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304 Multibody Systems Approach to Vehicle Dynamics
1.0
C 1.6 E 0.5 0.5 * sgn(x)
0.5
y/D
0.0
0.5
1.0
10
Fig. 5.59
NOT TO SCALE
8 6 4 2 0 2 4 6 8 10
BCx
Generation of an asymmetric curve
Table 5.1
Pure slip equations for the ‘Magic Formula’ tyre model (Monte Carlo version)
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
S v vertical shift
B x BCD x /C x D x
B (dy/dx (x0) )/CD S hx b 9 F z b 10
C (2/) arcsin(y s /D) S vy 0
D y max
E (Bx m tan(/2C))/(Bx m arctan(Bx m ))
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
y a 1 F z a 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 10 ||)
E y a 6 F z a 7
B z BCD z /C z D z
B y BCD y /C y D y S hz c 11 c 12 F z c 13
S hy a 8 a 9 F z a 10 S vz (c 14 F 2 z c 15 F z )c 16 F z c 17
S vy a 11 F z a 12 F z a 13
This relationship is illustrated in Figure 5.60 where the slope at zero vertical
load is taken as 2a 3 /a 4 .
This model has been extended to deal with the combined slip situation
where braking and cornering occur simultaneously. A detailed account of
the combined slip model is given in Pacejka and Bakker (1993). The equations
for pure slip only and as developed for the Monte Carlo model
(Bakker et al., 1989) are summarized in Table 5.1 and similarly for version
3 (Pacejka and Bakker, 1993) in Table 5.2. As can be seen a large number
of parameters are involved and great care is needed to avoid confusion
between each version.