## 2 Construction of FEM of a Rotating Tire
### Effects of Automobile Tire Inflation Pressure and Experimental Validation  
For various loads undertaken by tire, based on Table 3, the maximum acoustic pressure amplitudes change with the automobile tire inflation pressures, which are displayed in Figure 12.  
From Figure 12, the maximum acoustic pressure amplitude at 2.5 bar is the largest compared with that at 2.2 bar and 1.9 bar for the identical load and running speed. The cause is that the increase in the automobile tire inflation pressure leads to an increase in tire stiffness. Figure 13a shows distributions contour of acoustic pressure amplitude when the load is 3500 N and the running speed is 50 km/h for the automobile tire inflation pressure of 2.5 bar and Figure 13b is the corresponding distributions of the acoustic pressure amplitude in the polar coordinate. From Figure 13b, the acoustic pressure amplitude decreases with the angle of rotation at first, then increases to maximum, and decreases again. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**The** & **RMS Values** & **RMS Values** & **Relative Errors** \\
**Load (N)** & **Simulation (R1)** & **in Experiment (R2)** & \(E_{1}\)=\(\frac{|R1-R2|}{R1}\) **(\%)** \\ \hline
3000 & 66.44 & 66.84 & 0.60 \\
3500 & 75.91 & 85.59 & 12.75 \\
4000 & 88.34 & 90.85 & 2.84 \\
4500 & 102.56 & 103.82 & 1.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The RMS values between Figures 10b and 11b. Figure 11: The acoustic pressure amplitude distributions: (**a**) For the load of 4500 N; (**b**) For all these loads. Figure 14a,b display the acoustic pressure amplitudes distributions when the load is 3500 N and the running speed is 50 km/h with various automobile tire inflation pressures in the Cartesian coordinate. The influence of the phase is introduced, which causes the values in Figure 14a to appear to be either positive or negative. If the values of Figure 14a are taken in absolute values, Figure 14b is developed. From Figure 14a, there is one crest value and one trough, and Figure 14b indicates that there are two crest values and two troughs. In fact, the crest values arise near the positions of 0 degree and 150 degree, for the reason given in the previous section. From Figure 14b, the acoustic pressure amplitude distributions for various automobile tire inflation pressures have the analogical trend. Figure 12: Variation of maximum acoustic pressure amplitudes with inflation pressures: (**a**) For the load of 3000 N; (**b**) For the load of 3500 N; (**c**) For the load of 4000 N; (**d**) For the load of 4500 N.  
Figure 13: The acoustic pressure distribution for the inflation pressure of 2.5 bar: (**a**) Distributions contour of the acoustic pressure amplitude; (**b**) Distributions of the acoustic pressure amplitude in the polar coordinate. In the previous literature [18], the authors carried out an experimental study on TACRN where the acoustic pressure amplitudes distributions under the identical operating conditions of Figure 13 are displayed in Figure 15a,b, and the sound pressure distribution of Figure 13b is also shown in Figure 15a. Moreover, the RMS values between Figures 14b and 15b are shown in Table 5.  
The trend of the simulation consequences from Figure 15a,b, is consistent with that from Figures 13b and 14b, respectively. Besides, the relative errors of RMS values between Figures 14b and 15b are acceptable. Therefore, the simulation method has once again been validated by experiment. In short, the maximum acoustic pressure amplitude of simulation is generally lower than the maximum acoustic pressure amplitude in the experiment for two reasons: firstly, the tire in the experiment has patterns, which can cause the additional excitation input, resulting in larger input energy. On the contrary, the tire in the simulation is set as smooth for convenience, so the effect of pattern is not introduced. Secondly, there exists the friction between the tire and the drum in the experiment, but not in the simulation.