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Simulation output and interpretation 421 ‘transient understeer’ and ‘transient oversteer’ respectively – although the latter might not objectively be oversteer, it is certainly a greater yaw rate than might be expected. Objectively, the quantity of interest is the rate of change of body slip angle, known as . or ‘beta-dot’. The bulk yaw rate of the vehicle is made up of two components: the first is the yaw rate associated with a curved path – it might be thought of as the yaw rate that would be experienced by a stone on a string being swung around. Milliken and others refer to this as the ‘no-slip yaw rate’ – the yaw rate predicted without any body slip angle. The second is the beta-dot component. By inspection, it can be seen that: Ay ˙ (7.29) V Thus in general, beta-dot is available as a simple combination of vehicle states, particularly during multibody simulation work. For real vehicles, noise on accelerometer data and the difficulty of knowing the genuine forward speed of the vehicle under conditions of longitudinal tyre slip mean that beta-dot is difficult to discern in real time, although it is amenable to offline processing. With changes in suspension configuration, elasto-kinematic calibration and damper behaviour, beta-dot can be modified significantly. The overall behaviour of the vehicle may well only be slightly modified and so to concentrate merely on bulk yaw rate would be to underestimate the significance of the change from the driver’s perspective. In the steady state, the body slip angle is constant and so it is of little consequence except when something changes. In practice, real vehicles spend very little time in the steady state and so the short-lived events – described as ‘transients’ – are very important to the driver’s reaction to the vehicle because of the acute perception of beta-dot. This perception is not only in terms of absolute levels but also in terms of delays between driver requests and vehicle response. When the handwheel is in motion, the driver is implicitly requesting a change in body slip angle, since body slip angle is linked to lateral acceleration. If the change in body slip angle occurs in a way that is substantially connected to the change in steering, the driver feels reassured. If, however, the change in body slip angle occurs with perhaps a significant delay or perhaps with a characteristic shape different to the steer rate, the driver forms an impression that the vehicle has ‘a mind of its own’. This leads to poor subjective ratings for handling confidence whatever the objective measures might say. Thus the correlation between steer rate and beta-dot is a useful one in measuring improvements in vehicle behaviour using theoretical models and real vehicles alike. Transients are also important to the objective behaviour of the whole vehicle. Under circumstances of a steering reversal, particularly at or near the grip limit, there is a substantial increase in yaw rate transiently. To understand this, imagine the vehicle travelling in a steady curve with a body slip angle of, say, 5 degrees. Now imagine the steering is reversed to produce a lateral acceleration in the other direction with a corresponding slip angle of 5 degrees. If the adjustment of the body slip angle took place over a 1 second period, this would make a transient yaw rate of 10 degrees/second.
422 Multibody Systems Approach to Vehicle Dynamics Considering Figure 7.6, it can be seen this is a substantial proportion of the total available yaw rate at speeds over about 50 mph. If the steady state lateral acceleration produced a yaw rate of, say, 15 degrees/second (below the friction limit) then the addition of 10 degrees/second transient yaw rate would be more than the vehicle is capable of sustaining. For this reason alone, vehicles that converge (i.e. settle to a steady state solution) under normal conditions may well become unstable (spin) under conditions of steer reversal. Therefore any studies of transient vehicle dynamics must at some stage consider responses to steer reversals. Vehicle mass and particularly inertia properties are a modifier to vehicle behaviour. There is a popular belief that the minimum mass moment of inertia in yaw is the best. However, this is not necessarily so. To understand this, consider a vehicle manoeuvring in the ground plane, which may be described by the classical 2-degree-of-freedom formulation: C aC b˙ I f f r r zz C C mumv˙ f f r r (7.30) (7.31) At the moment of turn-in, yaw rate and rear slip angle r are both zero. Assuming a constant acceleration solution for the differential equations (a reasonable assumption for the first few moments of a turn-in event or following a disturbance at the front axle), the response of the builds in the manner given in equations (7.32) and (7.33): C f I zz f at (7.32) Cf ft v m (7.33) Combining these quantities gives an instant centre at a distance c behind the mass centre: v ( Cf ft)/ m c ( C at)/ I f f zz 2 2 Izz mk ma ma k a (7.34) Expressing the distance c as a fraction of the distance to the rear axle b leads to c b 2 k ab (7.35) This quantity may be recognized as the ‘Dynamic Index’ (DI) defined by the SAE and used to good effect by Olley, albeit in pitch rather than yaw. Its importance for vehicle behaviour is that a DI less than unity results in an increase in lateral velocity at the rear axle, and hence an increase in slip angle at the rear tyres (Figure 7.26). If DI is greater than unity then lateral velocity – and hence slip angles – are reduced. In a situation where the
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422 Multibody Systems Approach to Vehicle Dynamics<br />
Considering Figure 7.6, it can be seen this is a substantial proportion of the<br />
total available yaw rate at speeds over about 50 mph. If the steady state lateral<br />
acceleration produced a yaw rate of, say, 15 degrees/second (below the<br />
friction limit) then the addition of 10 degrees/second transient yaw rate would<br />
be more than the vehicle is capable of sustaining. For this reason alone,<br />
vehicles that converge (i.e. settle to a steady state solution) under normal<br />
conditions may well become unstable (spin) under conditions of steer<br />
reversal. Therefore any studies of transient vehicle dynamics must at some<br />
stage consider responses to steer reversals.<br />
Vehicle mass and particularly inertia properties are a modifier to vehicle<br />
behaviour. There is a popular belief that the minimum mass moment of<br />
inertia in yaw is the best. However, this is not necessarily so. To understand<br />
this, consider a vehicle manoeuvring in the ground plane, which may be<br />
described by the classical 2-degree-of-freedom formulation:<br />
C aC b˙<br />
I<br />
f f r r zz<br />
C C mumv˙<br />
f f r r<br />
(7.30)<br />
(7.31)<br />
At the moment of turn-in, yaw rate and rear slip angle r are both zero.<br />
Assuming a constant acceleration solution for the differential equations (a<br />
reasonable assumption for the first few moments of a turn-in event or following<br />
a disturbance at the front axle), the response of the builds in the<br />
manner given in equations (7.32) and (7.33):<br />
<br />
C<br />
f<br />
I<br />
<br />
zz<br />
f<br />
at<br />
(7.32)<br />
Cf<br />
ft<br />
v <br />
m<br />
(7.33)<br />
Combining these quantities gives an instant centre at a distance c behind<br />
the mass centre:<br />
v ( Cf<br />
ft)/<br />
m<br />
c ( C at)/<br />
I<br />
f f zz<br />
2 2<br />
Izz<br />
mk<br />
<br />
ma ma<br />
k<br />
a<br />
(7.34)<br />
Expressing the distance c as a fraction of the distance to the rear axle b<br />
leads to<br />
c<br />
b<br />
<br />
2<br />
k<br />
ab<br />
(7.35)<br />
This quantity may be recognized as the ‘Dynamic Index’ (DI) defined by<br />
the SAE and used to good effect by Olley, albeit in pitch rather than yaw.<br />
Its importance for vehicle behaviour is that a DI less than unity results in an<br />
increase in lateral velocity at the rear axle, and hence an increase in slip<br />
angle at the rear tyres (Figure 7.26). If DI is greater than unity then lateral<br />
velocity – and hence slip angles – are reduced. In a situation where the
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