In the previous sections the focus has been on analyzing and modeling the steady- state forces and moments. The tire is considered and modeled as a complex, non- linear function with multiple inputs and outputs as shown in Fig.4. In the steady-state approach any change of an input will result in an instantaneous change of the out- puts. As will be shown in this section, this is not how a real tire behaves in dynamic situations. Typically a tire responds with some delay on abrupt changes of the inputs.
A model to illustrate these effects is shown in Fig.33. The tire contact patch has a
Tire Characteristics and Modeling 97
Table 6 Self-aligning moment Magic Formula parameters
Constants Fz0= 5000N,r0= 0.315m
tp qBz1= 8.6117,qBz2= −0.8357,qBz3= −4.2173 qBz4= −0.2891,qBz5= 0.6218
qCz1= 1.1423
qDz1= 0.1231,qDz2= 0.0071,qDz3= 0,qDz4= 9.6467 qEz1= −3.1431,qEz2= 0.7236,qEz3= −6.4258 qEz4= −0.3275,qEz5= 0
qHz1= −0.0067,qHz2= 0,qHz3= 0.1209,qHz4= 0
Mzr qBz9= 0,qBz10= 0.2865
qDz6= −0.0037,qDz7= 0.0068,qDz8= 0.0610 qDz9= −0.0625,qDz10= 0,qDz11= 0 sz ssz1= 0.0363,ssz2= 0.0102,ssz3= −0.5962
ssz4= 0.2702
Model error Pure:𝜖= 7.82%, combined:𝜖= 17.12%
contact patch
V
−Vsy Fy
Vcp
εy
α α top view
−Vsy Fy
εy
A
S rear view
Fig. 33 Tire model with a lateral degree of freedom for the contact patch
lateral degree of freedom𝜀ywith respect to the wheel, the stiffness associated with this degree of freedom isky. The contact patch can slide with respect to the road.
The relative velocity of the contact patch ̇𝜀ywith respect to the wheel plane has to be added to the absolute sliding velocity of pointS,Vsy, to obtain the side slip angle 𝛼′of the contact patch. The side slip angle angle of the contact patch𝛼′becomes
tan(𝛼′) = −Vsy+ ̇𝜀y
|Vx| . (146)
Note that in the regular definition of the tire side slip angle𝛼(4) that the wheel is considered as a rigid disk. Assuming linear tire behavior and neglecting the mass of the contact path, the spring force and lateral force due to side slip have to be in
equilibrium,
Fy=CF𝛼𝛼′=ky𝜀y. (147)
Assuming that CF𝛼 and ky are independent of time, the following expression is obtained after differentiation with respect to time
̇𝜀y= CF𝛼
ky ̇𝛼′. (148)
Substituting (148) in (146) results in CF𝛼
ky 1
|Vx|̇𝛼′+ tan(𝛼′) = tan(𝛼). (149) The ratio between cornering stiffness CF𝛼 and lateral stiffnesskyis known as the relaxation length𝜎y, thus
𝜎y= CF𝛼
ky . (150)
The longitudinal velocity Vx is approximately equal to the time derivative of the traveled distancest. For a positive longitudinal velocity and linearization for small values of the side slip angle, the following expression is obtained
𝜎y
dt dst
d𝛼′
dt +𝛼′=𝜎y
d𝛼′
dst +𝛼′=𝛼. (151)
From this equation it becomes clear that the relaxation behavior of a tire is not depen- dent on time, but on the traveled distancest. Starting from an initial condition where 𝛼′and𝛼are equal to zero and applying a step in the tire side slip angle𝛼with mag- nitude𝛼step, the analytical solution of (151) equals
𝛼′= (1 −e−st∕𝜎y)𝛼step. (152) The lateral forceFybecomes
Fy=CF𝛼(1 −e−st∕𝜎y)𝛼step. (153) Note that this expression is only valid under the assumption of linear tire behavior and small side slip angles.
As the tire relaxation behavior is a function of the traveled distance, the forward velocity does not play a role when doing measurements to assess the tire relaxation behavior and as a result this behavior can be measured at very low speeds. A mea- surement device that allows to execute relaxation length measurements is the flat plank tire tester as shown in Fig.34. The wheel is mounted on a measurement hub
Tire Characteristics and Modeling 99
Fig. 34 TU/e flat plank tire tester
0 0.5 1 1.5 2 2.5
travelled distance st [m]
0 500 1000 1500
lateral force F y [N]
Fy,ss
y
Fy= 0.632 Fy,ss
measurement model
Fig. 35 Tire response to a step in side slip angle (𝛼step= 1◦,Fz= 4000N,𝜎y= 0.412m,CF𝛼= 1050N/deg)
that allows to measure the forces and moments that occur at the wheel center. The road is represented by a plank with a length of approximately 7 m, which can move in the horizontal plane with a maximum velocity of 0.05 m/s. The measurement hub can move vertically allowing to press the tire against the plank and thus giving con- trol over the vertical tire forceFz. Furthermore it can be rotated about the vertical axis allowing a steering angle to be applied to the wheel.
To apply a step in side slip angle the required steering angle is applied first, then the wheel is pressed against the plank and thereafter the plank is moved forward with a low, constant velocity. The lateral force as a function of traveled distance is shown in Fig.35. In the same figure (153) is plotted, showing that it can accurately represent the measurement results. Note that the relaxation length𝜎yfor this experiment equals 0.412 m. In general the relaxation length corresponds to the distance traveled since the start of the application of the step in side slip angle, when 63.2% of the steady- state value of the lateral forceFyis reached. Note that the relaxation behavior of the tire is not dependent on time, but traveled distance. This also means that when the forward velocity is increased, that the tire force will respond more quickly since the distance traveled in the same amount of time will increase.
Brush tire modelThe brush tire model can be used to gain more insight in various aspects of tire relaxation behavior. Figure36shows the response of the brush tire model to a small step in side slip angle. In this example it is assumed that no sliding of the bristles will occur and that the steady-state deformation pattern of the bristles
bristle deformation pattern - top view
V αstep
2a signals
α 0 Fy
0 Mz
0
travelled distancest
Fig. 36 Brush tire model transient behavior on a step in side slip angle
is triangular. Initially all bristles are undeformed and upon the application of the side slip angle the tire needs to move forward over a distance2ato obtain the steady state deformation pattern. As a result the lateral force and self-aligning moment will develop as a function of the traveled distance and relaxation effects become visible.
Various, qualitatively valid observations with respect to relaxation behavior can be made from the brush tire model step response:
∙ It can be noted that there is a difference in the response of the lateral forceFy and self-aligning momentMz. Upon application of the step in side slip angleFy starts to increase immediately, whereas the gradient ofMzis equal to zero. This behavior has also been observed for aircraft tires, for normal passenger car tires this phenomenon is hardly noticeable.
∙ For large side slip angles the tire will need to travel less than2a to obtain the steady state deformation pattern, as the bristles in the trailing part of the contact patch are already sliding. Thus the relaxation length is expected to decrease with increasing side slip angle.
∙ As the contact length2a increases with the vertical force, the relaxation length will also increase with the vertical forceFz.
∙ When the tire is running at a fixed side slip angle, a stepwise increase of the vertical force will also result in relaxation effects. The increase of the vertical force results in a longer contact length and increased maximum bristle deflections. The tire will need to travel over a certain distance before settling for a new steady-state deformation pattern.
∙ Since there is no fundamental difference between longitudinal and lateral bristle deformation, relaxation behavior will not only be present in the lateral direction but also in the longitudinal direction.
Tire Characteristics and Modeling 101 In the next section a pragmatic tire model will be developed, which includes all the aforementioned effects.