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Calculating rotation matrix of a line vector
Posted 07.06.2010, 06:52 GMT-4 4 Replies
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Hi All,
I have three points with known coordinates forming a triangle. Then two of the vertices move (like a rotation transition). My objective is to calculate the coordinate of the last point.
Can we do this in Comsol, like calculating the rotation matrix of the vectors?
Basis is to calculate the inverse of a matrix. How to give a coordinate matrix in comsol and find its matrix inverse? I guess that important matrix Matlab functions do not work in Comsol anymore.
I have three points with known coordinates forming a triangle. Then two of the vertices move (like a rotation transition). My objective is to calculate the coordinate of the last point.
Can we do this in Comsol, like calculating the rotation matrix of the vectors?
Basis is to calculate the inverse of a matrix. How to give a coordinate matrix in comsol and find its matrix inverse? I guess that important matrix Matlab functions do not work in Comsol anymore.
4 Replies Last Post 08.06.2010, 11:14 GMT-4
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Posted:
1 decade ago
Hi
If it is the related generalised displacement rigid body rotation you are after (those not gving any stress nor strain, see i.e. Elasticity Tensor, Dyadic and Engineering Approaches by ChiChou/Pagano, or Theory of Elasticity, by Timoshenko, or ...) they can be expressed as the antisymmetric displacement tensor:
in a v3.5a 2D carthesian case x,y the rotation wz=0.5*(vx-uy) in Comsol notations vx=dv/dx uy=du/dy, u,v displacements along x,y,
in 3D carthesian y,x,z you have all three rigid body rotations that can be expressed (with the "small displacement hypothesis" typical used for stress strain calculations)
wx = 0.5*(vz-wy)
wy = 0.5*(wz-uz)
wz = 0.5*(vx-uy)
This is in 3.5a notation, it changes in V4 because of the reference frame renaming (use upper case X,Y,Z)
But, you have the "rigid body connector" in V4 (still to be fully implemented, currently only partially there in 3D but to come in 2D, that gives you a more generalised value for the rotations, as I understand them, also valid for "large diplscement" rotations)
Hope this is what you were looking fore
Ivar
If it is the related generalised displacement rigid body rotation you are after (those not gving any stress nor strain, see i.e. Elasticity Tensor, Dyadic and Engineering Approaches by ChiChou/Pagano, or Theory of Elasticity, by Timoshenko, or ...) they can be expressed as the antisymmetric displacement tensor:
in a v3.5a 2D carthesian case x,y the rotation wz=0.5*(vx-uy) in Comsol notations vx=dv/dx uy=du/dy, u,v displacements along x,y,
in 3D carthesian y,x,z you have all three rigid body rotations that can be expressed (with the "small displacement hypothesis" typical used for stress strain calculations)
wx = 0.5*(vz-wy)
wy = 0.5*(wz-uz)
wz = 0.5*(vx-uy)
This is in 3.5a notation, it changes in V4 because of the reference frame renaming (use upper case X,Y,Z)
But, you have the "rigid body connector" in V4 (still to be fully implemented, currently only partially there in 3D but to come in 2D, that gives you a more generalised value for the rotations, as I understand them, also valid for "large diplscement" rotations)
Hope this is what you were looking fore
Ivar
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Posted:
1 decade ago
Hi Ivar,
Many thanks for your reply. Actually, my case is a bit simpler.
I have 3 mesh nodes creating 1 triangular mesh element. Now, it rotates - due to boundary moving - and I want to have the coordinate of the last vertex.
In the picture below, I know the coordinates of the triangle A. Now, after rotation I know the coordinates of blue and red points in triangle B, but I don't know the blue point coordinate in B. I want to find out the blue points' displacement in ALE coordinates.
Many thanks for your reply. Actually, my case is a bit simpler.
I have 3 mesh nodes creating 1 triangular mesh element. Now, it rotates - due to boundary moving - and I want to have the coordinate of the last vertex.
In the picture below, I know the coordinates of the triangle A. Now, after rotation I know the coordinates of blue and red points in triangle B, but I don't know the blue point coordinate in B. I want to find out the blue points' displacement in ALE coordinates.
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Posted:
1 decade ago
Hi
well if you know the start and ending points you have the values to resolve the ux uy uz vx vy vz and wx wy wz no ? as you are in 2D it simplfies too.
Its basically the curl or the determinant you need to get out
Hope this helps
Ivar
well if you know the start and ending points you have the values to resolve the ux uy uz vx vy vz and wx wy wz no ? as you are in 2D it simplfies too.
Its basically the curl or the determinant you need to get out
Hope this helps
Ivar
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Posted:
1 decade ago
Hi Ivar,
I am not an expert in mathematics, so please bear with me.
A better diagram: www.gliffy.com/pubdoc/2138013/L.png
I was thinking that if I can find the transformation matrix of the bottom of the triangle A (blue line, 1-2) to the blue line in B (blue line, 1'-2'), I can apply it to all other sides and find my desired coordinate (3'). The two triangles move and change size as well, making it bit more tricky; the coordinates in the picture are representative.
So far, I am stuck at transformation, translation and rotation matrices.
I am not an expert in mathematics, so please bear with me.
A better diagram: www.gliffy.com/pubdoc/2138013/L.png
I was thinking that if I can find the transformation matrix of the bottom of the triangle A (blue line, 1-2) to the blue line in B (blue line, 1'-2'), I can apply it to all other sides and find my desired coordinate (3'). The two triangles move and change size as well, making it bit more tricky; the coordinates in the picture are representative.
So far, I am stuck at transformation, translation and rotation matrices.