Page 334 - Automotive Engineering Powertrain Chassis System and Vehicle Body

Tyre characteristics and vehicle handling and stability C HAPTER 11.1

subsequently for the remaining real coordinates (for our through the height of the centre of gravity. We have, again
system only 4): for small angles:
0 2
2
d vT vT U ¼ ½ðc 41 þ c 42 Þ4 ½mgh 4 (11.1.33)
r ¼ Q u
dt vu vv The equations of motion are finally established by
d vT vT
þ r ¼ Q v using the expressions (11.1.31), (11.1.32) and (11.1.33)
dt vv vu (11.1.28) in the Eqs. (11.1.28). The equations will be linearised in
d vT vT vT the assumedly small angles 4 and d. For the variables u, v,
v þ u ¼ Q r
dt vr vu vv r and 4 we obtain successively:
d vT vT vU
0
0
þ ¼ Q 4 mð_ u rv h 4_ r 2h r _ 4Þ
dt v _ 4 v4 v4
¼ F x1 F y1 d þ F x2 (11.1.34a)
The generalised forces are found from the virtual work: 0 0 2
mð_ v þ ru h € 4 þ h r 4Þ
X (11.1.34b)
4
dW ¼ Q j dq j (11.1.29) ¼ F x1 d þ F y1 þ F y2
0
j ¼ 1 I z _ r þðI z q r I xz Þ€ 4 mh ð_ u rvÞ4
(11.1.34c)
with q j referring to the quasi coordinates x and y and the ¼ aF x1 d þ aF y1 þ M z1 bF y2 þ M z2
coordinates j and 4. Note that x and y can not be found ðI x þ mh Þ€ 4 þ mh ð_ v þ ruÞþðI z q r I xz Þ_ r
02
0
from integrating u and v. For that reason the term ‘quasi’ 02 2
coordinate is used. For the vehicle model we find for the ðmh þ I y I z Þr 4 þðk 41 þ k 42 Þ _ 4
0
virtual work as a result of the virtual displacements dx, þðc 41 þ c 42 mgh Þ4 ¼ 0 (11.1.34d)
dy, dj and d4 :
Note that the small additional roll and compliance
X X steer angles j i have been neglected in the assessment of
dW ¼ F x dx þ F y dy
X X the force components. The tyre side forces depend on
þ M z dj þ M 4 d4 (11.1.30) the slip and camber angles front and rear and on the tyre
vertical loads. We may need to take the effect of com-
where apparently bined slip into account. The longitudinal forces are either
X given as a result of brake effort or imposed propulsion
Q u ¼ F x ¼ F x1 cos d F y1 sin d þ F x2 torque or they depend on the wheel longitudinal slip
X which follows from the wheel speed of revolution re-
Q n ¼ F y ¼ F x1 sin d þ F y1 cos d þ F y2
X quiring four additional wheel rotational degrees of free-
Q r ¼ M z ¼ aF x1 sin d þ aF y1 cos d þ M z1 dom. The first equation (11.1.34a) maybeusedto
compute the propulsion force needed to keep the for-
bF y2 þ M z2
X ward speed constant.
Q 4 ¼ M 4 ¼ ðk 41 þ k 42 Þ _ 4 (11.1.31) The vertical loads and more specifically the load
transfer can be obtained by considering the moment
The longitudinal forces are assumed to be the same at the equilibrium of the front and rear axle about the re-
left and right wheels and the effect of additional steer
spective roll centres. For this, the roll moments M 4i
angles j i are neglected here. Shock absorbers in the (cf. Fig. 11.1-4) resulting from suspension springs and
wheel suspensions are represented by the resulting linear dampers as appear in Eq.(11.1.34d) through the terms
moments about the roll axes with damping coefficients with subscript 1 and 2 respectively, and the axle side
k 4i at the front and rear axles. forces appearing in Eq.(11.1.34b) are to be regarded. For
With the roll angle 4 and the roll axis inclination a linear model the load transfer can be neglected if initial
angle q r z (h 2 h 1 )/l assumed small, the kinetic energy (left/right opposite) wheel angles are disregarded. We
becomes: have at steady-state (effect of damping vanishes):

0
2
2
0
T ¼ ½mfðu h 4rÞ þðv þ h _ 4Þ g c 4i 4 þ F yi h i
DF zi ¼ (11.1.35)
2
þ ½I x _ 4 þ ½I y ð4rÞ 2 2s i
2 2
2
þ ½I z ðr 4 r þ 2q r r _ 4Þ I xz r _ 4 (11.1.32) The front and rear slip angles follow from the lateral
velocities of the wheel axles and the wheel steer angles
The potential energy U is built up in the suspension with respect to the moving longitudinal x axis. The lon-
springs (including the radial tyre compliances) and gitudinal velocities of the wheel axles may be regarded
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