4569846498

Kinematics and dynamics of rigid bodies 69
Z 3
Z 2
Body 2
X 3
X 2
Y 2 ,Y 3
O 2 ,O 3
Fig. 2.35
Vehicle body co-ordinate system
tank. The frame O 2 shown in Figure 2.35 is also positioned at the mass
centre and has its Y 2 -axis coincident with the Y 3 -axis of frame O 3 . The
frame O 2 represents the principal axes of the vehicle body and is obtained
by a transformation from O 3 represented by the rotation through an angle
about the Y 2 - and Y 3 -axes.
In determining the products of inertia for this body it can be seen that for
every element of mass with a positive y co-ordinate there exists an equivalent
element with a negative y co-ordinate. As a result we get
I I xy d m 0
xy
yx
I I yz d m 0
yz
zy
∫
∫
The inertia matrix for Body 2 [I 2 ] 2/3 measured from frame O 2 and referred
to O 3 is therefore
⎡Ixx
0 Ixz
⎤
[ I2] 2 / 3
⎢
0 I2
0
⎥
⎢
⎥
(2.215)
⎣⎢
Ixz
0 Izz
⎦⎥
From this it can be seen that the y-axis is a principal axis and is normal to
the plane of symmetry. The principal moment of inertia I 2 is therefore
equal to I yy . From section 2.2.7 we can see that the matrix [T 3 ] 2 that transforms
from frame O 3 to O 2 is given by
⎡cos 0 sin
⎤
[ T 3 ] 2
⎢
0 1 0
⎥
⎢
⎥
⎣⎢
sin 0 cos⎦⎥
(2.216)
From (2.206) we can see that using the transformation matrix [T 3 ] 2 given
for this particular case will lead to
[I 2 ] 2/2 [T 3 ] 2 [I 2 ] 2/3 [T 3 ] T 2