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74 Multibody Systems Approach to Vehicle Dynamics
Substituting in the numerical data for this problem gives
∑
∑
⎡M
⎢
⎢
M
⎣⎢
M
⎡M
⎢
⎢
M
⎣⎢
M
3x
3y
3x
3y
⎤
⎥
⎥
⎦⎥
3z
1/3
⎤
⎥
⎥
⎦⎥
3z
1/3
⎡ 0 0 100 ⎤ ⎡05
. 0 0 ⎤ ⎡ 10 ⎤
⎢
0 0 10
⎥ ⎢
0 10 . 0
⎥ ⎢
100
⎥
Nm
⎢
⎥ ⎢
⎥ ⎢ ⎥
⎣⎢
100 10 0 ⎦⎥
⎣⎢
0 0 05 . ⎦⎥
⎣⎢
0 ⎦⎥
⎡ 0 0 100 ⎤ ⎡50
. ⎤
⎢
0 0 10
⎥ ⎢
100
⎥
Nm
⎢
⎥ ⎢ ⎥
⎣⎢
100 10 0 ⎦⎥
⎣⎢
0 ⎦⎥
∑
⎡M
⎢
⎢
M
⎣⎢
M
3x
3y
⎤
⎥
⎥
⎦⎥
3z
1/3
⎡ 0 ⎤
⎢
0
⎥
Nm
⎢ ⎥
⎣⎢
500⎦⎥
(2.240)
As can be seen from the result in (2.240) the reaction torque on the wheel
bearing is about the z-axis and is due to gyroscopic effects as the wheel
spins about the y-axis and rotates about the x-axis.
It should be noted that in addition to the derivation of the equations of
motion based on the direct application of Newton’s laws, variational methods,
including, for example, Lagranges equations, provide an elegant alternative
and are often employed in MBS formulations. Many texts on classical
dynamics such as D’Souza and Garg (1984) include a thorough treatment
of these methods.
Variational methods are attractive for a number of reasons. Equations are
formulated using kinetic energy and work resulting in scalar rather than
vector terms. Solutions can also be more efficient since constraint forces
that do not perform work can be omitted. Variational methods also make
use of generalized rather than physical co-ordinates reducing the number
of equations required.
The theory and methods described in this chapter form a basis for the
multibody systems formulations covered in the next chapter. The vector
notation used here will be used to describe the part equations and the constraint
equations required to represent joints constraining relative motion
between interconnected bodies. In Chapter 4 the vector-based methods
described here will be used to carry out a range of analyses from first principles
on a double wishbone suspension system and to compare the calculated
results with those found using MSC.ADAMS.