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66 Multibody Systems Approach to Vehicle Dynamics
Y 2
Principal
planes
Y 2
dm
O 2
X 2
O 2
Z 2
Body 2
X 2 ,Y 2
X 2
dm
Z 2
O 2
Fig. 2.33
Principal axes for a body
Y 3
Y 2
O
Z Z 2 1
2
X 3
GRF
X
O 1
1 Y 1
Fig. 2.34 Transformation to principal axes
Z 3
O 3
Body 2
X 2
⎡H
⎢
⎢
H
⎣⎢
H
2x
2y
2z
⎤ ⎡I1
0 0⎤
⎡
⎥
⎥
⎢
0 I2
0
⎥ ⎢
⎢ ⎥ ⎢
⎦⎥
⎣⎢
0 0 I3⎦⎥
⎣⎢
2x
2y
2z
⎤ ⎡I
⎥
⎥
⎢
⎢
I
⎦⎥
⎣⎢
I
1
2
3
(2.201)
The principal axes and the principal moments of inertia may be obtained by
considering the two frames O 3 and O 2 both located at the mass centre in
Body 2 as shown in Figure 2.34. The axes associated with frame O 2 are again
taken to be the principal axes of the body. The angular momentum {H 2 } 1/2
referred to O 2 can be transformed from the angular momentum {H 2 } 1/3
referred to O 3 using the rotational transformation matrix [T 3 ] 2 as described
in section 2.2.7:
{H 2 } 1/2 [T 3 ] 2 {H 2 } 1/3 (2.202)
Since {H 2 } 1/2 [I 2 ] 2/2 { 2 } 1/2 and {H 2 } 1/3 [I 2 ] 2/3 { 2 } 1/3 it follows from
(2.202) that
[I 2 ] 2/2 { 2 } 1/2 [T 3 ] 2 [I 2 ] 2/3 { 2 } 1/3 (2.203)
If we pre-multiply a transformation matrix [T] by its transpose [T] T a property
of these matrices is that we produce an identity matrix:
2x
2y
2z
⎤
⎥
⎥
⎦⎥