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# Top view of tire contact area and its deformations ( , ) u v with respect to ( , , , ) C x y z . The position of the tread location relative to ( , , , ) O X Y Z can be expressed as the equation:

Source publication

Theoretical tire models are often used in tire dynamics analysis and tire design. In the past, scholars have carried out a lot of research on theoretical model modeling; however, little progress has been made on its solution. This paper focuses on the numerical solution of the theoretical model. New force and moment calculation matrix equations are...

## Contexts in source publication

**Context 1**

... the tire deformation is described, two coordinate systems are used; one is the road coordinate system ( , , , ) O X Y Z and the other is the contact patch coordinate system ( , , , ) C x y z , named as ISO W − . When the tire slips relative to the road surface, the friction forces makes the tire deform ( , ) u v relative to the contact patch coordinates ( , ) x y , as shown in Figure 1. P is the contact point at undeformed state and p is the vector of P relative to the ( , , , ) O X Y Z system. ...

**Context 2**

... model parameters are listed in Table 2, Figures 7-11 show the comparison between the model and the test and the test information is shown in Table 3. The contact pressure is performed on the TekScan. ...

**Context 3**

... should be noted that the response of normalized lateral force instead of lateral force with travel distance is used, which does not affect the assessment of transient behavior. As shown in Figure 11, the aligning moment is quite different from the experimental data, and a reasonable reason is whether the local camber of the carcass caused by the lateral force is not considered, which can lead to greater aligning moment if taken into consideration. ...

**Context 4**

... simulation results of step side slip angle with the proposed method and the reference method are as shown in Figure 12. ...

**Context 5**

... simulation results of sine side slip angle with the proposed method and the reference method are as shown in Figure 13. ...

**Context 6**

... simulation results of load variations at constant side slip angle with the proposed method and the reference method are as shown in Figure 14. Response of lateral force using the reference method. ...

**Context 7**

... simulation results of step turn slip with the proposed method and the reference method are as shown in Figure 15. ...

**Context 8**

... simulation results of sine longitudinal combined with a constant side slip with the proposed method and the reference method are as shown in Figure 16. ...

**Context 9**

... simulation results of sine turn combined with a constant side slip with the proposed method and the reference method are as shown in Figure 17. ...

**Context 10**

... simulation results of load variations at constant longitudinal and side slip with the proposed method and the reference method are as shown in Figure 18. of aligning moment using the proposed method. (f) Response of aligning moment using the reference method. ...

**Context 11**

... the side slip angle step condition, with the increase of the travel distance, the deformation of the tread element approximately changes shape from rectangular to trapezoidal and finally to triangular, as shown in Figure 19. ...

**Context 12**

... Deformation of the Tread Element under Load Variations at a Constant Side Slip Angle Figure 21 shows the that the lateral deformation of the proposed method is more regular and stable than reference. It explains why under load variations conditions at the same path frequency, when the traveling distance in an increment time is larger than the distance between adjacent tread elements, the lateral force and aligning moment of the reference method will be different. ...

**Context 13**

... 7 and 8 show that the contact patch parameters are reasonable, and the simulation results are consistent with the experiments. Figures 9, 10 and 11a show that the belt/carcass parameters are reasonable, and simulation and test results are consistent as well. However, Figure 11b shows that the aligning moment is quite different from the experimental data, and a reasonable reason is that the local camber of the carcass caused by the lateral force is not considered, which can lead to greater aligning moment if considered. Figure 6 shows that different iterative methods converge to the same value and that the iterative method in reference [54,55] has the fastest iteration speed, followed by the modified Richardson method proposed in this paper. ...

**Context 14**

... 9, 10 and 11a show that the belt/carcass parameters are reasonable, and simulation and test results are consistent as well. However, Figure 11b shows that the aligning moment is quite different from the experimental data, and a reasonable reason is that the local camber of the carcass caused by the lateral force is not considered, which can lead to greater aligning moment if considered. Figure 6 shows that different iterative methods converge to the same value and that the iterative method in reference [54,55] has the fastest iteration speed, followed by the modified Richardson method proposed in this paper. Table 1 shows that the modified Richardson has the best convergence and then method in reference [54,55] and the worst is the method in reference [3,51]. ...

**Context 15**

... 12, 13 and 16 show that whether the discrete method proposed or the references [52][53][54][55] are considered, the simulation results at different speeds with the same path frequency are only slightly different under side slip angle or longitudinal slip input. Fig- ures 14, 15, 17 and 18 show that for the discrete proposed method the simulation results still have slight differences, but the results of the method in the references [52][53][54][55] have significant differences at different speeds with the same path frequency, and the greater the speed difference, or more precisely the greater the ratio between the distance of the contact patch center traveling in space within the time interval and distance between adjacent tread elements in the contact patch, the greater the simulation difference. The calculation errors in Tables 4-6 also confirm the above analysis. ...

**Context 16**

... calculation errors in Tables 4-6 also confirm the above analysis. Figure 19 shows that the tread deformation varies nearly linearly at the leading edge of the contact patch under step side slip angle, but Figure 20 shows that the tread deformation varies nonlinearly in an approximate parabolic under step turn slip. Figure 21 shows that the tread deformation is more stable by using the proposed method under the load variations input. ...

**Context 17**

... 19 shows that the tread deformation varies nearly linearly at the leading edge of the contact patch under step side slip angle, but Figure 20 shows that the tread deformation varies nonlinearly in an approximate parabolic under step turn slip. Figure 21 shows that the tread deformation is more stable by using the proposed method under the load variations input. Tread deformation explains why the methods in the reference [52][53][54][55] have problems under the turn slip or load variations inputs, since the method performs a linear interpolation process at the leading edge which is not suitable for nonlinear deformation and deformation instability. ...