## 2 Construction of FEM of a Rotating Tire  
A tire of 185/60R15 is studied where the material properties are obtained from the tire manufacturer. Since the stiffness of the wheel is much greater than that of a tire, the automobile wheel is assumed as a rigid body in the simulation process. The tread pattern features are ignored while the cross-section characteristics of the tire are retained. Beside, a tire is very complex structure and it is formed by layers of parallel string reinforcements and rubber parts. In modeling of the tire, a two-dimensional (2D) FEM of automobile tire is built first, and then revolved into a three-dimensional (3D) FEM of automobile tire. The 2D FEM of tire with the size of 185/60R15 is built in Abaqus software, as show in Figure 1. In 3D FEM of tire, the sidewall, tread, apex, belt and bead are modeled according to the real automobile tire in Figure 2a. However, the inner liner is ignored because of its insignificant effect on TACRN. The acoustic medium (air) which fits the inner surface of the tire is modelled as displayed in Figure 2b, and there are a total of 97,440 elements in 3D FEM of tire. Considering that the large deformation of the tire and the non-linear contact behavior may occur in the simulation, the C3D8R and C3D6 elements are adopted to discretize the rubber parts, and the AC3D8R and AC3D6 elements are adopted to discretize the air part. The rubber is a typical nearly incompressible hyperelastic material, and the non-linear behavior of each rubber component in tire is described by Yeoh constitutive model. The hyperelastic properties of the rubber components determined from uniaxial test resultsare shown in Table 1[19]. In the tire model, the component of tire bead is simplified as an isotropic structure with steel material property. The validation of acoustic-structural model is performed by the tire structural modal and acoustic modal experiment and shown in our previous researches [14; 20; 21]. [PERSON] et al. [14] had simulated and tested the structural modal characteristics of the tire in inflated states and loaded states, and the simulation and test results showed that a good agreement and indicated the accuracy of the constructed model. [PERSON] et al. [20; 21] simulated the acoustic mode of tire cavity by sweep-frequency speed excitation, and the sweep-frequency speed excitation tests were performed in order to verify above simulation results as well as the validity of the FE model, and the simulation and test results showed that a good consistency. However, their research did not introduce the rotation of tire. According to the actual test equipment [18], a simplified but full scale three-dimensional drum model of the tire-test machine for TACRN is built in Abaqus software. The structural parameters of the finite element model are the same as those of the actual test tire with the test drum machine equipment. The wheel rim type is \(5.5\) J \(\times\) 15, and the diameter of the drum is \(1.7\) m. The wheel rim and drum are presumed to be rigid bodies, as indicated in Figure 3. The boundary conditions are that the surface of tire contacts the surface of the drum in order to simulate the load, and the constraint type of tire and air is tie, and same for the constraint type of tire and wheel rim. In inflation and load analysis step, the DOFs of drum and tire-wheel assembly model are all limited, but in rotating analysis step, the rotation DOF around axis of the drum model is released. The compound, tread pattern, the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parts** & **Density (t/mm\({}^{3}\))** & **C\({}_{10}\)** & **C\({}_{20}\)** & **C\({}_{30}\)** \\ \hline Tread & \(1.18\times 10^{-9}\) & 0.585 & \(-0.194\) & 0.077 \\ Belt & \(1.18\times 10^{-9}\) & 0.947 & \(-0.282\) & 0.118 \\ Sidewall & \(1.10\times 10^{-9}\) & 0.465 & \(-0.153\) & 0.045 \\ Apex & \(1.17\times 10^{-9}\) & 1.614 & \(-0.563\) & 0.282 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Rubber material parameters in tire. Figure 1: 2D FEM of tire.