To understand how changes in vehicle architecture affect the overall lateral
dynamics of the vehicle, it is useful to consider the basic principles of vehicle
dynamics. Utilising a simple vehicle model, such as the Bicycle model the basic
principles of vehicle dynamics can be introduced, these basic principles can then be
used to answer questions pertaining to why changes in vehicle architecture lead to
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Figure 4-3 Bicycle model
Figure 4-3 shows a representation of a simple vehicle. Its track has been
compressed to form a single track, it therefore does not include any weight transfer
effects, it is void of any suspension, and it has tyres with a linear cornering stiffness
which are connected via a completely rigid chassis. This is one of the simplest
vehicle models used for vehicle dynamics, due to its single track it is called the “Bicycle” model. It is a two degree of freedom model, lateral velocity and yaw
angle ( and ψ), longitudinal velocity (usually constant) and steer angle are used as inputs. When the vehicle is in a steady state turn, as shown in the previous
figure, it has a tangential velocity to the turn radius this acts at an angle (body slip angle) to the body, in practice the body slip angle is very small and it can be
assumed that = (the component of velocity along the vehicle longitudinal axis). It also has a lateral velocity which acts towards the turn centre perpendicular to
a b
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. The vehicle yaws about the turn centre and so has a yaw angle ψ. The force
produced to enable the vehicle to corner comes solely from the tyres cornering
stiffness and is equal and opposite to the centrifugal force acting at the vehicle Cog
perpendicular to . Its motion can be described by two equations, one for forces
and one for moments. These have been developed time and again in numerous texts
(Milliken and Milliken, 1995, Olley et al., 2002, Gillespie, 1992, Wong, 2001). It is
possible to express these equations in a derivative form to illustrate how the forces
and moments are influenced by changes in the body slip angle , yaw rate , and steer angle . As such the two equations can be written as
where = , = , = = , = , =
Whilst these partial derivatives represent a very simplified linear vehicle they can
be used to pinpoint areas where handling differences may arise between the two
vehicles under investigation in this study.
The term is simply the sum of front and rear cornering stiffness and so relates
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body slip angle. Relating this to the problem at hand, if, and it is quite likely, that the cornering stiffness’s of the tyres are affected when the standard vehicle
undergoes changes it will result in different rates of lateral tyre force generation
between the two vehicles, which will directly affect response times. Inversely this
term also shows that if body slip angles are different between the two vehicles then
they will generate different levels of lateral force.
The
term is a measure of the forces resulting directly from the yaw rate, it is related to the lateral tyre forces that comprise the yaw moment, Milliken terms this
derivative the lateral force/yaw coupling derivative. Again it can be seen that any
change in mass distribution and/or cornering stiffness will effect this term.
The final force derivative, is the lateral force generated directly by the steer
angle at the front wheels, steer angles of the two vehicles in this study will be
different, for reasons that will be discussed shortly, also it has already been
mentioned that cornering stiffness’s are likely to be effected by the changes the SV
will undergo and so again there will be differences in the lateral force generated by
the two vehicles.
Moving onto the moment derivatives, the first of these, , which Milliken terms
the static directional stability derivative, is a measure of the moment produced about
the Cog arising from body slip angle, it follows that if is larger than then the moment is stabilising, the vehicle is always trying to straighten itself, this
effectively means that the vehicle is understeer. Both vehicles in this study have
understeer characteristics but both possess different weight distributions and cornering stiffness’s and so this term will vary in magnitude for the different
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vehicles, showing they intrinsically posses different levels of understeer, thus have
different steering responses and thus respond to steer inputs at different rates.
is the yaw damping derivative, it can be seen that due to changes that have taken place to the vehicles’ mass distributions, this term will be different, and so will
result in different yaw responses.
Finally illustrates the change in moment due to steering, it can be thought of
as the control moment, and again it can be seen that it is dependent on the weight
distribution and front cornering stiffness. Again as changes have been made to both
of these parameters, this term will differ between the two vehicles and so different
yaw responses will likely be obtained from steering inputs.
In summary, consideration of the partial derivatives outlined in this section can
give clues as to how and why the two vehicles’ handling will differ, however they
are derived from a simple linear model, and the parameters within them are co-
dependent, so whilst they provide a useful insight, the non-linear multi-body models
produced within this study will provide a more representative view of the handling
differences that will present themselves due to the changes that have been made to
the SV when producing the GTV.
