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338 Multibody Systems Approach to Vehicle Dynamics
If we consider first the kinematic effects due to suspension geometry we
can see that there are two variables that provide input to the model:
z is the change in wheel centre vertical position (wheel travel)
v is the change in steering wheel angle
The magnitude of the wheel travel z will depend on the deformation of
the surface, the load acting vertically through the tyre resulting from
weight transfer during a simulated manoeuvre and a representation of the
suspension stiffness and damping acting through the wheel centre. The
magnitude of the change in steering wheel angle v will depend on either
an open loop fixed time dependent rotational motion input or a closed loop
torque input using a controller to feed back vehicle position variables so as
to steer the vehicle to follow a predefined path. The modelling of steering
inputs is discussed in more detail later in this chapter. The dependent variables
that dictate the position and orientation of the road wheel are:
x is the change in longitudinal position of the wheel
y is the change in lateral position (half-track) of the wheel
is the change in steer angle (toe in/out) of the wheel
is the change in camber angle of the wheel
The functional dependencies that dictate how the suspension moves with
respect to the input variables can be obtained through experimental rig
measurements, if the vehicle exists and is to be used as a basis for the
model, or by performing simulation with suspension models as described
in Chapter 4. For example, the dependence of camber angle on wheel
travel can be derived from the curves plotted for Case study 1 in Chapter 4.
The movement of the suspension due to elastic effects is dependent on the
forces acting on the wheel. In their paper Scapaticci and Minen (1992)
describe the relationship using the equation shown in (6.1) where the functional
dependencies due to suspension compliance are defined using the
matrix F E :
⎡x⎤
⎡
⎢
y
⎥ ⎢
⎢ ⎥ ⎢F
⎢⎥
⎢
⎢ ⎥ ⎢
⎣⎦
⎣
E
⎤ ⎡ Fxt ⎤
⎥ ⎢
Fxb
⎥
⎥ ⎢ ⎥
⎥ ⎢ Fy ⎥
⎥ ⎢ ⎥
⎦ ⎣ Mz ⎦
(6.1)
and the inputs are the forces acting on the tyre:
Fxt is the longitudinal tractive force
Fxb is the longitudinal braking force
Fy is the lateral force
Mz is the self-aligning moment
Note that the dimensions of the matrix F E are such that cross-coupling
terms, such as toe change under braking force, can exist. The availability of
such data early in the design phase can be difficult but the adoption of such
a generalized form allows the user to speculate on such values and thus use
the model to set targets for acceptable behaviour.