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24 Multibody Systems Approach to Vehicle Dynamics
attaching the appropriate sign to their magnitudes:
⎡Px⎤
{ RP} 11 ⎢
/ Py
⎥
⎢ ⎥
⎣⎢
Pz⎦⎥
(2.1)
The use of brackets { } here is a shorthand representation of a column
matrix and hence a vector. Note that it does not follow any quantity that can
be expressed as the terms in a column matrix is also a vector.
In writing the vector {R P } 1/1 the upper suffix indicates that the vector is
measured relative to the axes of reference frame O 1 . In order to measure a
vector it is necessary to determine its magnitude and direction relative to
the given axes, in this case O 1 . It is then necessary to resolve it into components
parallel to the axes of some reference frame that may be different
from that used for measurement as shown in Figure 2.2.
In this case we would write {R P } 1/2 where the lower suffix appended to
{R P } 1/2 indicates the frame O 2 in which the components are resolved. We
can also say that in this case the vector is referred to O 2 . Note that in most
cases the two reference frames are the same and we would abbreviate
{R P } 1/1 to {R P } 1 .
It is now possible in Figure 2.3 to introduce the concept of a relative position
vector {R PQ } 1 . The vector {R PQ } 1 is the vector from Q to P. It can also
be described as the vector that describes the position of P relative to Q.
X 2
X 1 Y 2
Pz 2
Z 1
P
O 1 O 2
{R P } 1/2
Px 2
Y 1
Z 2
Py 2
Fig. 2.2
Resolution of position vector components
Z 1
{R P } 1
P
{R Q } 1
{R PQ } 1
Q
O 1
X 1
Y 1
Fig. 2.3
Relative position vector