# Optimal Longitudinal Texture on Concrete Pavement to Reduce Lateral Vibration of Vehicles
## 4 Finite Element Analysis  
Finite element analysis was performed to verify the result of the field and driving simulation test. The finite element analysis of the lateral vibration for tires and longitudinal textures was performed by measurement of tire profile, preparation of tire sample cut, layout drawing, modeling of two-dimensional and three-dimensional, generation of concrete textures with grooves, and analysis for wheel load application. Using the modeled tires and textures, structural analysis was performed through the four-step process of rim fitting, inflation, relocation of road surface, and loading, as shown in Figure 3.  
The result of the structural analysis was obtained by calculating the lateral reaction force (RF) generated from the tire through loading using Equation (1). The lateral RF result was obtained by conducting analysis while moving the tire in 1 mm increments in the lateral direction, applying the General Motors Uniform Test Standard (GMUTS) for each case, and indirectly comparing the lateral vibration level for each texture and tire type [17]. GMUTS is calculated using the Max-Min difference of the lateral force that causes lateral vibration and the standard deviation. A larger value of GMUTS indicates better driving performance with fewer lateral vibrations. The specific calculation process cannot be described in this work because the information is confidential and proprietary to General Motors. Thus, only the evaluation results are shown. \[\text{RF}=\text{Fm}+\text{Fs} \tag{1}\]  
* Fm: A lateral force is generated by a moment caused by the vertical load when the tire block contacts the pavement texture. The lateral force may not occur symmetrically and a greater lateral force may occur in one direction depending on the specification of the timing groove and the lateral shifting distance. Fm represents the total lateral forces caused by a moment. * Fs: When the tire contacts the pavement texture by load, a contact shear stress is generated depending on the specification of the timing groove and the position of the tire. Fs denotes the total contact shear stress generated when the tire contacts the pavement texture. Finite element analysis was performed for a total of 48 parameter combinations, including 8 tire types and 6 longitudinal texturing types, which are the same as in the field  
Figure 3: Steps of finite element analysis for tire-surface. (**a**) Rim Fitting. (**b**) Inflation. (**c**) Relocation of Road Surface. (**d**) Loading. test. Figure 4 shows an example of the finite element analysis result for the 2 \(\times\) 3 \(\times\) 19 mm. The results of structural analysis are summarized in Table 7 by longitudinal texture specification and tire type. First of all, in the comparison by longitudinal texturing specification, the 2 \(\times\) 3 \(\times\) 19 mm showed the lowest probability of lateral vibration. The 2 \(\times\) 3 \(\times\) 16 mm and 3 \(\times\) 3 \(\times\) 16 mm also showed relatively low possibilities, while the 3 \(\times\) 3 \(\times\) Random showed the highest probability of lateral vibration. When the center-to-center interval was examined, the specifications in ascending order of probabilities were 3 \(\times\) 3 \(\times\) 16 mm \(<\) 3 \(\times\) 3 \(\times\) 19 mm \(<\) 3 \(\times\) 3 \(\times\) Random. Furthermore, the result for the surface with a 2 mm width was 2 \(\times\) 3 \(\times\) 16 mm \(<\) 2 \(\times\) 3 \(\times\) 19 mm. Thus, it is difficult to claim that fewer lateral vibrations occurred based on the center-to-center interval. However, it was found that the longitudinal texturing specification with the smallest lateral vibrations at the 3 mm width was 3 \(\times\) 3 \(\times\) 16 mm and that at the 2 mm width was 2 \(\times\) 3 \(\times\) 19 mm, relatively. Regarding the correlation of the longitudinal texturing specifications with the tire type, Tire2 showed the lowest probability of lateral vibration among the small vehicle tires as well as Tire7 among the SUV tires.