4569846498

70 Multibody Systems Approach to Vehicle Dynamics
⎡I1
0 0⎤
⎡cos 0 sin
⎤
⎡Ixx
0 Ixz
⎤ ⎡ cos
0 sin ⎤
⎢
0 I2
0
⎥
⎢ ⎥
⎢
0 1 0
⎥ ⎢
0 I2
0
⎥ ⎢
0
⎢
⎥ ⎢
⎥
1 0
⎥
⎢
⎥
⎣⎢
0 0 I3⎦⎥
⎣⎢
sin 0 cos⎦⎥
⎣⎢
Ixz
0 Izz
⎦⎥
⎣⎢
sin 0 cos⎦⎥
⎡I1
0 0⎤
⎡cos 0 sin
⎤
⎡Ixx cos Ixz sin 0 Ixx sin Ixz
cos⎤
⎢
0 I2
0
⎥
⎢ ⎥
⎢
0 1 0
⎥ ⎢
0 I2
0
⎥
⎢
⎥ ⎢
⎥
⎣⎢
0 0 I3⎦⎥
⎣⎢
sin 0 cos⎦⎥
⎣⎢
Ixz cos Izz sin 0 Ixz sin Izz
cos⎦⎥
⎡I1
0 0⎤
⎢
0 I2
0
⎥
⎢ ⎥
⎣⎢
0 0 I3⎦⎥
2 2 2 2
⎡ Ixx cos 2 Ixz sin cos Izz sin 0 Ixx sin cosIxz cos Ixz sin Izz
sin cos⎤
⎢
⎥
⎢
0 I2
0
⎥
⎢
2 2
2 2
⎣
Ixx sin cos Ixz cos Ixz sin Izz
sin cos 0 Ixx sin 2 Ixz sin cos Izz
cos ⎥
⎦
(2.217)
Multiplying out the matrix equation in (2.217) leads to the following
expressions for the principal moments of inertia I 1 and I 3 :
I 1 I xx cos 2 2I xz sin cos I zz sin 2 (2.218)
I 3 I xx sin cos I xz cos 2 I xz sin 2 I zz sin cos (2.219)
Equating now the zero elements on the left-hand side of (2.217) with the
terms in either row 1 column 3 or column 1 row 3 gives
0 I xx sin cos I xz sin 2 I xz cos 2 I zz sin cos (2.220)
This can be rearranged to give
0 I xz (cos 2 sin 2 ) (I xx I zz ) sin cos (2.221)
From trigonometric addition formulae we can make use of
cos 2 cos 2 sin 2 and sin 2 2 sin cos
which leads to
0 I 1
xz cos 2 2
(2.222)
2 ( I xx I
zz ) sin
Rearranging (2.222) leads to an expression from which we can determine a
value for using the known values for I xx , I zz and I xz :
tan
2
1
2
Ixz
( I I )
zz
xx
(2.223)
Using the value obtained for in (2.223) it is now possible to substitute this
back into (2.218) and (2.219) and obtain values for the two unknown principal
moments of inertia I 1 and I 3 .