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Tyre characteristics and modelling 305
Table 5.2 Pure slip equations for the ‘Magic Formula’ tyre model (version 3)
General formula
Longitudinal force
y(x) D sin[C arctan{Bx E(Bx arctan(Bx))}] X x
Y(X) y(x) S v
Y x F x
x X S h
D x x F z
B stiffness factor x b 1 F z b 2
C shape factor BCD x (b 3 F 2 z b 4 F z ) exp(b 5 F z )
D peak factor C x b 0
S h horizontal shift E x (b 6 F 2 z b 7 F z b 8 ) (1 b 13 sgn
S v vertical shift ( S hx ))
B (dy/dx (x0) )/CD
B x BCD x /C x D x
C (2/) arcsin(y s /D) S hx b 9 F z b 10
D y max S vy b 11 F z b 12
E (Bx m tan(/2C))/(Bx m arctan(Bx m )) Brake force only (b 11 b 12 b 13 0)
Lateral force
Aligning moment
X y
X z
Y y F y
Y z M z
D y y F z D z (c 1 F 2 z c 2 F z )(1 c 18 2 )
y (a 1 F z a 2 )(1 a 15 2 ) BCD z (c 3 F 2 z c 4 F z )(1 c 6 ||) exp(c 5 F z )
BCD y a 3 sin(2 arctan(F z /a 4 ))(1 a 5 ||) C z c 0
C y a 0 E z (c 7 F 2 z c 8 F z c 9 )(1 (c 19 c 20 )**
E y (a 6 F z a 7 )(1 (a 16 a 17 ) sgn( S hy )) sgn( S hz ))/(1 c 10 ||)
B y BCD y /C y D y
B z BCD z /C z D z
S hy a 8 F z a 9 a 10
S hz c 11 F z c 12 c 13
S vy a 11 F z a 12 (a 13 F 2 z a 14 F z ) S vz c 14 F z c 15 (c 16 F 2 z c 17 F z )
BCD y (N/rad)
a 3
arctan(2a 3 /a 4 )
0 a 4
F z (N)
Fig. 5.60
Cornering stiffness as a function of vertical load at zero camber angle
Apart from implementing the model into a multibody systems analysis program
for vehicle simulation some method is needed to obtain the coefficients
from raw test data. In Sharp (1992) a suggested approach is to use an
appreciation of the properties of the ‘Magic Formula’ to fix C based on the
values suggested in Pacejka and Bakker (1993) for lateral force, longitudinal
force and aligning moment. For each set of load data it is then possible
to obtain the peak value D and the position at which this occurs x m . Using the
slope at the origin and the values for C and D it is now possible to determine