Page 336 - Automotive Engineering Powertrain Chassis System and Vehicle Body

Tyre characteristics and vehicle handling and stability C HAPTER 11.1

2
2
will assume the forward speed u (z V) to remain Imu€ r þfIðC 1 þ C 2 Þþ mða C 1 þ b C 2 Þg_ r
1
constant and neglect the influence of the lateral com- þ fC 1 C 2 l mu ðaC 1 bC 2 Þgr
2
2
ponent of the longitudinal forces F xi . The equations of u
_
motion of the simple model of Fig. 11.1-9 for v and r ¼ muaC 1 d þ C 1 C 2 ld
now read: (11.1.47)
mð_ v þ urÞ¼ F y1 þ F y2 (11.1.42a) Here, as before, the dots refer to differentiation with
respect to time, d is the steer angle of the front wheel and
(11.1.42b)
I_ r ¼ aF y1 bF y2 l (¼ a þ b) represents the wheel base. The equations may
be simplified by introducing the following quantities:
with v denoting the lateral velocity of the centre of gravity
and r the yaw velocity. The symbol m stands for the
C ¼ C 1 þ C 2
vehicle mass and I (¼ I z ) denotes the moment of inertia Cs ¼ C 1 a C 2 b
about the vertical axis through the centre of gravity. For Cq ¼ C 1 a þ C 2 b 2 (11.1.48)
2
2
the matter of simplicity, the rearward shifts of the points mk ¼ I
2
of application of the forces F y1 and F y2 over a length equal
to the pneumatic trail t 1 and t 2 , respectively (that is the Here, C denotes the total cornering stiffness of the
aligning torques), have been disregarded. Later, we come vehicle, s is the distance from the centre of gravity to the
back to this. The side forces are functions of the so-called neutral steer point S (Fig. 11.1-11), q is a length
respective slip angles: corresponding to an average moment arm and k is the
radius of gyration. Eqs.(11.1.46) and (11.1.47) now
F y1 ¼ F y1 ða 1 Þ and F y2 ¼ F y2 ða Þ (11.1.43)
2
reduce to:
and the slip angles are expressed by:
C Cs
mð_ v þ urÞþ v þ r ¼ C 1 d
1 1 u u (11.1.49)
a 1 ¼ d ðv þ arÞ and a 2 ¼ ðv brÞ cq 2 Cs
u u mk _ r þ r þ v ¼ C 1 ad
2
(11.1.44) u u
neglecting the effect of the time rate of change of the steer and with v eliminated:
angle appearing in Eq.(11.1.36). For relatively low- 2 2 2 2 2 2 2
frequency motions the effective axle characteristics or m k u € r þ mCðq þ k Þu_ r þðC 1 C 2 l mu CsÞr
_
2
effective cornering stiffnesses according to Eqs.(11.1.17) ¼ mu aC 1 d þ uC 1 C 2 ld (11.1.50)
and (11.1.22) may be employed.
When only small deviations with respect to the un- The neutral steer point S is defined as the point on the
disturbed straight-ahead motion are considered, the slip longitudinal axis of the vehicle where an external side
angles may be assumed to remain small enough to allow force can be applied without changing the vehicle’s yaw
linearisation of the cornering characteristics. For the side angle. If the force acts in front of the neutral steer point,
force the relationship with the slip angle reduces to the the vehicle is expected to yaw in the direction of the force;
linear equation: if behind, then against the force. The point is of interest
when discussing the steering characteristics and stability.
(11.1.45)
F yi ¼ C i a i ¼ C Fai a i
11.1.3.2.1 Linear steady-state cornering
where C i denotes the cornering stiffness. This can be solutions
replaced by the symbol C Fai which may be preferred in
more general cases where also camber and aligning We are interested in the path curvature (1/R) that results
stiffnesses play a role. from a constant steer angle d at a given constant speed of
The two linear first-order differential equations now travel V. Since we have at steady state:
read:
1 r r
1 1 ¼ z (11.1.51)
m_ v þ ðC 1 þ C 2 Þv þ mu þ ðaC 1 bC 2 Þ r ¼ C 1 d R V u
u u
1 2 2 1 the expression for the path curvature becomes using
I_ r þ ða C 1 þ b C 2 Þr þ ðaC 1 bC 2 Þv ¼ aC 1 d
u u (11.1.47) with u replaced by V and the time derivatives
(11.1.46) omitted:
1 C 1 C 2 l
After elimination of the lateral velocity v we obtain the ¼ 2 2 d (11.1.52)
second-order differential equation for the yaw rate r: R C 1 C 2 l mV ðaC 1 bC 2 Þ

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