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28 Multibody Systems Approach to Vehicle Dynamics
2.2.4 The scalar triple product
The scalar triple product D of the vectors {A} 1 , {B} 1 and {C} 1 is defined as
D {{A} 1 {B} 1 } • {C} 1 (2.18)
2.2.5 The vector triple product
The vector triple product {D} 1 of the vectors {A} 1 , {B} 1 and {C} 1 is
defined as
{D} 1 {A} 1 {{B} 1 {C} 1 } (2.19)
2.2.6 Rotation of a vector
In multibody dynamics bodies may undergo motion which involves rotation
about all three axes of a given reference frame. The new components
of a vector {A} 1 , shown in Figure 2.8, may be determined as it rotates
through an angle about the X 1 -axis, about the Y 1 -axis, and about the
Z 1 -axis of frame O 1 .
Consider first the rotation about O 1 X 1 . The component Ax is unchanged.
The new components Ax, Ay and Az can be found by viewing along an
X 1 -axis as shown in Figure 2.9.
By inspection it is found that
AxAx
AyAy cos Az
sin
AzAy sin Az
cos
(2.20)
Z 1
γ
Az
A
{A} 1
Ay
Y 1
O 1
β
Ax
α
X 1
Fig. 2.8
Rotation of a vector