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238 Multibody Systems Approach to Vehicle Dynamics
{F E21 } 1 {F F21 } 1 {F G24 } 1 m 2 {g} 1 m 2 {A G2 } 1 (4.324)
⎡F
⎢
⎢
F
⎣⎢
F
E21x
E21y
E21z
⎤ ⎡FF21x⎤
⎡FG24x⎤
⎡ 0 ⎤ ⎡ 0 ⎤
⎥ ⎢
F
⎥ ⎢
⎥ F21y
F
⎥
⎢ ⎥
⎢ G24y⎥
3.5
⎢
0
⎥
3.5
⎢
23.373
⎥
N
⎢ ⎥ ⎢ ⎥
⎦⎥
⎣⎢
FF21z⎦⎥
⎣⎢
FG24z⎦⎥
⎣⎢
9.81⎦⎥
⎣⎢
0.869
⎦⎥
(4.325)
The summation of forces in (4.325) leads to the first set of three equations:
Equation 1 F E21x F F21x F G24x 0 (4.326)
Equation 2 F E21y F F21y F G24y 81.806 (4.327)
Equation 3 F E21z F F21z F G24z 31.294 (4.328)
For the rotational equations it is convenient to refer the vectors to the reference
frame O 2 fixed in and rotating with Body 2. The rotational equations
of motion for Body 2 may be written as Euler’s equations of motion in vector
form as
∑ { M } I ] } ] I ] }
(4.329)
G2 [ 12 2 2 2{ 2 1 2[ 2
1 2[ 2 2 2{
2 1 2
Before progressing the angular velocity vector { 2 } 1 and angular acceleration
vector { 2 } 1 need to be transformed from reference frame O 1 to O 2 to
give { 2 } 1/2 and { 2 } 1/2 . By inspection it can be seen from Figure 4.77 that
the transformation is trivial and that due to the wishbone geometry and
constraints 2x and 2x in frame O 1 simply become 2z and 2z when referenced
to frame O 2 . The process of vector transformation described in
Chapter 2 will, however, be applied to illustrate the process for more general
geometries. In this case we have only two rotations to account for, the
first being 90 degrees about the z-axis followed by a 90 degrees rotation
about the x-axis. Thus for the angular velocity vector we have:
{ }
2 1 2
{ }
2 1 2
⎡
⎢
⎢
⎣⎢
⎡
⎢
⎢
⎣⎢
2x2
2y2
2z2
2x2
2y2
2z2
⎤ ⎡1
0 0 ⎤ ⎡ cos sin 0⎤
⎡2
x1⎤
⎥
⎢
0 cos sin
⎥ ⎢
sin
cos 0
⎥ ⎢
⎥
⎥
1 rad/s
⎢
⎥ ⎢
⎥ ⎢ 2y
⎥
⎦⎥
⎣⎢
0 sin cos ⎦⎥
⎣⎢
0 0 1⎦⎥
⎣⎢
2z1⎦⎥
(4.330)
⎤ ⎡1 0 0⎤
⎡ 0 1 0⎤
⎡12.266⎤
⎡ 0 ⎤
⎥
⎥
⎢
0 0 1
⎥ ⎢
1
0 0
⎥ ⎢
0
⎥
⎢
0
⎥
rad/s
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎦⎥
⎣⎢
0 1 0⎦⎥
⎣⎢
0 0 1⎦⎥
⎣⎢
0 ⎦⎥
⎣⎢
12.266⎦⎥
(4.331)
The transformation of the angular acceleration vector takes place in a similar
manner so that we have:
{ 2 } T 1/2 [0 0 12.266] rad/s
{ 2 } T 1/2 [0 0 10.642] rad/s 2