Further analysis of the bicycle model yields an expression for the understeer
gradient of the simple 2DOF model, in terms of the aforementioned derivatives this
is expressed as
USG =
Using this equation with parameters shown in Table 4-2 the understeer gradient can
be calculated for the two vehicles in question in this study, this is shown in Table
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Parameter Unit SV GTV
a (front axle to Cog) [m] 1.16 1.22
b (rear axle to Cog) [m] 1.5 1.44
V (forward Velocity) [m/s] 7.214 7.214
Mf (Mass Front) [Kg] 1196 1193.27
Mr (Mass Rear) [Kg] 940 1016.49
Cf (front tyres cornering stiffness) [N/rad] -175800 -172800 Cr (rear tyres cornering stiffness) [N/rad] -156400 -159600
Table 4-2 SV and GTV Parameters in linear region
Unit SV GTV % delta
Bicycle Model [deg/g] 0.51 0.32 37.3
Dymola Model (in linear region of tyre) [deg/g] 0.52 0.33 36.5
Table 4-3 Bicycle model and Dymola model understeer gradients in linear region
In the majority of conditions, understeer gradients calculated by the bicycle
model are not comparable to those obtained from the Dymola models, or the actual
vehicle under test. Calculating the understeer gradient from the bicycle model, as
has been done in this section, only accounts for weight distribution and tyre
cornering stiffness effects on the resulting gradient, there are however more
numerous factors effecting the overall understeer gradient of an actual vehicle, these
can be summarised as; tyre cornering stiffness, load transfer, lateral force
compliance steer, roll steer, steering compliance, aligning torque, and camber thrust
all of which effect the understeer gradient, and it is found that the contribution of the
tyre cornering stiffness is relatively small compared to the other factors (Dixit,
2009). However, comparable numbers can be obtained from the Dymola model if
the tyre model remains within the linear region, and if lateral forces remain low
enough as not to introduce the effects of suspension compliances, and lateral load
transfer. Within section 3.14, understeer gradients were obtained from the Dymola
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Figure 3-38 (a), between 0.1 and 0.17g, the vehicle is operating at a very low speed
within the linear region of the tyre, and as such understeer gradients shown here
correlate well with those obtained from the bicycle model, the two sets of figures are
summarised in Table 4-3.
There are some important speeds related to the understeer gradient of the vehicle,
the first of these which was proposed by General Motors as a way for expressing the
understeer of a car, is termed the Characteristic Speed (Milliken and Milliken,
1995). The characteristic speed is defined as the speed at which and understeer vehicle’s steering angle is twice the Ackermann angle, this can be defined as;
2 = As =
=
Unit SV GTV
[m/s] 17.34 21.77
Table 4-4 Characteristic Speeds for SV and GTV in linear region
As the two vehicles in this study have the same wheelbase, the GTV has a higher
characteristic speed than the SV due to its lower USG, this is important as it has been
shown by Milliken (1995) and Olley (2002), that the yaw rate response reaches a
maximum at a vehicles characteristic speed, meaning that the GTV will exhibit its
peak yaw rate at a higher speed than the SV. This is something that will be drawn
upon when analysing the handling results. The Characteristic speeds for SV and
GTV, calculated using the USG from the bicycle model are shown in Table 4-4.
Similarly to the characteristic speed for the understeer vehicle, the oversteer
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the speed at which responses of the oversteer car become divergent, small inputs to
steering result in very large lateral accelerations and yaw rates. Looking back to the
derivatives of the bicycle model derived earlier, the critical speed can be explained as the speed at which the under/oversteer derivative is equal to the yaw damping moment , as speed increases the yaw damping will decrease, and will no longer
balance the under/oversteer moment, leading to divergent responses. As the vehicles
under investigation here do not possess an oversteer characteristic the critical speed
will not be of use in further analysis. However it is interesting to note that the
characteristic speed for an understeer vehicle, the speed where peak yaw rate
responses are observed, is equal to the critical speed of an oversteer vehicle, where
the yaw rate responses approach infinity (Milliken and Milliken, 1995). The critical
speed can thusly be defined as;
= There is one more interesting speed that will prove useful in upcoming analysis
of the handling results, this is the Tangent Speed. The tangent speed is defined as
the speed at which the vehicle will operate with no body slip angle. When a vehicle
travels at low speed on a constant radius the rear wheels will track a smaller radius
than the front (nose out attitude), as lateral acceleration and slip angles increase there
will become a point where both front and rear axles will track the same radius, body
slip angle at the Cog will be zero, above this speed the rear axle will track a wider
radius than the front (nose in attitude). Tangent speed can also be defined as
(Milliken and Milliken, 1995);
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Tangent speeds for the SV and GTV (in the linear region) are shown in Table 4-5.
As the tangent speed dictates at what speed the vehicle changes form a nose out to a
nose in attitude in cornering, it has an effect of the magnitude of tyre slip angles,
which can prove useful in later sections when analysing the handling characteristics
of the two vehicles.
Unit SV GTV
[m/s] 15.8 15.1
Table 4-5 Tangent speeds for SV and GTV in linear region