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38 Multibody Systems Approach to Vehicle Dynamics
20. Differentiation of the cross product {A} m {B} m with respect to time t
follows the same rule as that for the dot product. Hence,
d
(2.72)
d ({ A } m { B } m ) { A
t
} m { B˙
} m { A˙
} m { B } m
2.3 Geometry analysis
2.3.1 Three point method
In order to establish the position of any point in space, vector theory can be
used to work from three points for which the co-ordinates are already established.
Consider the following example, shown in Figure 2.13.
In this example the positions of A, B and C are taken to be known as the
lengths AD, BD and CD. The position of D is unknown and must be solved.
In terms of vectors this can be expressed using the following known inputs:
{R A } 1 T [Ax Ay Az]
{R B } T 1 [Bx By Bz]
{R C } T 1 [Cx Cy Cz]
|R DA |
|R DB |
|R DC |
In order to solve the three unknowns Dx, Dy and Dz, which are the components
of the position vector {R D } 1 , it is necessary to set up three equations as
follows:
|R DA | 2 (Dx Ax) 2 (Dy Ay) 2 (Dz Az) 2 (2.73)
|R DB | 2 (Dx Bx) 2 (Dy By) 2 (Dz Bz) 2 (2.74)
|R DC | 2 (Dx Cx) 2 (Dy Cy) 2 (Dz Cz) 2 (2.75)
B
{R DB } 1
A
{R DA } 1
{R DC } 1
O1
Y 1
D
X 1
Z 1
C
Fig. 2.13
Use of position vectors for geometry analysis