The final simulation model of the rear axle system is shown in
Figure 14. It comprises 30 rigid bodies, 11 flexible bodies, 5 point masses, 41 ideal joints, 2 motions, and 2 FTire tire models, which results in 217 degrees of freedom. The rigid body part is connected to the ground to exclude pitch, yaw, and translational motion in the lateral direction.
For the simulations, a combination of three different driving maneuvers is defined. Because a single axle is being simulated, only straight-line driving events are considered. The first five seconds are the settle time of the model. After that, a rotational motion
is applied to the differential side of the half shafts, as shown in
Figure 14. The acceleration phase lasts ten seconds until the predefined longitudinal velocity of 100 km/h is reached. After ten seconds of driving with a constant velocity of 100 km/h, the deceleration phase begins. During deceleration, the applied rotational motion at the drive shafts is continuously reduced. The simulation model slows down within ten seconds until it stops, and the simulation ends. This velocity curve is used for all following studies. For evaluation, the driving maneuver is divided into its three load cases→.
5.1. Variation of Initial Wheel Alignment
At first, the influence of the initial toe and camber angles on the friction work in the tire footprint on a two-dimensional road model of ISO 8608 class C is analyzed. Therefore, the initial angles are varied separately, and the simulation described above is conducted. Starting at the series initial toe angle
, eleven different initial toe angle settings are examined. The maximum and minimum limits are defined by
The initial camber angle
is kept constant. The initial wheel alignment of each setup is listed in
Table 4.
For evaluation of the influence of the different suspension setups on the total friction work
in the tire contact patch, the normalized friction work
is plotted as a function of suspension setups in
Figure 15.
All graphs show a large influence of the initial toe angle on the total friction work of each maneuver. The initial toe angle for a minimum of friction work in the tire footprint is different for the three maneuvers. For the acceleration phase, the series angle shows a minimum. For constant driving at 100 km/h, the angle should be near to zero and more negative for deceleration.
The initial camber angles are also varied. Because of the larger absolute value of the series camber angle
, the angles are reduced and increased by double the absolute value, as follows:
In total, eleven suspension setups (12–22) with equal steps in camber angle variation are simulated. The initial wheel alignment of each setup is listed in
Table 5.
The simulation results are shown in
Figure 16. The results are again normalized by the maximum value of overall friction work of each driving maneuver and plotted as a function of the suspension setup. The initial camber angle also has a big influence on the friction work in the tire footprint. A small absolute value of initial camber angle of the axle model leads to small friction work in all examined driving events.
5.2. Variation of Kinematics of Wheel Travel
For variation of the kinematics of wheel travel, first, a research of the state-of-the-art of the kinematics of wheel travel from actual cars is carried out and presented in
Figure 17. The graphs show the change in toe and camber angle during in phase vertical wheel center displacement. There is a difference in the gradient of angle change, but the trend of the curves is the same. In both graphs, no initial angle is considered.
For analysis of the influence of different suspension kinematics on the friction work in the tire footprint and the tire wear, the kinematics of wheel travel of the axle model are changed [
29]. The changes are conducted by adjusting the position of several kinematic points. The various resulting suspension setups with different toe and camber gradients are shown in
Figure 18.
To get an isolated view of the impact of each angle gradient change, the changes are implemented separately. During variation of the toe angle gradient, the camber angle gradient stays almost the same. The same applies if the camber gradient is changed. For all simulations with these different suspension setups, the initial toe and camber angles of the production car were maintained. Because the mass of the vehicle body is not changed, the changes in suspension kinematics result in a different displacement of the wheel center relative to the vehicle body. This is compensated by adjusting the preload of the spring, so that the relative position of wheel center and body is equal for all simulations. All simulations are conducted on four different road classes defined by ISO 8608 [
44], which results in 64 simulations.
First, the influence of the kinematic toe change in combination with a negative camber gradient (setup 23–29) on the overall friction work in the contact patch between tire and road is evaluated. The simulation results for all three maneuvers are plotted in
Figure 19 and
Figure 20. For a better comparison, in
Figure 19, the scaling of the axis is equal for all plots. In all plots, the resulting total friction work is normalized by the maximum value of friction work on each road.
The influence of the suspension setup is higher at a constant velocity than at the other maneuvers. Additionally, it is noticeable that, with a higher positive toe gradient, the friction work decreases. In contrast to the results of driving at a constant velocity, for acceleration and deceleration, a negative toe gradient seems to be better regarding friction work.
