## 2 Simulation Method  
### Sound Pressure Load Modeling and Verification  
As shown in Figure 2, the two-dimensional and three-dimensional finite element models (2-D and 3-D FEMs) of a 185/60 R15 tire were established to analyze the sound field generated by TACR. The whole assembly model was built in ABAQUS software, which consisted of the tire with hyperelastic rubber and reinforcements, the air medium, rim and road surface. The material properties were provided by the tire manufacturers. The rubber part was simulated by continuum elements CGAX3H and CGAX4H, the air medium was meshed by ACAX3 elements, and the reinforcements were defined by rebar elements with the type of SFMGAX1. Since the tire cavity was mainly focused on in this work, the rim and road face were regarded as rigid bodies due to the higher stiffness than the tire and air medium. The rim and tire were fixed, and vertical displacement constraints and loads were applied through the road surface. In the whole assembly model, the number of elements was 36,001, and the number of nodes was 40,622. The modal characteristics of the tire cavity coupling FEM agreed well with the experimental test results. The specific modeling and verification process could refer to authors' previous work [24]. In the research about the TACR problem, the most concerning aspect is the first natural frequency of the tire cavity, which can be calculated in Equation (1) as  
\[f_{1}=\frac{c}{l} \tag{1}\]  
where \(c\) represents the speed of sound in the acoustic medium and \(l\) is the median circumference of the tire cavity. In order to simulate TACR conditions, modal analysis of the tire acoustic cavity was carried out first. The free acoustic modal shapes were obtained in the case of a 2.5 bar inflation pressure, as shown in Figure 3. The two modes differed from each other in the circumferential direction (90 degrees) but belonged to the same natural frequency of 237 Hz. It is well known that under the road load, the first natural frequency of the tire cavity splits into two values, low and high, corresponding to the fore-and-aft and vertical modal shapes, respectively, as shown in Figure 4. Since the resonance in the vertical direction is the major cause of the spindle vibration, only the TACR sound pressure distribution at the higher natural frequency was analyzed in the following study. Figure 2: FEM of a 185/60 R15 tire. (**a**) 2-D FEM. (**b**) 3-D FEM. The frequency response characteristics of the tire acoustic cavity were acquired by sweep frequency analysis. The excitation was set by applying a uniform panel velocity acoustic boundary condition (1 mm/s in \(Z\) direction) at the contact patch. The simulation conditions were set with different road loads and inflation pressures (road load range = 3000-4500 N; inflation pressure range = 1.9-2.5 bar). The dynamic equation of the acoustic structure coupling system is described as follows [24]:  
\[\left[ \begin{array}{cc}[M_{s}]&[0]\\ \rho[S]^{T}&\left[M_{f}\right]\end{array} \right]\left\{ \begin{array}{c}\left\{ \tilde{u}\right\}\\ \left\{\tilde{p}\right\}\end{array} \right\}+\left[ \begin{array}{cc}[K_{s}] &-[S]\\ \left[0\right]&\left[K_{f}\right]\end{array} \right]\left\{ \begin{array}{c} \left\{u\right\}\\ \left\{p\right\}\end{array} \right\}=\left\{ \begin{array}{c}\left\{f_{s} \right\}\\ \left\{0\right\}\end{array} \right\} \tag{2}\]  
where \([M_{s}]\) and \(\left[M_{f}\right]\) are the acoustic and structure mass matrices, respectively, \(p\) is the sound pressure, \(\left[K_{f}\right]\) and \([K_{s}]\) indicate the acoustic and structure stiffness matrices, respectively, and \([S]\) is the acoustic structure coupling matrix showing the interaction between the fluid and the structure.