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Tyre characteristics and modelling 303
y
Y
D
arctan(BCD)
y s
S v
x m
x
X
S h
Fig. 5.58
Coefficients used in the ‘Magic Formula’ tyre
fit a particular tyre or if suitable taken as the constants given in Bakker
et al. (1986):
1.30 – lateral force curve
1.65 – longitudinal braking force curve
2.40 – aligning moment curve
B – is referred to as a ‘stiffness’ factor. From Figure 5.58 it can be seen that
BCD is the slope at the origin, i.e. the cornering stiffness when plotting lateral
force. Obtaining values for D and C leads to a value for B.
E – is a ‘curvature’ factor that effects the transition in the curve and the
position x m at which the peak value if present occurs. E is calculated using:
E
Bxm
tan
2C
Bx arctan Bx
m
( )
( )
m
(5.52)
y s – is the asymptotic value at large slip values and is found using:
y s D sin(C/2) (5.53)
The curvature factor E can be made dependent on the sign of the slip value
plotted on the x-axis:
E E 0 E sgn(x) (5.54)
This will allow for the lack of symmetry between the right and left side of
the diagram when comparing driving and braking forces or to introduce the
effects of camber angle . This effect is illustrated in Pacejka and Bakker
(1993) by the generation of an asymmetric curve using coefficients
C 1.6, E O 0.5 and E 0.5. This is recreated here using the curve
shape illustrated in Figure 5.59. Note that the plots have been made nondimensional
by plotting y/D on the y-axis and BCx on the x-axis.
The ‘Magic Formula’ utilizes a set of coefficients a 0 , a 1 , a 2 , … as shown in
Tables 5.1 and 5.2. In Figure 5.60 it can be seen that at zero camber the
cornering stiffness BCD y reaches a maximum value defined by the coefficient
a 3 at a given value of vertical load F z that equates to the coefficient a 4 .