The influence of the road models with different spectral unevenness is most pronounced at the constant velocity simulation at 100 km/h. The influence at acceleration and deceleration is low. This can be explained by the different scale of the absolute values of the performed friction work in the three phases of simulation and the influence of the road (
Figure 20). For constant velocity, the impact of toe gradient on the friction work increases with the unevenness of the road model starting from road class A. The amount of work done by friction significantly varies from maneuver to maneuver. The lowest friction work is done at steady state straight-line driving at 100 km/h, and the most at deceleration phase. At acceleration, the work done is almost ten times higher than at constant velocity.
Afterwards, the influence of the kinematic camber change in combination with a positive toe gradient is examined. The results are pictured in
Figure 21. Again, the camber gradient produces the biggest change in friction work at a constant velocity of 100 km/h. Overall, it has to be noted that a more negative camber angle gradient in combination with a positive toe angle gradient causes a reduction of friction work in the tire footprint. During the deceleration phase, the effect is very small. The influence of the road unevenness on the results is also small during the acceleration and deceleration phases. At a constant velocity of 100 km/h, there is a significant dependence of the calculated results on the ISO 8608 road class.
For an exemplary calculation of the amount of tire wear, an average mileage of 40,000 km for a car tire is presumed. During its lifetime, a tire loses about 1 kg through wear on average [
2]. The wear law used in [
14] applies only to stationary operation conditions as present in the test rig. Regardless, this wear law is used as an example for an exemplary calculation of tire wear.
For evaluation of the saving potential, only the constant straight line driving event with 100 km/h on an ISO 8608 road model with unevenness of class C is used. The simulated results are projected to a total distance of 40,000 km to be able to make a statement regarding the lifetime of the tire. The calculation of the amount of tire wear is done with Equation (8) for all suspension setups with different kinematics (setup 23–38). In
Figure 22, only the setups with minimum and maximum tire wear of the left tire are shown.
Figure 22 shows the results for two suspension setups with different toe gradients and two suspension setups with different camber gradients in comparison with the results of the series suspension setup. On the horizontal axis, the distance to the wheel center plane is used to show the tires’ lateral width. The vertical axis indicates the tire wear of each strip of the FTire contact model. The horizontal lines represent the amount of tire wear in g per mm width of the tire tread. The spatial discretization of the FTire model, described in
Section 3.1, results in a tread strip width of about 10.32 mm. The total tire wear is the integral over the tire width. Both plots show more wear at the side of the tire that is oriented to the center of the vehicle. This is probably caused by the initial positive toe and negative camber angles.
Using the local wear law shown in
Figure 13, suspension setup 23 leads to an increased amount of tire wear of 575 g in comparison with the series setup with 470 g. Suspension setup 29 has the highest positive toe gradient and shows the minimum tire wear. The amount of tire wear is reduced by 51 g by making the toe gradient more positive, which results in a reduction of more than 10% compared with the series suspension setup.
Figure 22b shows that changing the camber gradient leads to a similar distribution of tire wear as the adapted toe gradients. With 560 g, suspension setup 38 results in the highest amount of tire wear. In comparison, suspension setup 30 results in 428 g, which means a reduction of 42 g or at least 9% compared with the series setup.
For further reduction of the tire wear during the described steady state straight line driving maneuver, both setups with the highest reduction (23 and 30) are combined, which means the toe gradient of the rear axle is changed as well as the camber gradient. This results in a significantly higher reduction of 58.1% in comparison with the series setup, which means a total tire wear of 197 g. However, the toe gradient does not fit the state-of-the-art shown in
Figure 17 anymore. Therefore, the toe gradient is adjusted to the positive limit of the state-of-the-art’s toe gradients. The lateral distribution of tire wear is shown in
Figure 23. With 204 g total tire wear, the produced quantity of tire wear of one wheel during a 40,000 km straight line ride with 100 km/h is reduced by 266 g (56.6%) in comparison with a wheel on a series axle.
The toe and camber angle changes during vertical wheel center displacement of the adjusted suspension setup are shown in
Figure 24 in comparison with the state-of-the-art from
Figure 17. Both curves lie inside the spanned area of the design of the kinematics of wheel travel from actual cars. Hence, it is understood that the resulting vehicle dynamics also stays inside the merchantable range.