rotation matrix

dirk_mec1

1. The problem statement, all variables and given/known data

The rotation matrix below describes a beam element which is rotated around three axes x,y and z. Derive the rotation matrix.





2. Relevant equations
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3. The attempt at a solution
I can see where the x-values (CXx CYx CZx) come from. They're just the projections of the rotated x-axes (the one with rotation alpha and beta). But I don't understand how the rest is derived can somebody help me?

HallsofIvy

Rotation about the x-axis through angle [itex]\alpha[/itex] is given by the matrix
[tex]\begin{bmatrix}1 & 0 & 0 \\ 0 & cos(\alpha) & -sin(\alpha) \\ 0 & sin(\alpha) & cos(\alpha)\end{bmatrix}[/tex]

Rotation about the y-axis through angle [itex]\beta[/itex] is given by the matrix
[tex]\begin{bmatrix}cos(\beta) & 0 & -sin(\beta) \\ 0 & 1 & 0 \\ sin(\beta) & 0 & cos(\beta)\end{bmatrix}[/tex]

Rotation about the z-axis through angle [itex]\gamma[/itex] is given by the matrix
[tex]\begin{bmatrix} cos(\gamma) & -sin(\gamma) & 0 \\ sin(\gamma) & cos(\gamma) & 0 \\ 0 & 0 & 1\end{bmatrix}[/tex]

The result of all those rotations is the product of those matrices. Be sure to multiply in the correct order.

dirk_mec1

I suspect that there's a minus sign somewhere wrongly placed in your matrices Halls, am I correct? I moved the minus sign in your second matrix to the lower sine but there's still something wrong for this is my result:

HallsofIvy

No, all of the minus signs are correctly placed. I am, of course, assuming that a positive angle gives a rotation "counterclockwise" looking at the plane from "above"- from the positive axis of rotation.

dirk_mec1

But the wiki page shows a different position for the minus sign of your second matrix:

http://en.wikipedia.org/wiki/Rotation_matrix.

D H

Look at your diagram. Are all of those rotations positive by the right hand thumb rule? (Hint: The answer is no.)