Page 348 - Automotive Engineering Powertrain Chassis System and Vehicle Body

Tyre characteristics and vehicle handling and stability C HAPTER 11.1

The difficulty we have to face now is the fact that these Since F y1 and F y2 are functions of a 1 and a 2 it may be
pneumatic trails t will vary with the respective slip
i easier to take a 1 and a 2 as the state variables. With
angles. We have if the residual torques are neglected: (11.1.44) we obtain:

M zi ða i Þ da 2 dv=dr b
t i ða i Þ¼ (11.1.87) ¼ (11.1.93)
F yi ða i Þ da 1 dv=dr þ a
a which becomes with (11.1.92):
Introducing the effective axle loads
2
2
2
0 b 0 0 a 0 da 2 F y2 ða Þ=F z2 ðd a 1 þ a ÞV =gl
F z1 ¼ l 0 mg; F z2 ¼ l 0 mg (11.1.88) da 1 ¼ F y1 ða 1 Þ=F z1 ðd a 1 þ a ÞV =gl (11.1.94)
2
2
yields for the lateral force balance instead of (11.1.80): For the sake of simplicity we have assumed I/m ¼ k ¼ ab.
2
By using Eq.(11.1.94) the trajectories (solution
F y1 ¼ F y2 ¼ a y (11.1.89) curves) can be constructed in the (a 1 , a 2 ) plane. The
F 0 z1 F 0 z2 g isocline method turns out to be straightforward and
simple to employ. The pattern of the trajectories is
or after some rearrangements:
strongly influenced by the so-called singular points. In
0 0 these points, the motion finds an equilibrium. In the
a F y1 b F y2 a y
¼ 0 ¼ Q (11.1.90) singular points the motion is stationary and consequently,
a F z1 b F g
z2 the differentials of the state variables vanish.
From the handling diagram, K/mg and l/R are readily
with
obtained for given combinations of V and d. Used in
0 0
l a b combination with the normalised tyre characteristics
Q ¼ z 1 (11.1.91) F y1 /F z1 and F y2 /F z2 the values of a 1 and a 2 are found,
0
l ab
which form the coordinates of the singular points. The
The corrected normalised side force characteristics as manner in which a stable turn is approached and from
indicated in (11.1.90) can be computed beforehand and what collection of initial conditions such a motion can or
drawn as functions of the slip angles and the normal cannot be attained may be studied in the phase-plane.
procedure to assess the handling curve can be followed. One of the more interesting results of such an in-
This can be done by taking the very good approximation vestigation is the determination of the boundaries of the
Q ¼ 1 or we might select a level of Qa y /g then assess the domain of attraction in case such a domain with finite
values of the slip angles that belong to that level of dimensions exists. The size of the domain may give in-
the corrected normalised side forces and compute dications as to the so-called stability in the large. In other
Q according to (11.1.91) and from that the correct value words the question may be answered: does the vehicle
of a y /g. return to its original steady-state condition after a dis-
turbance and to what degree does this depend on the
magnitude and point of application of the disturbance
Large deviations with respect to the impulse?
steady-state motion For the construction of the trajectories we draw
The variables r and v may be considered as the two state isoclines in the (a 1 , a 2 ) plane. These isoclines are
variables of the second-order non-linear system repre- governed by Eq.(11.1.94) with slope da 2 /da 1 kept con-
sented by the Eqs. (11.1.42). Through computer nu- stant. The following three isoclines may already provide
merical integration the response to a given arbitrary sufficient information to draw estimated courses of the
2
variation of the steer angle can be easily obtained. For trajectories. We have for k ¼ ab:
motions with constant steer angle d (possibly after a step vertical intercepts (da 2 /da 1 / N):
change), the system is autonomous and the phase-plane
representation may be used to find the solution. For that, gl F y1 ða 1 Þ
we proceed by eliminating the time from Eqs.(11.1.42). a 2 ¼ V 2 F z1 þ a 1 d (11.1.95)
The result is a first-order non-linear equation (using
2
k ¼ I/m): horizontal intercepts (da 2 /da 1 / 0):
dv 2 F y1 þ F y2 mVr
¼ k (11.1.92) gl F y2 ða Þ
2
dr aF y1 bF y2 a 1 ¼ þ a þ d (11.1.96)
2
V 2 F z2
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