure...

 

[hacksaw]

As an added foot note, you response will also change considerable when you include the "suppoprt" elasticity supporting your structures.

 

[Hussein Hussein]

GregLocock;
For the analysis results of case#1 structure(Plate only), there is no big worry about the small margin of error, since it could be rectified by either mesh refinement or different Gauss integration points used for evaluating the stiffness and mass matrices. My major concern is about the big margin of error for case#2. Regarding the changing of plate dimensions, my Matlab code is made for general inputs and the square plate was just an example.

hacksaw,
When I analysed a straight cantilevered plate (1.0 m X 0.5 m) using Ansys, I got results very close to those from my Matlab code shown for case#2 (folded plate), and this gives an indication that the code deals with the folded plate as a straight one. The Ansys results shown for case#2 in the attached file are different from the code results because of the effects of the folding, I think.
I'm desperately looking for involving the folding effects in my code !!!.

Thank you both for your replies

 

[hacksaw]

you already have the methodolgy worked out for the single plate, so it is a matter of attaching the second plate as you've shown and sorting out your boundary conditions for the interface (displacements, slopes, moment, shear) along the support interface.

 

[aerostress82]

This will sound "not so good", but trying to help you however I can. First of all I didn't do this in Matlab before, so I can't help much with that. But I sure did natural frequency and frequency response analysis. So I'm familiar at FEA level only.
Your first 4 modes actually correspond to:
1) x translation,
2) y translation,
3) z translation,
4) x rotation.

Always remember, your natural frequency gets better (higher) as you move your center of gravity of the structure closer to the boundaries (constraints). As you change your design and modify your structure's design such that you move the center of gravity closer to the constraints, you will have better (higher) natural frequencies - which will make you design stronger under vibrational fatigue and any other frequency related phenomena.

Now you know my study's outcome. As you look at those frequencies and remember the directions given above for the first 4 modes, (and good thing you put a good snapshot with the coordinate system) you can observe that your ANSYS and Matlab results are changing by; (approximately)
1) +30% (the deformation will look like a translation in x-direction. so your cog-center of gravity- change with the addition of the support is having a reverse effect compared to below results. the deformation in this direction is not affecting your cog a lot-meaning cog is shifting to the right with the addition of the plate and luckily your deformation is in that direction. So you are not gaining extra moment behavior by this deformation)
2) -40% (the deformation is translation in y-direction. the deformation is affecting the cog a lot - meaning the deformation direction is perpendicular to the "cog" shift. that's where problems are starting)
3) -40% (same as 2)
4) -40% (same as 2)

All above feels like you need a transformation matrix for the boundary of that plate only. I might be wrong, but I just wanted you to know how I feel about it knowing the FEA side of it. Hope it helps and you understand what I'm getting at. You may not understand it perfectly, but "cog shift" and "direction of the modes" are the most important subjects I wanted to introduce you on this subject. Now hopefully you'll have a more mature way of handling this problem. My gut feeling says it is the transformation matrix. Because of that, you might be seeing a different behavior in modes 2, 3 & 4. Please let me know the solution as well. This is a nice interesting case. Not that it would be applicable everywhere, but just knowing about the transformation matrix will give me more insight to the Matlab solution of this. Good luck!


Spaceship!!
Aerospace Engineer, M.Sc. / Aircraft Stress Engineer

 

[GregLocock]

"Your first 4 modes actually correspond to:
1) x translation,
2) y translation,
3) z translation,
4) x rotation."

Rubbish, in general. You may be thinking of a free free model. This one isn't. The plate is encastre at one edge, hence first mode is almost certainly bending, second is probably as well.



Cheers

Greg Locock


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[aerostress82]

The stress police got me and he is right. Let me try to clean up the garbage from my previous comment then. My above explanation about cog location, and the relation between each mode's behavior and your error between FEA and Matlab is still something you need to look into to understand what actually is missing in your Matlab code.

Sorry about the confusion. Got mixed up with aerospace FEM validation reports and actual automotive modal results evaluation for sensitivity & etc checks on the most damaging (fatigue) modes. Was a busy day today at work, and this is the worst mistake I ever made I must admit.

Spaceship!!
Aerospace Engineer, M.Sc. / Aircraft Stress Engineer

 

[Hussein Hussein]

First and for most, I'm very proud to be one of this community members. One will learn a lot in different aspects.

aerostress82,
Thanks a lot for your explanation of natural frequency-center of gravity relationship. My chosen rectangular element has 3 degrees of freedom as shown in the attached file, therefore, you classification in terms of mode shapes is not right, I'm afraid. Yes, many researchers talk about transformation matrix (TM) when dealing with such structures, but this TM is a mystery to me so far. Sure, I'll let you know about it as soon as I get it.

GregLocock,
For the time being, I can't consider any other boundary conditions since it is the only case I'm studying.

Glad to see your comments and you're kindly requested to widen the discussion area in order to get as many responses as possible.

 

[GregLocock]

Well to be honest I've suggested how I would debug your model which has awful credibility problems with the flat plate example. You've chosen to ignore that. Byeeee.

Cheers

Greg Locock


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