Diploma Thesis - Erich Schmid Institute
Diploma Thesis - Erich Schmid Institute
Diploma Thesis - Erich Schmid Institute
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Orthogonal tensors and rotation<br />
⎛ cos( ω)<br />
0 sin( ω)<br />
⎞<br />
⎜<br />
⎟<br />
2<br />
O = ⎜ 0 1 0 ⎟<br />
(equ. 5.6)<br />
⎜<br />
⎟<br />
⎝−<br />
sin( ω)<br />
0 cos( ω)<br />
⎠<br />
The body in position r’ can be turned back by a rotation around e3 axis in the<br />
negative direction. Thus the angle’s sign is negative and the body’s starting position<br />
is supposed to be r’.<br />
⎛x<br />
⎞ ⎛ cos( ϕ)<br />
⎜ ⎟ ⎜<br />
⎜ y⎟<br />
= ⎜ − sin( ϕ)<br />
⎜ ⎟ ⎜<br />
⎝ z ⎠ ⎝ 0<br />
sin( ϕ)<br />
cos( ϕ)<br />
0<br />
0⎞<br />
⎛ x'⎞<br />
⎟ ⎜ ⎟<br />
0⎟<br />
⎜ y'⎟<br />
1⎟<br />
⎜ ⎟<br />
⎠ ⎝ z'<br />
⎠<br />
A closer look shows that the orthogonal tensor is the transposed or the inverse<br />
version of O 3 .<br />
5.3. Euler angles<br />
3 T<br />
( O ) r'<br />
r = (equ. 5.7)<br />
In the previous chapter we only thought about a rotation in a plane. In practice more<br />
possibilities are needed to express the tilt between two coordinate systems. One<br />
possibility is the concept of Euler angles [8], that can be understood as three<br />
separated rotations around certain axes, similar like before. The first step is the<br />
rotation around the e3 axis so e1 and e2 will change their position. After that the new<br />
e1 is supposed to be the next rotation axis, which will lead to two new e2 and e3 axes,<br />
and the last step is like the first one.<br />
After the first rotation the new axis is e1.<br />
⎛cos(<br />
ϕ1)<br />
− sin( ϕ1)<br />
0⎞<br />
⎜<br />
⎟<br />
ei<br />
' = ⎜ sin( ϕ1)<br />
cos( ϕ1)<br />
0⎟<br />
e<br />
⎜ 0 0 1⎟<br />
⎝<br />
⎠<br />
⎛1<br />
0 0 ⎞<br />
⎜<br />
⎟<br />
ei<br />
'' =<br />
⎜0<br />
cos( φ)<br />
sin( φ)<br />
⎟ ei<br />
'<br />
⎜0<br />
sin( ) cos( ) ⎟<br />
⎝ − φ φ ⎠<br />
i<br />
13
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