Rotating a 6x6 matrix 3

[sartorbjk]

Hello,
I have a matrix called compliance matrix (6x6) in 3d-stress space.
It looks like:
x y z 0 0 0
x1 y1 z1 0 0 0
x2 y2 z2 0 0 0
0 0 0 a 0 0
0 0 0 0 b 0
0 0 0 0 0 c

I want to make a rotation. I should turn X axis 45° towards y-axis.

My question is, how can I form my 6x6 rotation matrix? Is there an already implemented code in excel?
I really appreciate your help.


Best Regards,
Sartor

 

[IRstuff]

A 6x6 rotation makes no physical sense. Are you perhaps refering to a transposition, instead?

TTFN

FAQ731-376

 

[sartorbjk]

This is a compliance matrix.
When I change the element axis, compliance matrix is changing.
I think it is rotation.
but unfortunately I am not sure.
can it be also composition of linear transformation?


Best Regards,
Sartor

 

[walz]

can you write the result for this 45 deg rotation. I want to see how it alters... then may be i can help you to do it with excel

 

[IRstuff]

Perhaps this is apropos:

http://www.efunda.com/formulae/solid_mechanics/composites/calc_ufrp_cs_arbitrary.cfm

You'll have to modify it for 3 dimensions

TTFN

FAQ731-376

 

[rb1957]

isn't this "just" axes rotation ?

X2 = 0.707X+0.707Y
Y2 = -0.707X+0.707Y

 

[sartorbjk]

IRStuff:
could you show me a way how I can modify it?
I found a table from a book (please see the attached file), however, I can not form myself a Matrix which will rotate my 6x6 compliance matrix.
rb1957:
yes, it is just the axis rotation.
unfortunately i still could not manage it for 6x6 matrix.

Best Regards,
Sartor

 

[sartorbjk]

actually,

http://www.efunda.com/formulae/soli...s/comp_lamina_arbitrary_detail.cfm#Compliance

in this address, there is half of what I want.
However,
S'13, S'14 and some of the other 6X6 matrix elements are missing.
How can I write that missing parts in terms of the old values (like given in the formulas)?

Best Regards,
Sartor

 

[IDS]

Does this give what you are lookin for?:

http://www.mse.mtu.edu/~drjohn/my4150/compositesdesign/cd3/cd1.html

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[sartorbjk]

Not really.
Because my 6x6 matrix is called compliance matrix in ansys.
However, it is not directly related to stiffness of the material.
But at the end of the day,
I have a 6x6 matrix, and I think I need to build a rotation matrix (6X6, composed of sin and cos values)to rotate that 6x6 compliance matrix. In the attachment, there are some already done ones(like for plane stress condition or reduced compliance) but that will not work for my case.
my matrix has values on 4x4, 5x5, 6x6. I have to rotate them also I think. I am quite confused at the moment.
Do you have any ideas how to solve this problem, i would really appreciate any road maps.

Best Regards,
Sartor

 

[IRstuff]

Is this for school?

TTFN

FAQ731-376

 

[sartorbjk]

No, the main theme is a project for some aerospace application.
But I have to write a report and tell them it is not working due to matrix rotation. But i just dont want to write them that but also show them.

Best Regards,
Sartor

 

[rb1957]

if it's "just" axis rotation and you're rotating in the XY plane, then you rotate the first two rows and columns.

i'm not too sure about columns/rows 4,5,6 ... this is bending stiffness (i think), i suspect it's the same (rotate rows/columns 4,5) ...

GL !

 

[IRstuff]

Can't you just rotate the stress vector? Seems like that would be a way easier thing to do.

TTFN

FAQ731-376

 

[kacarrol]

This is harkening back quite a long ways in my memory. But I don't think you can use that formula to rotate your compliance matrix. I believe the example you have is for rotating a 2D vector, which would be a 4x4 matrix. This article below talks about 3D and 2D matrices and rotations of the 3D matrix, but doesn't explain the math behind it:

http://www.ohiocae.com/piezo-paper.htm

You could perhaps use it as a check on your math when you are done though.

I was reading this wiki article (

http://gpwiki.org/index.php/3D:Matrix_Math)

and there is a link at the bottom that may provide more answers, but I can't say for sure because I can't access that page at work:

http://www.makegames.com/3drotation/

I feel that you may be able to do the rotations by using the matrices at the bottom of the wiki article by extending the matrix to a 6x6 by adding addition rows/columns of 0's with 1's on the diagonal. But, that idea needs to be backed up by someone else who remembers more than I do.

Further more, when I was doing a lot of work with some very large matrices I can tell you that excel should be able to handle basic matrix operations (multiplication etc.) for matrices upto 25x25.

One last thing to keep in mind. You can't just apply rotations at random, the order in which you apply the rotations is important, see the article below for a good (if not simplistic) explanation.
(under subheading "Successive Rotations")

http://www.euclideanspace.com/maths/algebra/matrix/orthogonal/rotation/index.htm

I don't know if this has been any help. But good luck!
K

 

[TERIO]

I think this link may help

http://solidmechanics.org/text/Chapter3_2/Chapter3_2.htm

Particularly, look at Section 3.2.11 Basis change formulas for anisotropic elastic constants.

 

[kacarrol]

I was thinking it was rotating a vector matrix, please ignore my previous post as I was way off topic. I've read the other articles you guys have posted.

I've put a spreadsheet together with the information, as I understand it, from the link by TERIO, but I don't quite grasp the math behind this stuff yet. If you put in a rotation angle of zero the equation doesn't seem to return the original compliance matrix.

Can anyone explain what I'm missing?

Thank you,
K

 

[IRstuff]

If you look at the diagonal elements of those rotation matrices, they all go to 1 with theta=0, and off-diagonal elements go to zero, resulting in the identity matrix

TTFN

FAQ731-376

 

[kacarrol]

I see, I used radians when I should have used degrees. Must be time for a coffee.

Thank you!

I've updated the spreadsheet for anyone interested. I would appreciate it if someone would check the math behind the result matrix to be sure I multiplied the matrices correctly.

K

 

[IDS]

kacarrol - I haven't looked at your spreadsheet in detail, but you were right the first time with radians, not degrees. Trig functions in Excel (and every other spreadsheet afaik) use radians, not degrees.

In the original spreadsheet the sin(radians) cells were reading the cos cell, not the radians cell. When you deleted the rows containing the conversion to radians you corrected the sin rows to read from the cells labelled as degrees, but the angle should be entered in radians.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/