Mode 1/0

[WebPilot]

While working on my other thread, dynamic modeling of a strip valve, I became uncertain what the next troublesome frequency would be. I suspect it is not the 2nd flexural, but rather the first torsional frequency.



This thread is going to become a 'branch' to my aforementioned thread.
In this thread, I am going to

• analyze a cantilevered flat thin plate,
• use the finite element method to determine the requisite eigenvalues and eigenvectors, and
• 'see' just how refined of a model I need to make before before the results coincide (within reason) with what has been published as the analytical solution to this problem.

With success, I plan to use this method to analyze my 'thin strip' and finally a DynaJet petal valve which has no analytical solution.

[WebPilot]

Here is a little background on the modal numbering system.

A node, as you may know from vibration theory, is a point on a one dimensional element, let us say a beam, where there is no vibratory motion.

A nodal line is naturally, a collection of nodes.

An eigenvalue is one of many natural frequencies.

An eigenvector is the mode shape at a corresponding eigenvalue.

Mode m/n is defined as a natural frequency where:

• m is the number of nodal lines which are parallel to the lengthwise edge of the model and
• n is the number of nodal lines which are parallel to the spanwise edge of the model.

Examples

There are NO nodal lines for the 1st flexural vibration of a cantilever plate so its designation is Mode 0/0.



For the 2nd flexural vibration of a cantilever plate there is one nodal line which is parallel to the spanwise edge of the model. So its designation is Mode 0/1



For the 1st torsional vibration of a cantilever plate there is one nodal line which is parallel to the lengthwise edge of the model. The dotted blue line in the following contour plot is this 1 nodal line. Thus, its designation is Mode 1/0.



Click here to see image in its entirety.

[GRIM]

Hey Forrest , I must confess to having got a bit lost in the math jungle in the other thread, but dont let that stop you !,

These graphics are brilliant , and it just so happens that I am currently building a valved engine that has valves very similar to your model , see pic

I had considered mode 1/0 , an undesireable but inevitable effect ,

I had not considered Mode 0/1, that will stop it from running,

Fascinating , please continue, can you do a kind of, flexural/torsional analisis, for DUMMIES?

[WebPilot]

Hi Grim,

OK, I can see why you're interested in this thread.

I'm glad to read that you enjoy my graphics, but am saddened to read I 'lost' you on my dynamic modeling of a strip valve. You must PM me and tell me where you strayed. I am going to return to that thread.

This subject is difficult. Analytical solutions exist only for the simplest of geometries. I've looked over those solutions that have been published and some of those expressions are quite tedious to perform. I am going to compare my present finite element analysis with one of those analytical (exact) solutions. It is quite simple, but is only good approximately for the first five frequencies and for a/b <= 5.0. The first five frequencies is more than enough, though, for our needs.

continuing ...

First I am going to do the plug and chug analytical determination of the first three frequencies. Then, I'll do the finite element analysis. Of course using the fem method for this problem, is more work than necessary. For the dynajet petal valve (a more complicated geometry), it may prove to be the best way.

[WebPilot]

[Ape]

I know it's probably just me, but isn't that strip rather thick? 0.1" just seems really. . well. . thick.
And yes, For Dummies would be nice.

[WebPilot]

Hi Ape,

The value of 0.1 is arbitrary and I'm just using it for an example. If you halve this thickness, you will halve the fundamental frequency.

Sorry, a worked out formula example is as easy as it gets.

For those of you having 'troubles' with this complicated of an expression on a calculator, try googling using calculator

http://www.google.com/search?hl=en&clie ... tnG=Search

One seemingly applicable site I found is Using your calculator. Especially read and work out from the middle to the end of the page.

To aid you, I just performed this calculation using the windows supplied calculator.

Launch it and click view-scientific (or press alt-v-s keys)
NOTE: the hyphens are NOT typed!

Enter the following keystrokes to evaluate the last angular frequency expression. I color coded them to match the keyboard in order to help find the keys. I use MS (memory save) a lot. This saves what I've done correctly up to that point, in case I 'goof' on the next. To start over, press CE C to clear the 'bad stuff' and to recall the last number saved press MR (memory recall).

• press CE and C
• 0.0283 / ( 2747.3 * 32.2 * 12 ) =
• MS
• check the Inv box, and press x^2
• MS
• * 4 = 1/x
• MS
• * 3.472 =
• MS
• / ( 2 * pi ) =
• MS

You should now have

846.093270793673329156983307514841

showing in the display.

In engineering, we are usually only accurate to 3 digits but I usually work with 4

Round this number off to 846.1 Hz for further use.

[WebPilot]

[WebPilot]

download to read ...

[WebPilot]

In summary, for an a/b ratio of 2:1, Barton's analytical work gives values of

• mode 1 (or mode 0/0), 846.1 Hz
• mode 2 (or mode 1/0), 3638 Hz
• mode 3 (or mode 0/1), 5266 Hz

Here are the results of my first finite element model. It consists of only 6 nodes and 2 quadrilateral elements.



Performance of 2 elements? Correlation is rather poor.

Difference between Barton formulation and present FE model

1. mode 1, (768.3 - 846.1/846.1)100 = -9.195 %
2. mode 2, (2204 - 3638 / 3638)100 = -39.42 %
3. mode 3, (4076 - 5266 / 5266)100 = -22.60 %

The eigenvectors are of the correct shape, though. How do I know this? Let's look at the data for mode 1, only.

Code: Select all

#Node     X            Y            Z- 
#        COORD        COORD      TRANSLATION         normalized

     6   1.000        2.000        0.15097E+03        0.5
     5   0.000        2.000        0.15097E+03        0.5
     4   1.000        1.000        0.47516E+02        0.1574
     3   0.000        1.000        0.47516E+02        0.1574
     2   1.000        0.000        0                  0
     1   0.000        0.000        0                  0

The z-translations must be normalized for plotting purposes. One way to do this is to take the largest value and divide all the others by it. I went one step further and multiplied by 0.5, too. If you look at just nodes 2, 4, and 6, you can envision an almost parabolic line upwards - mode 1!

Try it yourself for modes 2 and 3.

Anyways, mesh refinement is 'in order'.

[Mike Everman]

Nice engineering classes you've got going here, Forrest! Thanks, it's interesting stuff.

[Irvine.J]

Yep i'm in awe, can you sit my eng exams forest?

[Mike Everman]

Forrest, this is such a good diatribe, can you possibly post the actual pics, so that they are preserved for posterity here? I fear that your pic hosting will not always be alive, and it would ruin this fine thread.

[WebPilot]

Yep i'm in awe, can you sit my eng exams forest?

James, I don't think we look much alike. Don't they check ID's these days in the land down under? I may be able to help you with your homework if you get stuck someday. PM me ... but you must first learn how to spell my name correctly.

Forrest, this is such a good diatribe, can you possibly post the actual pics, so that they are preserved for posterity here? I fear that your pic hosting will not always be alive, and it would ruin this fine thread.

I was going to do as you wish. I changed some of the ones I could, but then did not like the looks of the thread - I found it too confining and restrictive! This post is still evolving which is why I've developed this technique of posting over the years. Besides, if I find an error I can correct it externally from the forum. I can't work under the conditions of a 24 hr. time frame window in order to edit my posts for the last and very final time. Sorry.

You got your wish for one of them. I changed it earlier today, but now I cannot change it back since I'm out of editing time.

PS diatribe You didn't mean to say this, did you?

di⋅a⋅tribe
–noun
a bitter, sharply abusive denunciation, attack, or criticism: repeated diatribes against the senator.

[Mike Everman]

Oops, guess I've been using that word wrong all my adult life!