4569846498

Kinematics and dynamics of rigid bodies 31
Note that the convention used here in Figure 2.10 is based on a set of Body
(X-Y-Z) rotations. It will be shown later that different conventions may be
used such as the Yaw-Pitch-Roll method based on a set of Body (Z-Y-X)
rotations or the Euler angle method used in MSC.ADAMS that is based on
a Body (Z-X-Z) combination.
2.2.7 Vector transformation
In multibody systems analysis it is often necessary to transform the components
of a vector measured parallel to the axis of one reference frame to
those measured parallel to a second reference frame. These operations
should not be confused with vector rotation. In a transformation it is the
magnitude and direction of the components that change. The direction of
the vector is unchanged. Consider the transformation of a vector {A} 1, or in
full definition {A} 1/1 , from reference frame O 1 to reference frame O 2 .
Figure 2.11 represents a view back along the X 1 -axis towards the origin O 1 .
The reference frame O 2 is rotated through an angle about the X 1 -axis of
frame O 1 .
From Figure 2.11 it can be seen that
Ax2 Ax1
Ay2 Ay1 cos Az1
sin
Az Ay sin
cos
2 1 1
(2.27)
Z 1
Az 1 sin α
Ay 2
Ay 1 cos α
Ay 1 sin α
α
A
{A} 1/2
Az 2
Az 1 cos α
Z 2
Ay 1
Az 1
Az 1 sin α
{A} 1/1
Y 1
α
Ay 1 sin α
Y 2
α
O 1
O 2
Fig. 2.11
Transformation of a vector