Page 339 - Automotive Engineering Powertrain Chassis System and Vehicle Body

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

2 2
2 2
2
2
last coefficient a 2 may become negative which corre- m k V l þ mCðq þ k ÞVl
sponds to divergent instability (spin-out without oscil- h
lations). As already indicated, this will indeed occur þ C 1 C 2 l 2 1 þ gl V 2 ¼ 0 (11.1.67)
when for an oversteered vehicle the critical speed
(11.1.57) is exceeded. The condition for stability reads: For a single mass-damper-spring system shown in
! Fig. 11.1-13 with r the mass displacement, d the
V 2 d forced displacement of the support, M the mass, D the
a 2 ¼ C 1 C 2 l 2 1 þ h ¼ C 1 C 2 l 2 > 0
gl l=R ss sum of the two damping coefficients D 1 and D 2 and K
the sum of the two spring stiffnesses K 1 and K 2 a dif-
(11.1.65) ferential equation similar in structure to Eq.(11.1.50)
arises:
with the subscript ss referring to steady-state conditions,
or
_
M€ r þ D_ r þ Kr ¼ D 1 d þ K 1 d (11.1.68)
s ffiffiffiffiffiffiffi
gl and the corresponding characteristic equation:
V< V crit ¼ ðh < 0Þ (11.1.66)
h
2
Ml þDl þ K ¼ 0 (11.1.69)
The next section will further analyse the dynamic
nature of the stable and unstable motions. When an oversteered car exceeds its critical speed,
It is of importance to note that when the condi- the last term of (11.1.67) becomes negative which
tion of an automobile subjected to driving or braking apparently corresponds with a negative stiffness K.An
forces is considered, the cornering stiffnesses front inverted pendulum is an example of a second-order
and rear will change due to the associated fore and system with negative last coefficient showing monoto-
aft axle load transfer and the resulting state of nous (diverging) instability.
combined slip. In expression (11.1.60) for the un- The roots l of Eq.(11.1.67) may have loci in the
dersteer coefficient h the quantities F zi represent complex plane as shown in Fig. 11.1-12. For positive
the static vertical axle loads obtained through values of the cornering stiffnesses only the last
Eqs.(11.1.59) and are to remain unchanged! In Sub- coefficient of the characteristic equation can become
section 11.1.3.4 the effect of longitudinal forces on ve- negative which is responsible for the limited types of
hicle stability will be further analysed. eigenvalues that can occur. As we will see in Subsection
11.1.3.3, possible negative slopes beyond the peak of the
Free linear motions non-linear axle characteristics may give rise to other
To study the nature of the free motion after a small types of unstable motions connected with two positive
disturbance in terms of natural frequency and damping, real roots or two conjugated complex roots with a posi-
the eigenvalues, that is the roots of the characteristic tive real part. For the linear vehicle model we may have
equation of the linear second-order system, are to be two real roots in the oversteer case and a pair of complex
assessed. The characteristic equation of the system de- roots in the understeer case, except at low speeds where
scribed by the Eqs.(11.1.49) or (11.1.50) reads after the understeered vehicle can show a pair of real negative
using the relation (11.1.54) between s and h: roots.

Fig. 11.1-12 Possible eigenvalues for the over and understeered car at lower and higher speeds.

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