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308 Multibody Systems Approach to Vehicle Dynamics
If |S| is less than S* the tyre is considered to be in an elastic deformation
state and
F x C s S (5.58)
If |S| is greater than S* the tyre is considered to be in a complete sliding
state and
F x sgn(S){F z [(F z ) 2 /4|S|C s ]} (5.59)
For the lateral force a critical slip angle * is calculated using:
* arctan |3F z /C | (5.60)
If || is less than * then the tyre is considered to be in a state of elastic
deformation and
H 1 C |tan |/3|F z | (5.61)
F y |F z |(1 H 3 ) sgn() (5.62)
If || is greater than * then the tyre is considered to be sliding and
F y |F z | sgn() (5.63)
The rolling resistance moment M y is given by:
M y C r F z (forward motion) (5.64)
M y C r F z (backward motion) (5.65)
For the aligning moment M z if || is less than * (elastic deformation state)
then:
H 1 C |tan |/3|F z | (5.66)
M z 2|F z |R 2 (1 H)H 3 sgn() (5.67)
If || is greater than * (complete sliding state) then:
M z 0.0 (5.68)
Note that the Fiala tyre model formulation does not allow for the possibility
of negative aligning moments occurring at high slip angles.
5.6.7 Tyre models for durability analysis
The modelling of suspension systems for durability analysis where
component loads are needed for follow-on stress and fatigue analysis was
discussed in Chapter 4. As discussed the models are developed to simulate
tests carried out on the proving ground to represent impacts with road
obstacles of the type shown in Figure 5.62. If an actual vehicle has been
taken on the proving ground and instrumented to take readings at the wheel
centre, then the recorded data can be used as input to a suspension or
vehicle model and the need for a durability tyre model is negated.
As vehicle design processes move towards more use of virtual prototypes
the need to carry out full dynamic simulations of proving ground procedures
requires more sophisticated tyre models to interact with the terrain. In
the extreme simulations recreating the testing of off-road vehicles involving
conditions of the type shown in Figure 5.63 requires a tyre model that