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56 Multibody Systems Approach to Vehicle Dynamics
The resultant force is represented by the vector {F} 1 , m is the mass and
{V} 1 is the velocity vector measured relative to an inertial reference frame O 1 .
The linear momentum of the body is m{V} 1 . The components of the vectors
in (2.151) are all resolved parallel to the axes of O 1 . Taking the mass to
be constant (2.151) may be written as
∑
d
{} F m {} V m{}
A
d t
1 1 1
(2.152)
The accelerations in (2.152) are also measured relative to the inertial reference
frame O 1 . Since the velocities and accelerations used here are measured
relative to a non-moving frame we refer to them as absolute. Expanding the
vector equation given in (2.152) gives
⎡Fx
⎤ ⎡Ax
⎤
⎢
F
⎥
y m
⎢
A
⎥
∑ ⎢ ⎥
⎢ y ⎥
⎣⎢
Fz
⎦⎥
⎣⎢
Az
⎦⎥
(2.153)
2.8 Linear momentum of a rigid body
As a body translates and rotates in space it will have linear momentum {L} 1
associated with translation and angular momentum {H} 1 associated with
rotation. For the rigid body, Body 2, shown in Figure 2.27 the mass centre is
located at G 2 by the vector {R G2 } 1 relative to the reference frame O 2 .
A small element of material with a volume V is located at P relative to O 2
by the position vector {R P } 1 . Assuming Body 2 to be of uniform density ,
we can say that the element of material has a mass m given by
m V (2.154)
and that as the element becomes infinitesimal the mass of the body m 2 is
given by
∫
m2 d V
d m
vol
∫
vol
(2.155)
GRF
Z 1
X 1
O 1
Y 1
{V P } 1
P
δV
{R P } 1
Z 2
Body 2
G 2
O 2 {R G2 } 1
Y
X 2
2
{ω 2 } 1
Fig. 2.27
Linear momentum of a rigid body