Register to reply

Rotation matrix

by dirk_mec1
Tags: matrix, rotation
Share this thread:
dirk_mec1
#1
Mar30-12, 03:57 AM
P: 686
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
-


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?
Phys.Org News Partner Science news on Phys.org
'Smart material' chin strap harvests energy from chewing
King Richard III died painfully on battlefield
Capturing ancient Maya sites from both a rat's and a 'bat's eye view'
HallsofIvy
#2
Mar30-12, 06:29 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,682
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
#3
Mar30-12, 09:49 AM
P: 686
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:

[                        cos(a)cos(b),               -sin(b),                           cos(b)sin(a)                ]
[ sin(a)sin(c) + cos(a)cos(c)sin(b)         cos(b)cos(c)         cos(c)*sin(a)sin(b) - cos(a)sin(c) ]
[ cos(a)sin(b)sin(c) - cos(c)sin(a)        cos(b)*sin(c)     cos(a)cos(c) + sin(a)sin(b)sin(c)       ]

HallsofIvy
#4
Mar30-12, 10:23 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,682
Rotation matrix

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
#5
Mar30-12, 10:36 AM
P: 686
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
#6
Mar30-12, 12:56 PM
Mentor
P: 15,201
Quote Quote by dirk_mec1 View Post
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.

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


Register to reply

Related Discussions
Rotation matrix Advanced Physics Homework 0
Shankar 12.4.4 - the rotation matrix vs. a rotation matrix (tensor operators QM) Advanced Physics Homework 2
Rotation matrix vs regular matrix Linear & Abstract Algebra 5
Construct a rotation matrix out of another rotation matrix General Math 2
How do you use a Rotation Matrix in 2-D? Introductory Physics Homework 3