When the tire rotates with the rollers on the brake testing platform, the elastic hysteresis loss and friction loss were produced by internal friction of the tire and the relative slip between the tire and the drum. The change relationship of the measured front axle friction loss varies with tire air pressure for the Xiali A+ sedan is shown in Figure 3.

It can be seen from Figure 3, with the increase of tire pressure, the tire deformation decrease, resulting in a drop in the frictional resistance between the wheel and the roller surface, so as the friction loss descends. Compared to the tire pressure of 0.36 MPa, that of 0.03 MPa displays a 78.8% reduce in the friction loss. When the tire pressure is less than 0.15 MPa, the tire deformation is great, the tire elastic hysteresis loss become larger, and the frictional resistance increases significantly during rolling. On the contrary, as the tire inflation pressure increases, the tire contacted with the hard roller deforms smaller, the hysteresis loss fell down, the rolling resistance diminishes, and the friction loss decreases when the tire pressure is greater than 0.15 MPa.

The change relationship of the measured friction loss of the front axle with the axle load of the Xiali A+ sedan is shown in Figure 4. The test was conducted with a tire pressure of 0.25 MPa and a roller spacing of 471 mm, and the load of front axle was varied by additional mass.

As shown in Figure 4, the relationship between the frictional resistance measured on the brake testing platform and the axle load is approximately linear. With the increase in axle load, the tire deformation and the friction loss both increases. As the 2.66 km/h roller speed of brake testing platform is very low, the rolling resistance coefficient f under different axle load conditions remains unchanged. According to the rolling resistance equation (1), it is known that the friction loss increases linearly with the increase in axle load, the slope is the rolling friction coefficient, which proves that the actual measured results are accurate and reliable. Where F_f is the friction loss, G is the gravity of the tested vehicle, f is the rolling friction coefficient.

The force condition of the inspected vehicle wheel during testing is shown in Figure 5. With O₁ and O₂ as the center of the circle is the main and assistant roller of the anti-force rolling brake test platform, and the rear roller is higher than the front roller arrangement form. With O as the center of the circle is the vehicle wheel (Ignoring the rolling resistance and the influence of non-test wheels on the tested wheels.)

The placement angle α is 1/2 of the angle between the center of the wheel and the line connecting the centers, the relationship of front placement angle α₁, rear placement angle α₂ and placement angle α of the new brake test platform can be described in the equations (2), (3), (4) and (5). Where θ is the angle formed by the height difference between the front and rear rollers (inter-drum angle), h is the height difference between the rollers, L is the roller spacing, R is the wheel radius and r is the roller radius.

The wheel placement angle can be described by the equations (5). The wheel placement angle α is determined by the roller radius r, the wheel radius R and the roller spacing L. Ones the brake testing platform and the testing vehicle have been determined, r of 122.5 mm and R of 270 mm can be identified, thus the placement angle can be changed by altering L. Figure 6 shows the change relationship of front axle friction loss with the placement angle changes of measured Xiali A+ sedan. During the test, the tire pressure was 0.25 MPa, the roller spacing was adjusted from 471 mm to 612 mm, corresponding to the placement angle adjustment range was 36.9°~51.3°.

As can be seen from Figure 6, with the increase of the placement angle, the tire deformation becomes more pronounced, the frictional resistance between the wheel and the roller surface strengthens, and the friction loss gets larger. That’s due to the normal support force of the rigid roller on the tire enhances, which in turn enlarges the friction loss inside the brake testing platform, leading to the increase of the frictional resistance. At the placement angle of 51.3°, the frictional resistance is 338 N, reaching 6.9% of the front axle load, which shows that the placement angle has a great influence on the frictional resistance.

In this experiment, the tire pressure was 0.25 MPa and the roller spacing was 471 mm. In order to increase the speed of testing vehicle, the decelerator of brake testing platform was replaced so that the reduction ratio was 1:2. The motor speed was controlled by the frequency converter, and the change relationship of the friction loss with vehicle speed of the brake testing platform was tested as shown in Figure 7.

It shows that the frictional resistance and friction loss power of the brake testing platform rises with the increase of vehicle speed. The friction loss power at the speed of 66 km/h is 132 times of that at the speed of 2 km/h. That can be explained as follows. One hand, with the improvement of vehicle speed, the rolling resistance coefficient gets larger, and the friction loss between the wheel and the roller surface swells. On the other hand, with the increase in roller speed of the brake testing platform, the bearing friction coefficient enlarges, taking the internal friction loss of the brake testing platform up. The combination of the above two aspects results in the higher vehicle speeds and greater friction losses on the brake testing platform. Normally, when the vehicle is tested for power on the dynamometer chassis, the speed point is higher, usually above 40 km/h. To ensure the accuracy of chassis dynamometer measurement, the output power of the chassis dynamometer should be revised. While, in the brake performance testing, due to the low and constant roller speed of the brake testing platform, about 2.5 km/h, the effect of vehicle speed on the friction loss of the brake testing platform can be neglected.

