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88 Mutibody Systems Approach to Vehicle Dynamics
In summary there are now a set of kinematic position and velocity variables
for the nth part with components measured in the GRF and also a set of orientation
and angular velocity variables measured about the Euler-axis frame:
{Rn} 1 [Rnx Rny Rnz] T (3.7)
{Vn} 1 [Vnx Vny Vnz] T (3.8)
{n} e [n n n] T (3.9)
{n} e [nn n] T (3.10)
There is also a set of kinematic equations associated with the part which
may be simply stated as:
d
{ Vn} 1 { Rn}
1
dt
(3.11)
d
{ n} e { n}
e
(3.12)
dt
The remaining part variables and equations are those obtained by considering
the equations of motion for a rigid body. Each part can be considered to
have a set of six generalized co-ordinates given by
q j [Rnx, Rny, Rnz, n, n, n] (3.13)
The translational co-ordinates are the translation of the centre of mass
measured parallel to the axes of the ground reference frame while the rotational
co-ordinates are provided by the Euler angles for that part. For any
part the translational forces are therefore summed in the X, Y and Z directions
of the GRF while the summation of moments takes place at the centre
of mass and about each of the axes of the Euler-axis frame. Using a form of
the Lagrange equations this can be shown as:
d
dt
⎛ ∂T
⎞ ∂T
⎜
Qj
⎝ ∂q
⎟
˙
j⎠
∂q
j
n
∑
i1
∂i
i 0
∂q
j
(3.14)
The kinetic energy T is expressed in terms of the generalized co-ordinates
q j and is given by
1 T 1 T T
T { Vn} 1m{ Vn} 1 { n} e [ B] [ In][ B]{ n}
e
(3.15)
2
2
The mass properties are specified by m which is the mass of the part and
[I n ] which is the mass moment of inertia tensor for the part and given by
⎡Ixx Ixy Ixz
⎤
⎢
⎥
[ In]
⎢Iyx Iyy Iyz
⎥
(3.16)
⎢I I I ⎥
⎣ zx zy zz ⎦
The terms and represent the reaction force components acting in the
direction of the generalized co-ordinate q j . The term Q j represents the sum
of the applied force components acting on the part and in the direction of the
generalized co-ordinate q j . The equation can be simplified by introducing