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46 Multibody Systems Approach to Vehicle Dynamics
points P and Q are fixed in the same rigid body, Body 2. As such there can
be no component of motion along the line PQ and any relative motion must
therefore be perpendicular to this line.
This relationship can be expressed mathematically in two ways. The first of
these uses the vector dot product to enforce perpendicularity as shown in
equation (2.98):
{V PQ } 1 • {R PQ } 1 0 (2.98)
The second method uses the vector cross product as shown in equation
(2.99):
{V PQ } 1 { 2 } 1 {R PQ } 1 (2.99)
More often than not we use the cross product as this will yield the angular
velocity vector { 2 } 1 . This may be required for a later acceleration analysis.
In developing the equations to solve the velocities in a system of interconnected
rigid bodies the triangle law of vector addition can also be used
as shown in equation (2.100):
{V P } 1 {V Q } 1 {V PQ } 1 (2.100)
2.5 Acceleration analysis
Given that acceleration is the time rate of change of velocity we can
develop the equations required for an acceleration analysis using the vectors
shown in Figure 2.19.
It is possible to develop equations that would yield the relative acceleration
vector {A PQ } 1 using the angular velocity vector { 2 } 1 and the angular
acceleration vector { 2 } 1 as follows:
d
{ APQ} { VPQ}
dt
1 1
(2.101)
{ A }
d
RPQ
d t
{{ 2
} { } }
PQ 1 1 1
(2.102)
{V PQ } 1
{ω 2 } 1
{A PQ } 1
{α 2 } 1
P
O1
X 1
Z 1
Y 1
Q
{R PQ } 1
Body 2
Fig. 2.19
Relative acceleration vector