Deriving the Rotation Matrix of a Beam Element

In summary, the conversation is about deriving a rotation matrix for a beam element rotated around three axes (x,y,z). The matrices for rotation about each axis are given and the result of all three rotations is the product of these matrices. There may be a discrepancy in the placement of minus signs in the matrices, and it is important to consider the direction of rotation when using the right hand thumb rule.
  • #1
dirk_mec1
761
13

Homework Statement



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

http://img194.imageshack.us/img194/3351/60039512.png [Broken]

http://img808.imageshack.us/img808/159/64794596.png [Broken]

Homework Equations


-

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?
 
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  • #2
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.
 
  • #3
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:

Code:
[                        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)       ]
 
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  • #4
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.
 
  • #6
dirk_mec1 said:

Homework Statement



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

http://img194.imageshack.us/img194/3351/60039512.png [Broken]

Look at your diagram. Are all of those rotations positive by the right hand thumb rule? (Hint: The answer is no.)
 
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1. What is a rotation matrix in the context of a beam element?

A rotation matrix is a mathematical tool used to describe the rotation of a beam element in three-dimensional space. It is a square matrix that contains elements representing the cosine and sine of the rotation angles around each axis.

2. How is a rotation matrix derived for a beam element?

A rotation matrix for a beam element is derived by considering the coordinate system of the beam and the rotation angles around each axis. The matrix is constructed using trigonometric functions, such as cosine and sine, to represent the rotation around each axis.

3. What are the applications of a rotation matrix for a beam element?

A rotation matrix for a beam element has various applications in structural analysis, robotics, and computer graphics. It is used to describe the orientation of a beam element in three-dimensional space and is an essential tool in analyzing the behavior of complex structures.

4. Can a rotation matrix be used for any type of beam element?

Yes, a rotation matrix can be used for any type of beam element as long as the element's rotation can be described using three rotation angles around the three axes. It is a versatile tool that can be applied to various types of beams, including straight beams, curved beams, and beams with varying cross-sections.

5. Are there any limitations to using a rotation matrix for a beam element?

One limitation of using a rotation matrix for a beam element is that it assumes the beam to be rigid and does not account for any deformations due to bending or shear. Additionally, the rotation matrix can become more complex for beams with more than three rotation angles, making it challenging to apply in certain cases.

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