rotation matrix


by dirk_mec1
Tags: matrix, rotation
dirk_mec1
dirk_mec1 is offline
#1
Mar30-12, 03:57 AM
P: 664
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
SensaBubble: It's a bubble, but not as we know it (w/ video)
The hemihelix: Scientists discover a new shape using rubber bands (w/ video)
Microbes provide insights into evolution of human language
HallsofIvy
HallsofIvy is offline
#2
Mar30-12, 06:29 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,898
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
dirk_mec1 is offline
#3
Mar30-12, 09:49 AM
P: 664
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
HallsofIvy is offline
#4
Mar30-12, 10:23 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,898

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
dirk_mec1 is offline
#5
Mar30-12, 10:36 AM
P: 664
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
D H is offline
#6
Mar30-12, 12:56 PM
Mentor
P: 14,476
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