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32 Multibody Systems Approach to Vehicle Dynamics
In matrix form this can be written:
{} A
12 /
⎡Ax2
⎤ ⎡1 0 0 ⎤
⎢
Ay
⎥
2
⎢
0 cos sin
⎥
⎢ ⎥ ⎢
⎥
⎣⎢
Az2
⎦⎥
⎣⎢
0 sin
cos⎦⎥
⎡Ax1⎤
⎢
Ay
⎥
⎢ 1⎥
⎣⎢
Az1
⎦⎥
(2.28)
This equation may be expressed as
{A} 1/2 [T 1 ] 2 {A} 1/1 (2.29)
Consider next the transformation of the vector {A} 1/1 from reference frame
O 1 to reference frame O 3 . The reference frame O 3 is rotated through an
angle about the Y 1 -axis of frame O 1 . Following the same procedure as
before we get:
{} A /
13
⎡Ax
⎢
⎢
Ay
⎣⎢
Az
3
3
3
⎤ ⎡cos 0 sin⎤
⎥
⎢
0 1 0
⎥
⎥ ⎢
⎥
⎦⎥
⎣⎢
sin
0 cos
⎦⎥
⎡Ax1⎤
⎢
Ay
⎥
⎢ 1⎥
⎣⎢
Az1
⎦⎥
(2.30)
{A} 1/3 [T 1 ] 3 {A} 1/1 (2.31)
Finally consider the transformation to frame O 4 where O 4 is obtained from
a rotation of about the Z 1 -axis of frame O 1 :
⎡Ax4
⎤ ⎡ cos sin 0⎤
{ A} 14 /
⎢
Ay
⎥
4
⎢
sin cos 0
⎥
⎢ ⎥ ⎢
⎥
⎣⎢
Az4
⎦⎥
⎣⎢
0 0 1⎦⎥
⎡Ax1⎤
⎢
Ay
⎥
⎢ 1⎥
⎣⎢
Az1
⎦⎥
(2.32)
{A} 1/4 [T 1 ] 4 {A} 1/1 (2.33)
The square transformation matrices [T 1 ] 2 , [T 1 ] 3 and [T 1 ] 4 are the inverses of
the rotation matrices developed in section 2.2.6. This is to be expected
since the method here is the reverse of that shown previously where the
vector, rather than the frame, was rotated. It should be noted that a transformation
matrix [T m ] p , which transforms a vector from frame m to frame
p, has a transpose [T m ] T p that is also its inverse [T m ] 1 p .
In general terms the transformation of a vector from one frame m to another
frame p may be written as
{A} n/p [T m ] p {A} n/m (2.34)
2.2.8 Differentiation of a vector
The differentiation of a vector {A} 1 with respect to a scalar variable, such as
time t, results in another vector given by
d
{} A
dt
1
⎡dAx
/ dt⎤
⎢
dAy
/ dt
⎥
⎢ ⎥
⎣⎢
dAz
/ dt⎦⎥
(2.35)