4569846498

Kinematics and dynamics of rigid bodies 45
In the limit t approaches zero and we can write
d
dt
⎡PQx⎤
⎡ 0 d / dt
0⎤
⎡PQx⎤
⎢
PQy
⎥
⎢d
/ d t 0 0⎥
⎢
PQy
⎥
⎢ ⎥ ⎢
⎥ ⎢ ⎥
⎣⎢
PQz⎦⎥
⎣⎢
0 0 0⎦⎥
⎣⎢
PQz⎦⎥
(2.94)
which can be written
⎡V
⎢
⎢V
⎢
⎣V
PQx
PQy
PQx
⎤ ⎡ 0
⎥
⎥
⎢
⎢
⎥
⎦ ⎣⎢
0
z
z 0⎤
⎡PQx⎤
0 0
⎥ ⎢
PQy
⎥
⎥ ⎢ ⎥
0 0⎦⎥
⎣⎢
PQz⎦⎥
(2.95)
Note that generally rotations cannot be represented as vector quantities
unless they are very small, as in finite element programs. Hence angular
velocities obtained by differencing rotations over very small time intervals
are in fact vector quantities.
If the rigid link also undergoes small rotations about the X-axis and
about the Y-axis then the full expression is
⎡V
⎢
⎢V
⎢
⎣V
PQx
PQy
PQz
⎤ ⎡ 0
⎥ ⎢
⎥ ⎢ z
⎥ ⎢
⎦ ⎣
y
0
x
z
y
⎤ ⎡PQx⎤
⎥
⎢
x⎥
PQy
⎥
⎢ ⎥
0 ⎥
⎦ ⎣⎢
PQz⎦⎥
(2.96)
Note that this matrix is the skew-symmetric form of the angular velocity
vector [x y z] T . In general terms we write
{V PQ } 1 { 2 } 1 {R PQ } 1 (2.97)
The direction of the relative velocity vector {V PQ } 1 is perpendicular to the
line of the relative position vector {R PQ } 1 as shown in Figure 2.18. The two
X 1
Z 1
O 1
Y 1
Body 2
Q
{V PQ } 1
{ω 2 } 1
P
{R PQ } 1
Fig. 2.18
Relative velocity vector