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Tyre characteristics and modelling 249
understanding of these force and moment characteristics is essential before
introducing the various mathematical tyre models available and describing
the methods used to implement these with MBS vehicle models.
Before a computer simulation can be performed the tyre force and moment
characteristics must be estimated or obtained from experimental tests.
A traditional approach is to test the tyre using a tyre test machine and to measure
the resulting force and moment components for various camber angles,
slip angles and values of vertical force. The measured data is set up in tabular
form, which is interpolated during the computer simulation in order to
transfer the forces to the full vehicle model. Alternatively mathematical
functions are used to fit equations to the measured test data. These equations
provide a mathematical tyre model that can be incorporated into the full
vehicle model. This method requires the generation of a number of parameters
that must be derived from the measured data before the simulation can
proceed. The quality of the model will be a compromise between the accuracy
of the fit, relevance of the parameters, and the availability of methods to
generate the parameters.
5.2 Tyre axis systems and geometry
5.2.1 The SAE and ISO tyre axis systems
To assist with the description of the forces and moments generated by a
tyre, an axis system shown in simplified form in Figure 5.2 has been defined
by the SAE (1976). In this system the X-axis is the intersection of the wheel
plane and the road plane with the positive direction taken for the wheel
moving forward. The Z-axis is perpendicular to the road plane with a positive
direction assumed to be acting downwards. The Y-axis is in the road plane,
its direction dictated by the use of a right-handed orthogonal axis system.
The angles and represent the slip angle and camber angle respectively.
The SAE system will be used throughout this text unless stated.
It should be noted that not all practitioners adhere rigidly to this system in
their publications and another system, the ISO tyre system, is gaining
favour. This system, shown simplified in Figure 5.3, is likely to become the
standard for future tyre models.
It will be seen later that distortions in the tyre carcass will cause the contact
patch to move away from the rigid wheel plane shown in Figures 5.2 and
5.3. Although the components of force are still assumed to act through the
contact point P the distortions will introduce offsets and additional components
of moment also acting about the point P.
5.2.2 Definition of tyre radii
The definition of tyre radii is important for the formulation of slip in the
contact patch. In general we consider a tyre to have an unloaded radius, a
loaded radius and an effective rolling radius. The unloaded tyre radius, R u ,
is straightforward to comprehend and is shown in Figure 5.4. For a rigid
disc with radius R u rolling forward with no sliding (fully geared to the
road), during one revolution the disc will move forward a distance 2R u .