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64 Multibody Systems Approach to Vehicle Dynamics
that the terms in the matrix are transformed to frame O 2. Since O 2 and O 3
are parallel the matrix [I 2 ] 3/3 would be identical to [I 2 ] 3/2.
The positions of the frames O 2 and O 3 relative to G 2 , the mass centre of
Body 2, can be given by
⎡x2
⎤
{ RO2G2} 2
⎢
y
⎥
⎢ 2 ⎥
, { RO G}
⎣⎢
z2
⎦⎥
(2.192)
where according to the triangle law of vector addition if a, b and c are the
components of the relative position vector {R O3O2 } 2 we can write
⎡a⎤
{ RO O}
⎢
b
⎥
, { RO G}
⎢ ⎥
⎣⎢
c⎦⎥
3 2 2 3 2 2
(2.193)
On this basis it is possible to relate a moment of inertia, for example I 2 x 3 x 3
for frame O 3 to I 2 x 2 x 2 for frame O 2 :
∫
∫
∫
2
2 3 3 3
Ixx ( y z)
dm
[( y b)
( z c) ] dm
[( y2
2 y2bb ) ( z2
2 z2cc )]
dm
2 2
Ixx 2 2 2 2 b∫
y2 d m2 c∫ z2
d m( b c ) m
2 2
I x x 2 m ( by cz ) m ( b c
)
(2.194)
If we take the situation where O 2 is coincident with G 2 , the mass centre of
Body 2, such that x 2 , y 2 and z 2 are zero, then (2.194) can be simplified to
I 2 x 3 x 3 I 2 x 2 x 2 m 2 (b 2 c 2 ) (2.195)
In a similar manner it is possible to relate a product of inertia, for example
I 2 y 3 z 3 for frame O 3 to I 2 y 2 z 2 for frame O 2 :
∫
∫
∫
2
2
Iyz yz dm
2 3 3 3 3
2
3
2
( y b)( z c) dm
2 2
3 2 2
2
2
⎡x3⎤
⎢
y
⎥
⎢ 3⎥
⎣⎢
z3
⎦⎥
⎡x
⎢
⎢
y
⎣⎢
z
2 2 2 2 2 2 2
y2z2 cy2 bz2
bc dm
I y z m ( cy bz ) m bc
2 2 2 2 2 2 2
2
2
2
2
a⎤
b
⎥
⎥
c⎦⎥
2
(2.196)
Taking again O 2 to lie at the mass centre G 2 we can simplify (2.196) to
I 2 y 3 z 3 I 2 y 2 z 2 m 2 bc (2.197)
On the basis of the derivation of the relationships in (2.195) and (2.197) we can
find in a similar manner the full relationship between [I 2 ] 3/2 and [I 2 ] 2/2 to be
2
2
[ I ] / [ I ] / m
2 3 2 2 2 2 2
2 2
⎡b c ab ac
⎤
⎢
2 2 ⎥
⎢ ab c a bc
⎥
⎢
2 2
⎣
ac bc a b
⎥
⎦
(2.198)