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236 Multibody Systems Approach to Vehicle Dynamics
Rotate ψ 90d about Z 1 Rotate θ 90d about X 2 Rotate φ 0d about Z 2
Z 1
Z 1
Y 2 Y 2
X 2
Z 2
ψ
Y
X 1
Y 1
X 2
X 2
O 1 φ
O 2
θ
Z 2
Fig. 4.77
Definition of body principal axis system using Euler angle rotations
mounts and joints connecting the linkages. Mass centre positions have not
been provided. For a dynamic analysis the mass centre locations of all
moving bodies are required in order to set up the equations of motion. For
this example using Body 2 the position of the mass centre G 2 relative to the
inertial reference frame O 1 is defined by the position vector {R G2O1 } 1 and
assumed to be
{R G2O1 } T 1 [7 500 85] mm
The mass of Body 2, m 2 , is taken to be 3.5 kg. It should also be noted from
Figure 4.76 that the principal axes of Body 2 are located at the mass centre
G 2 and are defined by the reference frame O 2 . The transformation from reference
frame O 1 to O 2 is obtained through a set of three Euler angle rotations
as shown in Figure 4.77.
The mass moments of inertia for Body 2, measured about the principal axes
of the body O 2 , are taken to be for this example:
I 21 I 2xx 1.5 10 3 kg mm 2
I 22 I 2yy 38 10 3 kg mm 2
I 23 I 2zz 38 10 3 kg mm 2
The X 2 Y 2 plane of O 2 is taken to be a plane of geometric symmetry
for the part so that all cross products of inertia are zero. The inertia matrix
for Body 2 [I 2 ] 2/2 measured from and referred to reference frame O 2 is
therefore
3
1.5 10
⎢
I 2 22
[ ]
⎡
⎢
⎢
⎣
0 0
0
3
38
10 0
0 0 38
10
From the previous velocity and acceleration analysis we also have
{ 2 } T 1 [12.266 0 0] rad/s
(4.314)
{ 2 } T 1 [10.642 0 0] rad/s 2
Before progressing to set up the equations of motion we need to do one
more calculation to find the acceleration {A G2 } 1 of the mass centre for
Body 2:
{A G2 } 1 {A G2E } 1 { 2 } 1 {V G2 } 1 { 2 } 1 {R G2E } 1 (4.315)
3
⎤
⎥
⎥ kg mm
⎥
⎦
2