4569846498

Kinematics and dynamics of rigid bodies 43
I
J
I J
Z 1
Y
O 1
1
X 1
Steer change
Fig. 2.16
Using vectors to determine bump steer
new positions that can be designated C, E and G. It should be noted that
for the analysis using vectors here C and E are not the actual final positions
but are only used to find the magnitudes of |R DC | and |R DE | so that the
analysis may progress using the sequence shown in Figure 2.15. Having
calculated the new positions of all the movable nodes the movement of the
tyre contact patch, in this case taken to be point H, could be used to establish,
for example, the lateral movement or half track change. The change in
orientation of the wheel will also be of interest. The new positions I and J
can be compared with the undisplaced positions I and J to determine the
change in steer angle as shown in Figure 2.16.
The bump steer can be determined by finding the angle between the projection
of IJ and IJ onto the global X 1 Y 1 plane. The projection is achieved
by setting the z co-ordinates of all four vectors to zero and then rearranging
the vector dot product as shown in equation (2.86):
{ RIJ
}
cos { RIJ} 1 •
(2.86)
| R || R |
IJ
IJ
2.4 Velocity analysis
Consider the rigid body, Body 2, shown in Figure 2.17. In this case we are
initially only interested in motion in the X 1 Y 1 plane. The body moves and
rotates through an angle , measured in radians, about the Z 1 -axis.
The vector {R PQ } 1 moves with the body to a new position {R PQ } 1 . The
new vector {R PQ } 1 is defined by the transformation
{R PQ } 1 [A]{R PQ } 1 (2.87)
where [A] is the rotation matrix that rotates {R PQ } 1 onto {R PQ } 1 . Expanding
this gives
⎡PQx
⎤ ⎡cos
sin
⎢
PQy
⎥
⎢
sin cos
⎢ ⎥ ⎢
⎣⎢
PQz ⎦⎥
⎣⎢
0 0
0⎤
⎡PQx⎤
0
⎥ ⎢
PQy
⎥
⎥ ⎢ ⎥
1⎦⎥
⎣⎢
PQz⎦⎥
(2.88)