FEA - Rotation Matrix of Angular Deflection

[huyrich]

I am trying to use FEA with space frame element. I know that for rotating an angle a around the z-axis, the translational displacements of the local and global coordinates are related through the rotation matrix:

[tex]\begin{bmatrix}cos(a) & sin(a) & 0 \\ -sin(a) & cos(a) & 0 \\ 0 & 0 & 1\end{bmatrix}[/tex]

But how about angular displacement (deflection), I thought the rotation matrix for them would be:

[tex]\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}[/tex]

But it turns out it is the same with the first rotation matrix (or is it not?). Can anyone give me some hints how to derive or verify this?

 

[Achy47]

The rotation matrix you have mentioned is correct for rotational displacements (deflections) around the z-axis. This is because the rotation matrix represents the transformation between the local and global coordinates, and in this case, the local and global coordinates are the same since the rotation is only happening around the z-axis.

To understand this concept better, let's consider a simple example. Imagine a rectangular beam with one end fixed and the other end attached to a rotating motor. If we apply a rotational displacement around the z-axis, the beam will rotate around its fixed end. In this case, the local and global coordinates are the same, and the rotation matrix you have mentioned will apply.

However, if we apply a rotational displacement around the x-axis or y-axis, the beam will not only rotate but also deflect in the other two directions (translational displacements). In this case, the local and global coordinates are different, and the rotation matrix will be different as well.

To derive or verify the rotation matrix for such cases, you can use the basic principles of mechanics and apply them to the specific geometry and loading conditions of your space frame element. You can also refer to textbooks or online resources for further guidance.

I hope this helps. Good luck with your analysis!