## 2 Simulation Method
### Sound Pressure Load Modeling and Verification
Taking the 2.5 bar inflation pressure and 3000 N road load as an example, the simulation results of the TACR sound pressure amplitude distribution under the road load are shown in Figure 5a, and the sound pressure amplitude distribution in the circumferential direction of the tire cavity can be more intuitively represented in the polar coordinate system, shown in Figure 5b. The typical TACR generates two antinodes which are 180 degrees out of phase to each other and located at the top and bottom in the vertical direction. Considering the sound pressure amplitude change trend along the circumferential direction, a cosine function was selected to approximate the sound field in the tire cavity. In this paper, we only considered the standing wave generated in the vertical direction and  
Figure 4: Acoustic modal shapes of the tire cavities under road load. Figure 3: Free acoustic modal shapes of the tire cavity. ignored the mode deflection effect caused by rotation. The sound pressure distribution function resulting from the cavity resonance is approximately expressed as  
\[P_{s}=A_{pf}\cos\theta\sin(2\pi f_{1}t) \tag{3}\]  
where \(P_{s}\) is the sound pressure at each node, \(A_{pf}\) is the maximum sound pressure amplitude of the sound field in the simulation process, \(\theta\) is the angular position of the node in a polar coordinate system and \(f_{1}\) is the first natural frequency of the tire's acoustic cavity. In fact, the standing wave sound field caused by the cavity resonance was also the load distribution acting on the rim. To obtain the TACR sound pressure amplitude in the actual working conditions, the tire bench test was performed. The tire and wheel assembly was fixed on the test base and driven to rotate by the drum. Excitation was applied to the tire tread through a rotating drum. A pressure-tolerable microphone installed inside the cavity rotated with the tire and transmitted the measured sound pressure value to the signal acquisition and processing system through a wireless telemetry instrument. The specific experimental equipment device parameters are shown in Table 1, and the detailed test process can be referred to in [25]. The angular position of the sound pressure sensor was determined based on its time history data, obtained by the wheel angle sensor. Then, with both the pressure data and angular position data, the sound cavity pressure distribution could be processed. The test equipment is shown in Figure 6. In the case of the 2.2 bar inflation pressure and the vehicle speed being 50 km/h, the time history of the sound pressure in the tire cavity resulting from TACR is shown in Figure 7. It was the bandpass filtering for the original signal in a range of 210-260 Hz which covered the first natural frequency of the tire's acoustic cavity. From the plot, it can be seen that the period of the curve was 0.14 s, and the highest sound pressure was about 100 Pa. The \(A_{pf}\) in the load model could be determined by the experimental test results. \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Instructions** \\ \hline Tire & 185/60 R15 88H \\ Wireless telemetry system & [M3870 \\ The customized sound pressure sensor & CHZ-212 \\ Tire-testing machine & C-YLSN-1112 \\ Wheel & 5.5 J \(\times\) 15 \\ Battery & 12 V 1. 3 AH \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experiment devices. Figure 5: Sound pressure amplitude distribution in tire cavity under the road load. (**a**) Sound pressure amplitude distribution in the FEM. (**b**) Sound pressure amplitude distribution in the polar coordinate system. For the 185/60 R15 tire, the finite element simulation results and fitting load model of the sound pressure were drawn in the same polar coordinate system, as shown in Figure 8. It can be seen that they were in good agreement with each other; that is to say, the sound pressure distribution under the first TACR frequency excitation was consistent with the cosine function. The load model was verified and could be used to calculate the power flow in the wheels.