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266 Multibody Systems Approach to Vehicle Dynamics
Slip angle = 0
Camber angle = 0
Braking force F x (N)
F z = 8 kN
F z = 6 kN
F z = 4 kN
F z = 2 kN
φ
Longitudinal stiffness
C s = tan φ
0.0 Slip ratio
1.0
Fig. 5.19
Braking force versus slip ratio
It should be noted that slip ratio has been subject to various definitions by
researchers and research groups in tyre companies, several are listed in
Milliken and Milliken (1995). Some tyre models use R l instead of R e when
formulating slip ratio which may require careful consideration when using a
general-purpose MBS program to model ABS (Ozdalyan and Blundell, 1998).
Plotting curves of braking force, for convenience shown positive here,
against slip ratio for a range of tyre loads will generally produce curves of
the type shown in Figure 5.19.
Examination of the curves in Figure 5.19 reveals that at each vertical load
the braking force increases rapidly in a linear manner to reach a peak value
that, depending on tyre design and road conditions, would typically occur
at a slip ratio anywhere between 0.15 and 0.3. After this point the braking
force will level out or reduce as the wheel approaches the fully locked
situation. Examination of curves such as these, sometimes called ‘mu-slip’
curves, facilitates an understanding of ABS operation where cycling the brake
pressure maintains a slip ratio near the peak braking force position for each
wheel on the vehicle. This is desirable not only to maximize braking effort
but also to maintain a rolling wheel for cornering and directional stability.
An important property of each curve is the slope at the origin, referred to as
the longitudinal stiffness, C s . It can be seen that this is not a constant but
increases with load which is significant when considering the capability of
any tyre model to be used in braking simulations. In Figure 5.19 the curves
are shown to pass through the origin. In practice a small vertical offset in
longitudinal force will be apparent for a free rolling tyre, this being the
rolling resistance discussed earlier.
It is important to reiterate that frictional forces are not the property of the
tyre alone. The effects of road material and texture, or contamination with
water and ice, are also significant. Figure 5.20 demonstrates typical curves
of braking force against slip ratio, at a given tyre load for various road conditions
(Phillips, 2000). These curves demonstrate that on wet roads peak