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Tyre characteristics and modelling 301
5.6.5 The ‘Magic Formula’ tyre model
The tyre model which is now most well established and has generally
gained favour is based on the work by Pacejka and as mentioned earlier is
referred to as the ‘Magic Formula’. The ‘Magic Formula’ is not a predictive
tyre model but is used to represent the tyre force and moment curves
and is undergoing continual development. The early version (Bakker et al.,
1986, 1989) is sometimes referred to as the ‘Monte Carlo version’ due to
the conference location at which this model was presented in the 1989
paper. The tyre models discussed here are based on the formulations
described in Bakker et al. (1989) and a later version (Pacejka and Bakker,
1993) referred to as version 3 of the ‘Magic Formula’. Other authors have
developed systems based around the ‘Magic Formula’. The BNPS model
(Schuring et al., 1993) is a particular version of the ‘Magic Formula’ that
automates the development of the coefficients working from measured test
data. The model name BNPS is in honour of Messrs Bakker, Nyborg and
Pacejka who originated the ‘Magic Formula’ and the S indicates the particular
implementation developed by Smithers Scientific Services Inc.
In the original ‘Magic Formula’ paper Bakker et al. (1986) discuss the use
of formulae to represent the force and moment curves using established
techniques based on polynomials or a Fourier series. The main disadvantage
with this approach is that the coefficients used have no engineering
significance in terms of the tyre properties and as with interpolation
methods the model would not lend itself to design activities. This is also
reflected in Sitchen (1983) where the author describes a representation
based on polynomials where the curves are divided into five regions but
this still has the problem of using coefficients which do not typify the tyre
force and moment characteristics.
The general acceptance of the ‘Magic Formula’ is reinforced by the work
carried out at Michelin and described in Bayle et al. (1993). In this paper
the authors describe how the ‘Magic Formula’ has been tested at Michelin
and ‘industrialized’ as a self-contained package for the pure lateral force
model. The authors also considered modifications to the ‘Magic Formula’
to deal with the complicated situation of combined slip.
The ‘Magic Formula’ model is undergoing continual development, which
is reflected in a further publication (Pacejka and Besselink, 1997) where
the model is not restricted to small values of slip and the wheel may also run
backwards. The authors also discuss a relatively simple model for longitudinal
and lateral transient responses restricted to relatively low time and
path frequencies. The tyre model in this paper also acquired a new name
and is referred to as the ‘Delft Tyre 97’ version.
The ‘Magic Formula’ has been developed using mathematical functions
that relate:
(i) The lateral force F y as a function of slip angle
(ii) The aligning moment M z as a function of slip angle
(iii) The longitudinal force F x as a function of longitudinal slip
When these curves are obtained from steady state tyre testing and plotted
the general shape of the curves is similar to that indicated in Figure 5.57.