speed on level ground without head-wind. For vehicles with automatic transmissions or CVTs the certain time set
can allow automatic gear-shift (or ratio-change) or not. The reserve can also be measured in propulsion force.
In general terms, the lowest consumption is found in high gears. However, the vehicle will then tend to have a very small reserve in acceleration. It will, in practice, make the vehicle less comfortable and less safe to drive in real traffic, because one will have to change to a lower gear to achieve a certain higher acceleration. The gear shift gives a time delay.A characteristic of electric propulsion systems is that an electric motor can be run at higher torque for a short time than stationary, see
Figure 3-3. On the other hand, the stationary acceleration reserve is less gear dependent, since an electric motor can work at certain power levels in large portions of its operating range.
One can calculate the acceleration reserve at each time instant over a driving cycle. However,
integration of acceleration reserve, as we did with fuel, emissions and wear, makes less sense. Instead, a mean value of acceleration reserve tells something about the vehicleβs driveability. Minimum or maximum values can also be useful measures.
Acceleration reserve was above described as limited by gear shift strategy. Other factors can be limiting, such as energy buffer state of charge for parallel hybrid vehicles or how much overload an electric machine can take short term, see right part of Figure 3-32.
Lo n gi tudi n al for ce, ππ Speed, π 0 0
gear 1
gear 2
0 0gear 1
Vehicle with conventional propulsion system Vehicle with electric propulsion system
MassβAcceleration Reserve on gear 1
gear 2
Short term Acceleration Reserve MassβAcceleration Reserve on gear 2 MassβAcceleration Reserve on gear 1 MassβAcceleration Reserve on gear 2 Lo n gi tudi n al for ce, ππ Speed, πFigure 3-32: Acceleration reserves for different gears. Large dots mark assumed operating points, each with its acceleration reserve shown.
3.3.5 Load Transfer with rigid suspension
Longitudinal load transfer redistributes vertical force from one axle to the other. The off-loaded axle can limit the traction and braking. This is because the propulsion and brake systems are normally designed such that axle torques cannot always be ideally distributed.
For functions over longer events it is often reasonable to consider the suspension as rigid. We start with the free-body diagram in Figure 3-33, which includes acceleration, .
x
z
y
LONGITUDINAL DYNAMIC Moment equilibrium, around rear contact with ground:
β + β β (π β ππ ( ) + β β π π( )) β β β β β = ; β β = β ( β
π β ππ ( ) + β β π π( )
ββ) β β Moment equilibrium, around front contact with ground:
+ β β β (π β ππ ( ) β β π π( )) β β β β β = ; β β = β ( β π β ππ ( ) β β π π( ) + ββ) + β β [3.21]
These equations confirm what we know from experience, the front axle is off-loaded under acceleration with the load shifting to the rear axle. The opposite occurs under braking.
The load shift has an effect on the tyreβs grip. If one considers the combined slip conditions of the tyre (presented in Chapter 2), a locked braking wheel limits the amount of lateral tyre forces. The same is true for a spinning wheel. This is an important problem for braking as the rear wheels become off- loaded. This can cause locking of the rear wheels if the brake pressures are not adjusted appropriately. See more in Section 3.4.5.
3.3.5.1 Varying road pitch
The model in 3.4.5..3.4.7 assumes flat but not level road, i.e. is constant. An example where
varies is when passing a crest or a hollow, or meeting uphill or meeting downhill, see Figure 3-34. If negotiating a curve at the same time as a crest, a vehicle can loose vertical force under tyres so that lateral grip is affected.
crest
hollow
Μ
β’ The variable s is the distance along the road.
β’ Road gradient versus inertial coordinate system, β , is a function of s
β’ where =π ππ ; is a function of s. β’ Vertical acceleration in inertial coordinate
system, Μ β ;, β’ where =πππ 22; is a function of s. s s = = s π¦
Figure 3-34: Free Body Diagram for driving over arbitrary vertical road profile.
Moment equilibrium, around rear and front wheel contact with ground gives: = β (( + β³ ) β π β cos( ) + h β sin( ) ββ L) β β L ; = β (( + β³ ) β π β cos( ) h β sin( ) + ββ) + β β ; [3.22]
Note that this model is assuming that vertical variations of road are larger than wheel base and track width and same on left and right side of the road/vehicle. Else the variation would be called road unevenness, which will be more treated in Chapter 5.
If models with body vertical and pitch motion and suspension springs, such as in Sections 3.4.6 and 3.4.7 it is often suitable to express the vertical fictive force, Μ with π instead of β³ . The fictive force downwards will then be π = π instead. This can be understood from basic geometry, β³β π , where π is the road pitch curvature [1 π π π‘ββ ], see Figure 3-34.
3.3.6 Acceleration
Acceleration performance like, typically, 0-100 km/h over 5..10 s, will be addressed in this section. These accelerations are relatively steady state (vehicle pitch and heave is relatively constant), so the suspension compliance is not considered.
Accelerations will also be covered in Section 3.4, as being shorter events. The vehicle pitch and heave varies more and consequently, the suspension compliance becomes important to model. This
modelling is also more suited for braking, which typically involve suspension more than propulsion.