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Kinematics and dynamics of rigid bodies 29
In matrix form this can be written:
⎡Ax'
⎤ ⎡1 0 0 ⎤
⎢
Ay'
⎥
⎢
0 cos sin
⎥
⎢ ⎥ ⎢
⎥
⎣⎢
Az' ⎦⎥
⎣⎢
0 sin cos
⎦⎥
⎡Ax⎤
⎢
Ay
⎥
⎢ ⎥
⎣⎢
Az⎦⎥
(2.21)
In a similar manner when {A} 1 is rotated through an angle about the Y 1 -axis
the components of a new vector {A} 1 can be obtained from
⎡Ax'
⎤ ⎡ cos
0 sin⎤
⎢
Ay'
⎥
⎢
0 1 0
⎥
⎢ ⎥ ⎢
⎥
⎣⎢
Az' ⎦⎥
⎣⎢
sin 0 cos⎦⎥
⎡Ax⎤
⎢
Ay
⎥
⎢ ⎥
⎣⎢
Az⎦⎥
(2.22)
After a final rotation of {A} 1 through an angle about the Z 1 -axis the new
components of {A} 1 are given by
⎡Ax'
⎤ ⎡cos sin 0⎤
⎢
Ay'
⎥
⎢
sin cos 0
⎥
⎢ ⎥ ⎢
⎥
⎣⎢
Az' ⎦⎥
⎣⎢
0 0 1⎦⎥
⎡Ax⎤
⎢
Ay
⎥
⎢ ⎥
⎣⎢
Az⎦⎥
(2.23)
Y 1
Az sin α
Ay cos α
α
O 1
|Az|
Az
α
{A} 1
{A} 1
α
α
A
Ay
α
A
|Ay |
Ay
Z 1
Fig. 2.9
Az cos α
Ay sin α
Az
Rotation of a vector viewed along the X 1 -axis