4569846498

58 Multibody Systems Approach to Vehicle Dynamics
the reverse of sign for {V O2 } 1 . If for Body 2 frame O 2 is positioned at either
the mass centre so that ∫ vol {R P } 1 dm {0} 1 , or at a point of attachment to the
non-moving ground, where {V O2 } 1 {0} 1 , then (2.163) reduces to the more
convenient form which we will assume from now on:
(2.164)
If we now take the general case for any body (ignore body subscripts) and
expand the vectors into their full form we get
Applying the vector cross product, making use of the skew-symmetric form
of a vector in the normal manner, leads to
⎡ 0 z
y ⎤ ⎡ 0 z y
⎤ ⎡x⎤
{} R 1 {{ } 1 { R} 1}
⎢
z 0 x
⎥ ⎢
⎥
z
0
⎢
x y
⎥
⎢
⎥ ⎢
⎥ ⎢ ⎥
⎣⎢
y
x 0 ⎦⎥
⎢
y x 0 ⎥
⎣
⎦ ⎣⎢
z⎦⎥
⎡ 0 z
y ⎤ ⎡zy
yz⎤
⎢
z 0 x
⎥ ⎢
⎥
⎢
⎥ ⎢zxxz⎥
⎣⎢
y
x 0 ⎦⎥
⎢
⎣
yxxy⎥
⎦
Substituting (2.165) into (2.164) gives the general expression for the angular
momentum {H} 1 of a body where for simplicity the integral sign is now
taken to indicate integration over the volume of the body:
{ H} { R}
{{ } { R} } d m
(2.166)
∫
∫
{ H } { R } {{ } { R } } dm
2 1 vol P 1 2 1 P 1
⎡Hx
⎤ ⎡x⎤
⎡
x ⎤
{ H} 1
⎢
H
⎥
y , { R} 1
⎢
y
⎥
⎢
and { }
1
⎢
⎥
⎥ ⎢ ⎥
⎢ y ⎥
⎣⎢
Hz
⎦⎥
⎣⎢
z⎦⎥
⎣⎢
z ⎦⎥
1 1 1 1
⎡y( xyyx) z( zxxz)
⎤
⎢
⎥
⎢z( yzzy) x( xyyx)
⎥
⎢
⎣
x( zxxz) y( yzzy)
⎥
⎦
(2.165)
2
Hx x ∫ ( y z 2 ) d my ∫ xyd mz
∫ xzdm
∫
2
y x y 2 ) z
∫
H xyd m ( x z d m
yzdm
∫
(2.167)
(2.168)
H xzd m yzd m ( x y dm
z x y z
(2.169)
It is now possible to substitute into (2.167) to (2.169) the following general
terms for the moments of inertia I xx , I yy and I zz of the rigid body:
∫
2 2 )
Ixx ( y z d m
∫
∫
∫
2 2 )
(2.170)
∫
2 2 )
Iyy ( x z d m
(2.171)