4569846498

44 Multibody Systems Approach to Vehicle Dynamics
Assuming that is small so that we can take cos 1 and sin
(rads) leads to
⎡PQx
⎤ ⎡ 1 0⎤
⎡PQx⎤
⎢
PQy
⎥
⎢
1 0
⎥ ⎢
PQy
⎥
⎢ ⎥ ⎢
⎥ ⎢ ⎥
⎣⎢
PQz ⎦⎥
⎣⎢
0 0 1⎦⎥
⎣⎢
PQz⎦⎥
The change in relative position {R PQ } 1 is given by
(2.89)
{R PQ } 1 {R PQ } 1 {R PQ } 1 (2.90)
or
⎡PQx⎤
⎡ 1 0⎤
⎡PQx⎤
⎡1
⎢
PQy
⎥
⎢
1 0
⎥ ⎢
PQy
⎥
⎢
0
⎢ ⎥ ⎢
⎥ ⎢ ⎥ ⎢
⎣⎢
PQz⎦⎥
⎣⎢
0 0 1⎦⎥
⎣⎢
PQz⎦⎥
⎣⎢
0
This gives
⎡PQx⎤
⎡ 0
⎢
PQy
⎥
⎢
⎢ ⎥ ⎢
⎣⎢
PQz⎦⎥
⎣⎢
0
0⎤
⎡PQx⎤
0 0
⎥ ⎢
PQy
⎥
⎥ ⎢ ⎥
0 0⎦⎥
⎣⎢
PQz⎦⎥
If this change takes place in time t then
0 0⎤
⎡PQx⎤
1 0
⎥ ⎢
PQy
⎥
⎥ ⎢ ⎥
0 1⎦⎥
⎣⎢
PQz⎦⎥
(2.91)
(2.92)
t
⎡PQx⎤
⎡ 0 / t
0⎤
⎡PQx⎤
⎢
PQy
⎥
⎢
/ t
0 0
⎥ ⎢
PQy
⎥
⎢ ⎥ ⎢
⎥ ⎢ ⎥
⎣⎢
PQz⎦⎥
⎣⎢
0 0 0⎦⎥
⎣⎢
PQz⎦⎥
(2.93)
δγ
P
{R P Q } 1
Y 1
Q P
{R PQ } Body 2
1
Q
X O1 1
Fig. 2.17
Motion of a vector attached to a rigid body