4569846498

240 Multibody Systems Approach to Vehicle Dynamics
∑
{ M }
G2 1 2
⎡I2xx
0 0 ⎤ ⎡2
x2
⎤
⎢
0 I2yy
0
⎥ ⎢
⎢
⎥ ⎢
⎥
2y2
⎥
⎣⎢
0 0 I2zz
⎦⎥
⎣⎢
2z2
⎦⎥
⎡ 0
⎢
⎢ 0
⎢
⎣
2y2 2x2
0
2z2 2y2
2z2 2x2
3
⎡1.5 10 0 0
⎢
3
⎢ 0 38
10 0
⎢
⎣
0 0 3810
⎤ ⎡I2xx
0 0 ⎤ ⎡
⎥ ⎢
0 I2yy
0
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥
⎦
⎢⎣
0 0 I2zz
⎦⎥
⎣⎢
3
⎤ ⎡ 0 ⎤
⎥ ⎢
⎥ 0
⎥
⎢ ⎥
⎥
⎦ ⎣⎢
10.642⎦⎥
⎤
⎥
⎥
Nm
⎦⎥
3
⎡ 0 12.266 0⎤
⎡1.5 10
0 0
⎢
12.266 0 0
⎥ ⎢
3
0
⎢
⎥ ⎢ 38 10 0
⎣⎢
0 0 0⎦⎥
⎢
⎣
0 0 38
10
(4.335)
Equating (4.333) with (4.335) yields the rotational equations of motion for
Body 2:
Equation 4
(115F E21y2 5F E21z2 115F F21y2 5F F21z2 4F G24y2 ) 10 3 0
(4.336)
Equation 5
(115F E21x2 155F E21z2 115F F21x2 155F F21z2
F G24x2 0.120F G24z2 ) 10 3 0 (4.337)
Equation 6
(5F E21x2 155F E21y2 5F F21x2 155F F21y2
120F G24y2 ) 10 3 404.396 10 3 (4.338)
At this stage the observant reader will note the inertial terms in (4.335) have
only yielded a numerical value for the moment balance about the principal
Z 2 -axis. This makes sense as the Z 2 -axis has been chosen to be parallel to the
fixed axis of body rotation through points E and F. In the absence of components
of angular velocity or acceleration about X 2 and Y 2 equations
(4.336) and (4.337) above simplify to a static moment balance. It should
also be noted that when rotation is constrained about a single principal axis
the right-hand part of (4.334), [ 2 ] 1/2 [I 2 ] 2/2 [ 2 ] 1/2 , is entirely zero to indicate
a lack of gyroscopic terms in the absence of rotational coupling. Choosing a
body centred axis system for the upper wishbone Body 3, with an axis parallel
to an axis through points A and B, would yield a similar formulation.
This would not, however, be the case, for example, if we continued to set up
the equations for Body 4 where there is no single fixed axis of rotation.
Before leaving the area of formulating equations of motion for dynamic
analysis we should also ensure that the impression is not given that the use
2x2
2y2
2z2
3
⎤ ⎡ 0 ⎤
⎥ ⎢
⎥ 0
⎥
Nm
⎢ ⎥
⎥
⎦ ⎣⎢
12.266⎦⎥