Page 335 - Automotive Engineering Powertrain Chassis System and Vehicle Body

CHAP TER 1 1. 1 Tyre characteristics and vehicle handling and stability

the same left and right and equal to the vehicle longitu- (11.1.13) – (11.1.15). Furthermore, the resulting com-
dinal speed u. This is allowed when s i jrj«u. Then the pliance steer and roll steer coefficients for i ¼ 1 or 2 have
expressions for the assumedly small slip angles read: been introduced:

v þ ar ed _
a 1 ¼ d þ j
1
u (11.1.36) c sfi e i þt i C Fai
v br c sci ¼ c ji ;
a ¼ j
2
2
u 1 c sfi e i þt i C Fai
c ji
The additional roll and compliance steer angles j i and (11.1.39)
the wheel camber angles g i are obtained from 3 i þ s i c sfi e i þt i C Fgi
c ji
Eq.(11.1.21) with (11.1.13)–(11.1.15) or corresponding c sri ¼
non-linear expressions. Initial wheel angles are assumed 1 c sfi e i þt i C Fai
to be equal to zero. The influence of the steer angle ve- c ji
locity appearing in the expression for the front slip angle
is relatively small and may be disregarded. The small where the steer stiffness at the rear c j2 may be taken
products of the caster length e and the time rate of equal to infinity. Furthermore, we have the roll axis in-
change of j i have been neglected in the above clination angle:
expressions.
Eqs. (11.1.34) may be further linearised by assuming h 2 h 1
that all the deviations from the rectilinear motion are q r ¼ l (11.1.40)
small. This allows the neglection of all products of vari-
able quantities which vanish when the vehicle moves The relaxation length denoted by s i is an important
straight ahead. The side forces and moments are then parameter that controls the lag of the response of the side
written as in Eq.(11.1.5) with the subscripts i ¼ 1or2 force to the input slip angle. For the Laplace transformed
provided. If the moment due to camber is neglected and version of the Eqs. (11.1.38) with the Laplace variable s
the pneumatic trail is introduced in the aligning torque representing differentiation with respect to time, we
we have: may introduce tyre lag by replacing the slip angle a i by
the filtered transient slip angle. This may be accom-
F yi ¼ F yai þ F ygi ¼ C Fai a i þ C FYi g i plished by replacing the cornering stiffnesses C Fai
M zi ¼ M zai ¼ C Mai a i ¼ t i F yai ¼ t i C Fai a i appearing in (11.1.38) and (11.1.39) by the ‘transient
stiffnesses’:
(11.1.37)
The three linear equations of motion for the system of C Fai
Fig. 11.1-4 with the forward speed u kept constant finally C Fai / 1 þ ss i =u (11.1.41)
turn out to read expressed solely in terms of the three
motion variables v, r and 4: A similar procedure may be followed to include the
tyre transient response to wheel camber variations. The
0
mð_ v þ ur þ h € 4Þ relaxation length concerned is about equal to the one used
_
¼ C Fa1 fð1 þ c sc1 Þðud þ ed v arÞ=u þ c sr1 4g for the response to side slip variations. At nominal vertical
load the relaxation length is of the order of magnitude of
4g
þ C Fa2 fð1 þ c sc2 Þð v þ brÞ=u þ c sr2
the wheel radius. A more precise model of the aligning
s
þðC Fg1 s 1 þ C Fg2 2 Þ4 (11.1.38a) torque may be introduced by using a transient pneumatic
I z _ r þðI z q r I xz Þ€ 4 ¼ða t 1 ÞC Fa1 fð1 þ c sc1 Þ trail with a similar replacement as indicated by (11.1.41)
_
ðud þ ed v arÞ=u þ c sr1 4g but with a much smaller relaxation length approximately
ðb þ t 2 ÞC Fa2 fð1 þ c sc2 Þð v þ brÞ=u þ c sr2 4g equal to half the contact length of the tyre.
þðaC Fg1 s 1 bC Fg2 2 Þ4
s
(11.1.38b)
11.1.3.2 Linear analysis of the
02
0
ðI x þ mh Þ€ 4 þ mh ð_ v þ urÞ two-degree-of-freedom model
þðI z q r I xz Þ_ r þðk 41 þ k 42 Þ _ 4
0
þðc 41 þ c 42 mgh Þ4 ¼ 0 (11.1.38c) From the Eqs. (11.1.34b) and (11.1.34c) the reduced
set of equations for the two-degree-of-freedom model
In these equations the additional steer angles j i have can be derived immediately. The roll angle 4 and its
been eliminated by using expressions (11.1.21) with derivative are set equal to zero and furthermore, we

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