## 2 Simulation Method
### Vibration Energy Transmission Characteristics  
Direct-solution steady state dynamic analysis was applied to analyze the vibration response and structure intensity distribution of the wheels under TACR excitation. The vibration energy generated by TACR first impacted the wheel rim, then the wheel spokes, and then went through the bolts at the hub into the suspension system, and how this energy propagated in the wheels was focused on in this work. The cylindrical coordinate with the wheel center as the origin was used for analysis. As Figure 10 shows, along the spoke is the R direction, and along the rim circumference is the \(\theta\) direction. Considering the TACR energy transmission path, simulation of the connection between the wheel and suspension system at the hub was required. The radial spring and damping were added in the center of the hub to simulate the stiffness and damping of the suspension system. All freedoms of the bolts were constrained except the R direction in the cylindrical coordinate system, and the radial springs were set to simulate the action \begin{table}
\begin{tabular}{c c c} \hline
**Modal Shapes and** & **Experiment Test** & **Finite Element Simulation** \\
**Natural Frequencies** & **Experiment Test** & **Finite Element Simulation** \\ \hline
2 nd mode & & \\
1070 Hz & & \\
1062 Hz (\(-\)0.7\%) & & \\
3 rd mode & & \\
1731 Hz & & \\
1704 Hz (\(-\)1.6\%) & & \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
Figure 10: TACR sound pressure load model acting on the wheel. of the suspension system on the wheel, since the force from sound pressure in the cavity resulting from the acoustic cavity resonance was normal to the surface of the rim; that is, it was normal along the R direction. The four parallel spring dampers with uniform circumferential distribution were set as shown in the Figure 11.  
Based on vibration theory, the suspension damping can be expressed as  
\[\xi=\frac{C}{2\sqrt{KM}} \tag{4}\]  
where \(\xi\) is the damping ratio of the suspension system (generally about 0.2-0.45), \(C\) represents the equivalent damping coefficient of the suspension shock absorber, \(K\) is the suspension stiffness and \(M\) indicates the unsprung mass loaded on the suspension system. An ordinary working condition was chosen where \(K\) was 40 kN/m, C was 1386 Ns/m and \(M\) was 300 kg, referring to [26]. In the future, we will establish different kinds of suspension systems to cover more details. Based on above load models and constraint conditions, simulations of the 14-inch and 15-inch aluminum wheels under sound pressure excitation were performed. Figure 12 shows high stress values near the spokes and bolt holes, as the green-colored areas illustrate. For the 14-inch and 15-inch wheels, the maximum von Mises stresses were 89.39 Pa and 152.8 Pa, which were both located at the bolt holes, and the high structure intensity and power flow at these areas could be inferred as well. The power flow represents the ability of external forces to work or structures to dissipate energy per unit of time. The instantaneous power flow is defined as the product of the force and the velocity in the same direction and phase:  
\[P=F(t)\cdot V(t)\]  
Figure 11: Springs and dampers simulating the suspension’s stiffness and damping. Figure 12: Stress distributions of the two wheels: (**a**) 14-inch wheel and (**b**) 15-inch wheel. where \(P\) represents the power flow, \(F(t)\) is the external force and \(V(t)\) indicates the response velocity. For vibration analysis, the average power flow in a certain period of time can better reflect the energy intensity of the excited structure and can be expressed as follows:  
\[P=\frac{1}{T}\lim_{T\rightarrow\infty}\int_{0}^{T}F(t)\cdot V(t)dt \tag{6}\]  
For complex excitation \(F(t)=Fe^{i\omega t}\), the corresponding velocity response is \(V(t)=Ve^{i\omega t}\).