As mentioned in section 3.4, the impact of vehicle speed on friction loss can be ignored because of its low and constant characteristics in the vehicle braking performance test on anti-force brake testing platform. Importantly, tire pressure, placement angle and vehicle axle load have a large impact on the friction loss, which in turn directly affects the braking force test results. However, on account of the diversity of testing vehicles, there are significant differences in tire pressure, placement angle and vehicle axle load. In order to effectively measure the braking performance of the vehicle brake, the braking force results need to be corrected during the vehicle braking performance test.

Here, the relative slipping S was defined as equation (6), it integrally reflects the elastic hysteresis loss generated by the tire deformation and the friction loss generated by the relative slip between the wheel and the roller. Where S is the relative slip, ν_m is the vehicle speed corresponding to the main and assistant roller speed, ν_t is the vehicle speed corresponding to the third roller speed.

As can be seen in Figure 8, in the effect of tire pressure and axle load on the friction loss, when the relative slipping S>0, the friction loss increases along with the increase of the slip amount; when the relative slipping S<0, the friction loss increases significantly with the decrease of the slip amount. Because, when the tire pressure <0.07 MPa, which was small, the third roller support spring stiffness was larger. With the tire pressure decreased, the third roller support spring compression increased, so do as the spring force. Besides, the tire and the third roller contact surface was small, while the deformation was greater, leading to the third roller corresponding vehicle speed lower than that of the main and assistant roller speed. Therefore, on the condition of S<0, the friction loss is too large. However, according to the National Standard of the People’s Republic of China GB38900-2020, the braking performance test in the practical application needs to be carried out when the tire pressure is normal. In this case, the tire pressure bigger than 0.15 MPa and the relative slipping S>0. When the placement angle is constant and the tire pressure is normal, the relative slipping S>0, the friction loss increases with the rising of relative slipping. When the tire pressure or axle load changes, there is the same correspondence between the friction loss and the relative slipping. After adjusting the center distance of the main and assistant roller to enlarge the placement angle, when the relative slipping is the same, the friction loss increases greatly. As the placement angle gets larger, the relative slipping decreases slightly, and the friction loss caused by the slip between the wheel and the roller falling slightly too. But, after turning up the placement angle, the main and assistant roller normal support force increases, making the internal friction loss (including bearing friction loss, etc.) of the anti-force brake testing platform increases obviously. At the same time, the internal friction loss of the brake testing platform plays a dominant role, the combined effect of the two factors makes the friction loss of the testing platform increases with the increase of the placement angle. Consequently, the effects of relative slipping and placement angle need to be taken into account when considering friction loss corrections for new brake testing platform.

Considering the fitting accuracy and practical needs, the data were fitted using the measured friction loss with the variation of tire pressure and axle load at tire pressure >0.15 MPa to obtain the model of the variation of friction loss with relative slipping, which is ascribed as equation (7). Where, F_f1(N) is the friction loss caused by the change of relative slipping, S(%) is the relative slipping.

According to equation (7), after subtracting the data of the relationship between the measured friction loss and the variation of the relative slipping from the data of the relationship between the friction loss and the variation of the placement angle, data fitting is performed to obtain the model of the variation of the friction loss with the placement angle, that is described as the following equation (8). Where F_f2(N) is the friction loss caused by the change in placement angle, and α is the placement angle.

When the experimental vehicle is actually tested, the friction loss needs to be calculated according to the equation (7) and (8), and the vehicle braking force should be revised after summing of them.

As mentioned above, different axial loads could lead to different friction losses. To eliminate the influence of axial loads on friction losses, equations (7) and (8) are divided by the axial loads respectively, the friction coefficients ϕ_f1 varied with the relative slipping and ϕ_f2 varied with the placement angle are put forward, as shown in equations (9) and (10).

In order to verify the constructed friction loss model, a total of four types of vehicles, namely Changan microbus, Dongfeng light general truck, Isuzu light van and Jiefang heavy warehouse truck, were selected for the test. The test vehicle tire pressure was normal. Due to the greater impact of the placement angle on the vehicle braking force test results, a brake test platform applied has the function of automatically detecting the wheel diameter and setting the placement angle by adjusting the axle distance of the roller. During the test, the placement angle of the operating vehicle was set as the optimum placement angle. The relevant studies showed that the optimum placement angle can be calculated by equation (11) [8]. Where ϕ is the roller surface adhesion coefficient.

Here, ϕ was taken to be 0.85 and the corresponding placement angle α can be calculated to be 40.4°. In view of the hydraulic cylinder stroke limit for adjusting the axis distance of the main and assistant roller by pushing the assistant roller group to move by the hydraulic station was 250 mm, under the condition of the maximum main and assistant roller axis distance, the placement angle a was 35.36° for the Jiefang heavy warehouse truck, and that of the rest three types of vehicles were close to the optimal placement angle of 40.4°. The front axle load, placement angle, relative slipping and friction loss of the four types of vehicles were tested and calculated. Among them, the calculated and measured values of friction loss were received according to equations (9) and (10), and all data are showed in Table 1. The comparison results show that the maximum relative error between the calculated and measured friction losses of the brake testing platform is 2.2%, indicating that the constructed friction loss model is effective and reliable, and can accurately predict the friction losses of the experimental brake testing platform.