4569846498

68 Multibody Systems Approach to Vehicle Dynamics
The process used to find the determinant is documented in standard texts
dealing with the mathematical manipulation of matrices but for a general three
by three square matrix may be summarized as follows:
A B C
D E F A( EI HF)
B( DI GF) C( DH GE)
G H I
(2.211)
The solution of (2.210) leads to a cubic equation in I 1 with three positive real
roots, these being the three principal moments of inertia I 1 , I 2 and I 3 . If each
of these is substituted in turn into equations that equate all three columns
on either side of (2.207) we get
I
1
⎡T
⎢
⎢
T
⎣⎢
T
11
12
13
⎤ ⎡I I I
⎥ ⎢
⎥
⎢I I I
⎦⎥
⎢
⎣
I I I
xx xy xz
xy yy yz
xz yz zz
⎤ ⎡T
⎥ ⎢
⎥ ⎢
T
⎥
⎦ ⎣⎢
T
11
12
13
⎤
⎥
⎥
⎦⎥
(2.212)
I
2
⎡T
⎢
⎢
T
⎣⎢
T
21
22
23
⎤ ⎡I I I
⎥ ⎢
⎥
⎢I I I
⎦⎥
⎢
⎣
I I I
xx xy xz
xy yy yz
xz yz zz
⎤ ⎡T
⎥ ⎢
⎥ ⎢
T
⎥
⎦ ⎣⎢
T
21
22
23
⎤
⎥
⎥
⎦⎥
(2.213)
I
3
(2.214)
The solution of (2.212) to (2.214) thus yields all the terms in [T 3 ] 2 , the transformation
matrix from frame O 3 to O 2 . In summary I 1 , I 2 and I 3 are the
eigenvalues of the inertia matrix [I 2 ] 2/3 and are also the principal moments
of inertia for Body 2, these being the diagonal terms in the matrix [I 2 ] 2/2 . The
three column matrices in (2.212) to (2.214) are the eigenvectors of [I 2 ] 2/3 :
⎡T
⎢
⎢
T
⎣⎢
T
⎡T
⎢
⎢
T
⎣⎢
T
11
12
13
⎤
⎥
⎥
⎦⎥
31
32
33
⎤ ⎡I I I
⎥ ⎢
⎥
⎢I I I
⎦⎥
⎢
⎣
I I I
⎡T
⎢
⎢
T
⎣⎢
T
21
22
23
xx xy xz
xy yy yz
xz yz zz
⎤
⎥
⎥
⎦⎥
⎡T
⎢
⎢
T
⎣⎢
T
31
32
33
⎤
⎥
⎥
⎦⎥
⎤ ⎡T
⎥ ⎢
⎥ ⎢
T
⎥
⎦ ⎣⎢
T
31
32
33
⎤
⎥
⎥
⎦⎥
If each vector is now normalized so that the length of the vector is unity, we
get the direction cosines between each of the axes of O 2 , the principal axes
of Body 2, and O 3 .
We can now consider a practical application of this with regard to vehicle
dynamics where the body of a vehicle will generally be the largest and
most significant mass in the model. For the vehicle body, Body 2, shown in
Figure 2.35 we can take frame O 3 to be positioned at the mass centre and
orientated so that the x-axis is along the centre line and pointing to the rear
of the vehicle and the z-axis is vertical. The X 3 Z 3 plane is thus a plane of
symmetry. It should be noted that in reality this assumption involves some
approximation due to the asymmetry of the masses that may be lumped
with the vehicle body such as the engine, battery, exhaust system and fuel