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72 Multibody Systems Approach to Vehicle Dynamics
Expanding (2.228) gives
⎡M
⎢
⎢
M
⎣⎢
M
x
y
z
⎤ ⎡I I I
⎥ ⎢
⎥
⎢I I I
⎦⎥
⎢
⎣
I I I
xx xy xz
xy yy yz
xz yz zz
⎤ ⎡
⎥ ⎢
⎥ ⎢
⎥
⎦ ⎣⎢
2x
2y
2z
⎤ ⎡ 0 2z
2y
⎤ ⎡H
⎥ ⎢
⎥
⎥
⎢
⎢
2z
0 2x⎥
⎢
H
⎦⎥
⎢
⎥
⎣
2y
2x
0
⎦ ⎣⎢
H
(2.229)
Substituting in terms for H 2x , H 2y and H 2z now leads to the equations given
in (2.230) to (2.231). For convenience we can drop the subscript for Body 2:
M I I ( ) I
( )
x xx x xy y x z xz z x y
2 2
(
Izz Iyy ) yz Iyz( y z
)
2x
2y
2z
⎤
⎥
⎥
⎦⎥
(2.230)
M I ( ) I I
( )
y xy x y z yy y yz z x y
2 2
xx zz x z xz z x
(
I I ) I
( )
(2.231)
M I ( ) I
( )
z xz x y z yz y x z
2 2
zz z yy xx x y xy x y
I (
I I ) I
( )
(2.232)
In summary the rotational equations of motion for Body 2 may be written
in vector form as
Σ{ M (2.233)
G2} 12 / [ I2] 22 / { 2} 12 / [ 2] 12 / [ I2] 22 / { 2}
12 /
Hence we can see that in setting up the equations of motion for any rigid
body the translational equations for all bodies in a system may conveniently
be referred to a single fixed inertial frame O 1 . The rotational equations,
however, are better referred to a body centred frame, in this case O 2 .
A considerable simplification in these equations will result if frame O 2 is
selected such that its axes are the principal axes of the body (I 1 I xx ,
I 2 I yy , I 3 I zz ) and the products of inertia are zero. The equations that
result are known as Euler’s equations of motion:
M x I 1 x (I 3 I 2 ) y z (2.234)
M y I 2 y (I 1 I 3 ) x z (2.235)
M z I 3 z (I 2 I 1 ) x y (2.236)
The equations given in (2.234) to (2.236) become even simpler when the
motion of a body is constrained so that rotation takes place in one plane
only. If, for example, rotation about the x- and y-axes are prevented then
equation (2.236) reduces to the more familiar form associated with twodimensional
motion:
M z I 3 z (2.237)
The following example also demonstrates how gyroscopic effects associated
with three-dimensional motion may be identified. If we consider the
swing arm suspension system shown in Figure 2.37 we can take the suspension
arm Body 2 to be constrained by a revolute joint to rotate with a
constant angular velocity of 10 rad/s about the axis of the joint as shown.
While this motion is in progress the road wheel, Body 3, is also rotating