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348 Multibody Systems Approach to Vehicle Dynamics
θ
δ r
φ
a
L r
Fig. 6.24
Calculation of roll stiffness due to the anti-roll bar
The roll stiffness contribution due to the road springs K Ts at the end of the
vehicle under consideration is given by
K Ts M s /k s L 2 s /2 (6.6)
In a similar manner the contribution to the roll stiffness at one end of the
vehicle due to an anti-roll bar can be determined as shown in Figure 6.24.
In this case if the ends of the anti-roll bar are separated by a distance L r and
the vehicle rolls through an angle , the relative deflection of one end of
the anti-roll bar to the other r is given by
r aL r (6.7)
The angle of twist in the roll bar is given by
TL r
GJ
(6.8)
where as discussed earlier G is the shear modulus of the anti-roll bar material,
J is the polar second moment of area and T is the torque acting about
the transverse section of the anti-roll bar. Note that in this analysis we are
ignoring the contribution due to bending. The forces acting at the ends of
the anti-roll bar F r produce an equivalent roll moment M r given by
M r F r L r TL r /a GJ/a L r GJ/a 2 (6.9)
The roll stiffness contribution due to the anti-roll bar K Tr at the end of the
vehicle under consideration is given by
K Tr M r /L r GJ/a 2 (6.10)
The contribution of both the road springs and the anti-roll bar can then be
added, ignoring suspension bushes here, to give the roll stiffness K T :
K T K Ts K Tr (6.11)
Note that current practice in vehicles is to have relatively soft springs and fit
stiffer anti-roll bars than was the norm some years ago. If vehicles achieve a
large proportion of their roll stiffness from anti-roll bars, the subjective phenomenon
of ‘roll rock’ (also known as ‘lateral head toss’) becomes problematic.
A rule of thumb is that such phenomena begin to emerge when the